A network topology (spanner) is said to be power efficient if given any pair of ... for the unit disk graph UDG(V ), and the spanner is sparse and can be ...
EFFICIENT LOCALIZED TOPOLOGY CONTROL FOR WIRELESS AD HOC NETWORKS
BY YU WANG
Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Computer Science in the Graduate College of the Illinois Institute of Technology
Approved Advisor
Chicago, Illinois May 2004
c Copyright by YU WANG May 2004
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ACKNOWLEDGMENT I would like to thank lots of people for having contributed, in one way or another, to the completion of this thesis. First and foremost, I would like to thank Professor Xiang-Yang Li for being a wonderful advisor and friend during the past four years. This dissertation could not have been written without his insights and invaluable help and guidance. His visions in the area set the beginning of this thesis, and his involvement in working through vague ideas produced building blocks to the work. I am grateful to other members of my thesis committee, Professor Ophir Frieder, Professor Peng-Jun Wan, Professor Gruia Calinescu, and Professor Jinqiao Duan, for their service, interests and help. The special thanks go to Professor Ophir Frieder for his continuous help, such as funding my conference travel, rehearsing my talks, polishing my papers. I would also like to thank Professor Peng-Jun Wan, Professor Gruia Calinescu and Professor Ivan Stojmenovic for their help, suggestion, and joint work on my PhD research. To past and present members of our research group, Wei Huang, Hong Bai, ChihWei Yi, Wen-Zhan Song, Weizhao Wang, Kousha Moaveni-Nejad, and Khaled Alzoubi, I thank them all for useful meetings and discussions. There are so many friends I would like to thank for wonderful times together at Chicago. I can not list all of their names here, but Xinghua Shi, Bo Liu, Lei Huang, Yong Yang, Kyung Yoon Jeon, and Min Yu are definitely among them. Thanks you all. Last but not least, I thank my parents Shoutan Wang and Lubo Chen, and my brother Xiaosong Wang for their love and support, their unconditional encouragement and belief during these twenty-eight years.
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TABLE OF CONTENTS Page ACKNOWLEDGEMENT . . . . . . . . . . . . . . . . . . . . . . . . . .
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LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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LIST OF SYMBOLS
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CHAPTER 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . .
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1.1 Wireless Ad Hoc Networks . . . . . . . . . . . . . . . . . 1.2 Topology Control and Distributed Structures . . . . . . . . . 1.3 Organization of This Thesis . . . . . . . . . . . . . . . . .
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2. PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . .
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ABSTRACT
2.1 2.2 2.3 2.4
Modelling Wireless Ad Hoc Networks . Topology Control and Desirable Features Prior Art . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . .
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4. PLANAR SPANNER: LOCALIZED DELAUNAY . . . . . . . . .
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3. BOUNDED DEGREE STRUCTURES 3.1 3.2 3.3 3.4 3.5 3.6
4.1 4.2 4.3 4.4 4.5
Bounded Degree Yao Graph Sparsified Yao Graph . . . Symmetric Yao Graph . . . High-degree Yao Graph . . Ordered Yao Graph . . . . Summary . . . . . . . . .
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Definition of Localized Delaunay . . . . . . Properties of Localized Delaunay . . . . . . Localized Algorithms for Localized Delaunay Localized Delaunay on Virtual Backbone . . Localized Routing on Localized Delaunay .
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Page 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . .
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5. BOUNDED DEGREE AND PLANAR SPANNER . . . . . . . . .
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5.1 5.2 5.3 5.4 5.5
Previous Results . . . . . . . . . . . . Centralized Algorithm for Point Set . . . Centralized Algorithm for Unit Disk Graph Localized Algorithm for Unit Disk Graph . Summary . . . . . . . . . . . . . . . .
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72 74 81 87 96
6. EXPERIMENTAL RESULTS . . . . . . . . . . . . . . . . . . .
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6.1 Structural Properties . . . . . . . . . . . . . . . . . . . . . 6.2 Routing Performance . . . . . . . . . . . . . . . . . . . . 6.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . .
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7. CONCLUSION AND FUTURE WORK . . . . . . . . . . . . . .
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7.1 Localized Topology Control . . . . . . . . . . . . . . . . . 7.2 Other Related Topics . . . . . . . . . . . . . . . . . . . . 7.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . .
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BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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LIST OF TABLES Table
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Properties of Different Previous Topologies . . . . . . . . . . . . . .
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Stretch Factors and Maximum Node Degrees of Bounded Degree Graphs
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Quality Measurements of Different Topologies . . . . . . . . . . . .
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Performances and Communication Costs of BP S(V ) . . . . . . . . .
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Delivery Rates of Different Routing Methods . . . . . . . . . . . . .
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Maximum Spanning Ratio of Different Routing Methods
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Average Spanning Ratio of Different Routing Methods
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Properties of Different Topologies Proposed Here . . . . . . . . . . .
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LIST OF FIGURES Figure
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Illustration of Mobile Ad Hoc Networks . . . . . . . . . . . . . . .
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Examples of Wireless Sensor Nodes
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Examples of Unit Disk Graphs . . . . . . . . . . . . . . . . . . .
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Previous Structures Used in Wireless Networks . . . . . . . . . . .
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Definitions of RN G, GG, Y G and Del . . . . . . . . . . . . . . .
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Euclidean Minimum Spanning Tree with Large Stretch Factor
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Illustrations of the Difference Between θ-graph and Yao Graph . . . .
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Two Cases for the Angle ∠uwv . . . . . . . . . . . . . . . . . . .
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Dominators and Connectors Form the Backbone of the Network . . . .
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An Example of CDS Cannot Be Planar . . . . . . . . . . . . . . .
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Node u Has Degree (or In-degree) n − 1
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Illustrations of Bounded Degree Yao Graph . . . . . . . . . . . . .
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Node u Chooses the Shortest Links . . . . . . . . . . . . . . . . .
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Shaded Region Must Be Empty for uv ∈ Y Sk (V ) . . . . . . . . . .
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Y Sk (V ) with Large Stretch Factor . . . . . . . . . . . . . . . . .
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Y Dk (V ) Is Different from Y Sk (V ) and Y Yk (V ) . . . . . . . . . . .
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Relation Among Bounded Degree Structures
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Disk D2 Touches a Node w from N2 (u) ∪ N2 (v) . . . . . . . . . . .
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Two Cases in The Proof . . . . . . . . . . . . . . . . . . . . . .
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Illustration of First Case . . . . . . . . . . . . . . . . . . . . . .
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Illustration of Second Case . . . . . . . . . . . . . . . . . . . . .
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Constructing BP S0 for Point Set: Process Node u . . . . . . . . . .
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Shortest Path in Polygon P . . . . . . . . . . . . . . . . . . . . .
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Figure
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Constructing BP S1 for U DG: Process Node u
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No New Edges Can Be Added by Other Nodes to Intersect si si+1 . . .
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Illustrations of Planar Proof
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Illustrations of Delaunay Neighbors of u
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All Subcases in Case 2 Do Not Exist . . . . . . . . . . . . . . . .
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Different Localized Topologies Generated from Same Unit Disk Graph
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Results When Number of Wireless Nodes Increasing . . . . . . . . .
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Various Localized Routing Methods
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LIST OF SYMBOLS Symbol β πG (u, v) ρH (G)
Definition the real constant in the power-attenuation model, between 2 to 4 a path connecting u and v in graph G the power stretch factor of a graph H with respect to G
BP S0 (V )
the bounded degree and planar spanner of point set V
BP S1 (V )
the 1st centralized bounded degree & planar spanner of U DG(V )
BP S2 (V )
the 2nd centralized bounded degree & planar spanner of U DG(V )
BP S(V )
the localized bounded degree & planar spanner of U DG(V )
Cdel CDS Del(V ) disk(u, v) disk (u, v, w) |G| GG(V ) G = (V, E) ICDS LDelk (V ) M CDS M IS M ST (V )
the length stretch factor of Delaunay triangulation = the connected domination set the Delaunay triangulation of point set V the disk with diameter uv the circumcircle of u, v and w the size of graph G, i.e. the number of edges in G the Gabriel graph of graph U DG(V ) a graph G with point set V and edge set E the induced connected domination set graph the k-localized Delaunay graph of graph U DG(V ) the minimum connected domination set the maximum independent set the minimum spanning tree of graph U DG(V )
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√ 4 3 π 9
≈ 2.42
Symbol Nk (u) P LDel(V ) RN G(V ) tH (G)
Definition the k-hop neighbors of node u the planarized localized Delaunay graph of graph U DG(V ) the relative neighborhood graph of graph U DG(V ) the length stretch factor of a graph H with respect to G
U Del(V )
the unit Delaunay triangulation of graph U DG(V )
U DG(V )
the unit disk graph of point set V
||uv||
the Euclidean distance between nodes u and v
uv
a path connecting u and v
uvw
the triangle with endpoints u, v and w
∠uvw
the angle between the two rays vu and vw
uvw
the closed infinite area inside ∠uvw, also referred to as a sector
Y Dk (V )
the high-degree Yao graph of graph U DG(V )
−−→ Y Gk (V )
the Yao graph of graph U DG(V )
−−→∗ Y Gk (V )
the Yao and sink graph of graph U DG(V )
Y Ok (V )
the odered Yao graph of graph U DG(V )
Y S k (V )
the symmetric Yao graph of graph U DG(V )
−−→ Y Y k (V )
the Yao Yao graph U DG(V )
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ABSTRACT There are no wired infrastructures or cellular networks in ad hoc wireless network. Each mobile node has an adjustable transmission range. Node v can receive the signal from node u if node v is within the transmission range of the sender u. Otherwise, two nodes communicate through multi-hop ad hoc wireless links by using intermediate nodes to relay the message. Wireless ad hoc networks are also called packet radio networks in the early 70’s. While many fundamental ideas existed about twenty to thirty years ago, recent years we see tremendous research activity in wireless ad hoc networks due to its applications in various situations such as battlefield, emergency relief, and so on. Mobile wireless networking enjoys a great advantage over the wired networking counterpart because it can be formed in a spontaneous way for various applications. Hundreds of protocols that take the unique characteristics of wireless ad hoc networks, such as energy efficiency, routing and MAC layer protocols, have been developed recently. In this thesis, I focus on discussing one of the central challenges in the design of ad hoc networks: efficient localized topology control. I study how to construct a sparse spanner efficiently as the network topology for a set of static or quasi-static wireless nodes such that, for any given pair of nodes, there is a power-efficient path. I first review previous results for topology control when the network is modelled by unit disk graph. Then I propose several new localized structures: some bounded degree structures (Chapter 3), one planar spanner (Chapter 4), and one planar spanner with bounded degree (Chapter 5). Finally, I show some simulation results and discuss some possible future works as a conclusion of this thesis.
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1 CHAPTER 1 INTRODUCTION
In recent years computer users have demanded increasing mobility, with traditional desktop computing giving way to new and innovative mobile computing solutions. Notebook computers, personal digital assistants (PDAs), and palmtop devices have facilitated the ability to be mobile within our environment. Even though the portable computers are small and mobile, their usefulness are somewhat limited if the network services are unavailable. Wireless networks add the final piece of the puzzle. Compared with classical wired networks, wireless networks have the advantages: mobility, flexibility and cost saving. Typically, wireless networks can be classified into two categories: (a) cellular networks, and (b) ad hoc networks. A cellular network has a fixed network of limited range base-stations (BS), and mobile devices can access the network by establishing a wireless connection to the base station. This kind of network is also referred to as infrastructured model. The advantage of the infrastructured model for wireless communications is that existing wired networks can leveraged to support access for mobile users without modifications to the network’s control structure. The disadvantage is that it requires a fixed infrastructure which constrains node mobility, limits network deployability, and increases installation and management costs. Infrastructured wireless networks are not well suited for rapid network deployment, temporary networking for mobile devices, or for environments where it is difficult to achieve adequate BS station coverage, or the installation of fixed infrastructure is not feasible. To address these shortcoming, wireless ad hoc networks (also called mobile ad hoc networks MANETs) is emerging as a flexible and powerful wireless architecture that does not rely on a fixed networking infrastructure. 1.1 Wireless Ad Hoc Networks There are no wired infrastructures or cellular networks in wireless ad hoc network.
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Figure 1.1. Illustration of Mobile Ad Hoc Networks Each mobile node has an adjustable transmission range. Node v can receive the signal from node u if node v is within the transmission range of the sender u. Otherwise, two nodes communicate through multi-hop ad hoc wireless links by using intermediate nodes to relay the message. Consequently, each node in the wireless network also acts as a router, forwarding data packets for other nodes. Examples of such networks are wireless local areas networks (WLAN), sensor networks and packet radio networks. Figure 1.1 illustrate an ad hoc network formed by several personal wireless devices. Unlike traditional wired network, a single transmission by a node can be received by all nodes within its transmission range in the wireless network. In this thesis, we assume that each node has a low-power Global Position System (GPS) receiver, which provides the position information of the node itself. Wireless ad hoc networks are also called packet radio networks in the early 70’s. While many fundamental ideas existed about twenty to thirty years ago, recent years we see tremendous research activity in wireless ad hoc networks due to its applications in various situations such as battlefield, emergency relief, and so on. Mobile wireless networking enjoys a great advantage over the wired networking counterpart because it can be formed in a spontaneous way for various applications. Hundreds of protocols [Bos01, Chl99, Das00,
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Figure 1.2. Examples of Wireless Sensor Nodes Joh96, Ko97, Li02a, Mal99, Per97, Ram96, Roy99, Sto01, Wan02c, Zar01] 1 that take the unique characteristics of wireless ad hoc networks, such as energy efficiency, routing and MAC layer protocols, have been developed recently. 1.2 Topology Control and Distributed Structures Unlike wired networks, in wireless ad hoc networks, each node can move and thus change the topology of the network. In this case, we need to adjust the transmission power or selected neighbors to keep some properties of the network topology such as connectivity or power efficiency. The lifetime of a wireless network, which depending on battery power, usually is restricted because of limited capacity and resources on each node. For example, in sensor networks, a wireless sensor node is a tiny device (like the ones in Figure 1.2 [Gli03, Hil01]) only powered by small battery. Thus a main goal of topology control is to increase the longevity of such networks which can be obtained by designing power efficient algorithms [Bah02, Hu91, Hua02, Li01a, Raj02, Ram00, R¨uh03, Wat01]. Also there are many other desirable features for the wireless topology, such as spareness, bounded degree, etc.. We will introduce them in Chapter 2. 1
Correspond to coded references.
4 One effective approach [Gao01a, Gr¨u02, Hu93, Jia03, Li01a, Li01b, Li02b, Li04b, Raj02, R¨uh03, Wan02b, Wan02c, Wat01] is to maintain only a linear number of links using a localized construction method. In other words, we construct a sparse distributed structure as network topology for the wireless network. However, this sparseness of the constructed network topology should not compromise too much on the power consumptions on communications. In addition, we hope this construction method is a distributed or even localized method, since it is adaptive to topology changes and easy to update. In this thesis, we review and study how to construct a spanner (a sparse network topology) efficiently for a set of static wireless nodes such that every unicast route in the constructed network topology is power efficient. Here a route is power efficient for unicasting if its energy consumption is no more than a constant factor of the least energy needed to connect the source and the destination. A network topology (spanner) is said to be power efficient if given any pair of nodes there is a power efficient route to connect them. 1.3 Organization of This Thesis The rest of this thesis is organized as follows. In Chapter 2, we first give some definitions and concepts necessary for topology control. Specifically, we specify how the wireless network is modelled, review some well known geometry structures, define the graph spanners, and introduce some desirable features for wireless network topology. In addition, several previous results for topology control (some distributed structures and their spanner properties) are reviewed in detail in Chapter 2. Then we propose our new results for localized topology control: some bounded degree structures (Chapter 3), one planar spanner (Chapter 4), and one bounded degree planar spanner (Chapter 5). Finally, in Chapter 6 we give our current experimental results for most of our new network topologies, and we conclude this thesis in Chapter 7 by discussing some possible future work.
5 CHAPTER 2 PRELIMINARIES
In this chapter, we first review some definitions and concepts necessary for discussing topology control. We specify how the wireless network is modelled in a geometry view, introduce some desirable features for wireless network topology, and define the spanner. Then we review several previous results for topology control including some distributed structures and their spanner properties. 2.1 Modelling Wireless Ad Hoc Networks 2.1.1 Unit Disk Graph We consider a wireless ad hoc network consisting of a set V of wireless nodes distributed in a two-dimensional plane. Each node has some computational power. By a proper scaling, we assume that all nodes have the maximum transmission ranges equal to one unit. These wireless nodes define a unit disk graph U DG(V ) in which there is an edge between two nodes if and only if their Euclidean distance is at most one. See Figure 2.1 (a). Hereafter, we always assume that U DG(V ) is a connected graph. Otherwise, the wireless network is not connected, i.e., some nodes cannot communicate with some other nodes. However, we can show that given a set of points uniformly and randomly distributed in an area if the transmission range satisfies some value then the U DG(V ) is connected with high probability [Gup98, Ram00, San99]. This result will be shown in Subsection 2.1.4. We call all nodes within a constant k hops of a node u in the unit disk graph U DG(V ) as the k-local nodes of u, denoted by Nk (V ). Usually, here the constant k is 1 or 2, which will be omitted if it is clear from the context. The unit disk graph gives us all possible communication links in the network. However, the size of the unit disk graph could be as large as the square order of the number of network nodes. It means that the network
6 topology can be very dense when using U DG(V ) as network topology, such as Figure 2.1 (b). To keep all links in this U DG(V ) as network topology will cost a lot of power, this is bad in wireless ad hoc networks. So in this thesis we want to construct a subgraph (spanner) for the unit disk graph U DG(V ), and the spanner is sparse and can be constructed locally in an efficient way.
