Int. J. of Computers, Communications & Control, ISSN 1841-9836, E-ISSN 1841-9844 Vol. III (2008), Suppl. issue: Proceedings of ICCCC 2008, pp. 465-469
Concepts of Graph Theory Relevant to Ad-hoc Networks M. A. Rajan, M. Girish Chandra, Lokanatha C. Reddy, Prakash Hiremath
Abstract: The issues in Mobile ad-hoc networks (MANETs) always bring the attention of research community. The fundamental issues of connectivity, scalability, routing and topology control in MANETS is worth to study. Graph theory plays an important role in the study of these fundamental issues. This paper highlights the concepts of graph theory that are employed to address these fundamental issues. Keywords: graphs,connectivity,graph spanners,proximity,MANET
1
Introduction
Ad-hoc networks are decentralized, self-organizing networks capable of forming a communication network without relying on any fixed infrastructure. Each node in an ad-hoc network is equipped with a radio transmitter and receiver, which allow it to communicate with other nodes over wireless channels. All nodes can function, if needed, as relay stations for data packets to be routed to their final destination. A special kind of ad-hoc network is the sensor network where the nodes forming the network do not or rarely moves. Further, the nodes of a sensor network are similar. Salient Features of mobile ad-hoc networks [1] are, 1) Use of ad-hoc networks can increase mobility and flexibility, as ad-hoc networks can be brought up and brought down in a very short time. 2) Ad-hoc networks can be more economical in some cases, as they eliminate fixed infrastructure costs. 3) Ad-hoc networks can be more robust than conventional wireless networks because of their non-hierarchical distributed control and management mechanisms. 4) Because of multi-hop support in ad-hoc networks, communication beyond the Line of Sight (LOS) is possible at high frequencies. In the study of MANETs two areas are of great importance[3-6]: 1) Understanding the fundamental issues like connectivity, scalability and routing and 2) Network modeling and simulation. Since a network can be modeled mathematically as a graph (there exists a bijection between network topology and graph), the Graph Theory concepts play an important role in analyzing these fundamental issues. Also, the problems associated with ad-hoc networks can be explained mathematically. Going further, graphs can be algebraically represented as matrices and hence the study of the network can be automated through algorithms. A lot of research about ad-hoc networks is carried out using the mathematical models and their simulation rather than experimenting on real mobile ad-hoc networks. Several issues like node density, mobility of the nodes, link formation between nodes and packet routing between the nodes needs to be simulated. To simulate MANETs concepts of graph theory (particularly random graph theory) are utilized. The very basic purpose of any network is to facilitate exchange of information between any two nodes. This can happen only when the network is connected. Hence the connectivity is one of the fundamental and most important issues of the MANETs. The important factor, which affects the connectivity, is the transmission range of the nodes and the mobility of the nodes. The majority of research work related to connectivity has been done by considering the static network, wherein nodes will be stationary. Some of the concepts of graph theory that are extensively used to study the connectivity issues are graph spanners, proximity graph sparsifications and spectral graph theory. Another important issue is the scalability. Scalability is the study of network stability, whenever the number of nodes changes; the topology of the network changes.. This is one of the important issues in ad-hoc networks, because of the mobility of the nodes in the network. Addition of nodes to the network may cause the network be disconnected to start with. This necessitates topology control. Some of the fundamental questions that arise during topology change are how the performance of the network and routing will be affected? A lot of work has been done related to topology control utilizing the graph theory concepts like graph clustering, graph partitioning, and graph evolution[5,12,16]. One of the issues in transporting the data packets among the nodes of a MANET or even to outside is routing. The factors which can affect the routing are connectivity, mobility of the nodes and the traffic of the network[3,4,5,6]. Routing protocols in mobile ad-hoc networks are more complex than in static networks. One of the easiest routing techniques is flooding, where packets are simply delivered from source to all its neighbor nodes. The neighbor nodes pass the packets to their neighbor nodes and the process continues until the packet reaches the destination node. But, this affects the throughput of the network as flooding introduces congestion. Hence more optimal routing techniques are needed. A lot of research has been done utilizing the concepts of graph theCopyright © 2006-2008 by CCC Publications - Agora University Ed. House. All rights reserved.
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ory in devising routing algorithms; some of the concepts that are explored are sparse graphs, graph planarity and proximity graphs[5,13,15,16]. Based on the discussion so far, it is apparent that the graph theory concepts can play a great deal in the study and design of ad-hoc networks. This paper is an attempt to succinctly capture many such concepts scattered in the literature, outlining their applications in the study of ad-hoc networks. In order to present these, the paper is organized as follows: in Part 2, a very brief introduction of few important terminologies and notations of graph theory is given Part 3 presents the graph theory concepts related to connectivity, routing and topology control issues. Conclusions are given in Part 4.
