Energy efficient multicasting problem in wireless ad-hoc networks∗

Proceedings of the 10th WSEAS International Conference on COMPUTERS, Vouliagmeni, Athens, Greece, July 13-15, 2006 (pp469-481)

Energy efficient multicasting problem in wireless ad-hoc networks∗ Manki Min Panos M. Pardalos Dept. of Industrial and Systems Engineering University of Florida Gainesville, FL 32611, USA. {mkmin, pardalos}@ufl.edu

Abstract

and wide deployment. Some of example applications of wireless ad-hoc networks include emergency Wireless communication consumes significant search-and-rescue operations, data acquisition operaamount of energy. In an ad-hoc network where each tions in inhospitable environments such as battlefield, device runs on its own battery, the energy consump- flooding area, etc. Important features of a wireless tion should be minimized to extend the network ad-hoc network include resource limitation (computlifetime and hence the energy consumption is one of ing, communicating, battery power, etc), multi-hop the most important factors in designing a wireless communication, dynamic topology, and lack of inad-hoc network. In this paper, we survey literature frastructure. for the energy efficient multicasting problem in Due to scarceness of wireless link and resources, wireless ad-hoc networks. Due to the NP-hardness mobile hosts communicate indirectly with each other of the problem, many heuristics have been studied. in a multi-hop fashion. In such a scenario, each However, the heuristics in the literature suffer from host should cooperate with others by relaying packets coarse performance ratio. Relatively small amount from the source host to a far away destination host. of research has been done in optimization approach. As a result, the mobile hosts should act as routers by It is important to have knowledge of the optimal themselves and ad-hoc routing schemes are required solution structure in order to develop more efficient for unicasting, multicasting, and broadcasting. In a heuristics and algorithms. We also summarize the wireless ad-hoc network, the network topology may optimization approach for the problem and compare be dynamic due to the mobility of hosts and physical the computational results. In addition, we also link failures. Multiple access, background noise, and review literature for slightly different approaches to interference from other transmissions make an active extend the network lifetime, since minimizing the link between two hosts unavailable. Thus, the comtotal energy does not always lead to a longer network munication link may be unreliable and retransmission lifetime. may be required for reliable services. The unreliaKey-Words: Energy efficiency, Multicasting, bility of communication links leads to frequent upBroadcasting, Wireless ad-hoc networks, Optimizadates of topology or route information which should tion also be performed in a cooperative manner by mobile hosts. Among the challenging properties, the battery power is the most critical issue in wireless 1 Introduction ad-hoc networking. In order to transmit, a device A wireless ad-hoc network is a self-organizing sys- must consume the transmission energy (significantly tem in which mobile hosts communicate each other larger amount than the energy required for computausing a shared, scarce wireless channel. One of the tion only). However, for unicasting, we can use the advantage of wireless ad-hoc networking is its quick traditional shortest path methods to find the minimum energy consuming paths. The same approach does ∗ Research is partially supported by NSF and Air Force grants. not work out for multicasting or broadcasting. 1

Proceedings of the 10th WSEAS International Conference on COMPUTERS, Vouliagmeni, Athens, Greece, July 13-15, 2006 (pp469-481)

The minimum energy broadcasting problem in wireless ad-hoc networks has been shown to be NPhard [5], and many heuristics have been proposed. Relatively small amount of research has been done in the optimization approach. Heuristics usually find good solutions in moderate computation time but the worst-case performance can be quite bad. On the contrary, optimization techniques can find actual optimal solutions by consuming long computation time and large amount of memory. In order to develop a more efficient heuristic, wireless communication intrinsic nature such as WMA (wireless multicast advantage [44]) should be adopted. To find more such properties, we need a way to compute an optimal solution in a moderate time. Existing ad-hoc routing protocols can be categorized as table-driven (proactive) and demand-driven (reactive) [31]. Topology information (in proactive routing) and route information (in reactive routing) are updated and maintained by flooding request packets. Flooding means the network-wide broadcasting. Network-wide flooding may cause unreliable flooding [32] and broadcast storm problem [39]. [32] studies the inefficiency of local broadcasts. The experimental results show that even in moderately sparse graphs, a specific broadcast message is expected to be reachable by around 80% of nodes only in the network. The reachable area of a single broadcast message decreases as the network load increases. Furthermore, the experiments exhibit the degradation of data delivery rate of local broadcasts, which confirms the inefficiency of local broadcasts. The broadcast storm problem is caused by network-wide flooding, which is common in ad-hoc routing for the update of topology information or route information. More specifically, it may result in excessive redundancy, contention, and collision which cause high protocol overhead and interference with other ongoing communication traffic. [39] provides theoretical analysis on redundancy, contention, and collision of a single packet that is relayed by flooding mechanism. Based on the analysis, a rebroadcast can, on average, provide only 41% additional coverage beyond the coverage induced by the previous transmission. This redundancy becomes even worse when more than one rebroadcast happen; when a host rebroadcasts a packet more than three times, the expected additional coverage is below 5%. The expected probability of contention between two hosts is about 59% and the probability that n hosts experience contention increases over 80% as n ≥ 6.