(a)
(b)
Figure 2.1. Examples of Unit Disk Graphs
2.1.2 Power-attenuation Model Energy conservation is a critical issue in ad hoc wireless network for the node and network life, as the nodes are powered by batteries only. Each mobile node typically has a portable set with transmission and reception processing capabilities. To transmit a signal from a node to the other node, the power consumed by these two nodes consists of the following three parts. First, the source node needs to consume some power to prepare the signal. Second, in the most common power-attenuation model, the power required to support the transmission between two nodes is dependent on their distance. Normally, the power needed to support a link uv is defined as uvβ , where uv is the distance between u and v, β is a real constant between 2 and 5 dependent on the wireless transmission environment. Finally, when a node receives the signal, it needs consume some power to receive, store and then process that signal [Rod98]. The power cost p(e) of a link e = (u, v) is then defined as the power consumed for transmitting signal from u to node v and possibly the energy consumed by node u and node v to process the signal. Many
7 researchers [Fee01] showed that the power for transmission is much higher than the one for receiving. Therefore, for our model using in this thesis, we only consider the power consumed for transmitting signal and ignore the other parts. 2.1.3 Broadcast Model Moreover, in this thesis, we assume that each wireless node has an omnidirectional antenna. This is attractive for a single transmission of a node can be received by many nodes within its vicinity which, we assume, is a disk centered at the node. We call this single transmission a local broadcast. Later, in this thesis, we count the number of these local broadcasts to measure the communication cost of our topology protocols. Notice that, since the identity of a node in a network with n nodes can be represented by O(log n) bits, we assume that each local message has the size of O(log n). 2.1.4 Statistic Results for Connectivity As power is a scarce resource in wireless networks, it is important to save the power consumption without losing the network connectivity. The universal minimum power used by all wireless nodes such that the induced network topology is connected is called the critical power. Determining the critical power was studied by several researchers [Gup98, Ram00, San99] recently when the wireless nodes are statically distributed. For simplicity, we can assume that the wireless devices are distributed in a unit area square (or disk) according to some distribution function, e.g., uniform distribution or Poisson process. Additionally, assuming that the movement of wireless devices still keeps them the same distribution (uniform or Poisson process). Gupta and Kumar [Gup98] showed that there is a critical power almost surely when the wireless nodes are randomly and uniformly distributed in a unit area disk. The result by Penrose [Pen97] implies the same conclusion. Moreover, Penrose [Pen97] gave the probability of the network to be connected if the transmission radius is set as a positive real number r and n goes to infinity. As fault-
8 tolerance is imperative for wireless networks, recently we [Li03f] also consider how to set the trasmission radios for a set of n nodes V to achieve k-connectivity of U DG(V ) when V is randomly distributed. 2.2 Topology Control and Desirable Features 2.2.1 Topology Control A main goal of topology control is to increase the longevity of wireless ad hoc networks which can be obtained by designing power efficient algorithms [Bah02, Hu91, Hua02, Li01a, Raj02, Ram00, R¨uh03, Wat01]. Some topology control protocols use power control (adjust transmission power or turn off some nodes)to save energy, for example they may restrict communication only to nearby nodes with reduced transmission range or schedule the wireless node to turn on/off their communication sometimes and yet can maintain global network connectivity. However, in this thesis, we assume the transmission range of each node is fix (so that the network is modelled by a U DG) and nodes are always awake. We consider another effective approach, which maintains only some of links from U DG using a localized construction method. Recall that keeping all links in U DG(V ) as network topology will cost a lot of power, which is not power efficient in wireless ad hoc networks. So we want to construct a distributed sparse structure as the network topology for wireless ad hoc network. However, this sparseness of the constructed network topology should not compromise too much on the power consumptions on communications. Recently, many researchers proposed several topology control protocols [Gao01a, Gr¨u02, Hu93, Jia03, Li01a, Li01b, Li02b, Li04b, Raj02, R¨uh03, Wan02b, Wan02c, Wat01]. Typically, there are two kinds of structures using for topology control: flat geometry structures and hierarchical structures. Flat geometry structures are normally geometry subgraphs of U DG, constructions of them need geometry position information. But hierarchical structures usually select some subset of nodes as the backbone nodes forwarding messages for other nodes, they do not need position information. We will review all these structures in
9 Section 2.3. 2.2.2 Desirable Features Clearly, not all sparse structures are good for topology control in wireless networks. We hope to construct a sparse network topology, i.e., a subgraph of U DG(V ), which has the following desirable features. Connectivity Connectivity is the most basic feature of the network topology, it guarantees that there exist at least one path from one node to any other nodes. Notice that here we require that the subgraph of U DG(V ) is connected if U DG(V ) is connected. Sparseness. The topology should be a sparse graph, i.e., the total number of links in this network is linear with the total number of wireless nodes. This enables most of algorithms, e.g., routing algorithm based on the shortest path, running on this topology more efficiently in term of both time and power consumption. Spanner. We want the subgraph to be a spanner of U DG(V ) in terms of power consumption. In other words, we hope every unicast route in the constructed network topology is power efficient, which means its power consumption is no more than a constant factor (power stretch factor) of the minimum power needed to connect the source and the destination. The formal definition of spanner will be presented in next subsection. Bounded degree. It is also desirable that the node degree in the constructed topology is small and bounded from above by a constant. A small node degree reduces the MAC-level contention and interference, also may help to mitigate the well known hidden and exposed terminal problems. In addition, a structure with small degree will improve the overall throughout [Kle78]. Planarized. The topology is a planar graph (no two edges cross each other in the graph). Some routing algorithms ask the topology be planar, such as right hand rout-
10 ing, Greedy Perimeter Stateless Routing (GPSR) [Kar00], Greedy Face Routing (GFG) [Bos01], Adaptive Face Routing(AFR) [Kuh02]. and Gready Other Adaptive Face Routing (GOAFR) [Kuh03]. Efficient Localized Construction. Due to the limited resources of the wireless nodes, it is preferred that the underlying network topology can be constructed and maintained in a localized manner. Here a distributed algorithm constructing a graph G is a localized algorithm if every node u can exactly decide all edges incident on u based only on the information of all nodes within a constant hops of u (plus a constant number of additional nodes’ information if necessary). More importantly, we expect that the time complexity of each node running the algorithm to construct the underlying topology is at most O(d log d), where d is the number of 1-hop or 2-hop neighbors. In addition, we require the total communication cost (the number of local broadcast messages) of our algorithms is O(n), where n is the number of nodes in the network. This implies the average number of message sent by each node is at most a constant. 2.2.3 Spanners Spanners have been studied intensively in recent years [Ary95, Bos02a, Jar92, Luk99b]. Let G = (V, E) be a n-vertex connected graph. The distance in G between two vertices u, v ∈ V is the length of the shortest path between u and v and it is denoted by dG (u, v). A subgraph H = (V, E ), where E ⊆ E ,is a t-spanner of G if for every u, v ∈ V , dH (u, v) ≤ t · dG (u, v). The value of t is called the stretch factor. Spanners for Euclidean graphs is called Geometric Spanners or Euclidean Spanners. It means the distance dG (u, v) in Euclidean graph G between u and v is the Euclidean distance between vertices u and v. Geometric spanners were introduced in computational geometry community by Chew [Che86]. Now they have numerous applications in computer science, such as VLSI, robotics motion planning, distributed systems and communi-
11 cations networks. In this thesis, we focus on the application in wireless networks. All previous algorithms that construct a t-spanner of the Euclidean complete graph K(V ) in computational geometry are centralized methods. The rapid development of the wireless communication presents a new challenge for algorithm designing and analysis. Distributed algorithms are favored than more traditional centralized algorithms. As in the definition of t−spanner, the length stretch factor
2
tH (G) of a graph H
with respect to G is defined as the maximum ratio of the total edge length of the shortest path connecting any pair of nodes in H to the total edge length of the shortest path between them in G. tH (G) = max u,v∈V
dH (u, v) . dG (u, v)
In wireless networks for the nodes’ limited resource, we want every unicast route in the constructed network topology is power efficient. Here a route is power efficient for unicasting if its energy consumption is no more than a constant factor of the least energy needed to connect the source and the destination. So here we introduce the definition of another stretch factor, the power stretch factor. Consider any unicast path π(u, v) in G (could be directed) from a node u ∈ V to another node v ∈ V : π(u, v) = v0 v1 · · · vh−1 vh , where u = v0 , v = vh and h is the number of hops of the path π. The total transmission power p(π) consumed by this path π is defined as p(π) =
h
vi−1 vi β .
i=1
Let pG (u, v) be the least energy consumed by all paths connecting nodes u and v in G. The path in G connecting u, v and consuming the least energy pG (u, v) is called the least-energy path in G for u and v. When G is a U DG(V ), we will omit the subscript G in pG (u, v). 2
Some researchers call it dilation ratio, spanning ratio.
12 Let H be a subgraph of G. The power stretch factor of a graph H with respect to G is then defined as ρH (G) = max u,v∈V
pH (u, v) . pG (u, v)
If G is a unit disk graph, we use ρH (V ) instead of ρH (G). For any positive integer n, let ρH (n) = sup ρH (V ). |V |=n
When the graph H is clear from the context, it is dropped from notation. In this section, we present some basic properties of the power stretch factor.
Lemma 1 For a constant δ, ρH (G) ≤ δ iff for any link vi vj in graph G but not in H, pH (vi , vj ) ≤ δvi vj β .
P ROOF. The necessary part is obvious. We only concentrate on the sufficient part. Consider any two nodes u and v. Assume πG (u, v) = v0 v1 · · · vh−1 vh , where u = v0 , v = vh , is the path with the least power consumption among all paths connecting u and v in G. Then for each link vi vi+1 , there is a path πH (vi , vi+1 ) in H with energy consumption at most δvi vi+1 β . Consider the path formed by concatenating all paths πH (vi , vi+1 ), i = 0, · · · , h − 1. Its power consumption is at most h
p(πH (vi , vi+1 )) ≤
i=1
h
(δvi vi+1 β ) = δ · pG (u, v).
i=1
Then the lemma follows.
The above lemma implies that it is sufficient to analyze the power stretch factor of H to each link in G but not in H.
Lemma 2 For any H ⊆ G with length stretch factor δ, its power stretch factor is at most δ β for any graph G.
13 P ROOF. From Lemma 1, it is sufficient to show that pH (u, v) ≤ δ β uvβ for any link uv in G but not in H. There is a path πH (u, v) in H with length at most δuv. Then the lemma follows from p(πH (u, v)) =
e∈πH (u,v)
eβ ≤
β e ≤ (δuv)β = δ β uvβ .
e∈πH (u,v)
Therefore a geometry structure H with a constant length stretch factor δ implies that its power stretch factor is no more than δ β . In particular, a graph with a constant bounded length stretch factor must also have a constant bounded power stretch factor. But the reverse is not true. Finally, the power stretch factor has the following monotonic property.
Lemma 3 If H1 ⊂ H2 ⊂ G then the stretch factor ρH1 (G) ≥ ρH2 (G).
2.3 Prior Art In wireless communication area many geometry structures are used as network topology or routing underlying graph recently [Bos99, Bos01, Gao01b, Kar00, Kar00, Sto01, Wat01]. For example, Bose and Morin [Bos99] showed that several localized routing protocols guarantee to deliver the packets if the underlying network topology is the Delaunay triangulation of all wireless nodes. They also gave a localized routing protocol based on the Delaunay triangulation such that the total distance traveled by the packet is no more than a small constant factor of the distance between the source and the destination. Bose et al.[Bos01] and Karp et al. [Kar00] proposed to use the Gabriel graph [Gab69] as network topologies that can be constructed efficiently in a distributed manner. Besides these geometry structures, there are another set of structures, hierarchical structures, are used in wireless networks. Instead of all node are involved in relaying packets for other
14 nodes, the hierarchical routing protocols pick a subset of nodes that server as the routers, forwarding packets for other nodes. In other words, these picked nodes form a virtual backbone for the network. The structure used to build this virtual backbone is usually the connected dominating set [Alz02c, Das97, Sto02, Wu01]. We will review all these wellknown flat geometry structures and the hierarchical structures in this section. Figure 2.2 summaries these two kind of structures. Structures for Network Topology
Flat Structures
MST
RNG
GG
Hierarchical Structures
Yao
Del
CDS
Figure 2.2. Previous Structures Used in Wireless Networks
2.3.1 Flat Geometry Structures Several flat geometrical structures have been studied recently both by computational geometry scientists and network engineers. Here we review the definitions and properties of some of them which could be used in the wireless networking applications. Let G = (V, E) be a geometric graph defined on V . For wireless ad hoc networks, the graph G is a unit disk graph U DG. 2.3.1.1 Minimum Spanning Tree (MST) The minimum spanning tree of G, denoted by M ST (G), is the tree belong to E that connects all nodes and whose total edge length is minimized. If G is a U DG, we use M ST (V ) instead of M ST (G) to denote the minimum spanning tree. M ST (V ) is obviously one of the sparsest possible connected subgraph, but its length and power stretch factor can be as large as n − 1.
15 Wan et al.[Wan01] showed that the minimum energy cost of broadcasting or multicasting is related to the total energy cost of all links in the Euclidean minimum spanning tree M ST . However, it is impossible to build M ST (V ) using only local information. Therefore, Li et al. [Li03a, Li04c] proposed several localized structures to approximate M ST (V ) for broadcasting in wireless ad hoc networks. 2.3.1.2 Relative Neighborhood Graph and Gabriel Graph The relative neighborhood graph, denoted by RN G(G), is a geometric and graph theoretic concept proposed by Toussaint [Jar92, Tou80]. It consists of all edges uv ∈ E such that there is no point w ∈ V such that there exist edges uw and wv in E which are satisfied with uw < uv and wv < uv. Thus, an edge uv is included if the intersection of two circles centered at u and v and with radius uv do not contain any vertex w from the set V such that edges uw and wv exist in E. If G is a U DG, then, we only have to test if the lune(u, v) contains other node inside, also we use RN G(V ) to denote the graph. See Figure 2.3 (a) for an illustration, the shaded area is empty of nodes inside. Let disk(u, v) be the disk with diameter uv. Then, the Gabriel graph [Gab69] GG(G) contains edge uv ∈ E if and only if disk(u, v) contains no other point w ∈ V such that there exist edges uw and wv in E which are satisfied with uw < uv and wv < uv. If G is a U DG, an edge uv ∈ GG(V ) if and only if disk(u, v) does not contain any node inside. See Figure 2.3 (b) for an illustration, the shaded area is empty of nodes inside. It is easy to show that RN G(V ) is a subgraph of the Gabriel graph GG(V ). Both GG(V ) and RN G(V ) are connected and contain the minimal spanning tree of G, when G is a U DG. Both GG(V ) and RN G(V ) are planar grpahs, which also implies their sparseness: |RN G(V )| ≤ 3n and |GG(V )| ≤ 3n. From the definitions, it is easy to see that GG(V )
16 u
v
u
v
u
(a) RNG
(b) GG
11111 00000 00000 11111 00000 11111 00000 u 11111 00000 11111 111111111 000000000
v
(c)YG
w
(d)Del
Figure 2.3. Definitions of RN G, GG, Y G and Del and RN G(V ) can be constructed locally. So they are used in wireless ad hoc networks by many researchers [Bos99, Bos01, Kar00]. However, the same analysis by Bose et al. [Bos02a] implied that the length stretch factor of RN G(V ) is at most n − 1 and the length √
stretch factor of GG(V ) is at most 4π 32n−4 . Recently, Wang et al. [Wan03c] showed that √ the factor of GG(V ) is precisely n − 1 actually. In this section, we will analyze their power stretch factors. Since the relative neighborhood graph has the length stretch factor as large as n − 1, then the Lemma 2 implies that its power stretch factor is at most (n − 1)β . In this section, we show that it is actually n − 1.
Theorem 1 [Bos02a, Jar91] ρRN G (n) = n − 1.
P ROOF.
First we prove that ρRN G (n) is at most n − 1. Consider the path between u
and v in M ST (V ). This path contains at most n − 1 edges and each edge has length at most uv. Therefore its total power consumption is at most (n − 1)uvβ . Notice that M ST (V ) ⊂ RN G(V ) if U DG(V ) is connected. From Lemma 1, ρRN G (n) ≤ n − 1. Then we show that ρRN G (n) ≥ n − 1 − ε for any small positive ε by constructing an example illustrated in Figure 2.4. We consider two cases. We first consider even n, say n = 2m. The construction of
17 v2m-1 θ α θ
v5
v3
v1
θ α θ
θ α θ
θ
θ
θ
θ
α
α
θ
θ αθ
α θ α
θ α
v2m-1
v2m
αθ
θ
v5
v6
v3
v4
v1
v2
v2m-2
θ θv α 6
θ α θ
θ θ v α 4
θ α θ
θ
θ α
θ
θ
v2
(2)
(1)
Figure 2.4. Euclidean Minimum Spanning Tree with Large Stretch Factor the point set V is shown in Figure 2.4 (1), which was used in [Bos02a]. Let α=
π π + 2δ, θ = − δ, 3 3
where δ is a sufficiently small positive number which will be fixed later. The m points with odd subscripts v1 , v3 , v5 , · · · , v2m−1 are collinear, so are the m points with even subscripts v2 , v4 , v6 , · · · , v2m . As proved in [Bos02a], RN G (V ) is a path v1 , v3 , v5 , · · · , v2m−1 , v2m , · · · , v6 , v4 , v2 . As δ −→ 0, the length of each edge in RN G (V ) tends to v1 v2 from below, which implies that pRN G (u, v) −→ n − 1, p (u, v) So we can find a sufficiently small δ > 0 such that pRN G (u, v) > n − 1 − , p (u, v)
18 which implies that ρRN G (n) > n − 1 − . When n is odd, the construction is shown in Figure 2.4 (2) and the existence can be proved by a similar argument.
The above proof also shows that any graph contains the Euclidean minimum spanning tree has the power stretch factor at most n − 1. Notice that the length stretch factor of some connected subgraphs of U DG(V ) could be arbitrarily large. Bose, et al. [Bos02a] proved that the length stretch factors of the Gabriel graph √
√
has length stretch factor in between 2n and 4π 32n−4 . Recently, Wang et al. [Wan03c] √ showed that the factor is precisely n − 1 actually. More worse, it was shown by Bose et al. [Bos02a] that the length spanning ratio of Gabriel graph on a uniformly random n points set in a square is almost surely at least O( log n/ log log n). Thus, no matter how good the routing method is, the length spanning ratio achieved by applying the method on Gabriel graph GG(V ) (also RN G(V )) is at least O( log n/ log log n) almost surely.
Theorem 2 [Bos02a] If n points are drawn uniformly and at random from the unit square [0, 1]2 , and S is the spanning ratio of the induced Gabriel graph, then a log n 1−12a−o(1) P {S < c } ≤ 2e−2n log log n for any constant c. Thus, for a < 1/12, with probability tending exponentially quickly to one,
S≥c
a log n . log log n
Since Gabriel graph has length stretch factor (n − 1), from Lemma 2, its power β √ n − 1 . From Lemma 3, its power stretch factor ρGG (n) is also stretch factor is at most at most ρRN G (n) = n − 1. We will show that ρGG (n) = 1.
19 Theorem 3 [Li02b, Li01b] The power stretch factor of any Gabriel graph is one. P ROOF. Consider any link uv in any least energy path in U DG(V ). Then uv itself is the least energy path between u and v. Therefore, the open disk using u and v as diameter is empty of wireless nodes. It implies that edge uv remains in the Gabriel graph GG(V ).
2.3.1.3 Yao Graph and θ-Graph −−→ The Yao graph [Yao82] with an integer parameter k ≥ 6, denoted by Y Gk (G), is defined as follows. At each node u, any k equal-separated rays originated at u define k cones. In each cone, choose the shortest edge uv among all edges from u, if there is any, → Ties are broken arbitrarily or by ID. The resulting directed and add a directed link − uv. graph is called the Yao graph. See Figure 2.3 (c) for an illustration, the shaded area is empty of nodes inside. Let Y Gk (G) be the undirected graph by ignoring the direction of −−→ → instead of the link − → the graph is denoted by each link in Y Gk (G). If we add the link − vu uv, −−→ ←−− Y Gk (G), which is called the reverse of Yao graph. Again, if G is a U DG, we use Y Gk (V ) to denote the Yao graph. If ignore the direction of edges in Yao graph, we use Y Gk (V ). Some researchers used a similar construction named θ-graph [Kei92, Kei88, Luk99b], the difference is that it chooses the edge which has the shortest projection on the axis of each cone instead of the shortest edge in each cone (See Figture 2.5). For more detail, please refer to [Kei92, Kei88, Luk99b].
u
(a) θ-graph
u
(b) Yao Graph
Figure 2.5. Illustrations of the Difference Between θ-graph and Yao Graph Yao graph [Yao82] and θ-graph [Bos02b, Kei92, Kei88] have been studied and
20 known well as a spanner in geometry area for a long time. Clearly, Yao graph is not planar, −−→ but it is sparse since |Y Gk (V )| ≤ kn. Several papers showed that the Yao graph Y Gk (V ) has a length stretch factor at most 1−2 1sin π . Then from the Lemma 2, we know that its power k
stretch factor is no more than
( 1−2 1sin
π k
) . [Li01b] prove a stronger result as following. β
Theorem 4 [Li02b, Li01b] The power stretch factor of the Yao graph Y Gk (V ) is at most 1 1−(2 sin
π β ) k
.
P ROOF. From Lemma 1, it is sufficient to show that for any nodes u and v with uv ≤ 1, there is a path connecting u and v in Y Gk (V ) with energy consumption at most 1 uvβ . 1 − (2 sin πk )β Let δ =
1 1−(2 sin
π β ) k
. We construct a path u v connecting u and v in Y Gk (V ). If link
vu ∈ Y Gk (V ), then set the path u v as the link uv. Otherwise, there must exist another node w in the same cone as v, which is a neighbor of u in Y Gk (V ). Then u v is set as the concatenation of the link uw and path w v. Notice that the angle θ of each cone section is
2π . k
When k > 6, then θ < π3 . It is easy to show that wv < uv. Consequently, the
path u v is a simple path, i.e., each node appears at most once. We prove by induction that the path u v has cost p(u v) at most δuvβ on the number of its edges. Obviously, if there is only one edge in u v, p(u v) = uvβ < δuvβ . Assume that the claim is true for any path with l edges. Then consider a path u v with l + 1 edges, which is the concatenation of edge uw and the path w v with l edges. We consider two cases.
21 Case 1: the angle ∠uwv is not acute. We have uw2 + wv2 ≤ uv2 . Notice that
uw uv
≤ 1 and
wv uv
≤ 1. It implies that
uw uv
β +
wv uv
β ≤
uw uv
2 +
wv uv
2 ≤ 1.
Therefore uwβ + wvβ ≤ uvβ for any β ≥ 2. Notice that wv < uv ≤ 1, which implies that we can apply induction on the path w v also. See the following Figure 2.6 (a).
w
w v u
v u
(a) ∠uwv ≥
π 2
(b) ∠uwv
ID(− → ID(− uv) pq) if 1. uv > pq or 2. uv = pq and ID(u) > ID(p) or 3. uv = pq, u = p and ID(v) > ID(q). → of each directed link − → is its Correspondingly, the rank, denoted by rank(− uv), uv order in the sorted directed links. Notice that, we only have to consider the links with length no more than one.
30 −−→ The Yao graph Y Gk (V ) can be constructed as follows. Algorithm: Constructing-YG
1. Each node u divides the space by k equal-sized cones centered at u. We generally assume that the cone is half open and half-close. Node u chooses a node v from each → has the smallest ID(− → among all directed links − → cone so the directed link − uv uv) uv i −−→ with vi in that cone, if there is any. Let Y Gk (V ) be the union of all chosen directed links.
3.1 Bounded Degree Yao Graph Arya, et al. [Ary95] presented an innovative technique to generate a bounded degree graph with a constant length stretch factor. In [Li01b], the authors applied the same technique to construct a sparse network topology with a bounded degree and a bounded power stretch factor. 3.1.1 Constructing Y G∗k (V ) The technique is to replace the directed star consisting of all links toward a node u by a directed tree T (u) with u as the sink. Tree T (u) is constructed recursively. The algorithm is as follows. Algorithm: Constructing-YG∗ −−→ 1. First, construct the graph Y Gk (V ). Each node u will have a set of in-coming nodes −→ →∈− I(u) = {v | − vu Y Gk (V )}. 2. For each node u, use the following Tree(u,I(u)) to build tree T (u).
Algorithm: Constructing-T (u) Tree(u,I(u))
31 1. To partition the unit disk centered at u, we choose k equal-sized cones centered at u: C1 (u), C2 (u), · · · Ck (u). 2. Node u finds the nearest node yi ∈ I(u) in Ci (u), for 1 ≤ i ≤ k, if there is any. Link − y→ i u is added to T (u) and yi is removed from I(u). For each cone Ci (u), if I(u) ∩ Ci (u) is not empty, call Tree(yi ,I(u) ∩ Ci (u)) and add the created edges to T (u).