2
Some Basic Definitions from Graph Theory
A graph consists of number of vertices and edges, where an edge is an association between two vertices. As mentioned earlier, there is a bijection between a graph and a network. With respect to the network, a vertex is a node and an edge is a link between two nodes. Mathematically, a graph G is a triplet consists of Vertex Set V(G) , Edge Set E(G) and a relation that associates two vertices with each edge. An edge between two nodes i and j is represented as (i,j) and by using usual notation, E(G) can be written as E(G) ⊆ {(i,j) |∀ i,j ε V and (i,j)=(j,i)} . Two vertices are said to be adjacent to each other, if there exist an edge between them. Two edges are said to be adjacent to each other, if the one of the end vertex of the edges are same. If each edge of a graph is associated with some specific value (weight), graph is said to be weighted graph. The number of edges associated with the vertex is called degree of any vertex v is denoted by d(v). The minimum degree of a graph is the least degree of a vertex of a graph denoted by δ (G) and the maximum degree of a graph is the maximum degree of any vertex of a graph denoted by ∆(G). A graph G is regular if and only if △( G) = δ (G). A graph G is said to be connected, if for every pair of vertices u, v of G, there exist a path, otherwise Graph is disconnected. A disconnected graph has number of components; each component being a connected graph. A planar graph is a graph which can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges may intersect only at their endpoints.A dual graph of a given planar graph G is a graph which has a vertex for each plane region of G, and an edge for each edge in G joining two neighboring regions, for a certain embedding of G.
3 Graph theory concepts related to Connectivity, Routing and Topology Control This section describes the concepts of graph spanners, proximity of graphs(UDG,NNG,RNG,DT and Voronoi diagram) that can be applied to answer the fundamental issues connectivity, routing and topology control issues one by one. Graph Spanners[10,15]:. A spanner of a graph is a sub-graph that preserves approximate distances between all pairs of vertices. Formally, given t ≥ 1 , a t-spanner of a graph G is a sub-graph S of G such that for each pair of vertices the distance in S is at most t times the distance in G, t is referred to as the multiplicative stretch factor of the spanner. That is dS (u, v) ≤ tdG (u, v), ∀ u, v ε V and S is called multiplicative t-spanners of G[10]. If r ≥ 0 and dS (u, v) ≤ tdG (u, v) + r , ∀ u, v ε V , S is called additive r-spanner of G and r is called additive stretch factor of G. Several graph spanners may exist for a given graph. So between any two nodes these spanners can be different paths between the nodes.The routing algorithms can use these spanners concept to device efficient algorithms as the spanners provide several alternate paths and also throughput of the network can be increased in case of congestion. By using graph spanners one can determine several graph spanners,which are useful in designing certain class of routing algorithms, study of network clustering, partitioning and network topology control. One of the difficulties in dealing with graph spanners in ad-hoc network is how the algorithm can be made distributed with less complexity? The concept of proximity is also related to graph spanners. A lot of work is done in devising algorithms to construct the graph spanners locally. Graph spanners also finds application in many areas including computational geometry, computational biology, and robotics and distributed computing. Proximity: With respect to mobile ad-hoc network, the study of proximity graphs plays an important role in topology control, connectivity of the network. Proximity represents the neighbor relationship between nodes. Two nodes are joined by a link if they are deemed close by some proximity measure. This certainly affects the network connectivity. It is the measure that determines the type of a graph that results. Many different measures of proximity have been defined, giving rise to many different types of proximity graphs. One such measure is spanning
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(u,v) ratio. It is defined as Maximum({ ηΩ(u,v }), where η (u, v) is the length of the shortest path between two nodes u and v, where the edge length is measured by Euclidean distance and Ω(u, v) is the direct Euclidean distance. There are different types of proximity graphs available. The important ones are Unit Distance Graph (UDG), Nearest Neighbor Graphs (NNG), Minimum Spanning Trees (MST), Relative Neighborhood Graphs (RNG), Delaunay Triangulation (DT), and Gabriel Graphs[10,15]. Unit Disk Graph (UDG): is a graph UDG (V, E) with set of nodes V and a link (i,j) between nodes i and j belongs to link set E if and only if their Euclidean distance is less than 1. UDG can be directed in which link is also directed. Most of the ad-hoc network modeling uses UDG. A more generalized UDG is one, in which the link between two nodes is possible if their Euclidean distance is less than r , where r is the radius of the circular transmission range of the antenna of the mobile node. This model is well suited for ad-hoc network, as it is almost realistic to ad-hoc network, where the geodesic of the transmission range of the radio signals coverage is almost circular. One of the issue related to this models is to find out the minimum threshold range of the transmitters of the nodes so that network is connected. Nearest Neighbor Graph(NNG)[10]: of a graph G(V,E)is a directed graph, denoted