Collisions that are caused by various reasons may result in even more inefficient rebroadcasting. Unlike unicasting communications where the shortest path between the source host and the destination host is easily computed, multicasting and broadcasting communications have many optimization problems which are often NP-hard. When the objective is to minimize the number of relaying hosts in a multicasting session, the problem can be modeled as a Steiner tree problem. The problem of minimizing the number of relaying hosts in a broadcasting session can be modeled as a minimum connected dominating set problem. When we consider the power, or energy to be consumed for a broadcasting session, we can model the problem as minimum broadcast cover problem [5]. Many heuristics or approximation schemes have been proposed for these NP-hard problems. We will mainly focus on the multicasting problems in terms of minimum or efficient consumption of transmission energy. The rest of this paper is organized as follows. Section 2 provides the definition of the minimum energy broadcasting problem and reviews heuristics and IP approaches in the literature for the minimum energy multicasting (including broadcasting) problem. Section 3 surveys energy efficient multicasting schemes which consumes low energy and typically of which goal is to lengthen the network lifetime.

2 Minimum Energy Multicasting In wireless networks, the connectivity of the pair of hosts, or nodes, depends on the transmission power consumed by the sender host. The received signal power varies as r−α , where r is the distance between the pair of hosts and α is a parameter that typically takes on a value between 2 and 4, depending on the characteristics of the communication medium. Based on this model, the transmission power Pij required for a link from the node i to j separated by the distance r is proportional to rα . Ignoring the constant coefficient, we get Pij = rα .

2.1 Minimum Energy Broadcasting [5] provides the definition of the Geometric Minimum Broadcast Cover (GMBC) problem and shows its NPcompleteness. GMBC is described as follows: Given a set of nodes in two-dimensional Euclidean metric space with power level for each node and cost 2

Proceedings of the 10th WSEAS International Conference on COMPUTERS, Vouliagmeni, Athens, Greece, July 13-15, 2006 (pp469-481)

for each edge which is proportional to its length, determine whether there is a power assignment for each node so that any node has a path from the source node in the induced graph using at most a constant amount of total power, i.e. sum of all the transmitting powers of nodes. GMBC is proven to be NP-complete by reduction from the Planar 3-SAT problem, which is known to be NP-complete [18]. Note that GMBC is equivalent to asking whether there exists a broadcast tree that requires at most a constant total cost in Euclidean metric spaces. The mobile devices in a wireless ad-hoc network operate on their batteries. As a result, energy efficiency is one of the most important factors in designing such networks. For unicast communications, it is best to transmit at the lowest possible power level to save transmission energy. Throughout this chapter, we use power and energy interchangeably. In case of broadcast communications, it may not be the best strategy to use the lowest possible power level. Unlike wired networks, wireless networks have “node-based” nature of communications [44], which arises from the broadcast nature of wireless channels. Hence using higher power level than the minimum, a single transmission can reach more hosts. As a result, the overall energy consumption can be reduced.