Notice that, node u constructs the tree T (u) and then broadcasts the structure of T (u) to all nodes in T (u). Figure 3.1 (a) illustrates a directed star centered at u and Figure 3.1 (b) shows the directed tree T (u) constructed to replace the star with k = 8. The union −−→ of all trees T (u) is called the sink structure Y G∗k (V ). Notice that in [Gr¨u02, R¨uh03], they −−→ redefine Y G∗k (V ) structure, and call it Bounded Degree Yao graph or Yao and Sink graph. −−→ Notice that the communication cost of the first step to build Y Gk (V ) is O(n). Then for the second step, each node u call Tree(u,I(u)) to build tree T (u), it will take at most O(du ) messages, where du is the degree of u. So totally the second step will still cost O(n) messages. Consequently, the total communication cost of this algorithm is O(n).
u
u
(a) Star formed by links toward to u
(b) Directed tree T (u) sinked at u
Figure 3.1. Illustrations of Bounded Degree Yao Graph
3.1.2 Correctness The algorithm uses a directed tree T (u) to replace the directed star for each node u. Therefore, it does not change the connectivity of the structure. It means if nodes u and
32 −−→ −−→ v are connected by a path in Y Gk , they are also connected by a path in Y G∗k . We already −−→ −−→ know that Y Gk is strongly connected if UDG(V) is connected, so does Y G∗k . 1 2 Then we will prove that its power stretch factor is at most ( 1−(2 sin and its π β) ) k
2
degree is bounded by (k +1) −1 (its in-degree is bounded by k(k +1) while the maximum out-degree is k.) . −−→ Theorem 5 [Li01b] The maximum degree of the graph Y G∗k (V ) is at most (k + 1)2 − 1. P ROOF. Notice that the sink geometry structure does not change the out-degree of a node. → implies that there is also one directed edge − → in the sink T (u) for One directed edge − vu vw some w ∈ I(u). Moreover, because each node v participates in at most k + 1 sink trees (it will at most participate k sink trees for some other nodes and itself will also have one sink tree T (v)) and v participates in one sink tree will introduce at most k in-degree, the total in-degree is therefore at most k(k + 1). Consequently, the total degree is at most k(k + 1) + k = (k + 1)2 − 1. −−→ 1 2 Theorem 6 [Li01b] The power stretch factor of the graph Y G∗k (V ) is at most ( 1−(2 sin π β) . ) k
−−→ P ROOF. In [Li01b], we already proved that the power stretch factor of Y Gk (V ) is at most 1 1−(2 sin
. It means that for any nodes u and v with uv ≤ 1, there is a path connecting −−→ 1 β u and v in Y Gk (V ) with power consumption at most 1−(2 sin π β uv . Using the same ) π β ) k
k
argument, we can show that for each node v ∈ I(u), there is a direct path ΠT (u) (v, u) in 1 β T (u) such that the power consumption of ΠT (u) (v, u) is no more than 1−(2 sin π β vu . It ) k −−→ 1 2 implies that the power stretch factor of the graph Y G∗k (V ) is at most ( 1−(2 sin π β ) from ) k
Lemma 1. For detailed proof, please refer [Li01b].
Notice that the sink structure and the Yao graph structure do not need to have the same number of cones, and the cones do not need to be aligned. For setting up a power-
33 −−→ efficient wireless networking, each node u finds all its neighbors in Y Gk (V ), which can be done in linear time proportional to the number of nodes within its transmission range. −−→ However, the construction of Y G∗k (V ) is actually more complicated and the perfor−−→ mance gain compared with Y Y k (V ) (discussed next) is not so obvious in practice as shown by our experimental results in Chapter 6. 3.2 Sparsified Yao Graph We begin this section by giving the algorithm constructing Sparsified Yao graph −−→ Y Y k (V ). Then we prove the correctness of this algorithm by showing the connectivity and spanner property (in civilized graph) of the new topology. 3.2.1 Constructing Y Y k (V ) Algorithm: Constructing-YY −−→ 1. First, construct the graph Y Gk (V ). → has 2. Node u chooses a node v from each cone, if there is any, so the directed link − vu → among all directed links computed in the first step in that cone. the smallest ID(− vu)
u
Figure 3.2. Node u Chooses the Shortest Links
The union of all chosen directed links in the second step is the final network topol−−→ ogy, denoted by Y Y k (V ). If the link directions are ignored, the graph is denoted as −−→ −−→ Y Yk (V ). Compared with Y G∗k (V ), Y Y k (V ) replaces the directed tree T (u) by a directed
34 star (See Figure 3.2) consisting of at most k links toward a node u. It is easy to show that both steps of this algorithm take at most O(n) messages. So the total communication −−→ cost of this algorithm is O(n). Notice that in [Gr¨u02, R¨uh03], they reinvestigate Y Y k (V ) structure, and call it Sparsified Yao graph or Yao Yao graph. 3.2.2 Correctness −−→ It is obvious that both the out-degree and in-degree of a node in Y Y k (V ) are −−→ bounded by k. This implies that Y Y k (V ) is a sparse graph. From the construction al−−→ −−→ gorithm, we also know Y Y k (V ) is a subgraph of Y G∗k (V ), because all the links selected −−→ by node u in the second step are in the directed tree T (u) built by node u in Y G∗k (V ). −−→ Theorem 7 [Wan02b] The directed graph Y Y k (V ) is strongly connected if U DG(V ) is connected and k ≥ 6.
P ROOF. Notice that it is sufficient to show that there is a directed path from u to v for −−→ any two nodes u and v with uv ≤ 1. Notice that the Yao graph Y Gk (V ) is strongly −→ → in − connected. Therefore, we only have to show that for any directed link − uv Y Gk (V ), there −−→ is a directed path from u to v in Y Y k (V ). We prove the claim by induction on the ranks of −−→ all directed links in Y Gk (V ). −−→ → has the smallest rank among all links of Y → If the directed link − uv Gk (V ), then − uv will obviously survive after the second step. So the claim is true for the smallest rank. −−→ Assume that the claim is true for all links in Y Gk (V ) with rank at most r. Then −−→ −→ → in Y → = r + 1 in − → survives consider a directed link − vu Gk (V ) with rank(− vu) Y Gk (V ). If − vu → can only be removed by the node in the second phase, then the claim holds. Otherwise, − vu → survived with a smaller u in the second phase. Then, there must exist a directed link − wu → In addition, the angle ∠wuv is less than θ = identity in the same cone as − vu.
2π . k
Here
(wu, ID(w), ID(u)) < (vu, ID(v), ID(u)). Therefore wu ≤ vu. Because
35 ∠wuv
1, such that for any directed link − v− i vj in the graph Y Gk (V ), −−→ the minimum power consumption path in Y Y k (V ) from vi to vj is no more than δvi vj β . In other words, we show that − → p− (v , v ) ≤ δvi vj β . Y Y k (V ) i j
→ 2. There is a constant ρ > δ, such that for any directed link − v− i vj not in the graph −−→ −−→ Y Gk (V ), the minimum power consumption path in Y Y k (V ) from vi to vj is no more than ρvi vj β . In other words, we show that − → p− (v , v ) ≤ ρvi vj β . Y Y k (V ) i j
−−→ For the directed link with the rank one, it is in Y Y k (V ), therefore the first claim holds. Assume that the claims are true for all links with rank at most r. Then consider the − → directed link ut with rank r + 1. −−→ − → (u, t) = Case 1: link ut does not belong to Y Gk (V ). Then there is a directed path ΠY−−→ Gk −−→ q1 q2 · · · qh from u to t in graph Y Gk (V ), where q1 = u and qh = t. Let v be node q2 . Then we have → = (− → ID(u), ID(v)) rank(− uv) uv, − → − → < rank( ut) = ( ut, ID(u), ID(t))
37 because of the selection method of the first step. Similarly, − → − → rank( vt) = ( vt, ID(v), ID(t)) − → − → < rank( ut) = ( ut, ID(u), ID(t)) − → − → → − → → and − because vt < ut. Then we can apply the induction on − uv vt. Notice that here vt −−→ may not belong to Y Gk (V ). Consequently, we have − → p− (u, v) ≤ δuvβ , Y Y k (V ) − → p− (v, t) ≤ ρvtβ . Y Y k (V )
Therefore, we have − → − → − → p− (u, t) ≤ p− (u, v) + p− (v, t) Y Y k (V ) Y Y k (V ) Y Y k (V )
≤ δuvβ + ρvtβ . There are two subcases here. Either ∠uvt is acute or not. Subcase 1.1: the angle ∠uvt is not acute. Then we have uvβ + vtβ ≤ utβ . It implies that δuvβ + ρvtβ < ρ(uvβ + vtβ ) ≤ ρutβ . Consequently, we have − → p− (u, t) ≤ δuvβ + ρvtβ < ρutβ . Y Y k (V )
In other words, the claims hold for this subcase as long as δ < ρ. Subcase 1.2: the angle ∠uvt is acute. Then we have π θ vt ≤ 2 sin ut = 2 sin ut. 2 k
38 Consequently, we have δuvβ + ρvtβ ≤ δutβ + ρvtβ
β π ≤ δutβ + ρ 2 sin ut k
π β utβ . = δ + ρ 2 sin k Therefore, if
π β δ + ρ 2 sin ≤ρ k
then we have − → (u, t) ≤ δuvβ + ρvtβ p− Y Y k (V )
π β utβ ≤ δ + ρ 2 sin k
≤ ρutβ . In other words, the claims hold for this subcase. −−→ → with rank r + 1 does belong to Y Case 2: a link − uv Gk (V ). Then we know that −−→ − → (u, v) = v1 v2 · · · vh from u to v in Y Y k (V ), where v1 = u there is a directed path Π− YYk and vh = v. Let w = vh−1 . If w is u then we have − → p− (u, v) ≤ uvβ < δuvβ . Y Y k (V )
→ and − → are at a same cone So the claims hold. Otherwise, because the directed links − uv wv centered at v, we have → < rank(− → rank(− wv) uv) due to the selection method in the second phase. Notice that ∠uvw
0, we can transfer these conditions to δ δ − 1 δ − λβ ). ≤ ρ ≤ min( , 1−α α 1 − λβ β So for a given small λ, if we select k such that α = 2 sin πk < λβ then the existence of δ and ρ is guaranteed. For example, we can choose α = λβ /2, then δ = ρ=
δ 1−α
2−λβ 2−2λβ
and
= 2. Then we can get the bounded stretch factor.
−−→ Here we only prove the spanner property of Y Y k (V ) in civilized graph. Our experimental results show that this sparse topology has a small power stretch factor in practice −−→ (see Chapter 6). We conjecture that Y Y k (V ) also has a constant bounded power stretch factor (even length stretch factor) theoretically in any general graph. −−→ Recently Jia et al. [Jia03] claim that they prove that Y Y k (V ) also has a constant bounded power stretch factor theoretically in general graphs. However we doubt their result since it seems there are some bugs in their proof. So the proof of the conjecture or the construction of a counter-example remain a future work. 3.3 Symmetric Yao Graph In [Li04a], we also considered another undirected structure, called symmetric Yao graph Y Sk (V ), which guarantees that the node degree is at most k. 3.3.1 Constructing Y S k (V ) Symmetric Yao graph Y Sk (V ) is constructed as follows. Algorithm: Constructing-YS −−→ 1. First, construct the graph Y Gk (V ). And each node tell its neighbor whether the link −−→ is selected in Y Gk (V ).
42 → and − → 2. An edge uv is selected to graph Y Sk (V ) if and only if both directed edges − uv vu −−→ are in Y Gk (V ).
It is easy to show that both steps of this algorithm take at most O(n) messages. So the total communication cost of this algorithm is O(n). 3.3.2 Correctness For Y Sk (V ), we can prove the following theorems:
Theorem 9 [Wan03b] The symmetric Yao graph is a subgraph of the Yao Yao graph, i.e. Y Sk (V ) ⊆ Y Yk (V ). → and − → are in P ROOF. Consider any edge uv ∈ Y Sk (V ). Then both directed links − uv vu −−→ Y Gk (V ). So no nodes lies in the shaded region shown in Figure 3.3. Then in the cone −→ → is the shortest directed link to u. It implies that − →∈− I of node u, the link − vu vu Y Y k (V ). −→ →∈− Similarly, − uv Y Y k (V ). Therefore, uv ∈ Y Yk (V ).
II
u
I θ
θ
v
Figure 3.3. Shaded Region Must Be Empty for uv ∈ Y Sk (V )
Theorem 10 [Wan02b] The graph Y Sk (V ) is strongly connected if U DG(V ) is connected and k ≥ 6.
43 P ROOF. In [Gr¨u02], Grunewald, et al. gave a proof of the connectivity of Y Sk (V ). It used an similar induction method in the proof of Theorem 7. Here, we breifly review it. We prove the theorem by an induction over the distance ||uv|| between nodes u and v. First, note that the closest pair of nodes form an edge of the Y Sk (V ). Now observe for two nodes u and v: Either there is an edge from u to v or there is a node w with ||uw|| < ||uv|| and uw and uv are in the same cone of node u (or symmetrically ||vw|| < ||vu|| and vw and vu are in the same cone of node v). Because the angle ∠wuv is less than θ =
2π k
≤ π3 , we have
||vw|| < ||uv||. By induction there is a path from u to w and a path from w to v. Therefore a path from u to v exists.
In [Gr¨u02], Grunewald, et al. constructed a counter example to show that Y Sk (V ) is not a spanner for length or power. The basic idea of the counter example is similar to the counter example for RNG proposed by Bose et al. [Bos02a]. For the completeness of the presentation, we still review the counter example here.
Theorem 11 [Gr¨u02] Y Sk (V ) is not a spanner for length and power.
P ROOF. Let nodes v1 and v0 have distance half unit from each other. Assume the ith cone of v1 contains v0 , and the i th cone of v0 contains v1 . Then draw two lines l1 = v1 v3 and l2 = v0 v2 such that both the angles ∠v3 v1 v0 and ∠v2 v0 v1 are
π 2
− α, where α is a very
small positive number. We first consider even n, say n = 2m. Figure 3.4 illustrates the construction of the point set V . The node v2j is placed on l2 in the ith cone of v2j−1 and it is very close to the upper boundary of the ith cone of v2j−1 . The node v2j+1 is placed on l1 in the i th cone of v2j close to the upper boundary of that cone. Using this method, we place all nodes from v2 to v2m in order. Then it is easy to show that the Y Sk (V ) does not contain any edge v2j v2j+1 and v2j+1 v2j+2 for 0 ≤ j ≤ m − 1. The nearest neighbor of v2j is v2j+1 ,
44 but for v2j+1 , the nearest neighbor is v2j+2 . So although in Y Sk (V ) there is a path from v1 to v2 , its length is v1 v2m−1 + v2m−1 v2m + v2m v2 . So when α is appropriately small, the length stretch factor of Y Sk (V ) cannot be bounded by a constant. Similarly, its power stretch factor cannot be bounded also. When n is odd, the construction is similar. α
l2 α
l1
v2m v2m-1 v6 v5 v4 v3 i’ v1
v2 θ
v0 i
Figure 3.4. Y Sk (V ) with Large Stretch Factor
Nevertheless, our experiments show that Y Sk (V ) has a small power stretch factor in practice. 3.4 High-degree Yao Graph In [Li04a], we proposed a new undirected structure, called High-degree Yao graph Y Dk (V ), to bound the node degree of topology for bluetooth networks. 3.4.1 Constructing Y Dk (V ) The basic idea is that Yao construction is applied to all nodes (each at appropriate iteration) with an oder related to node degree. An edge remains in the structure if and only if both endpoints selected it in their respective applications of the Yao construction, otherwise it is deleted from the structure. This iterative approach is to divide the process
45 into iterations, and apply Yao structure to several nodes in each iteration. Each node creates a key for comparison with neighboring nodes. We consider the key as (degree, ID), where degree is the node’s degree of the original U DG(V ). This method consists of several iterations. Algorithm: Constructing-YD
1. Initially all nodes are undecided. 2. In each iteration, undecided nodes with higher keys than any of their undecided neighbors (we shall refer to such nodes as active nodes in the sequel) apply Yao structure to limit the degree, and inform its neighbors the decisions. (a) The active node u chooses a node v from each cone, if there is any, so the undirected edge vu has the smallest ID(vu) among all remaining edges in that cone. Notice that we do not consider the edges removed by previous iterations. (b) If an edge uv is selected by u, u will inform v that it wants to keep this edge. Otherwise, u will inform v deleting the edge uv. (c) The active node u then switches to a decided state. 3. At the end of each iteration, for every node v, if it receives deleting edge uv message from node u, node v deletes that edge vu from its own list. 4. Repeat iterations until all nodes are decided.
The union of all remaining edges at last is the final network topology, denoted by Y Dk (V ). Note that the elimination of any such edge uv by a node u immediately reduces the degree of v, i.e., node v has to remove link uv also. However, in order to avoid excessive information exchange between neighbors, the originally decided keys (that is, original degrees) are used in all comparisons.
46 Notice that for each node u, the algorithm only apply Yao structure and send out its decision once when it is processing. So the total communication cost of this algorithm is O(n). 3.4.2 Correctness We shall prove that the graph remains connected after this phase.
Theorem 12 [Li04a] The graph Y Dk (V ) is strongly connected if U DG(V ) is connected and k ≥ 6.
P ROOF. First of all, Y S k (V ) ⊆ Y Dk (V ) since any edge uv from Y S k (V ) will not be removed by either node u or node v in the construction of Y Dk (V ). See the left figure of Figure 3.5. Since we already know that Y S k (V ) is strongly connected if U DG(V ) is connected and k ≥ 6. Thus, its supergraph Y Dk (V ) is connected also.
x w
y
x
u
v u
YS ⊆YD
w
YD
y
x
v u
w
v
YY
Figure 3.5. Y Dk (V ) Is Different from Y Sk (V ) and Y Yk (V ) This structure Y Dk (V ) is different from all previous structures. First of all, we already know Y S k (V ) ⊆ Y Dk (V ). See the left figure of Figure 3.5. The graph Y Dk (V ) (represented by four solid lines) is different from Y S k (V ) (represented by three thick solid lines). Here the dashed lines define some cones around the nodes. Then it is also not
47 difficult to construct an example, e.g., the middle and right figures of Figure 3.5, such that Y Y k (V ) = Y Dk (V ). Here the node degrees are in decreasing order as w, u, v, x, and y. The experimental results reported in Chapter 6 show that this sparse topology has a small power stretch factor in practice. We conjectured that Y Dk (V ) also has a constant bounded power (or length) stretch factor theoretically in any unit disk graph. The proof of this conjecture or the construction of a counter-example remain a future work. 3.5 Ordered Yao Graph In [Bos02b], Bose et al. study a variant of θ-graphs called ordered θ-graphs. An ordered θ-graph of V is obtained by inserting the points of V in some order. When a point p is inserted, we draw the same cones around p and connect p to its closest previously-inserted neighbor in each cone. An ordered θ-graph of V is dependent on the order imposed on V ; different orderings of V can produce different graphs. Nevertheless, in [Bos02b], they show that ordered θ-graphs are also spanners, regardless of the ordering used. In same way, we can generally define an ordered Yao graph Y Ok (V ) by some order π imposed on V . Define any total order π on V so that πv is the rank of v in this order. We can prove the following theorem using same arguments in [Bos02b].
Theorem 13 The power stretch factor of the π-ordered Yao graph Y Ok (V ) is at most 1 1−(2 sin
π β ) k
, for any ordering π.
P ROOF. From Lemma 1, it is sufficient to show that for any nodes u and v with uv ≤ 1, there is a path connecting u and v in Y Ok (V ) with energy consumption at most 1 uvβ . 1 − (2 sin πk )β Let δ =
1 1−(2 sin
π β ) k
. We use induction on max{πu , πv }. Without loss of generality, assume
πu > πv . If πu = 2 then πv = 1 and there is a edge from u to v so the claim is trivial.
48 Otherwise, consider the cone c of u that contains v and let w be the neighbor of u in c. If w = v then we are done. Otherwise, u v is set as the concatenation of the link uw and path w v. Notice that the angle θ of each cone section is
2π . k
When k > 6, then θ < π3 .
It is easy to show that wv < uv. Consequently, the path u v is a simple path, i.e., each node appears at most once. Case 1: the angle ∠uwv is not acute. We have uw2 + wv2 ≤ uv2 . Notice that
uw uv
≤ 1 and
wv uv
uw uv
≤ 1. It implies that
β +
wv uv
β ≤
uw uv
2 +
wv uv
2 ≤1
Therefore uwβ + wvβ ≤ uvβ for any β ≥ 2. Since πw < πu , the inductive hypothesis states that p(w v) ≤ δwvβ . Then p(u v) = uwβ + p(w v) ≤ uwβ + δwvβ ≤ δuvβ .
Case 2: the angle ∠uwv is acute. We bound the length wv respecting to uv. Notice that uw ≤ uv and ∠wuv < θ because of the definition of the neighbors in the narrow region graph. The maximum length of vw is achieved when uw| = uv because the angle ∠uwv is acute. Therefore θ π wv ≤ 2 sin uv = 2 sin uv. 2 k By induction, we have p(u v) = |uw|β + p(w v) ≤ uwβ + δwvβ ≤ uvβ + This finishes the proof.