by NNG(V’,E’) is a spanning sub-graph of a graph G with V=V’ and there exist a directed edge e between any two nodes i and j if and only if node j is the nearest neighbor of node i(the Euclidean distance is the measure for selecting nearest neighbor). This graph gives the all-pair shortest paths of a given network.For the given network, if one can able to find its NNG, packet routing will become simple and straight forward and also the throughput of the network will be increased, because of reduced delay in the packet delivery. But the failure of a link , which belongs to NNG , makes it disconnected. Hence a more generalized NNG is required to make it fault tolerant. NNG can be generalized to include more than one neighbor. The K-NNG is a generalized NNG, which represents the number of edges from any vertex to K nearest neighbors. There can be more than one NNG of a given graph. NNG can also be a undirected graph; is a spanning sub graph of Minimum Spanning Tree(MST).MST of a weighted graph G is a spanning tree of G which is acyclic, connected with the property that the sum of the weights of the edges of the resulting tree is minimum. Since the edges of a graph could contain the same weights, any particular graph G could have multiple MSTs. A variation of NNG is Relative Neighborhood Graph (RNG); of a graph G (V, E)is a graph RNG (V, E’)in which there exist an edge between nodes i and j and if for all nodes k in V, d(i,j) ≤ Maximum(d(u, k), d(k, j)). Gabriel Graph (GG)[15]: is a graph which contains a link between nodes i and j if a disk with radius ij2 , contains no other node. Formally p GG (V, E)is node set V and link set E , such that there exist a link between nodes i and jif and only if d(i,j) ≤ d(i, k)2 + d(k, j)2 , ∀ k ε V, k 6= i and k 6= j. A Gabriel Graph can be constructed in time O(n log n) by first finding the Delaunay Triangulation and Voronoi Diagram for the set of points. Then for each edge in the triangulation, if the edge intersects its Voronoi edge, it is added as an edge to the GG. Voronoi Diagram: Given a set P = {P1 , P2 , ......, Pn } of n points in ε , where ε = Em , E is an edge set in an affine space, it is often useful to find a partition of the space into regions each containing a single point of P.The Dirchlet-Voronoi diagram V(P) of P is the family of subsets of ε consisting of the sets Vi = ∩ j 6= i H(pi , p j ) and all of their intersections.Dirchlet-Voronoi diagrams are also called Voronoi diagrams, Voronoi tessellations or Theissen polygons. Delaunay Triangulations (DT): A very interesting undirected graph can be obtained from the Voronoi diagram is : The vertices of this graph are the points pi (each corresponding to a unique region of V(P) ), and there exists an edge between pi and pj if and only if, the regions Vi and Vj share an edge. The resulting graph is called a Delaunay triangulation(DT) of the convex hull of P. A triangulation T is called a DT. if and only if a circle which contains any triangle of T, whose vertices fall on the circle’s edge, does not contain any other points in it’s interior. Figure 1 describes the Voronoi diagram and its DT. DT is used very well in one of the routing protocol called Location Aware Routing(LAR).LAR is a simple Greedy algorithm in which packet is forwarded along the route which uses only DT(Here the neighbor node is chosen for forwarding the packet, which is geometrically nearest to the destination node in a DT). Note that NNG(V ) ⊆ MST (V ) ⊆ RNG(V ) ⊆ GG(V ) DT (V ). There are few interesting properties involving Delaunay Triangulations.
• The Delaunay graph of a planar node set is a planar graph and any angle-optimal triangulation of a node set P is a DT • Any DT of P maximizes the minimum angle over all triangulations of P and DT is also a dual graph of the Voronoi diagram. It has an edge between any two Voronoi cells which share a Voronoi edge. DT can be created by creating a Voronoi diagram of P, then creating the dual of this diagram and also using randomized algorithm with complexity O(n log n). The steps involved in creating it is as follows
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Figure 1: Voronoi diagram and its DT 1. Begin with an initial large triangle, which covers all nodes in set P 2. Incrementally insert all nodes one by one while maintaining DT properties explained above. • Insert a point into an already legalized DT • Triangulate by adding 2 or 3 edges from this node • Legalize all possible illegal edges recursively Repeat steps until all nodes in set P have been triangulated 3. Remove the initial large triangle There are lots of applications of Voronoi diagrams and Delaunay triangulations like ; Finding the 1. NNG of a given graph and 2. MST Restricted Delaunay Graph (RDG (G)): It is well known that DT(G) is a spanning sub-graph of complete graph, but cannot be constructed locally and may contain long edges. To over come these disadvantages RDG concept is evolved. A RDG is a graph that contains all the short edges of G , which is planar and also it is defined as Euclidean spanner of UDG. RDG finds applications related to routing like face routing methods[17] and also in memory less routing algorithm that combines greedy forwarding and local minimum recovery based on face routing. Face routing algorithms applicable only on planar graphs. It is enough, by devising an algorithm to convert the given graph (non-planar) into planar graph by eliminating (or dropping) some links of the graph. In short graph theory plays a vital role in determining, whether a network is planar or non-planar and also to convert non-planar graphs into planar graphs.