the WMA property. BLU uses shortest unicast paths to construct broadcast trees by superposing the paths to individual destinations. The resulting broadcast tree can be inefficient, or require more total transmission power, since while some nodes can receive a transmission without help of relaying nodes, BLU may create unnecessary links for those redundant relays. BLiMST is based on the use of the standard MST algorithm where link cost is assigned for each pair of nodes. Again the WMA property is ignored in this algorithm. Hence, the resulting tree may be inefficient in some cases. The inefficiency of the previous two algorithms comes from the usage of a link-based approach. To improve energy efficiency, BIP constructs a tree rooted at the source node by adding nodes with minimum additional cost. This algorithm is based on Prim’s algorithm with iterative modifications of the link costs. The link costs will be updated at each step as follows: Pij0 = Pij − P (i)

where Pij is the link-based cost of a transmission between the node i and j, and P (i) is the power level at which the node i is already transmitting. This approach does not guarantee minimum cost spanning trees but experimental results show improved performance over a wide range of examples [44]. All three algorithms (BLU, BLiMST, and BIP), are 2.2 Heuristics for broadcasting centralized algorithms and their time complexities are Due to its computational complexity, many heuristics O(N 2 ), O(N 3 ), and O(N 3 ), respectively. Here, N have been proposed to compute good (but not opti- is the number of nodes in the network. The performance can be slightly improved by the sweep promal) solutions [44, 40, 5, 28, 29]. [44] presents three heuristics: BLU (Broadcast cedure [44] which eliminates unnecessary transmisLeast-Unicast-cost algorithm), BLiMST (Broadcast sions with additional O(N 2 ) time complexity. The Link-based MST algorithm), and BIP (Broadcast In- sweep operation utilizes WMA property to remove cremental Power algorithm). The authors focused on redundancy. Hence if the difference of the transthe difference between wired and wireless networks. mission power before and after the sweep operation In wired networks, link-based approaches such as a is small, then we can say that the algorithm (withminimum cost spanning tree can be used as a broad- out sweep operation) effectively utilizes the WMA cast tree. On the other hand, wireless networks have property. The authors also described how those althe wireless multicast advantage (WMA) property gorithms can be applied to multicast routing. which enables a single transmission (broadcast in na[40] provides the proof of the approximation ratios ture) to be reachable by every node within the trans- of BLU, BLiMST and BIP. In addition to the three mission range from the source node . Hence, the to- algorithms, the authors proposed a variation of BIP tal transmission power to reach a set of nodes from a based on Chvatal’s algorithm [7] for the Set Cover source node is the maximum power to reach any node problem. The variation is named Broadcast Average in the set. This observation leads to a node-based ap- Incremental Power, BAIP for short. In BAIP, a set proach. BLU and BLiMST are link-based approaches of new nodes with minimal average incremental cost which are based on conventional networking tech- will be added to the spanning tree. Minimal avernologies and BIP is node-based approach adopting age incremental cost for a set of new nodes is defined 3

Proceedings of the 10th WSEAS International Conference on COMPUTERS, Vouliagmeni, Athens, Greece, July 13-15, 2006 (pp469-481)

MLU

BAIP 4n , ln n−o(1)

MLiMST

MIP 13 3 ,

EWMA

SPF

MIPF

OMEGa

SOR

–, –

–, –

–,

13 3

–, –

–, –

lower bound

n n 2, 2

upper bound

–, –

–, –

12, –

12, –

12, –

–, 2c

–, 2c

6, –

6, –

complexity

O(N 2 )

–

O(N 3 )

O(N 3 )

O(d4 )m2

–

–

O(N 3 log N )

O(N 4 )

–

6, n − 1

n − 2 − o(1)

n is the number of receiving nodes N is the number of nodes in the network d is the maximum degree, m is the number of transmitting nodes c is a constant between 6 and 12

Table 1: Comparison of MLU, BAIP, MLiMST, MIP, EWMA, SPF, MIPF, OMEGa, and SOR. Values of lower and upper bound are presented for broadcast, multicast, respectively. to be the average (over the set) of minimum additional power increase to reach the nodes in the set. Despite BAIP’s similarity to BIP, its approximation ratio is quite different from that of BIP. Here the approximation ratio of a heuristic is defined to be the maximum ratio of the energy required to broadcast a message using the tree resulting from the heuristic to the least power required by any spanning tree. They used geometric arguments to analyze the lower and upper bounds of approximation ratios of the four algorithms.