1 π β 1 β β π β (2 sin ) uv = π β uv . 1 − (2 sin k ) k 1 − (2 sin k )
49 However the node degree of Y Ok (V ) is not bounded by k, since for node u after applying Yao structure, other node can still add more edges to node u. The reason we list Y Ok (V ) in this chapter is that Y Ok (V ) using the same techniques (Yao structure) as all bounded degree structures proposed above. The communications cost of this algorithm is O(n), if an ordering is given. Notice that we can use a local order π instead of the global order π in above algorithm. Since we can use O(n) message to build a local order, like we do in the localized algorithm in Chapter 5, Y Ok (V ) can be built using O(n) messages. In [Bos02b], Bose et al. also study different properties that can be obtained by choosing orderings of V . They show that every point set has an ordering such that the ordered θ-graph has maximum degree O(k log n); and for every point set there exists an ordering such that, in the resulting ordered θ-graph, there is a
1 -path cos θ−sin θ
with O(log n)
edges between every pair of nodes. We say that such a graph has O(log n) spanner diameter. Notice that in ordered Yao graph, when you process a node, it only considers all previous processed nodes. If we change it to only consider all unprocessed nodes, the spanner proof still holds. The high-degree Yao consider both processed and unprocessed nodes, it is not a kind of ordered Yao graph. 3.6 Summary In this chapter, we present several efficient localized algorithms to construct network topologies with bounded node degrees for wireless ad hoc networks. We showed −−→ −−→ that Y G∗k (V ), Y Y k (V ) have bounded power stretch factors, while Y S k (V ) does not have. −−→ Notice that the power stretch factor of Y Y k (V ) is only true in civilized graph. We also proposed another structure Y Dk (V ), though we do not know whether it is a spanner yet. We summarize the properties of these topologies in Table 3.1, and the relations among all these subgraphs of U DG(V ) are shown in Figure 3.6. Here G → H means H is a sub-
50 graph of G. Notice Y G∗k (V ) may not be a subgraph of Y Gk (V ) and Y Dk (V ) may not be a subgraph of Y Yk (G). RNG YS YD
YY YG
YG*
Figure 3.6. Relation Among Bounded Degree Structures
Table 3.1. Stretch Factors and Maximum Node Degrees of Bounded Degree Graphs Power Stretch Factor Length Stretch Factor Max Node Degree YG Y G∗
1 1−(2 sin 1 1−(2 sin
π β ) k
2
π β ) k
1 1−2 sin 1 1−2 sin
π k π k
2
n−1 (k + 1)2 − 1
YY
2
?
2k
YD
?
?
k
YS
O(n)
O(n)
k
YO
1 1−(2 sin
π β ) k
1 1−2 sin
π k
n−1
51 CHAPTER 4 PLANAR SPANNER: LOCALIZED DELAUNAY
Given a set of points V , it is well-known that the Delaunay triangulation Del(V ) is a planar t-spanner of the completed graph K(V ). This is first proved by Dobkin, Friedman and Supowit with constant t = bound on t to be
√ 4 3 π 9
√ 1+ 5 π 2
≈ 5.08 . Then Keil and Gutwin improved the upper
≈ 2.42. However, it is not appropriate to require the construction
of the Delaunay triangulation in the wireless communication environment because of the possible massive communications it requires. Therefore, we consider the unit Delaunay triangulation U Del(V ), which is a subset of the Delaunay triangulation. Given a set of points V , let U Del(V ) be the graph by removing all edges of Del(V ) that are longer than one unit, i.e., UDel (V ) = Del (V ) ∩ UDG(V ). In [Gao01b, Li03f], It was proved that U Del(V ) is a t-spanner of the unit disk graph U DG(V ) as the following theorem.
Theorem 14 For any two vertices u and v of V , ||πU Del(V ) (u, v)|| ≤
√ 1+ 5 π·||πU DG(V ) (u, v)||. 2
Notice that, Keil and Gutwin [Kei92] showed that the Delaunay triangulation is a t-spanner for a constant t =
√ 4 3 π 9
≈ 2.42. They proved this using induction on the order of
the lengths of all pair of nodes (from the shortest to the longest). We can show that the path connecting nodes u and v constructed by the method given in [Kei92] also satisfies that all edges of that path is shorter than uv. Consequently, we know that the unit Delaunay triangulation UDel (V ) is a
√ 4 3 π-spanner 9
of the unit disk graph UDG(V ).
We then give a localized algorithm that constructs a graph, called localized Delaunay graph LDel (1) (V ). We prove that LDel (1) (V ) is a t-spanner by showing that it is also a supergraph of UDel (V ). Additionally, we prove that LDel (1) (V ) has thickness two, i.e., it can be decomposed to two planar graphs. We then show how to make the graph LDel (1) (V )
52 planar efficiently without losing the spanner property. The total communication cost of our approach is O(n log n) bits or O(n) messages, which is optimal within a constant factor. Notice that every node has to send at least one message to its neighbors to notify its existence in any protocol, which implies that the communication cost is at least n log n bits for any protocol. Here, assume a node ID can be represented by log n bits. Previously, there were some approaches proposed to approximate the Delaunay triangulation locally. Hu [Hu91] used the Delaunay triangulation to configure the wireless network topology such that a planar graph with bounded node degree is computed. A major step in his method is that each node u computes all Delaunay edges whose length is no more than the transmission range. It used the Voronoi diagram of node u to compute all such Delaunay edges. However, this approach will not work always. A simple observation is to determine whether an edge uv belongs to the Delaunay triangulation, we have to check whether there is an empty circle passing u and v. Here a circle is empty if it does not contain any wireless node inside. Obviously, in the worst case, the circumradius of this empty circle could be infinity even when the edge length uv is bounded, implying that we may have to check all nodes. Almost at the same time as [Li02a], Gao et al. [Gao01b] also proposed another structure, called restricted Delaunay graph RDG and showed that it has good spanning ratio properties and is easy to maintain locally. They called any planar graph containing U Del(V ) as a restricted Delaunay graph. They described a distributed algorithm to maintain a RDG such that at the end of the algorithm, each node u maintains a set of edges E(u) incident to u. Those edges E(u) satisfy that (1) each edge in E(u) has length at most one unit; (2) the edges are consistent, i.e., an edge uv ∈ E(u) if and only if uv ∈ E(v); (3) the graph obtained is planar; (4) U Del(V ) is in the union of all edges E(u). Their algorithm works as follows. First, each node u acquires the position of its 1-hop neighbors N1 (u) and computes the Delaunay triangulation Del(N1 (u)) on N1 (u), including u
53 itself. In the second step, each node u sends Del(N1 (u)) to all of its neighbors. Let E(u) = {uv | uv ∈ Del(N1 (u))}. For each edge uv ∈ E(u), and for each w ∈ N1 (u), if u and v are in N1 (w) and uv ∈ Del(N1 (w)), then node u deletes edge uv from E(u). They proved that when the above steps are finished, the resulting edges E(u) satisfy the four properties listed above. The communication cost could be as large as Θ(n2 ), and the computation cost could be as large as Θ(n3 ), which are much higher than ours. 4.1 Definition of Localized Delaunay We first introduce some geometric structures and notations to be used in this section. All angles are measured in radians and take values in the range [0, π]. For any three points p1 , p2 , and p3 , the angle between the two rays p1 p2 and p1 p3 is denoted by ∠p3 p1 p2 or ∠p2 p1 p3 . The closed infinite area inside the angle ∠p3 p1 p2 , also referred to as a sector, is denoted by p3 p1 p2 . The triangle determined by p1 , p2 , and p3 is denoted by p1 p2 p3 . An edge uv is called Gabriel edge if uv ≤ 1 and the open disk using uv as diameter does not contain any node from V . It is well known [Pre85] that the constrained Gabriel graph is a subgraph of the Delaunay triangulation. Recall that a triangle uvw belongs to the Delaunay triangulation Del (V ) if its circumcircle disk (u, v, w) does not contain any other node of V in its interior. Here we often assume that there are no four nodes of V co-circumcircle. It is easy to show that nodes u, v and w together can not decide if they can form a triangle uvw in Del (V ) by using only their local information. We say a node x can see another node y if xy ≤ 1. The following definition is one of the key ingredients of our localized algorithm.
Definition 1 [Li02a, Li03c] A triangle uvw satisfies k-localized Delaunay property if the interior of the circumcircle disk (u, v, w) does not contain any node of V that is a kneighbor of u, v, or w; and all edges of the triangle uvw have length no more than one unit. Triangle uvw is called a k-localized Delaunay triangle.
54 Triangle uvw is called localized Delaunay if it is a k-localized Delaunay triangle for some constant integer k ≥ 1.
Definition 2 [Li02a, Li03c] The k-localized Delaunay graph over a node set V , denoted by LDel (k) (V ), has exactly all Gabriel edges and edges of all k-localized Delaunay triangles.
When it is clear from the context, we will omit the integer k in our notation of LDel (k) (V ). The original conjecture was that LDel (1) (V ) is a planar graph and thus we can easily construct a planar t-spanner of UDG(V ) by using a localized approach. Unfortunately, as we shown in [Li02a, Li03c], the edges of the graph LDel (1) (V ) may intersect. While LDel (1) (V ) is a t-spanner, its construction is a little bit more complicated than some other non-planar t-spanners, such as the Yao structure [Yao82] and the θ-graph [Kei92]. But we can make LDel (1) (V ) planar efficiently, a result we describe later. Notice the k-localized Delaunay graph LDel (k) (V ) over a node set V satisfies a monotone property: LDel (k+1) (V ) is a subgraph of LDel (k) (V ) for any positive integer k. 4.2 Properties of Localized Delaunay The following theorem was proved in [Li02a, Li03c].
Theorem 15 [Li02a, Li03c] Graph UDel (V ) is a subgraph of the k-localized Delaunay graph LDel (k) (V ). Since the unit Delaunay triangulation UDel (V ) is a
√ 4 3 π-spanner 9
of the unit disk
graph UDG(V ), this theorem implies that LDel (k) (V ) is a t-spanner. However LDel (1) (V ) may be non-planar. The definition of the 1-localized Delaunay triangle does not prevent two triangles from intersecting or prevent a Gabriel edge from intersecting a triangle. Figure 4.1 gives such an example with 6 nodes {u, v, w, x, y, z} that
55 LDel (1) (V ) is not a planar graph. Here uv = xy = 1, and uy = vy > 1. Node x is out of circumcircle of disk (u, v, w). Triangle uvw is a 1-localized Delaunay triangle. If the node z does not exist, edge xy is an Gabriel edge. The triangle uvw intersects the Gabriel edge xy if z does not exist, otherwise it intersects the 1-localized Delaunay triangle xyz. The example illustrated by Figure 4.1 also implies that a triangle in LDel (1) (V ) can intersect many other edges (by creating equal-length Gabriel edges x1 y1 , x2 y2 , · · · , which are parallel to Gabriel edge xy).
u
w
x z v
y
Figure 4.1. LDel (1) (V ) Is Not Planar In [Li03c], we also claim that LDel (1) (V ) has thickness two, or in other words, its edges can be partitioned in two planar graphs. From Euler’s formula, it follows that a simple planar graph with n nodes has at most 3n − 6 edges, and therefore LDel (1) (V ) has at most 6n edges. The proof of the following theorem is in [Li03c].
Theorem 16 [Li02a, Li03c] Graph LDel (1) (V ) has thickness 2.
Although we gave example to show that LDel (1) (V ) is not always a planar graph, we will show that LDel (k) (V ), k ≥ 2, is always planar. Theorem 15 implies that each edge uv of UDel (V ) is either a Gabriel edge or forms a 1-localized Delaunay triangle with some edges from UDel (V ). Obviously, any two edges in UDel (V ) do not intersect. Thus, each possible intersection in LDel (k) (V ) is caused by at least one edge of some localized Delaunay triangle. We begin the proof that LDel (k) (V ), k ≥ 2, is planar by giving some simple facts and lemmas.
56 R EMARK: If a Gabriel edge uv intersects an edge xy, then xy does not belong to UDel (V ).
Lemma 4 [Li02a, Li03c] If a Gabriel edge xy intersects a localized Delaunay triangle uvw, then x and y can not be both outside the circumcircle disk (u, v, w).
P ROOF. Let c be the circumcenter of the triangle uvw. Then at least one of the u, v, and w must be on the different side of line xy with the center c; Let’s say u. If both x and y are outside, then ∠yux > π2 . Thus, u is inside disk (x, y), which contradicts that xy is a Gabriel edge.
Theorem 17 [Li02a, Li03c] Assume two triangles uvw and xyz introduced to LDel (k) (V ), k ≥ 1, intersect, then either disk (u, v, w) contains at least one of the nodes of {x, y, z} or disk (x, y, z) contains at least one of the nodes of {u, v, w}.
See [Li03c] for the proof. The above theorem guarantees that if two k-localized Delaunay triangles uvw and xyz intersect, then either disk (u, v, w) or disk (x, y, z) violates the Delaunay property by just considering the nodes {u, v, w, x, y, z}. We then show that LDel (2) (V ) is a planar graph.
Theorem 18 [Li02a, Li03c] LDel (2) (V ) is a planar graph.
P ROOF.
Notice that two Gabriel edges do not intersect. Then every intersection must
involves a localized Delaunay triangle. Assume that an edge xy of LDel (2) (V ) intersects a localized Delaunay triangle uvw on an edge uv. Edge xy is either a Gabriel edge or an edge of a localized Delaunay triangle, say xyz. If xy is a Gabriel edge, then Lemma 4 implies that either x or y is inside the disk (u, v, w), say y. If xy is an edge of a
57 localized Delaunay triangle xyz, then Theorem 17 implies that either x or y is inside the disk (u, v, w), say y. The triangle inequality implies that xu + yv < xy + uv ≤ 2. The existence of the 2-localized Delaunay triangle uvw implies that y ∈ / N1 (u)∪N1 (v)∪ N1 (w). Thus, yv > 1, which implies that xu < 1. In other words, x ∈ N1 (u). Consequently, y ∈ N2 (u) because of the path yxu in the unit-disk graph UDG(V ), which contradicts to the existence of 2-localized Delaunay triangle uvw. The theorem follows.
We defined a sequence of localized Delaunay graphs LDel (k) (V ), where 1 ≤ k ≤ n. All graphs are t-spanner of the unit-disk graph with the following properties: • UDel (V ) ⊆ LDel (k) (V ), for all 1 ≤ k ≤ n; • LDel (k+1) (V ) ⊆ LDel (k) (V ), for all 1 ≤ k ≤ n; • LDel (k) (V ) are planar graphs for all 2 ≤ k ≤ n; • LDel (1) (V ) is not always planar.
4.3 Localized Algorithms for Localized Delaunay In this section, we study how to locally construct a planar t-spanner of UDG(V ). We assume that the identity of a node u can be represented by O(log n) bits and its location can be represented by O(1) bits. Although the graph UDel (V ) is a t-spanner for UDG(V ), it cannot be constructed locally. In this sections, we proposed two different methods to construct a planar spanner based on localized Delaunay. In our first method, we first construct LDel (1) (V ), then extract a planar graph PLDel (V ) out of LDel (1) (V ). In the second method, we directly construct LDel (2) (V ), which is guaranteed to be a planar spanner of
58 UDel (V ). However, the total communication cost for the second method could be much larger than the first method, due to collecting the information of 2-hop neighbors. 4.3.1 Planarized 1-localized Delaunay Recall that LDel (1) (V ) is not guaranteed to be a planar graph. We propose an algorithm that constructs LDel (1) (V ) and then makes it a planar graph efficiently. The final graph still contains UDel (V ) as a subgraph. Thus, it is a t-spanner of the unit-disk graph UDG(V ). In the following, the order of three nodes in a triangle is immaterial. We assume that when a node sends out a message, all neighboring nodes will receive this message immediately. Algorithm: Localized Unit Delaunay Triangulation
1. Each wireless node u broadcasts its identity and location to its neighbors N1 (u) and listens to the messages from other nodes in N1 (u). 2. Assume that node u gathered the location information of N1 (u). It computes the Delaunay triangulation Del (N1 (u)) of its 1-neighbors N1 (u), including u itself. 3. For each edge uv of Del (N1 (u)), let uvw and uvz be two triangles incident on uv. Edge uv is a Gabriel edge if both angles ∠uwv and ∠uzv are less than π/2. Node u marks all Gabriel edges uv, which will never be deleted. 4. Each node u finds all triangles uvw from Del (N1 (u)) such that all three edges of uvw have length at most one unit. If angle ∠wuv ≥
π , 3
node u broadcasts a
message proposal(u, v, w) to N1 (u) to form a 1-localized Delaunay triangle uvw in LDel (1) (V ), and listens to the messages from its neighboring nodes. 5. When a node u receives a message proposal(u, v, w), u accepts the proposal of con-
59 structing uvw if uvw belongs to Del (N1 (u)) by broadcasting accept(u, v, w) to N1 (u); otherwise, it rejects the proposal by broadcasting reject(u, v, w) to N1 (u). 6. A node u adds the edges uv and uw to its set of incident edges if the triangle uvw is in Del (N1 (u)) and both v and w have sent either accept(u, v, w) or proposal(u, v, w).
We first claim that the graph constructed by the above algorithm is LDel (1) (V ). Indeed, for each triangle uvw of LDel (1) (V ), one of its interior angle is at least π/3 and uvw is in Del (N1 (u)), Del (N1 (v)) and Del (N1 (w)). So one of the nodes among {u, v, w} will broadcast the message proposal(u, v, w) to form a 1-localized Delaunay triangle uvw and the other two nodes will accept the proposal. Thus, LDel (1) (V ) is a subgraph of the constructed graph. Obviously, the constructed graph is also a subgraph of LDel (1) (V ) by definition, which in turn implies that they are the same. As Del (N1 (u)) is a planar graph, and a proposal is made only if ∠wuv ≥ π3 , node u broadcasts at most 6 proposals to its neighbors N1 (u). And each proposal is replied by at most two nodes. Therefore, the total communication cost is at most O(n) messages (in other words O(n log n) bits). The above algorithm also shows that LDel (1) (V ) has O(n) edges, which we know from Theorem 16. Putting together the arguments above, we have:
Theorem 19 [Li02a, Li03c] The above algorithm constructs LDel (1) (V ) with total communication cost O(n log n) bits.
We then propose an algorithm to extract from LDel (1) (V ) a planar subgraph without violating the property of constant spanning ratio. Algorithm: Planarize LDel (1) (V )
60 1. Each wireless node u broadcasts the Gabriel edges incident on u and the triangles uvw of LDel (1) (V ) and listens to the messages from other nodes. Each edge is sent at most 4 times. 2. Assume node u gathered the Gabriel edge and 1-local Delaunay triangles information of all nodes from N1 (u). If a triangle uvw in LDel (1) (V ) intersects a Gabriel edge xy, then node u removes triangle uvw. For two intersected triangles uvw and xyz from LDel (1) (V ) known by u, node u removes the triangle uvw if its circumcircle contains a node from {x, y, z}. 3. Each node u broadcasts all the triangles incident on u which it has not removed in the previous step, and listens to the broadcasting by other nodes. Each edge is sent at most 4 times. 4. Node u keeps the edge uv in its set of incident edges if it is a Gabriel edge, or if there is a triangle uvw such that u, v, and w have all announced they have not removed the triangle uvw in Step 2.
We denote the graph extracted by the algorithm above by PLDel (V ). We first show that PLDel (V ) is indeed a planar graph. Theorem 20 [Li02a, Li03c] PLDel (V ) is a planar graph. P ROOF. This proof is almost the same as the proof that LDel (2) (V ) is planar. Notice every intersection must involves a localized Delaunay triangle. Assume that an edge xy intersects a 1-localized Delaunay triangle uvw on an edge uv. Edge xy is either a Gabriel edge or an edge of another 1-localized Delaunay triangle, say xyz. If xy is a Gabriel edge, obviously triangle uvw does not belong to UDel (V ). Thus, removing uvw will not affect the spanning ratio bound. We then show that one
61 of the end-points of uvw will detect this intersection. Lemma 4 implies that either x or y is inside the disk (u, v, w), say y. The triangle inequality implies that xu + yv < xy + uv ≤ 2. The existence of the 1-localized Delaunay triangle uvw implies that y ∈ / N1 (u) ∪ N1 (v) ∪ N1 (w). Thus, yv > 1, which implies that xu < 1. In other words, x ∈ N1 (u). Thus, node u will know the existence of a Gabriel edge xy from x. Then node u will decide to remove the triangle uvw. If xy is an edge of a localized Delaunay triangle xyz, then Theorem 17 implies that either x or y is inside the disk (u, v, w), say y. Then we know that triangle uvw cannot exist in the unit Delaunay triangulation UDel (V ). Consequently, if we remove uvw, the graph UDel (V ) is still a subgraph of the final graph. Similar to the previous case, we know that x ∈ N1 (u). Thus, node u will know the existence of a triangle xyz, which will provide a node y as witness for removing uvw. Then node u will decide to remove the triangle uvw. Note that any triangle of LDel (1) (V ) not kept in the last step of the planarization algorithm is not a triangle of LDel (2) (V ), and therefore PLDel (V ) is a supergraph of LDel (2) (V ). Thus, by using Theorem 15, we have: UDel (V ) ⊆ LDel (2) (V ) ⊆ PLDel (V ) ⊆ LDel (1) (V ). The total communication cost to construct the graph PLDel (V ) is a O(log n) bits times the number of edges of the graph LDel (1) (V ), since each edge is broadcasted at most 8 times. Putting together all the arguments above, we have:
Theorem 21 [Li02a, Li03c] PLDel (V ) is planar
√ 4 3 π-spanner 9
of UDG(V ), and can be
constructed with total communication cost 48n log n = O(n log n) bits.