4
Conclusions
In this paper we have presented the importance of graph theory to address the fundamental issues in MANET. The paper focuses on the key issues(connectivity,routing,topology control,scaling) and brings an insight of graph theory concepts like graph spanners and proximity graphs . We have extensively studied the graph sparfications,spectral graph theory,random graph models, graph coloring concepts, graph partitioning to address the issues in MANETS[19].
References [1] ChipElliot and Bob Heile Self-Organizing, Self-Healing Wireless Networks Technical report, BBN Technologies, Cambridge, MA, 2001. [2] C. F. Huang, Y. C, Tseng, S. L. Wu, and J. P. Sheu, “Increasing the throughput of multihop packet radio networks with power adjustment,” Procedings of International conference on Computer Communications and Netwoks, pp. 220-225, 2001. [3] J. Broch, D. A. Maltz, D. B. Johnson, Y.-C. Hu, and J. Jetcheva, A performance comparison of multi-hop wireless ad hoc network routing protocols, in Proceedings of the Fourth Annual ACM/IEEE International Conference on Mobile Computing and Networking(Mobicom98), ACM, October 1998.
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[4] Christian Bettstetter, "On the minimum node degree and connectivity of a wireless multihop network," in Proceedings of the 3rd ACM international symposium on Mobile Computer Science Department, UCLA, 1976. [5] G. Di Battista and R. Tamassia. On-line Maintenance of TriconnectedComponents with SPQR-trees.Algorithmica,15:302318, 1996. [6] T. K. Philips, S. S. Panwar, and A. N. Tantawi, "Connectivity properties of a packet radio network model," IEEE Trans. Inform. Theory, vol. 35, pp. 1044-1047, Bollob´as, B., Erdos, P., [7] Graphs of extremal weights, Ars Combinatoria 50 (1998),225-233, 1989. [8] F. Harary, Graph Theory, Narosa Publishing House [9] Charles E. Perkins. Ad-hoc networking. Addison-Wesley, Boston, 2001. [10] D. Peleg and A. A. Sch¨affer, Graph Spanners, Journal of Graph Theory,13:99-116, 1989. [11] E. W Dijkstra, A note on two problems in connexion with graphs, Numerische Mathematik, 1959. [12] M. D. Penrose. Random Geometric Graphs. Oxford University Press, 2003. [13] O. H¨aggstr¨om and R. Meester, "Nearest neighbor and hard sphere models in continuum percolation," Random Structures and Algorithms, vol. 9, pp. 295-315, 1996. [14] Dragos M. Cvetkovic Michael Doob Horst Sachsp, "Spectra of Graphs, Theory and Application", Academic Press [15] X.-Y. Li and I. Stojmenovic, Partial Delaunay triangulation and degree limited localized Bluetooth scatternet formation, in: Proc. AD-HOC NetwOrks and Wireless (ADHOC-NOW), Fields Institute,Toronto, 2002. [16] R. Albert and A. L. Barab´asi, Statistical mechanics of complex networks, Rev. Modern Physics, 74, 47-97, 2002. [17] P. Bose, P. Morin, I. Stojmenovic, and J. Urrutia. Routing with guaranteed delivery in ad hoc wireless networks. Wireless Networks, 7(6):609-616, 2001. [18] M.A.Rajan, M.Girish Chandra, Lokanatha C. Reddy and Prakash Hiremath, A Study Of Connectivity Index of Graph Relevant to Adhoc Networks, IJCSNS VOL.7 No.11, November, 198-204, 2007. [19] M.A.Rajan, M.Girish Chandra, "A Study of Graph Thoery Concepts Relevant to MANETS", Technical Report, TCSL, October 2006.
M.A.Rajan1 , M.Girish Chandra2 ,Lokanatha C. Reddy3 and Prakash Hiremath4 Research Scholar Dravidian University(DU)and Tata Consultancy Services Limited(TCSL)1 , Consultant TCSL2 , Professor DU3 , Professor, Gulbarga University4 Department Of Computer Science1 ,Embedded Systems Lab2 Kuppam, Andhra Pradesh and TCSL, Bangalore, INDIA1 , Bangalore2 , Kuppam, Andhra Pradesh3 , Gulbarga, Karnataka4 1 2 3 E-mail:
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