OMEGa significantly improves the quality of solutions. In SOR, with a little bit higher computational complexity, it further improves the quality of solutions. Through iterations, the tree structure is repeatedly changed by removing redundant transmissions and the tree is always maintained as tight as possible. Heuristic approaches are quite efficient in terms of computation time (up to O(n4 ), where n is the number of the nodes). However, heuristic approaches have very coarse worst-case performance ratio (6 to 12) to optimal solutions, although some produce very Another heuristic based on the WMA property is good solutions. If we can find the optimal solution proposed in [5]. The heuristic Embedded Wireless structure, we may develop a near-optimal heuristic. Multicast Advantage, EWMA for short, is based on the link-based MST but iteratively modifies the tree 2.3 Heuristics for multicasting by excluding some transmitting nodes so that the total transmission power decreases. EWMA consists The minimum cost multicast tree problem can be of two phases. In the first phase, it constructs an modeled as an NP-hard Steiner tree problem. HowMST, and in the second phase, it incrementally builds ever in this subsection, the minimum energy mula broadcast tree starting from a single (source) node ticasting problem is modeled as a generalized vertree. For the selection of transmitting nodes, gain is sion of minimum energy broadcasting problem. Mulused as a metric. The gain of a node v is defined ticasting requires a set of destination nodes, which as the decrease in the total transmission power of a is called as the multicast group, and by setting the broadcast tree by excluding some of the transmitting multicast group to be the whole nodes set, the mulnodes in the MST and instead increasing the trans- ticasting becomes the broadcasting. Heuristics for mission power of v. EWMA also has 12 as its upper broadcasting can be used for multicasting through a bound of approximation ratio, but the lower bound is process called pruning in which unnecessary transnot clear. Its time complexity is O(d4 )m2 , where d is mission are pruned out. the maximum degree and m is the number of trans[44] proposes the applications of three broadcast mitting nodes in the tree. tree heuristics to multicast tree construction; MultiMin et al presented two heuristics named OMEGa (Optimistic Most Energy Gain) [28] and SOR (Shrinking Overlapped Range) [29]. In OMEGa, optimistic energy gain (lower bound or upper bound of lower bound of actual energy gain, both are used to estimate the actual energy gain) of a transmission is used as a measure for the selection of transmissions. With the time complexity comparable to that of BIP,

cast Incremental Power (MIP) algorithm, Multicast Least-Unicast-cost (MLU) algorithm, and Multicast Link-based MST (MLiMST) algorithm. In MIP, first a broadcast tree is constructed by BIP and then to obtain the multicast tree, redundant transmissions are pruned from the tree. A transmission is redundant if all the nodes in the multicast group can receive the multicast message without the transmission. Sim4

Proceedings of the 10th WSEAS International Conference on COMPUTERS, Vouliagmeni, Athens, Greece, July 13-15, 2006 (pp469-481)

ilarly, MLiMST prunes the broadcast tree obtained from BLiMST to get a multicast tree. The redundancy of a transmission is defined the same as MIP. MLU is almost identical to BLU, the only difference is that MLU finds unicast paths from the source to only the destinations in the multicast group. The multicast tree is formed by superposition of the unicast paths. Experimental results are provided in [44] in which MIP, MLU and MLiMST (BIP, BLU and BLiMST for broadcast tree, respectively) are simulated. Performance metric is the total power of the multicast (or broadcast) tree. Since a multicast tree is constructed as a subtree of the corresponding broadcast tree, multicast tree heuristics can be expanded to generate a broadcast tree by setting the entire hosts as the multicast group. When the multicast group size is small, MLU performs the best. This is a natural result since MLU computes optimal paths to individual destinations. However when the multicast group size becomes large, the performance of MLU degrades while the performance of MLiMST and MIP improves. Especially MIP outperforms both MLiMST and MLU for large multicast group. This improvement comes from the fact that MIP successfully exploit the “node-based” nature of wireless (broadcast) communications. [24] presents a heuristic named as MIP3S (Multicast Incremental Power with Potential Power Saving). MIP3S finds the multicast tree by repeatedly selecting the transmissions that increase the total transmission energy the least. Here, the increase of the transmission energy is computed as the sum of the increased transmission power of the newly selected node and the potential power saving of nodes that were already covered by other nodes and also covered by the newly selected node. However, MIP3S does not have the theoretical upper bound for the performance ratio, even for the broacasting case. [42] proposes two heuristics, SPF (Shortest-Path First) and MIPF (Minimum Incremental Path First), for minimum power multicast tree problem based on BLiMST and BIP, respectively. They showed none of MIP, MLU and MLiMST has constant approximation ratio and their two heuristics have constant approximation ratios. Instead of pruning branches in the broadcast tree, both SPF and MIPF iteratively add paths from the source node to the required destinations. In SPF, a tree T rooted at the source node is maintained during the procedure. Starting from a single (source) node, T is iteratively grown by one