Notice that the constants here are all artifacts of rough estimation, which can be reduced by careful implementation, and can be much smaller practically.
62 4.3.2 2-localized Delaunay Recently, C˘alinescu [Cˇal03a] presented a localized method such that all wireless nodes collectively find the 2-hop neighbors N2 (u) for every node u in time O(n) by assuming that the geometric location of every node is known. Notice that, knowing the 2-hop neighbors information enables us to quickly construct the 2 local Delaunay triangulation LDel 2 (V ) using O(n) messages communication cost. However, the hidden constant using N2 (u) is much larger than the presented method in previous section. The method [Wan03a] to construct LDel(2) (V ) using O(n) messages is as follows: Algorithm: Construct LDel(2) Locally
1. Every node u collects the location information of N2 (u) based on an efficient method described later. It computes the Delaunay triangulation Del (N2 (u)) of its 2-neighbors N2 (u), including u itself. 2. For each edge uv of Del (N2 (u)), let uvw and uvz be two triangles incident on uv. Edge uv is a Gabriel edge if both angles ∠uwv and ∠uzv are less than π/2 and uv ≤ 1. Node u marks all Gabriel edges uv, which will never be deleted. 3. Each node u finds all triangles uvw from Del (N2 (u)) such that all three edges of uvw have length at most one unit. If angle ∠wuv ≥
π , 3
node u broadcasts a
message proposal(u, v, w) to N1 (u) to form a localized Delaunay triangle uvw in LDel (2) (V ), and listens to the messages from its neighboring nodes. 4. When a node u receives a message proposal(u, v, w), u accepts the proposal of constructing uvw if uvw belongs to Del (N2 (u)) by broadcasting accept(u, v, w) to N1 (u); otherwise, it rejects the proposal by broadcasting reject(u, v, w) to N2 (u). 5. A node u adds the edges uv and uw to its set of incident edges if the triangle
63 uvw is in Del (N2 (u)) and both v and w have sent either accept(u, v, w) or proposal(u, v, w).
First, we review the communication efficient method proposed by Calinescu [Cˇal03a] to collect N2 (u) for every node u when the geometry information is known. Computing the set of 1-hop neighbors with O(n) messages is trivial: every node broadcasts a message announcing its ID. Computing the 2-hop neighborhood is not trivial, as the UDG can be dense. The broadcast nature of the communication in ad hoc wireless networks is however very useful when computing local information. The approach by Gruia [Cˇal03a] is based on the specific connected dominating set introduced by Alzoubi, Wan, and Frieder [Alz02b]. This connected dominating set is based on a maximal independent set (MIS). In the algorithm, each node uses its adjacent node(s) in the MIS to broadcast over a larger area relevant information. Listening to the information about other nodes broadcast by the MIS nodes enables a node to compute its 2-hop neighborhood. The algorithm uses heavily the nodes in the connected dominating set, an example in [Cˇal03a] shows that overloading certain nodes might be unavoidable. We start from the moment the virtual backbone is already constructed, and every node knows the ID and the position of its neighbors. The idea of the algorithm is for every node to efficiently announce its ID and position to a subset of nodes which includes its 2hop neighbors. The responsibility for announcing the ID and position of a node v is taken by the MIS nodes adjacent to v. Each such MIS node assembles a packet containing: , with the ID and position of v, and a counter variable being set to 2. The MIS node then broadcasts the packet. A connector node is used to establish a link in between several pairs of virtuallyadjacent MIS nodes, and will not retransmit packets which do not travel in between these pairs of MIS nodes. The connector node will rebroadcast packets with nonzero counter
64 originated by one of the nodes in a pair of virtually-adjacent MIS nodes, thus making sure the packet advances towards the other MIS node in the pair. Recall that the path in between a pair of virtually-adjacent MIS nodes has one or two connector nodes. When receiving a packet of type , an MIS node checks whether this is the first message with this ID, and if yes decreases the counter variable and rebroadcasts the packet. A node listens to the packets broadcasted by all the adjacent MIS nodes (here it is convenient to assume a MIS is adjacent to itself), and, using its internal list of 1-hop neighbors, checks if the node announced in the packet is a 2-hop neighbor or not - thus constructing the list of 2-hop neighbors. We then prove the following lemma which will be used in Chapter 5.
Lemma 5 [Wan03a] An edge uv is in LDel (2) (V ) iff there is a disk passing through u and v, which does not contain node from N2 (u) ∪ N2 (v) inside. P ROOF. It is trivial that if an edge uv is in LDel (2) (V ) then that kind disk exists, since either uv is a Gabriel edge or uv is an edge from a 2-localized Delaunay triangle. Then we prove the other direction. Assume that there is a disk D1 passing through u, and v, and there is no node from N2 (u) ∪ N2 (v) inside this circle D1 . If uv is a diameter of circle D1 , then it is a Gabriel edge which must be in LDel (2) (V ). Otherwise, let D3 be the disk whose diameter is uv (with center c3 ). Disk D3 must contain some node, say w, inside as shown in Figure 4.2. Disk D1 cannot contain w inside. Assume D1 has center c1 . Let D be a disk centered at some point c on the segment c1 c3 and passing through u and v. Then we can move the center c of disk D along c1 c3 from c1 to c3 and set the radius of D be cu, until the disk touches the first node w from N2 (u) ∪ N2 (v) or becomes D3 . If the disk becomes D3 , then uv is a Gabriel edge and in LDel (2) (V ). Otherwise, the
65
w
D1
c1
u
c3
c2
D2 D3 v
Figure 4.2. Disk D2 Touches a Node w from N2 (u) ∪ N2 (v) disk D touches some node w, which is shown in Figure 4.2 as disk D2 . Then D becomes the circumcircle disk (u, v, w) of u, v and w. Since D2 does not contain any node from N2 (u) ∪ N2 (v) inside, we only need show it is empty from N2 (w) to prove that uvw is a 2-localized Delaunay triangle and thus uv is in LDel (2) (V ). We prove this by contradiction. Assume that there is a node y from N2 (w) inside disk (u, v, w). Clearly, node y cannot be from N2 (u) ∪ N2 (v). Node y must be two hops away from w, otherwise y ∈ N2 (u). In addition, node y cannot be inside the cap defined by arc uwv since uw > 1 and uv > 1. Assume that a node x is one hop neighbor of both y and w. Notice that x cannot be one hop neighbor of u or v, otherwise, y will become the two-hop neighbor of u or v, which is a contradiction to the property of disk D. Then we know that edges uw, uv, vw, xy and xw are shorter than one unit, while edges uy, vy, wy, xu and xv are longer than one unit. There are two cases about the location of node x: on the different side of uv as y and on the same side of uv as y, as shown in Figure 4.3. Clearly, node x is outside of the disk D, otherwise, D will contain a 2-hop neighbor x of u inside (through path uwx). For the first case, we divide the half-space bounded by line uv, which contains w and excludes the cap uwv, into three regions as shown in Figure 4.3 (a). If x is inside the region I, see Figure 4.4 (a) for an illustration. Since xw ≤ 1, uw ≤ 1, and xu > 1, we have ∠xwu > π/3. Thus, ∠xuw < 2π/3. Since xy ≤ 1, xu > 1, and uy > 1, we have ∠yux < π/3. Thus, ∠wuy = 2π − ∠xuw − ∠yux > π,
66
x
I
II
I
u y
y
u
w
c w
II v
III
v
IV
III (a) Different side
(b) Same side
Figure 4.3. Two Cases in The Proof which is impossible. If x is inside the region II, see Figure 4.4 (b) for an illustration. Since xu > 1, yu > 1, and xy ≤ 1, we have ∠xuy < π/3. Similarly, we have ∠uxv < π/3, ∠xvy < π/3, and ∠xvy < π/3. Thus, 2π = ∠xuy + ∠uxv + ∠xvy + ∠xvy < 4π/3, which is a contradiction. When node x is inside region III, the proof is the same as it is in region I.
I
I
x u y
u y
x w
w
II
II
(a) Subcase 1
v
v
III
III (b) Subcase 2
Figure 4.4. Illustration of First Case
For the second case, we further divide it into four subcases when node x is inside
67 region I, II, III, or V. Obviously, ∠uyv + ∠uwv > π and ∠uyv < π/3. Thus, ∠uwv > 2π/3, which implies ∠uvw < π/3. If node x is inside the region I, see Figure4.5 (a) for an illustration. Since ∠uwv > 2π/3, we have ∠wuv < π − ∠uwv < π/3. Notice that ∠wux + ∠wuv > π, so ∠wux > 2π/3. This implies that 1 ≥ wx > ux > 1. It is a contradiction. If node x is inside the region II, see Figure4.5 (b) for an illustration. Here c is the circumcenter of the disk D. Thus, ∠wux > π/2. This implies that 1 ≥ wx > ux > 1. It is a contradiction. When node x is inside the region III, or V, the proofs are similar to the cases II, or I respectively. x
x
II
I
II
I u w
y u
c
III
v
y c
v
IV (a) Subcase 1
III
w
IV (b) Subcase 2
Figure 4.5. Illustration of Second Case Then we know the circumcircle disk (u, v, w) of the triangle uvw does not contain any node from N2 (u) ∪ N2 (v) ∪ N2 (w) inside. Thus uv is in LDel (2) (V ). This finishes the proof.
Notice the property in this lemma is similar to the one of Delaunay triangulation which says an edge uv is in Del(V ) iff there is a disk passing through u and v which does
68 not contain other nodes. 4.4 Localized Delaunay on Virtual Backbone Remember that in Chapter 2 we give a example to show that the virtual backbone CDS maybe a non-planar graph. However, some routing algorithms, such as right hand routing and Greedy Perimeter Stateless Routing (GPSR) [Kar00], ask the topology be planar. Applying the above localized Delaunay triangulation algorithms on the induced CDS (denoted by ICDS), we can get a planar graph called LDel(ICDS) which can be used as new virtual backbone. Moreover, in [Wan02c], we prove that ICDS has a bounded node degree and so does LDel(ICDS). It is proved above that LDel(G) is a spanner if G is a unit disk graph. Notice that ICDS is a unit disk graph defined over all dominators and connectors. Consequently, LDel(ICDS) is a spanner in terms of length. Then in [Wan02c] we prove that the hops stretch factor of LDel(ICDS) is also bounded by a constant. In summary, we combines the connected dominating set and the local Delaunay graph to form the backbone of wireless network. This new topology has these following attractive properties: (1) the backbone is a planar graph; (2) the node degree of the backbone is bounded from above by a positive constant; (3) it is a spanner both for hops and length; moreover, we show that, given any two nodes u and v, there is a path connecting them in the backbone such that its length is no more than 6 times of the length of the shortest path and the number of links is no more than 3 times of that of the shortest-hops path; (4) it can be constructed locally and is easy to maintain when the nodes move around; (5) the total computation cost is at most O(n log n), and the communication cost of each node is bounded by a constant (which means total communication cost is bounded by O(n) messages). Simulation results [Wan02c] show this topology has good practical performance. 4.5 Localized Routing on Localized Delaunay Recently, we are interested in studying the performances of several localized rout-
69 ing protocols on localized Delaunay triangulation. In [Li03e], we prove that the localized Delaunay triangulation almost surely is same as the Delaunay triangulation of the set n of randomly distributed wireless nodes when the transmission range rn satisfies nπrn2 ≥ 4 ln n+c(n) , where c(n) → ∞ as n goes infinity. Notice that, Gupta and Kumar n [Gup98] showed that the unit disk graph is connected with high probability if the transmission range rn satisfies π · rn2 ≥
ln n+c(n) n
for any c(n) with c(n) → ∞ as n goes infinity.
When the unit disk graph is connected, then with high probability, we can construct the Delaunay triangulation Del(V ) by constructing the localized Delaunay triangulation instead. In our experiments [Li03e], several simple local routing heuristics, applied on the localized Delaunay triangulation, have always successfully delivered the packets, while other heuristics were successful in over 90% of the random instances. Moreover, because the constructed topology is planar, a localized routing algorithm using the right hand rule guarantees the delivery of the packets from source node to the destination when simple heuristics fail. The experiments also show that several localized routing algorithms (notably, compass routing [Kra99] and greedy routing) also result in a path whose length is within a small constant factor of the shortest path; we already know such a path exists since the localized Delaunay triangulation is a t-spanner. Bose and Morrin [Bos99] have proposed a method to route the packets using the Delaunay triangulation. Their routing strategy is based on a remarkable proof by Dobkin, Friedman and Supowit [Dob90]. They proved that the Delaunay triangulation is a t-spanner by constructing a path Πdf s (u, v) in Del (V ) with length no more
√ 1+ 5 πuv. 2
The con-
structed path consists of at most two parts: one is some direct DT paths, the other is some shortcut subpaths. Routing the packets along the direct DT path is a localized routing method, but it is not competitive on its own for all Delaunay triangulations. Bose and Morin [Bos99] presented an example such that the distance traveled in this approach could be arbitrarily larger than the minimum. The routing strategy by Bose and Morin uses the direct DT path as long as it is above the x-axis. When the direct DT path lead us to an
70 edge bi bi+1 that intersects uv, it either continue to use the direct DT path or the shortcut to node bj . The difficulty occurs as the strategy does not know prior which of these two paths is shorter. Their solution is to simulate exploring both paths in a parallel manner whenever the first one reaches node bj . However, there are plenty of technique details left to be discussed. In [Li03e], we present a completed localized routing method using the Delaunay triangulation. Basically, we answer the following questions for implementation: (1) how to find the neighbor in the direct DT path locally, (2) how to find the neighbor in the shortcut path locally, and (3) how to determine whether node bj is reached. We call the routing method Delaunay triangulation based routing, denoted by DTR. This routing method guarantees that the distance traveled by the packet is no more than a small constant factor
√ 9(1+ 5) π 2
of the minimum using the property of Delaunay triangulation. It is proved
by Morin [Mor01]. The routing algorithm works as following. Let v0 = u and i = 0. Let node vi+1 be the node returned by E XPLORE(vi ). If vi+1 is not node v, then increase i by one and continue E XPLORE(vi ). The following is the detailed description of the algorithm E XPLORE(vi ). Algorithm: Explore(vi ) Let p0 be the next neighbor of vi in the direct DT path, and q0 be the next neighbor of vi in the shortcut path. Let j = 0 and l0 = min(vi p0 , vi q0 ). Repeat the following exploring until a node, which is on the direct DT path and is above the segment uv, is reached. We denote such node by vi+1 . If vi p0 ≤ vi q0 , we explore the direct DT path first. Otherwise, we have to explore the shortcut path first.
1. E XPLORE DIRECT DT PATH: Route the packet along the direct DT path from node vi until reaching node vi+1 or reaching a node, say pj+1 , such that the distance traveled from p0 to pj+1 is larger than 2lj for the first time.
71 If node vi+1 is reached, return vi+1 and quit. Otherwise, set j = j + 1 and lj be the distance traveled from p0 to pj+1 , and then return to node vi . 2. E XPLORE SHORTCUT PATH: Route the packet along the shortcut path from node vi until reaching node vi+1 or reaching a node, say qj+1 , such that the distance traveled from q0 to qj+1 is larger than 2lj for the first time. If node vi+1 is reached, return vi+1 and quit. Otherwise, set j = j + 1 and lj be the distance traveled from q0 to qj+1 , and then return to node vi .
In [Li03e], We also study the performance of this localized routing method by some simulations in which results show the delivery is guaranteed and the ratio of the length traveled by packet to the minimum is small. 4.6 Summary It is well-known that Delaunay triangulation Del (V ) is a t-spanner of the completed graph K(V ). In this chapter, we define several new structures and then gave a localized algorithm that constructs a graph, namely PLDel (V ). We proved that PLDel (V ) is a planar graph and it is a t-spanner by showing that UDel (V ) is a subgraph of PLDel (V ) (It contains all edges that are both in the unit-disk graph and the Delaunay triangulation of all nodes.). The total number of messages sent by all nodes in our algorithm is O(n log n) bits. Then we show that by applying PLDel (V ) on the connected dominating set we can get a planar virtual backbone with spanner properties for wireless ad hoc network. Also in [Li03e], we prove that the localized Delaunay triangulation almost surely contains the Delaunay triangulation of the set n of randomly distributed wireless nodes. Our experiments [Li03e] showed that the delivery rates of existing localized routing protocols are increased when localized Delaunay triangulation is used instead of several previously proposed planar topologies.
72 CHAPTER 5 BOUNDED DEGREE AND PLANAR SPANNER
Considering the limited storage of the wireless node, we prefer to construct a structure that has a bounded node degree. The structures in Chapter 3 have a bounded degree but it is not guaranteed to be planar graph. The structure localized Delaunay triangulation is a planar graph but it does not have a constant bounded node degree. We then want to combine them to get a planar spanner with bounded node degree. However it is much more difficult than we expect. 5.1 Previous Results The idea to construct a planar spanner with bounded node degree for wireless ad hoc networks has been studied in [Hu91]. Hu [Hu91] used the Delaunay triangulation to configure the wireless network topology such that a planar graph with bounded node degree is computed. A major step in his method is that each node u computes all Delaunay edges whose length is no more than the transmission range. It used the Voronoi diagram of node u to compute all such Delaunay edges. However, this approach will not work always. A simple observation is to determine whether an edge uv belongs to the Delaunay triangulation, we have to check whether there is an empty circle passing u and v. Here a circle is empty if it does not contain any wireless node inside. Obviously, in the worst case, the circumradius of this empty circle could be infinity even when the edge length uv is bounded, implying that we may have to check all nodes. Recently Bose et al. [Bos02c] proposed a centralized O(n log n)-time algorithm that constructs a plan t-spanner for a given nodes set V , for t 10.02, such that the node degree is bounded from above by 27. As we knew, this algorithm is the first method to compute a plane spanner of bounded degree. Their algorithm consists of the following
73 steps.
1. First, it computes the Delaunay triangulation of V , Del(V ), and a degree-3 spanning subgraph BDS(V ) of Del(V ) that includes the convex hull CH(V ) of V . This graph BDS(V ) partitions CH(V ) into (possibly degenerate) simple polygons, such that each node of V is on the boundary of at most three polygons. 2. Then, for each polygon P in the above partition, their algorithm processes the nodes of Del(V ) ∩ P in geometry breadth-first order, and prunes this part of the Delaunay triangulation such that each node of P has low degree. The resulting graph is a plane spanner for the nodes of P in the sense that any two nodes u and v of P are connected by a path whose length is at most a constant times the length of a shortest path between u and v that is completely contained in P . By combining all the spanners for each of the polygons, we get a plane spanner of bounded degree. 3. Run a greedy algorithm in [Gud02] on the plane spanner with bounded degree to bound the total weight from the weight of M ST (V ).
They show that the length stretch factor of the final graph is
(π+1)2π (3 cos π/6)(1+)
and node
degree is at most 27. The running time of their algorithm is O(n log n). However, their method is impossible to have a localized even distributed version, since they use BFS and many operations on polygons (such as degree-3 partitions). Notice that breadth-first-search may take O(n2 ) communications. Hence, we will give a new method for constructing a plane spanner with bounded node degree for a point set V (or completed graph K(V )) in Section 5.2. The basic idea of our method is to combine Delaunay triangulation and the ordered Yao structure [Bos02b]. Then we also give the centralized algorithm for building the spanner for U DG(V ) in Section 5.3, and show that it can be converted to a localized method in Section 5.4.
74 5.2 Centralized Algorithm for Point Set The basic idea of the new algorithm is combining Delaunay triangulation and ordered Yao structure. We propose the algorithm first, then prove its properties. 5.2.1 Construction Algorithm Algorithm: Construct Planar Spanner with Bounded Degree for Point Set V
1. First, it computes the Delaunay triangulation of a set V of n nodes, Del(V ). Let NDel (u) be the neighbors of node u in the Delaunay triangulation Del(V ), and du be the degree of node u in Del(V ). By proper data structure, NDel (u) and du can be achieved in time O(n). 2. Find an order π of V as follows. Let G1 = Del(V ) and dG,u be the node degree of u in graph G. Remove the node u with the smallest value of (dGi ,u , ID(u)) from Gi , let πu = n − i + 1, and call the remaining graph Gi+1 . Repeat this procedure for 1 ≤ i ≤ n. Let πun = 1. Let Pv denote the predecessors of v in π, i.e., Pv = {u ∈ V : πu < πv }. Notice since Gi is always a planar graph, we know that the smallest value of dGi ,u is at most 5. Then, in ordering π, node u at most have 5 edges to its predecessors Pu in Del(V ). 3. Let E be the edge set of Del(V ), E be the edge set of the desired spanner. Initialize E to be empty set and all nodes in V are unprocessed. Then, for each node u in V , following the increasing order π, run the following steps to add some edges from E to E (we only consider the Delaunay neighbors NDel (u) of u): (a) We use v1 , v2 , · · · , vk to denote the predecessors of node u (see Figure 5.1). Notice that u can have at most 5 edges to its predecessors (processed Delaunay neighbors) in E, i.e., k ≤ 5. Then there are k ≤ 5 open sectors at node u whose boundaries are rays emanated from u to the processed neighbors vi of u
75 in Del(V ). For each such sector at u, we divide it into a minimum number of open cones of degree at most α, where α ≤ π/2 is a parameter. (b) For each such cone, let s1 , s2 , · · · , sm be the geometrically ordered neighborhood NDel (u) of u in this cone. That is, s1 , s2 , · · · , sm are all unprocessed nodes that are connected by some edges of E to u in this cone. For this cone, we first add the shortest edge in E that is connected to u to the edge set E , then add to E all the edges (sj , sj+1 ), 1 ≤ j < m. (c) Mark node u processed. Repeat this procedure in the increasing order of π, until all nodes are processed. The final graph formed by edges E is denoted by BP S0 (V ). 4. Run the greedy spanner algorithm by [Gud02] to bound the weight of the graph.