path from the source node to a destination which is not in T yet. First the entire tree T is collapsed into a temporary node and then the algorithm computes all the shortest paths from the temporary node to all the destinations which are not in T . Among them, a path with the least total transmission power is added. MIPF is based on BIP and the execution is similar to SP F with the exception that MIPF uses incremental power instead of total transmission power as the metric of a tree. Starting from a single (source) node, the path from a node in T to a destination which is not in T yet with the least incremental power is added iteratively. The incremental power of a path is defined as the total transmission power minus the transmission power of the starting node in the path. OMEGa (Optimistic Most Energy Gain) [28] and SOR (Shrinking Overlapped Range) [29] can be used to find the minimum energy multicast tree. In order to be used for multicasting, OMEGa method defines a penalty function for a transmission to a node which is not in the multicast group. In this way, a small opportunity is left for a node not in the multicast group to be added. SOR also defines a penalty function for a transmission to a node not in the multicast group. The basic idea is that for the node not in the multicast group, two-hop paths to a multicast group node are considered. The experimental results show not only the lower energy consumption but also the lower overlaps of transmissions and shorter path lengths. SOR outperforms the other algorithms especially when the multicast group size increases.

2.4 MIP Approach In the pursuit of optimal solutions to the minimum energy broadcasting problem, MIP (Mixed Integer Programming) has been used [14, 30, 47, 27]. The difficulty of solving the minimum energy broadcasting problem mainly comes from the constraints of connectivity. In [30], the problem is a little bit different in a sense that bidirectional links are used to construct connectivity of the graph and no source node is assumed. However, their computational results show asymptotically exponential growth in the computation time as the problem size (the number of nodes) grows. One interesting point is that their iterative exact algorithm outperforms the traditional approach (solving the MIP by feeding the formulation into the solver). The computational time of the iterative algorithm reduces up to 1 % of that of IP approach. 5

Proceedings of the 10th WSEAS International Conference on COMPUTERS, Vouliagmeni, Athens, Greece, July 13-15, 2006 (pp469-481)

[14] presents three MIP formulations A, B, and C. In the formulation A, the authors introduced auxiliary variables to denote a link at a step. These auxiliary variables can guarantee a tree structure in any feasible solution, but too many (O(n3 ), where n is the number of nodes) variables and constraints are required. One interesting point in the formulation A is that they used the transmission power constraints to restrict the transmission power of a node to be chosen only from its possible transmission powers. In the formulation B, the authors used subtour prevention constraints for TSP problem [25]. These constraints can prevent subtours and hence can guarantee the tree structure in any feasible solution. It requires O(n2 ) variables and constraints. Finally in the formulation C, network flow constraints are used to guarantee the tree structure. The difference between the sum of the incoming flows and the sum of the outgoing flows at each node is defined in a different way depending on whether the node is a destination node or not. If a node is a destination node, then the difference is 1, and 0 otherwise. It also requires O(n2 ) variables and constraints. However the experimental results are not provided in the paper. [47] presents two MIP formulations, F1 and F2. F1 is similar to the formulation C in [14] with the exception that they didn’t separate the constraints of the difference of the flow sum. This makes sense when only broadcasting is considered. F1 also requires O(n2 ) variables and constraints. If F2, the author uses the transmission power constraints of the formulation in [14] and separates the flow variable of each link based on the source-destination pairs. One advantage of F2 against F1 is that it produces a tighter LP (linear programming) relaxation value than F1 does. However the experimental results in [27] shows that all those formulations find the optimal solutions very slowly even for small-sized problem instances with 10 or 20 nodes.