Notice that in the algorithm we use open sectors, which means that in the algorithm we do not consider adding the edges on the boundaries (any edge involved previously processed neighbors). For example, in Figure 5.1, the cones do not include any edges uvi . This guarantee the algorithm does not add any edges to node vi after vi has been processed. This approach, as we will show it later, bounds the node degree. 5.2.2 Analysis of Algorithm First, we prove the connectivity of this final topology. Theorem 22 [Li03d] The graph BP S0 (V ) is a connected graph. P ROOF. The proof is trivial. since the Del(V ) is a connected graph, we only need to show for each edge uv ∈ Del(V ), there is an path in BP S(V ) connecting u and v. Assume πu < πv , then node u will be processed before node v, and v is unprocessed neighbor of u. In the above algorithm, for node u, it connects its unprocessed neighbor si by adding edge
76
s 1 v4 s3 s 2 v5 v1
u
v3
v2
Figure 5.1. Constructing BP S0 for Point Set: Process Node u si u (choosing si as shortest neighbor in the cone), or adding path si , si+1 , ..., sj , u (when sj is the nearest neighbor). So there must be a path from u to v in BP S0 (V )
To show the node degree of the generated graph is bounded by a constant, we prove following theorem. . Theorem 23 [Li03d] The maximum node degree of BP S0 (V ) is at most 19 + 2π α
P ROOF.
Notice that for a node u there are two cases that an edge uv is added to the
BP S0 (V ), let us discuss them one by one. Case 1: When we process node u, some edges uv have already been added by some processed nodes w before. There are two subcases for this case. Subcase 1.1: The edge uv has been added by a processed node v (w = v). For example, in Figure 1 , node u has edges from v2 , v3 and v5 before it is processed. For each predecessor v, it only adds one edge to node u. Subcase 1.2: The edge uv has been added by processed node w (w is not v), node v is also an unprocessed node when processing w. For example, in Figure 1 , node s2 have
77 edges from s1 and s3 added by processing node u before node s2 is processed. Notice that both v and u are neighbors of this processed node w. For each predecessor w, it at most adds two edges to node u. Because for each u, it can only have at most 5 predecessor neighbors (processed neighbors), and each of predecessor can at most add 3 edges to it (either Subcase 1.1 or Subcase 1.2, or both). Thus, the number of this kind of edges (edges added by its predecessors before u is processed) is bounded by 15. Case 2: When node u is processed, we can add one edge uv for each cones. Since we have at most 5 sectors emanated from u and each cone must have angle at most α, it is cones at u. So the number of this kind of easy to show that we can at most have 4 + 2π α . edges is also bounded by 4 + 2π α Notice that after node u is processed, no edges will be added to it. Consequently, in the final structure. the degree of each node u is bounded by 19 + 2π α For example, when α = π/2, then the maximum node degree is at most 23; when α = π/3, then the maximum node degree is at most 25. Either case improves the previous bound 27 on the maximum node degree by Bose et al. [Bos02c]. It is trivial that BP S0 (V ) is a planar graph. Since Del(V ) is a planar graph and the algorithm only adds the Delaunay edges to BP S0 (V ). Notice that all edges si si+1 are also in Del(V ) since si and si+1 are consecutive Delaunay neighbors of node u. Finally, we prove that the graph BP S0 (V ) is a spanner. Theorem 24 [Li03d] BP S0 (V ) is a t-spanner, where t = max{ π2 , π sin α2 + 1} · Cdel . P ROOF. First, remember that Del(V ) is a spanner with a constant length stretch factor Cdel =
√ 4 3 π 9
≈ 2.42. Keil and Gutwin [Kei92] proved it using induction on the order of
78 the lengths of all pair of nodes (from the shortest to the longest). We can show that the path connecting nodes u and v constructed by the method given in [Kei92] also satisfies that all edges of that path is shorter than uv. So if we can prove this claim: for any edge uv ∈ Del(V ), there exists a path in BP S0 (V ) connecting u and v whose length is at most a constant times uv, then we know BP S0 (V ) is a · Cdel -spanner. Then we prove the above claim. Consider an edge uv in Del(V ). If uv ∈ BP S0 (V ), the claim holds. So assume that uv ∈ / BP S0 (V ). Assume w.l.o.g. that πu < πv . It follows from the algorithm that, when we process node u, there must exist a node v in the same cone with v such that uv > uv , uv ∈ BP S0 (V ), and ∠v uv < α ≤ π/2. Let v = s1 , s2 , · · · , sk = v be this sequence of nodes in the ordered unprocessed neighborhood of u from v to v. Same with the proof in [Bos02c], consider the polygon P , consisting of nodes u, s1 , · · · , sk . We will show that the path s1 s2 · · · sk has length that is at most a small constant factor of the length uv. Let us consider the shortest path from s1 to sk that is totally inside the polygon P . Let S(s1 , sk ) denote such path. This path consists of diagonals of P .For example, in 5.2 , S(s1 , sk ) = s1 s7 s9 . Assume that uv = x. Let w be the point on segment uv such that uw = uv . Assume that uv = y, then wv = y − x. Notice that node v is the closest Delaunay neighbors in such cone. Obviously, all Delaunay neighbors si in this cone is outside of the sector defined by segments uw and uv . We will show that such path S(s1 , sk ) is contained inside the triangle ws1 sk . First, if no Delaunay neighbors is inside ws1 sk , then S(s1 , sk ) = s1 sk . Thus, the claim trivially holds. If there is some Delaunay neighbors inside ws1 sk , then s1 will connect to the one Si forming the smallest angle ∠us1 sj . Similarly, node sk will connect to the one sj forming the smallest angle ∠usk sj . Obviously si and sj are inside ws1 sk , thus, the shortest path connecting them is also inside ws1 sk .
79 Since path S(s1 , sk ) is the shortest path inside the polygon P to connect s1 and sk , by convexity, the length of S(s1 , sk ) is at most v w + wv = 2x sin 2θ + y − x. Here θ = ∠v uv < α.
s1 (si ,v’) s2 x
u
s3
S P s7 (sj )
θ x
w
s4 s5 s6
D s8
s9 (sk , v)
y-x
Figure 5.2. Shortest Path in Polygon P
An edge si sj of S(s1 , sk ) has endpoints si and sj in the neighborhood of u. Let D(si , sj ) be the sequence of edges between si and sj in the ordered neighborhood of u, which are added by processing u. For example, in 5.2 , D(s1 , s7 ) = s1 s2 s3 s4 s5 s6 s7 . This path is in BP S(V ). We can bound the length of D(si , sj ) by π/2si sj by the argument in [Bos02c, Bos99]. In [Bos99], it is shown that the length of D(si , sj ) is at most π/2 times si sj , provided that (1) the straight-line segment between si and sj lies outside the Voronoi region induced by u, and (2) that the path lies on one side of the line through si and sj . In other words, we need D(si , sj ) to be one-sided Direct Delaunay path 5 [Dob90]. In [Bos02c], they showed 6 that both these two conditions hold when ∠si usj < π/2. This For any pair of nodes u and v, let u = w1 , w2 , · · · , wk = v be the sequence of nodes whose Voronoi region intersect segment uv and the Voronoi regions at wi and wj share a common boundary segment. Then the Direct Delaunay path DT (u, v) is w1 w2 · · · wk . 5
6
Firstly, the Voronoi region centered at u will not intersect the segment si sj . This can be proved by showing that up > max{si p, sj p} for any point p on segment si sj , which is due to ∠usi p + ∠usj p > ∠si up + ∠sj up = ∠si usj . Notice that ∠si usj < α ≤ π/2. Secondly, the path D(si , sj ) is on one-side of si sj because it is part of the shortest path connecting s1 and sk . Thirdly, the path D(si , sj ) is Direct Delaunay path DT (si , sj ). This can be proved by showing that V or(sq ) intersects the segment si sj for any i ≤ q ≤ j. This is obvious since the circumcenter (belonging to V or(sq )) of any triangle usq−1 sq is on the same side of si sj as u.
80 is trivially satisfied since ∠si usj < α ≤ π/2. Thus, we have a path us1 s2 · · · sk to connect u and v with length at most θ + y − x) · π/2 2 π x α π ≤y · ( + · (π sin − + 1)) 2 y 2 2 α π ≤y · max{ , π sin + 1} 2 2 x + (2x sin
Putting it all together, we know BP S0 (V ) is a spanner with length stretch factor at most max{ π2 , π sin α2 + 1} · Cdel . For example, when α = π/2, then the spanning ratio is at most (
√ 2π +1)·Cdel ; 2
when
α = π/3, then the spanning ratio is at most ( π2 +1)·Cdel ; when α = 2 arcsin( 12 − π1 ) 20.9o , then the spanning ratio is at most
π 2
· Cdel . We expect to further improve the bound on the
spanning ratio by using the following property: all such Delaunay neighbors si is inside the circumcircle of the triangle uvv ; see 5.2. Notice that, the method by Bose et al. [Bos02c] actually achieves the same spanning ratio as this one, although they did not prove this. However, the node degree of the graph generated by our method is smaller than that by [Bos02c]. Notice that the time complexity of our centralized algorithm is O(n log n) too. We can build Delaunay triangulation in O(n log n), and do ordering in time O(n log n) (using heap for the ordering based on degrees), and Yao structure in O(n) (each edge is processed at most a constant times and there are O(n) edges to be processed). When using heap for the ordering, initially building a heap needs O(n log n), then we remove one node and it has at most 5 adjacent edges, it needs at most 5 times updating the heap based on degree (each of which can be done in time O(log n)). So the ordering can be done in O(n log n). Consequently, the time complexity is O(n log n), same with the method by Bose et al. [Bos02c]. However, our algorithm has smaller bounded node degree, and (more
81 importantly) our algorithm has potential to become a localized version for wireless ad hoc networks application as we will describe later. 5.3 Centralized Algorithm for Unit Disk Graph In this section we give two centralized algorithms to construct planar spanner with bounded degree for U DG(V ). 5.3.1 Centralized Algorithm 1 Algorithm 1: Construct Planar Spanner with Bounded Degree for U DG(V )
1. Same with the algorithm for point set, first, compute Delaunay triangulation Del(V ). 2. Removing the edges whose length is longer than 1 in Del(V ). Call the remaining graph unit Delaunay triangulation U Del(V ). For every node u, we know its unit Delaunay neighbors NU Del (u) and its node degree du in U Del(V ). 3. Then, same with the algorithm for point set, find an order π of V as follows: Let G1 = U Del(V ) and dG,u is the node degree of u in graph G. Remove the node u with the smallest value of (dGi ,u , ID(u)) from Gi , let πu = n − i + 1, and call the remaining graph Gi+1 . Repeat this procedure for 1 ≤ i ≤ n. Obviously, in ordering π, node u at most have 5 edges to its predecessors Pu in U Del(V ). 4. Let E and E be the edge sets of U Del(V ) and the desired spanner. Initialize E = ∅ and all nodes in V are unprocessed. Then, same with the algorithm for point set, for each node u in V , following the increasing order π, run the following steps to add some edges to E : (a) Node u uses its predecessors (processed Unit Delaunay neighbors) in E to define at most 5 open sectors at node u (see Figure 5.3, v1 , · · · , v5 are the processed neighbors of node u in U Del(V )). For each sector, we divide it into a
82 minimum number of open cones of degree α, where α ≤ π/3. (b) For each cone, let s1 , s2 , · · · , sm be the ordered neighbors NU Del (u) of u in this cone. That is, s1 , s2 , · · · , sm are all unprocessed nodes that are connected by an edge of the unit Delaunay triangulation to u. For each cone, first add the shortest edge in E that is adjacent to u to the edge set E , then add to E all the edges sj sj+1 between its geometrically ordered unprocessed neighbors in this cone, 1 ≤ j < m. Notice that, here such edges sj sj+1 are not necessarily in U Del(V ). For example, when node u has a Delaunay neighbor x such that ux intersects edge si si+1 and ux > 1. (c) Mark node u processed. Repeat this procedure in order of π, until all nodes are processed. Let BP S1 (V ) denote the final graph formed by edge set E . s 1 v4 s3 s 2 v5 v1
u
v3
v2
Figure 5.3. Constructing BP S1 for U DG: Process Node u
5.3.2 Centralized Algorithm 2 Algorithm 2: Construct Planar Spanner with Bounded Degree for U DG(V )
1. Run the algorithm for point set to build BP S0 (V ) with parameter α ≤ π/3.
83 2. Removing the edges whose length is longer than 1 in BP S0 (V ). The final graph is denoted by BP S2 (V ). Notice that in both these algorithms for U DG(V ), we change the cone angle bound from π/2 to π/3. The reason is in the proof of spanner property we need to guarantee the edge si sj and vv must be in U DG(V ), i.e., si sj ≤ 1 and vv ≤ 1. The constructed graphs BP S1 (V ) and BP S2 (V ) could be different since (1) the ordering of nodes could be different; (2) BP S1 (V ) could add some edges (some si si+1 type edges) that do not belong to the unit Delaunay triangulation U Del(V ) = Del(V ) ∩ U DG(V ), while BP S2 (V ) always uses the edges from U Del(V ). 5.3.3 Analysis of Algorithm First, we prove the connectivity of both final topologies BP S1 (V ) and BP S2 (V ). Though later theorems about spanner properties will imply the following two theorems, for completeness, we give short proofs for them here.
Theorem 25 [Li03d] Graph BP S1 (V ) is connected if U DG(V ) is connected graph.
P ROOF. Because U Del(V ) is a connected graph when U DG(V ) is a connected graph [Li02a], we only need to show for each edge uv ∈ U Del(v), there is an path in BP S1 (V ) connecting u and v. Assume πu < πv , then node u will be processed before node v, and v is unprocessed neighbor of u. In the first algorithm, for node u, it either connects its unprocessed neighbor v by adding edge vu (if v is the closest neighbor in the cone), or adding path v = si , si+1 , · · · , sj , u (when sj is the nearest neighbor). Notice that, since the cone has angle at most π/3 and usi , usi+1 both have length at most one, then si si+1 ≤ 1. Thus, edge si si+1 is in U DG(V ) (although it may be not in U Del(V )). So there is a path from u to v in BP S1 (V ). This finishes the proof.
84 Theorem 26 [Li03d] Graph BP S2 (V ) is connected if U DG(V ) is connected graph.
P ROOF. We prove it by showing that for every uv ≤ 1 we can find a path in BP S2 (V ) to connect them. If uv ∈ BP S2 (V ), claim holds. Otherwise, there is a node v in the same cone with v in BP S2 (V ). Then we consider the spanner path D(v , v) constructed in BP S2 (V ) (in our spanner proof of Theorem 24), we show that it does not have edges longer than one unit. This is trivial. Notice that the angle of cone is at most π/3 here. Thus, for any two nodes si and sj , si sj < uv ≤ 1. From the proof given by Keil and Gutwin [Kei92], we know all the edges in the Directed Delaunay path D(si , sj ) constructed in BP S2 (V ) have length at most si sj . Consequently, they all have length at most one. Thus, path D(v , v) survives after removing long edges. This finishes the proof.
The bounded node degree properties of these two final structures are trivial. The proof is similar to the one for point set. Only difference is that the angle of open cone is α ≤ π/3 instead of α ≤ π/2. Notice that node degree is bounded by 25 if α = π/3.
Theorem 27 [Li03d] The maximum degrees of the graphs BP S1 (V ) and BP S2 (V ) are at most 19 + 2π . α Since BP S2 (V ) is a subgraph of planar graph BP S0 (V ), it must be a planar graph. So we only need to prove the graph BP S1 (V ) is a planar graph.
Theorem 28 [Li03d] The graph BP S1 (V ) is a planar graph.
P ROOF. Observe that U Del(V ) is a planar graph. When each node u is being processed, we add two kinds of edges: (1) edge usi , where si is the nearest unprocessed node in some
85
si w
u
Q
si+1 Figure 5.4. No New Edges Can Be Added by Other Nodes to Intersect si si+1 cone divided by u; (2) some edges si si+1 , when si and si+1 are consecutive unprocessed neighbors of u in graph U Del(V ). See Figure 5.3 for illustration. These edges usi belong to U Del(V ), so they will not intersect each other. If edge si si+1 is in U Del(V ), then it will not break the planar property of the graph also. Otherwise, the only possible reason / U Del(V ) is that there are some edges (such as uw in Figure 5.4) which makes si si+1 ∈ in Del(V ) between usi and usi+1 with length longer than 1. Then all such endpoints w of these long edges and si , sj , u will form a polygon, denoted by Q, in U Del(V ). We will show that after si si+1 is added no intersecting edges can be added in BP S1 (V ). Notice that all the edges which are possible to add in BP S1 (V ) must be diagonals of some polygons in U Del(V ). However, all the diagonals of polygon Q intersecting si si+1 are longer than 1, as uw is, i.e., they will never be considered by our algorithm. Consequently, adding edge si si+1 will not break the planar property. This finishes our proof.
Finally, we prove BP S1 (V ) and BP S2 (V ) are spanners.
Theorem 29 [Li03d] BP S1 (V ) is a · Cdel -spanner, where = max{ π2 , π sin α2 + 1}.
P ROOF. Keil and Gutwin [Kei92] showed that the Delaunay triangulation is a t-spanner for a constant Cdel =
√ 4 3 π 9
using induction on the increasing order of the lengths of all
pair of nodes. We can show that the path connecting nodes u and v constructed in [Kei92]
86 also satisfies that all edges of that path is shorter than uv. Consequently, for any edge uv ∈ U DG(V ) we can find a path in UDel (V ) with length at most a t =
√ 4 3 π 9
times uv,
and all edges of the path is shorter than uv. So we only need to show that for any edge uv ∈ U Del(V ), there exists a path in BP S1 (V ) between u and v whose length is at most a constant times uv. Then BP S1 (V ) is a · Cdel -spanner. Consider an edge uv in U Del(V ). If edge uv is in BP S1 (V ), the claim trivially holds. Then consider the case uv ∈ / BP S1 (V ). The rest of the proof is similar to the proof of Theorem 24. There must exist a node v in the same cone with v such that uv > uv , uv ∈ BP S1 (V ), and ∠v uv < α ≤ π/3. Let v = s1 , s2 , · · · , sk = v be the sequence of nodes in the ordered unprocessed neighborhood of u in U Del(V ) from v to v. Let v = w1 , w2 , · · · , wk = v be the sequence of nodes in the ordered unprocessed neighborhood of u in Del(V ) from v to v. Obviously, the set {s1 , s2 , · · · , sk } is a subset of {w1 , w2 , · · · , wk }. Similar to Theorem 24, we know that the length of the path uw1 w2 · · · wk to connect u and v with length at most max{ π2 , π sin α2 + 1} · uv, where w1 = s1 is the nearest neighbor of u in the cone, and wk = v. Since any such node wi is not inside the polygon Q (defined in the Figure 5.4 of proof for Theorem 28), the path us1 s2 · · · sk is not longer than the length of path uw1 w2 · · · wk . This finishes the proof.
Theorem 30 [Li03d] BP S2 (V ) is a · Cdel -spanner, where = max{ π2 , π sin α2 + 1}. P ROOF. Since BP S2 (V ) is a subgraph of BP S0 (V ), by removing edges longer than one, and BP S0 (V ) is a spanner, we only need to prove the spanner path D(v , v) constructed in BP S2 (V ) (in our spanner proof) does not have edges longer than one for each u and v if uv ∈ U DG(V ). This is trivial. Since the angle of cone is π/3 here, si sj < uv ≤ 1. From
87 the proof given by Keil and Gutwin [Kei92], we know all the edges in the spanner path D(si , sj ) constructed in BP S2 (V ) are bounded by si sj . Consequently, they all have length at most one. So the spanner path D(v , v) survives after removing long edges. This finishes the proof. Notice that the computation costs of both algorithms are O(n log n). 5.4 Localized Algorithm for Unit Disk Graph The centralized algorithms can be extended to a localized algorithm [Wan03a]. The basic idea is as follows: first construct the localized Delaunay triangulation (LDel),a planar spanner, for UDG; then build a local order based on node degree in LDel; finally apply the same technique in previous algorithms to bound the node degree following the local order. In Section 4.3.2, we have described a localized algorithm that can construct a planar spanner LDel(2) using O(n) messages for wireless ad hoc networks when every node has the same maximum transmission range. However, some node in structure LDel(2) could have degree as large as O(n). We then give an efficient method to bound the node degree. 5.4.1 Construction Algorithm Algorithm: Locally Construct Planar Spanner with Bounded Degree for U DG(V ) 1. First, compute the planar localized Delaunay triangulation LDel(2) (V ) using the algorithm in Section 4.3.2, so that every node u knows its neighbors NLDel(2) (u) and its node degree du in LDel(2) (V ). 2. Build a local order π of V as follows: (Every node u initializes πu = 0, i.e., unordered.) (a) For node u with πu = 0, if its degree du ≤ 5 then node u queries 7 each node v, 7
If all unordered neighbors with dv ≤ 5 has larger ID, we call such query round a
88 from its unordered neighbors, the current degree dv . If node u has the smallest ID among all unordered neighbors v with dv ≤ 5, node u sets πu = max{πv | v ∈ NLDel(2) (u)} + 1, and broadcasts πu to its neighbors NLDel(2) (u) in LDel(2) (V ). (b) If node u receives a message from its neighbor v saying that πv = k, it updates its du = du − 1 and also updates the order πv stored locally. So du represents how many neighbors are not ordered so far. If node u finds that du ≤ 5 and πu = 0, it goes to Step 2 (a). When node u finds that du = 0 and πu > 0, it can go to step 3. 3. Build structures based on local order π as follows: (Initialize all nodes unprocessed) (a) If a unprocessed node u has the highest local order in its unprocessed neighbors Nu in LDel(2) (V ), let k be the number of processed neighbors
8
of u in
LDel(2) (V ). Node u divides its transmission range to k open sectors cut by the rays from u to these processed neighbors. Then for each sector apexed at u, divide it into a minimum number of open cones of degree at most α with α ≤ π/3. For each cone, let s1 , s2 , · · · , sm be the ordered unprocessed neighbors of u in NLDel(2) (u). For this cone, node u first adds an edge usi , where si is the nearest neighbor in s1 , s2 , · · · , sm . Node u then tells s1 , s2 , · · · , sm to add all the edges sj sj+1 , 1 ≤ j < m. Node u marks itself processed, and tells all nodes in NLDel(2) (u) that it is processed. (b) If a unprocessed node v receives a message for adding edge vv from its neighbor u, it adds edge vv . failed round. Node u performs a new round of queries only if it finds that the unordered neighbors have been reduced from previous failed round. 8
There are at most 5 processed neighbors since graph LDel(2) (V ) is planar.
89 4. When all nodes are processed, the final network topology is denoted by BP S(V ).
5.4.2 Analysis of Algorithm We first show that the algorithm does process all nodes. First of all, the algorithm cannot stop at stage of ordering nodes locally. This can be shown by contradiction. Assume that some nodes are unordered. The graph formed by these unordered are planar, and thus it contains some nodes with at most 5 unordered neighbors. Among these nodes, the node with the smallest ID will perform step 2 (a), thus reducing the number of unordered nodes consequently. Notice that the ordering computed by our method is not a total ordering. Some nodes may have the same order. However, no two neighboring nodes in LDel(2) (V ) receive the same order. Thus, after all nodes are ordered, the algorithm will process all nodes. Observe that the algorithm does not process two neighboring nodes at the same time. Assume that there are two nodes, say u and v, are processed at the same time. Remember that we process a node only if it has the highest ordering among its unprocessed neighbors. Thus, nodes u and v must receive the same order, i.e., πu = πv , which is impossible in our ordering method. Additionally, remember that our algorithm checks if du ≤ 5 for computing an ordering locally. Here number 5 can be replaced by any integer larger than 5. Using larger integer may make the algorithm run faster, but on the other hand, it worsens the theoretical bound on the node degree. It is not difficult to show that the constructed topology is still connected and has bounded node degree. Proofs are similar with BP S0 (V ) and BP S1 (V ), which are omitted here due to space limit. Notice that, the algorithms [Bos02c, Li03d] always add the edges in the Delaunay
90 triangulation to construct a bounded degree planar spanner for a set of points. Thus, the planarity of the final structure is straightforward. The algorithm we proposed in Section 5.3 may add some edges (such as edges si si+1 added in step 4(b) of the first algorithm in Section 5.3) that do not belong to the U Del(V ). To prove the planarity of the structure BP S1 (V ), we show that no two added diagonal edges intersect. The property that edges, which possibly intersect si si+1 in the centralized algorithm, are all Delaunay edges is crucial in the proof of Theorem 28. This property does not hold anymore in the localized algorithm. We will show that BP S(V ) is a planar graph using a different approach.
Theorem 31 [Wan03a] Graph BP S(V ) is a planar graph. P ROOF. Notice that the algorithm in Section 5.4 only adds the edges in LDel(2) (V ) or edge si si+1 such that usi and usi+1 are edges of LDel(2) (V ) and si , si+1 are consecutive neighbors of u in LDel(2) (V ) and ∠si usi+1 < π/3. We call such edge si si+1 the diagonal edge of the graph LDel(2) (V ). Clearly, these diagonal edges cannot intersect any edge from LDel(2) (V ). Thus, the only possible intersections in BP S(V ) are caused by some diagonal edges. See Figure 5.5 (a) for an illustration of such two intersected diagonal edges uy and vx. Assume that ∠uyv < ∠uxv. Then y is outside of the circumcircle disk (u, v, x) of the triangle uvx. If the disk disk (u, v, x) does not contain a node from N2 (x) ∪ N2 (v) inside, then edge xv belongs to the graph LDel(2) (V ). This is a contradiction to the fact that edges vu and vy are neighboring edges in graph LDel(2) (V ). Thus, there must have some node, say z, from N2 (x) ∪ N2 (v) inside the disk disk (u, v, x). If the node z is inside the region II, then z cannot be from N2 (v). Otherwise, we cannot find an empty circle passing through u and v that is free of nodes of N2 (u) ∪ N2 (v) inside. This contradicts to the fact edge uv belongs to the graph LDel(2) (V ). Thus, node z must be from N2 (x), but not from N1 (x) (otherwise z ∈ N2 (v) again). Assume that there
91
x I u
y III
x
y
I v
III
u
c
z0
c II
II
v
D0
D (a) Two diagonal edges uy and vx intersect
(b) z0 belongs to the sector uvy
Figure 5.5. Illustrations of Planar Proof is a 2-hop path xwz connecting x and z. We then show that w ∈ disk (u, v, x). If node w is inside the region I or III, then uw ≤ 1. Thus, any circle passing through u and v will contain w or z inside. Since w ∈ N1 (u) and z ∈ N2 (u), edge uv cannot belong to graph LDel(2) (V ). It is a contradiction. Similarly, if node w is inside the region II, nodes x and w will cause a contradiction to the fact uv ∈ LDel(2) (V ). Thus node w ∈ / disk (u, v, x). Then similar to the proof of Lemma 5, we can show that it is impossible to have node z ∈ N2 (x) in region II. Similarly, region I cannot contain any node from N2 (u) ∪ N2 (x). Thus, only region III can possibly contain some node z inside. Then vz ≤ 1. This is proved as follows: if z is inside the triangle vux, it is obvious since the three sides of this triangle have length at most 1; if z is inside the cap defined by arc xv, vz ≤ vx since ∠vux < π/3. Let c be the circumcenter of disk disk (u, v, x). Let D be a disk passing through v with center on the segment vc. Clearly, D is inside the disk disk (u, v, x). Among all such disks, we find the largest disk D0 that is empty of node inside, i.e., the disk that passing through some node z0 , and node v. Then edge vz0 belongs to graph LDel(2) (V ). We then show that z0 must belong to the sector uvy. If z0 is inside the cap cut by segment vy, then any disk passing through v and y will contain u or z0 inside since ∠yuv + ∠yz0 v > π. It
92 contradicts to the existence of edge vy in graph LDel(2) (V ). As shown in Figure 5.5 (b), if z0 belongs to the sector uvy, and vz0 ∈ LDel(2) (V ), then y and u cannot be consecutive neighbors of v in LDel(2) (V ). It is a contradiction.
Theorem 32 [Wan03a] BP S(V ) is a t-spanner, where t = max{ π2 , π sin α2 + 1} · Cdel . P ROOF. We only need to show that for any edge uv ∈ U Del(V ), there exists a path in BP S(V ) between u and v with length at most uv. Then BP S(V ) is a · Cdel -spanner. We prove the above claim. Consider an edge uv in U Del(V ). If uv ∈ BP S(V ), the claim holds. So assume that uv ∈ / BP S(V ). Assume w.l.o.g. that πu > πv . It follows from the algorithm that, when we process node u, there must exist a node x in the same cone with v such that uv > ux, ux ∈ BP S(V ), and ∠xuv < α ≤ π/3. There is two cases: ux is in U Del(V ) or not. Case 1: ux ∈ U Del(V ). We will show that no edges other than Delaunay edges are added to u between ux and uv. Then we can use the same proof in Theorem 29 to prove that there is a path in BP S(V ) to connect u and v with length at most max{ π2 , π sin α2 +1}·uv. Let w1 , w2 , · · · , wm be the sequence of Delaunay neighbors of u in Del(V ) from v to x. See Figure 5.6 (a) as illustrations. First all the neighbors wi should be inside the circumcircle disk (u, v, x) of the triangle uvx, since otherwise any circle passing through u and wi will contain either x or v inside which is a contradiction with the fact uwi is Delaunay triangle. Then we prove all the edges wi wi+1 are shorter than one unit. Remember that uv ≤ 1, ux ≤ 1 and ∠xuv ≤ π/3, then we have xv ≤ 1. If wi and wi+1 are both inside the triangle vux or the cap cut by segment vx, wi wi+1 < 1. Therefore, the only case that edge wi wi+1 is longer than one unit is shown in Figure 5.6 (b). Assume that wi wi+1 > 1. Since xwi+1 < 1 and xwi < 1, we have ∠wi wi+1 x < π/2. Thus, ∠xuv + ∠wi wi+1 x < π/3 + π/2 < π. It implies node x is inside
93 the circumcircle disk (u, wi , wi+1 ). This is a contradiction and finishes the proof of no long edges among all the edges wi wi+1 . Thus, we know all edges wi wi+1 ∈ U Del(V ), in addition, they are also in LDel(2) (V ). Therefore we can not have an additional edge uy added to LDel(2) (V ) in sector vux, since such edge breaks the planar property of LDel(2) (V ). See Figure 5.6 (a) as illustrations. v
v w1 wi
u
y
w i+1
u
wi
wi+1
wm x
(a) Only Delaunay edges added to u
x
(b) No edge wi wi+1 longer than one
Figure 5.6. Illustrations of Delaunay Neighbors of u
Case 2: ux ∈ / U Del(V ). Assume ux is added to LDel(2) (V ) in the sector w1 uw2 , where w1 and w2 are consecutive Delaunay neighbors of node u. There are three cases for Delaunay edges w1 u and w2 u. We prove that all of them do not exist by contradiction. Subcase 2.1: both edges w1 u and w2 u are no more than one unit, shown in Figure 5.7 (a). From the property of Delaunay, x must be outside of the circumcircle disk (u, w1 , w2 ) of the triangle uw1 w2 . Thus, ∠uw1 x + ∠uw2 x > π. Any circle passing though u and x will contain either w1 or w2 inside. Notice that w1 , w2 ∈ N1 (u). It contradicts to the existence of edge ux in LDel(2) (V ). Subcase 2.2: both edges w1 u and w2 u are longer than one unit, shown in Figure 5.7 (b). Since uw1 > 1 ≥ ux, ∠uw1 x < π/2. Similarly, ∠uw2 x < π/2. Then we
94 v w1
1
>1
>1
uw2 . Therefore, there is no edge from U Del(V ) in downside of ux, which selects ux as the shortest neighbor. Then assume an edge uv ∈ U Del(V ) in upper-side is in the same cone as ux and is longer than ux. Since uv ≤ 1, ux ≤ 1 and ∠vux < π/3, we have vx ≤ 1. Notice that w1 ∈ / uvx because of uw1 > 1. Again from the property of Delaunay, v and x must be outside of the circumcircle disk (u, w1 , w2 ). It implies that ∠vw1 x + ∠vux > π. Thus, ∠vw1 x > π − ∠vux > 2π/3. Then 1 ≥ vx > xw1 > 1 causes a contradiction.
95 Therefore Subcase 2.3 shown in Figure 5.7 (c) does not exist also. Consequently, it is impossible that any node u will add an edge ux ∈ / U Del as the shortest link to BP S(V ) in a cone that has some edges uv from U Del. Together with proof of Case 1, it finishes our proof of spanner property of BP S(V ).
Theorem 33 [Wan03a] The algorithm in Section 5.4.1 uses at most O(n) messages, where each message has O(log n) bits.
P ROOF.
Notice that it was shown in [Cˇal03a] that we can collect the 2-hop neighbor
information for all nodes using total C1 · n messages. Constant C1 here is at most 675. This constant can be improved by a tighter analysis. The communication cost of building LDel(2) is C2 · n since every node only has to propose at most 6 triangles and each propose is replied by two nodes. Constant C2 here is at most 18. The second step (local ordering) takes C3 · n messages, since processing every node u only causes following broadcasts: (1) node u queries at most 5 times, when its d(u) is decreased and 1 ≤ d(u) ≤ 5; (2) some nodes v reply u’s queries, the total number of this kind of replies is at most i = 15 times, where 1 ≤ i ≤ 5 and (3) node u claims its new order after it was ordered. Notice that, since node u queries at most i ≤ 5 unordered nodes in its ith query, only these i nodes reply it in that query round. Constant C3 here is at most 5 + 15 + 1 = 21. The third step (bounded degree) also takes C4 · n messages, because every node only broadcasts two kind of messages: (1) tells its neighbors to add some edges, and (2) claims that it is processed. The total messages of telling neighbors to add some edges is 12n since the total added edges is at most 3n from the planar property of the final topology. Notice that each edge uv in the final topology can be added due to at most 4 messages of
96 adding edges (2 from the endpoints u and v, 2 from the two nodes beside the edge uv). Plus the second kind of messages (once per node), the constant C4 here is at most 12 + 1 = 13. Thus, the total communication cost is bounded by O(n) where the constant can be at most C1 + C2 + C3 + C4 = C1 + 18 + 21 + 13 = C1 + 52 = 675 + 52 = 727. Here most comes from the slack analysis of collecting N2 (u).
In addition, it is easy to show that the computation cost of each node is at most O(d2 log d2 ), where d2 is the number of its 2-hop neighbors in the original unit disk graph. This can be improved to O(d1 log d1 + d2 ), where d1 is the number of its 1-hop neighbors in the original unit disk graph. The improvement is based on the fact that we only need the triangles wuv in LDel(2) (V ) that has angle ∠wuv ≥ π/3. All such triangles are definitely in LDel(1) (V ). Thus, we can construct the Delaunay triangulation Del (N1 (u)) of N1 (u) in the first step of the algorithm in Section 4.3.2. Then check the candidate triangles to see if they contain any node from N2 (u) inside its circumcircle. If it does not, then it belongs to Del (N2 (u)) also. Observe that, after each node u collects the 2-hop neighbors N2 (u) (Step 1 of the algorithm in Section 4.3.2), our algorithms can be performed asynchronously. However, collecting N2 (u) need synchronized communication since otherwise, a node cannot determine if it indeed already collected N2 (u). 5.5 Summary In this chapter, we first propose a new structure which is a plane spanner with bounded node degree for any point set V . Then we show both centralized and localized algorithms to construct this structure for U DG(V ). We prove all the properties for centralized algorithms. The localized algorithm can be implemented using O(n) messages. The basic idea of this new method is using (localized) Delaunay triangulation to make planar graph then apply ordered Yao graph to bound node degree. However, it is carefully de-
97 signed to not lose all good properties when combining them. We also consider use YaoYao or other Yao relative structures on localized Delaunay triangulation or other planar structures to achieve planar and bounded degree spanner, but it is difficult to show the spanner properties. We leave it as a future work.
98 CHAPTER 6 EXPERIMENTAL RESULTS
We conducted extensive simulations to study the performances of our new structures in term of both structural properties and routing performances. 6.1 Structural Properties In a wireless network, each node is expected to potentially send and receive messages from many nodes. Therefore an important requirement of such network is a strong connectivity. In previous chapters, we have shown all these sparse topologies are strongly connected if the unit disk graph is connected. So in our experiments, we randomly generate a set V of n wireless nodes and its U DG(V ), and test the connectivity of U DG(V ). If it is connected, we construct different localized topologies from V by various algorithms. Then we measure the sparseness and the power efficiency of these topologies by the following criteria: the average and the maximum node degree, denoted by davg , dmax ; the average and the maximum length stretch factor, denoted by tavg , tmax . and the average and the maximum power stretch factor, denoted by ρavg , ρmax . Notice that, for a directed topology, we also measure the average and the maximum in-degree, denoted by Iavg , Imax . In the experimental results presented here, we choose total n = 50 wireless nodes; choose 8 cones when we construct any graph using the Yao structure (for example, Y G(V ), Y D(V ), Y S(V ), Y G∗ (V ) and Y Y (V )); set the angle parameter α = π/3 when we construct BP S(V ); set the power attenuation constant β = 2. We distribute these 50 wireless nodes randomly in a square area with radius 100 meters. Each node is specified by a random x, y coordinate, with transmission radius 70 meters. Then we generate 100 such vertex sets V (each with 50 vertices) and then construct the graphs for each of these 100 vertex sets. Figure 6.1 gives all nine different localized topologies defined or proposed in this thesis for the unit disk graph illustrated by the first figure of Figure 6.1.
99
U DG(V )
RN G(V )
GG(V )
Y G(V )
Y G∗ (V )
Y Y (V )
Y S(V )
Y D(V )
P LDel(V )
BP S(V )
Figure 6.1. Different Localized Topologies Generated from Same Unit Disk Graph
100 6.1.1 Node Degree Before we show the power efficiency of different topologies, we would like to understand the characteristics of the resulting topologies. The average node degree of the wireless networks should not be too large. Otherwise a node with a large degree has to communicate with many nodes directly. This increases the interference and collision, and increases the overhead at this node. The average node degree should also not be too small either: a low node degree usually implies that the network has a lower fault tolerance and it also tends to increase the overall network power consumption as longer paths may have to be taken. Thus, the average node degree is an important performance metric for the wireless network topology. Table 6.1 shows that Y S(V ), Y D(V ), and Y Y (V ) have a much less number of edges than the Y G(V ), since their average node degrees are lower. In other words, these graphs are sparser than the Yao graph, which is also verified by Figure 6.1. Notice that theoretically, the sink structure Y G∗ (V ) has the same number of edges as the −−→ Yao graph Y G(V ). However, the in-degree of each node of the sink structure Y G∗ (V ) is bounded from above by a constant. Here, Omax is the maximum node out-degree over all nodes and all directed graphs. Notice that RN G(V ), GG(V ), P LDel(V ) do not have large degree in this experiments. The reason is the nodes are distributed randomly in the area. In real life, the network maybe not distributed randomly, it is possible that it is similar with the distribution of Figure 2.9. Our bounded degree and planar spanner BP S(V ) has small node degree both theoretically and practically.