the computation time growth rate against the network size increase is the slowest among the formulations in the literature. The constants and variables used in our formulation are described below. • N = {1, ..., n} is the set of nodes. • s ∈ N is the source node. • D ⊆ N is the set of destination nodes. D = N for broadcasting. • Pij is the minimum transmission power required for the node i to directly transmit to the node j. • xi is the transmission power assigned at the node i. • uij is a binary variable indicating whether the node i transmits at power level of Pij . • Fij is a flow variable on each edge ij. (2) is used to maintain tighter LP relaxation and to speed up the computation. In their iterative algorithms, which will be presented in the next section, the efficiency (both in terms of tightness of LP relaxation and computation time) of (2) is extensively utilized. (5) is used to further speed up the computation since now we have additional constraints about the destination node coverage. (6) binds the flow variables and the transmission variables. (8, 9) are the flow sum constraints at each node and they can be used to guarantee the coverage of the destination nodes. At a glance, it looks redundant to have two different types of constraints which can be used to guarantee the coverage, i.e. (5) and (8, 9), but the experimental results show that it significantly improves the computation time. The MIP formulation has O(n2 ) variables and constraints.

2.6 Iterative Algorithms 2.5 MIP formulation

[27] presents two iterative algorithms. In both of the iterative algorithms, the relaxed MIP formulation (Figure 1 minus (6) – (9) and (12)) is used. At each iteration, cuts are added to transform the infeasible solution obtained from the relaxed MIP formulation into a feasible solution. The original cut constraints to guarantee the connectivity is as follows. XX uij ≥ 1 ∀ S ⊂ N (13)

Figure 1 is the MIP (Mixed Integer Programming) formulation for the minimum energy broadcasting problem [27]. The experimental results show that it is the most efficient formulation for the problem in the literature in terms of the computation time. Other formulations in the literature were unable to solve even 20 node cases within 30 minutes, on the other hand, the formulation in Figure 1 solved 20 nodes cases very quickly (within 3 seconds on average). Also

i∈S j ∈S /

6

Proceedings of the 10th WSEAS International Conference on COMPUTERS, Vouliagmeni, Athens, Greece, July 13-15, 2006 (pp469-481)

min

X

xi

i∈N

s.t.

(1) X

xi −

Pij uij = 0

∀i∈N

(2)

j∈N,j6=s,j6=i

X

uij = 1

if i = s

(3)

uij ≤ 1 ∀ i 6= s

(4)

j∈N,j6=s,j6=i

X

j∈N,j6=s,j6=i

X

X

uik ≥ 1 ∀ j ∈ D

(5)

uik − Fij ≥ 0 ∀ i, j ∈ N, j 6= s, j 6= i

(6)

i∈N k∈N,k6=s,k6=i,Pik ≥Pij

X

|D|

k∈N,k6=s,k6=i,Pik ≥Pij

X

Fij ≤ |D|

j∈N,j6=s

X

Fji −

j∈N,j6=i

X

(7)

Fij = 1

∀i∈D

(8)

Fij = 0

∀i∈ /D

(9)

j∈N,j6=s,j6=i

j∈N,j6=i

X

if i = s

Fji −

X

j∈N,j6=s,j6=i

xi ≥ 0 ∀ i ∈ N uij ∈ {0, 1} ∀ i, j ∈ N, j 6= s, j 6= i Fij ≥ 0 ∀ i, j ∈ N, j 6= s, j 6= i

(10) (11) (12)

Figure 1: MIP formulation where V 0 is the set of nodes in the current source-rooted tree.

Theorem 1 If the relaxed MIP formulation with a partial set of constraints (13) produces a feasible solution, then the solution is optimal.

2. Repetition prevention constraints : X X uij = 0

Constraints (13) guarantee the connectivity of the (15) resulting graph. But the problem with this type of i∈V 0 j∈N,uij =1 constraints lies in the exponential number of the constraints. However for any optimal solution, we don’t 3. Branch cutoff constraints : need every such type of constraints. Hence we can reX X duce the number of such constraints by running iterauij = 0 (16) tive algorithm and adding only necessary constraints. i∈V 0 j∈N,j6=s,j6=i,Pij ≥Piu In the first iterative algorithm (PathCuts1), a heuristic such as SOR is used to quickly achieve a Constraints (14) are used to branch on a specific better upper bound. Three additional types of cut node and compute an increased but not yet feasible constraints are used in the algorithm to prevent vis- solution. After an intermediate solution is computed, iting the same power assignments. this type of constraints is removed from the system. 1. Single node connection cut which connects a Whenever a feasible solution is found, (15) is used to prevent the solver from reaching the same solution node u to the source-rooted tree: again. This prevention helps faster termination by poX X uij ≥ 1 (14) tentially increasing the minimum objective values for 0 other branches. When a branch does not need to be i∈V j∈N,j6=s,j6=i,Pij ≥Piu 7