Notice that since here we set the node number and transmission range some special values, the density of the graph is a special case. If we change the density of the graph, the ranges of stretch factors of some structures, such as Gabriel graph, change also. Then we keep the transmission range at 70 meters, and vary the number of wireless nodes in the networks. The two upper figures in Figure 6.2 give us the results of node degree when the number of nodes are increased from 25 to 200. We observed that all structures keep small
101
Table 6.1. Quality Measurements of Different Topologies davg
dmax
Iavg = Oavg
Imax
Omax
ρavg
ρmax
tavg
tmax
U DG
14.20
30.00
-
-
-
1.000
1.000
1.000
1.000
RN G
2.29
4.00
-
-
-
1.061
2.668
1.313
4.462
GG
3.37
7.00
-
-
-
1.000
1.000
1.113
2.084
YG
7.73
17.00
5.87
17.00
8.00
1.001
1.479
1.036
1.696
YS
4.02
8.00
-
-
-
1.003
1.547
1.081
1.969
4.10
7.00
-
-
-
1.005
1.582
1.081
1.783
4.70
10.00
4.36
9.00
8.00
1.002
1.479
1.064
1.739
YY
4.52
9.00
4.27
8.00
8.00
1.003
1.479
1.067
1.912
P LDel
3.52
8.00
-
-
-
1.000
1.000
1.104
2.040
BP S
4.41
9.00
-
-
-
1.005
1.916
1.067
1.959
YD YG
∗
node degrees when the U DG(V ) becomes denser and denser. 6.1.2 Stretch Factor Besides strong connectivity, the most important design metric of wireless networks is perhaps the power efficiency, as it directly affects both the node and the network lifetime. Table 6.1 summarizes our experimental results on the power stretch factors of these topologies when the network has 50 nodes. It shows that the new proposed network topologies still have small power stretch factors. Notice that although the average and the maximum node degree of Y S(V ), Y D(V ), Y Y (V ) and BP S(V ) are much smaller than that of Y G(V ), the average and the maximum power stretch factors of these graphs are at the same level of that of Y G(V ). Especially, the power stretch factors of Y S(V ), Y D(V ), and Y Y (V ) are just a little bit higher than that of Y G(V ). Remember that Y S(V ), Y D(V ), Y Y (V ), and BP S(V ) have bounded node degrees while Y G does not have such a property. As we expected, P Del(V ) has power stretch factor as one. Table 6.1 also list the length stretch factors of these structures. Then we also vary the number of nodes in the region. We plotted the stretch factors
102 of all structures in Figure 6.2. Again we observed that all structures have small stretch factors even the network is very dense. Notice that though RN G(V ) has a little bit higher stretch factors than others, however, the maximum of its length stretch factors is still smaller than 5. The reason is the nodes are distributed randomly in the area and the number of nodes is not too large. In the simulation, the case similar with the distribution of Figure 2.4 seldom occurs. 6.1.3 Communication Cost In Chapter 5 we proved that the localized algorithm constructing BSP (V ) uses at most O(n) messages. In our simulations , we found that when the number of wireless nodes increases the average messages used by each node for constructing BP S(V ) is still in the same level. In this experiment, we generate from 50 to 300 random wireless nodes in a 10 × 10 square and run our localized algorithm to build BSP (V ). The transmission range is set as 3. The average and the maximum are computed over 20 vertex sets. All other parameters and settings are same with previous experiments. Table 6.2 summarizes our experimental results of the node degree, length and power stretch factors, and communication costs of BP S(V ). Here, davg (U DG)/dmax (U DG) is the average/maximum node degree for the original unit disk graph; tot msgavg /tot msgmax is the average/maximum total messages cost for constructing BP S(V ); nod msgavg /nod msgmax is the average/maximum messages cost in each node during the construction. Notice that here we do not count the messages used in building LDel(2) (V ), since in [Li03c] it was proved that the communication cost of building LDel(2) (V ) is O(n). In other words, we only consider the messages used in the second and third steps of the localized algorithm in Section 5.4.1. The first two rows of Table 6.2 show the network becomes more and more dense while the number of wireless nodes increases. Experimental results of communication costs on each node show that the localized method does not cost more messages on each node even the graph becomes more dense. Simulations in Table 6.2 also show that the performances of our new
103
60
100
UDG RNG GG YAO YAOS YAOYAO SNKYAO YAOD PDEL BPS
avg node degree
40
90
UDG RNG GG YAO YAOS YAOYAO SNKYAO YAOD PDEL BPS
80
70
max node degree
50
30
60
50
40
20 30
20
10 10
0 20
40
60
80
100 120 number of nodes
140
160
180
0 20
200
40
average node degree
1.25
UDG RNG GG YAO YAOS YAOYAO SNKYAO YAOD PDEL BPS
4.5
4
1.2
1.15
3.5
1.05
1.5
40
60
80
100 120 number of nodes
140
160
180
1 20
200
average length stretch factor
200
160
180
200
180
200
UDG RNG GG YAO YAOS YAOYAO SNKYAO YAOD PDEL BPS
40
60
80
100 120 number of nodes
140
3.5
UDG RNG GG YAO YAOS YAOYAO SNKYAO YAOD PDEL BPS
3
max power stretch factor
avg power stretch factor
1.05
180
maximum length stretch factor
1.08
1.06
160
2.5
2
1.07
140
3
1.1
1 20
100 120 number of nodes
maximum node degree
max length stretch factor
avg length stretch factor
1.3
80
5
1.4
1.35
60
1.04
1.03
2.5
UDG RNG GG YAO YAOS YAOYAO SNKYAO YAOD PDEL BPS
2
1.02
1.5 1.01
1 20
40
60
80
100 120 number of nodes
140
160
average power stretch factor
180
200
1 20
40
60
80
100 120 number of nodes
140
160
maximum power stretch factor
Figure 6.2. Results When Number of Wireless Nodes Increasing
104 topology BP S(V ) are stable when number of nodes changes. Table 6.2. Performances and Communication Costs of BP S(V ) num of nodes
50
100
150
200
250
300
davg (U DG)
16.52
34.99
51.81
68.15
85.87
103.85
dmax (U DG)
29
62
94
114
140
175
davg
4.19
4.39
4.54
4.60
4.58
4.63
dmax
8
9
11
11
9
9
tavg
1.094
1.101
1.100
1.098
1.099
1.096
tmax
1.958
1.968
1.949
1.978
1.995
1.977
ρavg
1.017
1.012
1.012
1.009
1.009
1.010
ρmax
1.918
1.937
1.900
1.932
1.916
1.937
tot msgavg
393
812
1229
1655
2076
2498
tot msgmax
398
821
1244
1670
2090
2512
nod msgavg
7.86
8.12
8.19
8.27
8.30
8.32
nod msgmax
12
13
15
14
16
14
6.2 Routing Performance In this section, we study the routing performance of some classical localized routing methods on these proposed structures. First, we review the localized routing methods for wireless ad hoc networks. 6.2.1 Classical Localized Routing Methods The geometric nature of the multi-hop ad-hoc wireless networks allows a promising idea: localized routing protocols. A routing protocol is localized if the decision to which node to forward a packet is based only on: 1. The information in the header of the packet. This information includes the source and the destination of the packet, but more data could be included, provided that its total length is bounded. 2. The local information gathered by the node from a small neighborhood. This infor-
105 mation includes the set of 1-hop neighbors of the node, but a larger neighborhood set could be used provided it can be collected efficiently.
Randomization is also used in designing the protocols. A routing is said to be memory-less if the decision to which node to forward a packet is solely based on the destination, current node and its neighbors within some constant hops. Localized routing is sometimes called in the literature stateless [Kar00], online [Bos00, Bos99], or distributed [Sto01]. We summarize some localized routing protocols proposed in the networking and computational geometry literature. Figure 6.3 shows the illustrations of these routing protocols. Shaded area is empty and v is next node.
• C OMPASS ROUTING (C MP ): Let t be the destination node. Current node u finds the next relay node v such that the angle ∠vut is the smallest among all neighbors of u in a given topology. See[Kra99]. • R ANDOM C OMPASS ROUTING (RC MP ): Let u be the current node and t be the destination node. Let v1 be the node on the above of line ut such that ∠v1 ut is the smallest among all such neighbors of u. Similarly, we define v2 to be nodes below line ut that minimizes the angle ∠v2 ut. Then node u randomly choose v1 or v2 to forward the packet. See[Kra99]. • G REEDY ROUTING (G RDY ): Let t be the destination node. Current node u finds the next relay node v such that the distance vt is the smallest among all neighbors of u in a given topology. See [Bos01]. • M OST F ORWARDING ROUTING (MFR): Current node u finds the next relay node v such that v t is the smallest among all neighbors of u in a given topology, where v is the projection of v on segment ut. See [Sto01].
106 • N EAREST N EIGHBOR ROUTING (NN): Given a parameter angle α, node u finds the nearest node v as forwarding node among all neighbors of u in a given topology such that ∠vut ≤ α. • FARTHEST N EIGHBOR ROUTING (FN): Given a parameter angle α, node u finds the farthest node v as forwarding node among all neighbors of u in a given topology such that ∠vut ≤ α. • G REEDY-C OMPASS (GC MP ): Current node u first finds the neighbors v1 and v2 such that v1 forms the smallest counter-clockwise angle ∠tuv1 and v2 forms the smallest clockwise angle ∠tuv2 among all neighbors of u with the segment ut. The packet is forwarded to the node of {v1 , v2 } with minimum distance to t. See [Bos99, Mor01] v
v1 t
u
v
t
u
t
u
v2
Compass
Random Compass
v
v u
Greedy
t
Most Forwarding
u
α
v t
Nearest Neighbor
u
α
t
Farthest Neighbor
Figure 6.3. Various Localized Routing Methods It was shown in [Bos01, Kra99] that the compass routing, random compass routing and the greedy routing guarantee to deliver the packets from the source to the destination if Delaunay triangulation is used as network topology. They proved this by showing that the distance from the selected forwarding node v to the destination node t is less than the distance from current node u to t. However, the same proof cannot be carried over when the network topology is Yao graph, Gabriel graph, relative neighborhood graph, and the localized Delaunay triangulation. When the underlying network topology is a planar graph, the
107 right hand rule is often used to guarantee the packet delivery after simple localized routing heuristics fail [Bos01, Kar00, Sto01]. Morin proved the following results in [Mor01]. The greedy routing guarantees the delivery of the packets if the Delaunay triangulation is used as the underlying structure. The compass routing guarantees the delivery of the packets if the regular triangulation is used as the underlying structure. Delaunay triangulation is a special regular triangulation. There are triangulations (not Delaunay) that defeat these two schemes. The greedy-compass routing works for all triangulations, i.e., it guarantees the delivery of the packets as long as there is a triangulation used as the underlying structure. Every oblivious routing method is defeated by some convex subdivisions. Although some of the localized routing protocols guarantee the delivery of the packet if some special geometry structures are used, none of these guarantees the ratio of the distance traveled by the packets over the minimum possible. Bose and Morrin [Bos99] proposed a method to bound this ratio using the Delaunay triangulation. Notice that constructing Delaunay triangulation in a distributed manner is communication expensive. It is easy to see that there is no memoryless routing method that works always in the unit disk graph. Localized routing protocols support mobility by eliminating the communication-intensive task of updating the routing tables. But mobility can affect the localized routing protocols, in both the performance and the guarantee of delivery. There is no work so far to design protocols with guaranteed delivery when the network topology changes during the routing. 6.2.2 Experimental Results We then present our experiments of various routing methods on different topologies. We keep the same configuration with the experiments of structural properties. We choose 50 nodes distributed randomly in a square area with radius 100 meters. Each node is specified by a random x, y coordinate, with transmission radius 70 meters. Figure 6.1
108 illustrates some discussed topologies. We randomly select 20% of nodes as source; and for each source, we randomly choose 20% of nodes as destination. The statistics are computed over 20 different node sets.
Table 6.3. Delivery Rates of Different Routing Methods RNG
GG
YG
YS
YY
YG∗
YD
NN
40.8
69.3
97.5
73.6
79.5
84.1
84.0
80.7
91.6
FN
38.2
68.8
95.1
81.6
87.4
90.7
83.8
66.7
86.9
MFR
66.2
86.7
100.0
92.9
93.2
95.1
89.6
93.5
96.8
Cmp
32.7
49.6
79.4
62.5
70.5
70.5
48.2
52.3
71.9
RCmp
49.2
79.2
96.5
88.9
88.3
91.3
85.0
77.3
93.7
Grdy
78.9
99.2
100.0
99.3
97.7
99.0
98.1
97.9
98.4
GCmp
35.0
56.5
86.1
74.2
71.4
78.3
70.2
61.1
81.0
PLDel BPS
Table 6.3 illustrates the delivery rates of different localized routing protocols on various network topologies. For nearest neighbor routing and farthest neighbor routing, we choose the angle α = π/3. In other words, we only choose the nearest node or the farthest node within π/3 of the destination direction. The results show all Yao relative structures have much better delivery rates than GG(V ) and RN G(V ). Remember that Y S(V ), Y Y (V ), Y G∗ (V ),Y D(V ) have bounded degree while Y G(V ) does not, though their delivery rates are a little lower than Y G. P LDel(V ) has similar delivery rate with GG(V ), but better than RN G(V ). BP S(V ) has similar delivery rate with all Yao relative structures, but better than RN G(V ) and GG(V ). The BP S(V ) graphs are preferred over the Yao graph because we can apply the right hand rule when previous simple heuristic localized routing fails. Both [Bos01] and [Kar00] use the greedy routing on Gabriel graph and use the right hand rule when greedy fails.
Table 6.4 and Table 6.5 illustrate maximum/average spanning ratios of Π(s, t)/st, where Π(s, t) is the path traversed by the packet using different localized routing protocols
109
Table 6.4. Maximum Spanning Ratio of Different Routing Methods RNG
GG
YG
YS
YY
YG∗
YD
NN
1.5
1.6
1.6
1.7
1.6
1.8
2.5
1.7
1.7
FN
1.8
2.2
3.4
2.6
2.7
3.0
2.1
2.2
2.2
MFR
2.4
2.4
7.7
2.2
3.3
3.0
2.4
2.2
2.3
Cmp
3.2
7.1
7.3
14.3
14.9
8.4
9.3
6.6
5.7
RCmp
36.7
57.6
32.0
79.9
52.4
88.0
39.4
33.2
77.9
Grdy
2.2
1.8
1.7
1.9
1.6
1.9
1.9
1.8
1.8
GCmp
3.2
3.3
2.8
2.1
2.4
2.2
2.4
4.4
2.4
PLDel BPS
on various network topologies from source s to destination t. The average spanning ratios of all the methods and topologies are small. However, the maximum spanning ratios have big differences. In our experiments, the compass routing, random compass routing have much larger maximum spanning ratios. The reason maybe our configuration of transmission ranges (the density of the graph) is in the ranges in which the compass routing, random compass routing do not work well.
Table 6.5. Average Spanning Ratio of Different Routing Methods RNG
GG YG
YS YY YG∗
YD PLDel BPS
NN
1.3
1.3
1.3
1.3
1.2
1.3
1.3
1.3
1.3
FN
1.3
1.4
1.4
1.3
1.4
1.4
1.4
1.3
1.4
MFR
1.4
1.4
1.4
1.3
1.4
1.4
1.4
1.3
1.3
Cmp
1.5
2.3
1.9
1.9
1.9
1.9
2.2
2.1
1.9
RCmp
7.2
6.0
5.0
5.8
6.8
5.2
6.8
6.1
4.9
Grdy
1.4
1.3
1.2
1.3
1.2
1.3
1.3
1.3
1.3
GCmp
1.6
1.5
1.3
1.5
1.4
1.4
1.4
1.4
1.4
6.3 Summary In this chapter, we construct some simulation for testing the performances of the new sparse and power efficient topologies proposed in this report. We randomly generate the set V of n wireless nodes and its U DG(V ), then construct different topologies by
110 various algorithms. We try to measure the spanner properties of these structures and test some existed localized routing methods on them. The results show that the new structures have good sparseness and power efficiency in practice, also work for the existed localized routing methods.
111 CHAPTER 7 CONCLUSION AND FUTURE WORK
7.1 Localized Topology Control In this thesis, we focus on discussing one of the central challenges in the design of ad hoc networks: efficient localized topology control. We study how to construct a sparse spanner efficiently as network topology for a set of static or quasi-static wireless nodes such that, for any given pair of nodes, there is a power-efficient path. We first review previous results for topology control when the wireless nodes have uniform transmission range. Then we propose some of our new results: some bounded degree structures using Yao structure, one planar spanner using localized Delaunay triangulation, and one planar spanner with bounded degree by carefully combining Yao structure and localized Delaunay. We show that most of them are spanner, while some of them are unknown or not spanner. All results are summarized by Table 7.1. Notice that the power stretch factor of Yao Yao graph is only true in civilized graph. Our experimental results show all these structures are good in practice. 7.2 Other Related Topics In this section, we review some topics which are also related to topology control. 7.2.1 Transmission Power Control In the previous sections, we have assumed that the transmission power of every node is equal and is normalized to one unit. We relax this assumption for a moment in this section. In other words, we assume that each node can adjust its transmission power according to its neighbors’ positions. A natural question is then how to assign the transmission power for each node such that the wireless network has some ”good” properties (in terms
112
Table 7.1. Properties of Different Topologies Proposed Here
Power Stretch Factor Length Stretch Factor YG Y G∗
1 1−(2 sin 1 1−(2 sin
π β ) k
2
π β ) k
1 1−2 sin 1 1−2 sin
π k
2
π k
Max Degree
Planar
Localized
n−1
no
yes
(k + 1)2 − 1
no
yes
YY
2
?
2k
no
yes
YD
?
?
k
no
yes
YS
O(n)
O(n)
k
no
yes
n−1
no
yes
n−1
yes
yes
20
yes
yes
YO P LDel BP S
1 1−(2 sin
1
√ ( 4 9 3 π(1
π β ) k
β
+ π))
1 1−2 sin π k √ 4 3 π 9 √ 4 3 π(1 + 9
π)
of network tasks such as disjoint paths, connectivity or fault-tolerance) with optimization criteria being minimizing the maximum (or total) transmission power assigned. The minimum energy connectivity problem was first studied by Chen and Huang [Che89], in which the induced communication graph is strongly connected while the power assignment is minimized. This problem has been shown by them to be NP-hard. Recently, this problem has been heavily studied and many approximation algorithms have been proposed when the network is modelled by using symmetric links or asymmetric links [Kir00, Cle00c, Blo02, Alt03, Ram00]. Along this line, several authors [Haj03, Cˇal03b, Che02] considered the minimum power assignment while the resulting network is k-strongly connected or k-connected. This problem has been shown to be NP-hard too. Solving this problem can improve the fault tolerance of the network. In [Cle00a, Cle00b, Cle00c], Clementi et al. also considered the minimum energy connectivity problem while the induced communication graph have a diameter bounded by a constant h. In [Llo02], Lloyd et al. proposed one general framework that leads to an approximation algorithm for minimizing total power assignment. Using the framework they proposed a new 2-connected approximation method for power assignment. In [Kru03], Krumke et al. also studied the minimum power assign-
113 ment so that networks satisfy specific properties such as connectivity, bounded diameter and minimum node degree. Other relevant work in the area of power assignment (or called energy-efficiency) includes energy-efficient broadcasting and multicasting in wireless networks. The problem, given a source node s, is to find a minimum power assignment such that the induced communication graph contains a spanning tree rooted at s. This problem was proved to be NP-hard. In [Wan02a, Cle01, Wie00, Hui], they presented some heuristic solutions and gave some theoretical analysis. Recently, Srinivas and Modiano [Sri03] also studied finding k-disjoint paths for a given pair of nodes while minimizing the total node power needed by nodes on these k-disjoint paths. In [Wan04a], Wang and Li also studied the power assignment such that the induced communication graph is a spanner for the original communication graph when all nodes have the maximum power. An excellent survey of some recent theoretical advances and open problems on energy consumption in ad hoc networks can be found in [Cle02]. 7.2.2 Low Weight Structures The power stretch factor we previously discussed is defined for the unicasting communications. However, in practice, we also have to consider the broadcast or multicast communications. Wan et al.[Wan01] showed that the minimum energy cost of broadcasting or multicasting is related to the total energy cost of all links in the Euclidean minimum spanning tree M ST . They proved that a broadcasting method based on the Euclidean minimum spanning tree rooted at the sender uses energy no more than 12 times the minimum energy cost of any broadcasting scheme. Unfortunately, the broadcasting (or multicasting) power stretch factor of any graph structures mentioned above (except M ST ) could be an arbitrarily large number theoretically [Li02b, Li01b]. A structure is called low weight if its total edge length is within a constant factor of the total edge length of the M ST . The best distributed algorithm [Fal95, Gal83] can compute MST in O(n) rounds using O(m + n log n) communications for a general graph with m edges and n nodes. Since
114 the relative neighborhood graph, the Gabriel graph and the Yao graph all have O(n) edges and contain the Euclidean MST, we can construct the minimum spanning tree of UDG in a distributed manner using O(n log n) messages. Unfortunately, even for wireless network modeled by a ring, the O(n log n) number of messages is still necessary for constructing MST of UDG. In [Li04c], Li et al. presented the first localized method to construct a bounded degree planar connected structure Incident MST and RNG Graph (IMRG) whose total edge length is within a constant factor of MST. The degree of each node is at most 6. The total communication cost of their method is at most 7n, and every node only uses its partial two-hop information to construct such structure. It was shown in [Li03b] that some two-hop information is necessary to construct any low-weighted structure. However, the IMRG is not a spanner. So we leave it as a possible future work to design an efficient localized algorithm achieving low weight, bounded degree, planar spanner. 7.3 Future Work Topology control is still a new topic in wireless ad hoc network. There are many interesting open problems. Some of them, we mentioned previously in this thesis.
1. Even the YaoYao graph Y Yk (V ) and High-degree Yao graph Y Dk (V ) have bounded degrees and small power stretch factors in practice, it is still an open problem whether they have a bounded power stretch factor theoretically. The proof of the conjecture or the construction of a counter-example still remain a future work. 2. Since we proved Ordered Yao graph is a spanner and different properties can be obtained by choosing orderings of V . So choosing some orderings to make Y Ok (V ) have nice properties is also an interesting problem. 3. Another interesting problem is how to do topology control when wireless nodes have nonuniform transmission range. We can model wireless networks as some disk graphs [Li02c] (mainly using ack-communication graph), instead of unit disk graph.
115 In [Li02c], we model the ack-based communication or symmetric communication in wireless networks by the ack-communication graph. We still consider a set V of wireless nodes distributed in a two-dimensional plane. Each node v ∈ V has a transmission radius rv . Two nodes u and v are connected if uv ≤ min(ru , rv ). Then all these structures in this report need to be redefined and studied. 4. Notice that when we discuss the communication cost for all algorithms, we only consider the sending communication cost. Here we assume that for each node listening does not cost anything. However, in real life, this is not true. If we consider the listening cost, we need redesign all the algorithms which construct spanner locally to make them efficient for the new cost.
With quick development of wireless networks, there are much more new topics raised in this field. We believe the localized spanner will have more applications in wireless networks.
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