Proceedings of the 10th WSEAS International Conference on COMPUTERS, Vouliagmeni, Athens, Greece, July 13-15, 2006 (pp469-481)

checked again, (16) cuts off the branch. The branch can be cut off on the following conditions:

reduction of protocol overhead; thus it greatly improves the network throughput. This is achieved by propagating control packets inside the virtual backbone, not the whole network. Other benefits include the support of broadcast/multicast traffic and the propagation of link quality information for qualityof-service (QoS) routing [34]. In the long run, the deployment of virtual backbones will be helpful in lengthening the network life time by reducing the communication overhead. The concept of a virtual backbone was initially studied for mobile packet radio networks [17, 19]. Those initial approaches aimed at finding a feasible interconnected set of clusters covering the entire node population without the objective of optimizing the size of the backbone. [17, 19] propose two algorithms for clustering : Lowest-ID Cluster Algorithm and Highest-Connectivity Cluster Algorithm. In the formation of the clusters and the clusterheads, the node with the lowest ID or the highest connectivity is selected to be a clusterhead. [19] gives experiments about the number of cluster changes upon a single host movement and the result shows that the lowest-ID algorithm requires less cluster changes. Both algorithms have the following two interesting properties: clusterheads are not directly linked each other, and any two nodes in the same cluster are at most two-hops away, since the clusterhead is directly linked to every other node in the cluster. The above two properties imply that the set of the clusterhead nodes form an independent set and this idea has been commonly used in an approximation of the minimum connected dominating sets with the benefit of a constant upper bound of the backbone size.

• Upper bound cutoff : when the intermediate objective value is greater than or equal to the current best objective value • Feasibility cutoff : when the intermediate solution is feasible • No available cut cutoff : when there is no more available single node connection cut (uij ’s may be set to be zero by constraints (15, 16))

In the second algorithm (PathCuts2), the basic idea is to shrink the transmission power of the source node from the maximum possible value to the minimum possible value at each iteration and to compute the minimum broadcast tree for each value at each iteration. This approach is based on the observation that optimal solutions have many transmissions in the outward directions from the source node in most cases. No additional cut constraints are used and only partial set of constraints (13) are added to the relaxed MIP formulation. The computational results show that iterative algorithms produce an optimal solution more efficiently than the MIP formulation (Figure 1) approach in terms of computation time and memory usage. Even though the MIP formulation approach uses relatively small branch tree than other MIP approach in the literature, when the problem size grows (to at least 30 nodes), the tree size becomes large and hence it takes long time to check every branch of the tree. However, iterative algorithms solve a small subproblem at each iteration and it uses smaller amount of memory for the branch than other MIP formulation. Comparing iterative algorithms, PathCuts2 is more efficient for bigger problems than PathCuts1. In addition, the 3.1.1 Virtual backbone-based communication computation time growth of PathCuts2 is less steep The main application of the virtual backbones in than that of PathCuts1. wireless ad-hoc networks is the virtual backbonebased routing [12, 34, 45, 32]. Virtual backbones can be used for broadcasting [35] or mobility man3 Energy Efficient Multicasting also agement [22]. [12] discusses a virtual backbone scheme with a 3.1 Virtual backbone schemes goal of minimizing the size of the virtual backbone In a wireless ad-hoc network, there are no fixed in- and introduced its application to routing. Their virfrastructure and central administration. This defi- tual backbone structure aims at supporting unicast, ciency aggravates the inefficiency of ad-hoc com- multicast, fault-tolerant routing in wireless ad-hoc munications. To alleviate these problems, virtual networks. The virtual backbone is constructed from backbone-based routing strategies have been intro- an approximation of minimum connected dominatduced [12, 33, 45, 32, 35]. The most important ben- ing set, MCDS for short. The backbone structure efit of virtual backbone-based routing is the dramatic will change upon node movement and the hosts in the 8

Proceedings of the 10th WSEAS International Conference on COMPUTERS, Vouliagmeni, Athens, Greece, July 13-15, 2006 (pp469-481)

backbone are used only for computing and updating the routes. In order to accommodate the dynamic nature of the virtual backbone, their approach splits the routing problem into two levels: find and update the virtual backbone, and then find and update the routes. The route computation is a traditional shortest path algorithm. Low communication overhead for the route computation is critical in wireless ad-hoc networks since the routes are computed frequently due to the dynamic topology. The local information-based route computation is more adaptable to dynamic environments than the global information-based route computation. [45] discusses a virtual backbone based routing as a solution of these requirements. They proposed a simple and message-efficient mechanism (marking process and 2 rules to reduce the size) to construct a virtual backbone, which is a connected dominating set. The nodes in the backbone are called gateway. Though their routing scheme is based on the shortest path scheme, they showed that the shortest path between any two nodes should consist of gateway nodes only. Using this property, the shortest path route can be computed only among gateway nodes. [34] proposes CEDAR, a core-extraction distributed ad-hoc routing algorithm, aiming at quality-ofservice (QoS) routing in wireless ad-hoc network environments. Their objective is to compute the unicast routes satisfying a minimum bandwidth requirement from the source to the destination. Due to the dynamic topology and the unreliable transmissions, their goal is to provide the routes that are highly likely to satisfy the bandwidth requirement of a route. CEDAR has three components: core extraction, link state propagation, and route computation. The set of core nodes is obtained by approximating minimum dominating sets using the local computation and the local state. The core nodes (the dominators) maintain the local topology of its domain and compute the routes on behalf of the nodes in its domain. For QoS purpose, the bandwidth availability information of the stable high bandwidth links is propagated to core nodes, while information about the dynamic or low bandwidth links is kept local. The routes are computed iteratively, starting from the core path, which consists of virtual links between the core nodes, between the dominator of the source and the dominator of the destination. Using the core path, the routes are computed by finding successive partial routes from the source to the furthest possible node in the core path satisfying the bandwidth requirement where the

intermediate destination becomes the new source for the next partial route. Virtual backbones can be used to enhance the existing routing schemes and a recent experimental result is found in [32]. The authors presented extensive simulation results on the effect of applying the existing routing schemes on the virtual backbone infrastructure. They used an adaptation of the core infrastructure (extracted by CEDAR algorithm) as a virtual backbone and the routes are computed by applying the existing routing schemes over the backbone. They chose DSR and AODV and carried out performance evaluation of the routing schemes over the backbone. The experimental results show that by restricting the exchange of the routing control packet to the backbone nodes, the performance of AODV and DSR is improved. The average number of messages per route request is significantly reduced to about 10% for both schemes. The protocol overhead decreases up to 50% for AODV scheme. For DSR, the overhead does not decrease and this is due to DSR’s aggressive caching policy. But in a highly dynamic scenarios, the cached information easily becomes stale, hence the result does not imply that virtual backbones are not suitable for DSR. [35] discusses about the application of virtual backbones to broadcasting. The authors proposed two schemes; backbone broadcasting scheme and neighbor elimination scheme. In backbone broadcasting scheme, rebroadcasting is limited to the backbone nodes. Compared to the flooding, this scheme requires significantly less retransmissions. Neighbor elimination scheme further reduces the number of retransmissions by removing redundant rebroadcasts. A backbone node will rebroadcast the received message only when it has a neighbor that may need the message. In order to compute the redundancy, it is required that each node knows the exact location of all its neighbors or maintains the list of its 2-hop neighbors. Every non-backbone node registers itself to one of its backbone neighbors and the backbone nodes decide whether there is a registered neighbor that will not receive the rebroadcast message from other backbone nodes. The authors conducted experiments on various broadcasting schemes and the result shows that the backbone broadcasting performs the best and with the neighbor elimination added, the performance is further improved. Another application of virtual backbone to mobility management is discussed in [22]. The authors proposed ad-hoc mobility management scheme utiliz9

Proceedings of the 10th WSEAS International Conference on COMPUTERS, Vouliagmeni, Athens, Greece, July 13-15, 2006 (pp469-481)

ing the location databases. These location databases are distributed on the virtual backbone nodes. The virtual backbone is constructed using a greedy algorithm for Minimum Set-Covering problem, hence the backbone nodes need not be connected via 1hop links, but the connectivity between the backbone nodes are maintained as multi-hop paths. Routing is performed on the flat network structure, contrary to the (hierarchical) backbone routing, which involves non-backbone nodes for more balanced loading on the nodes and the links. The virtual backbone nodes provide location information from the databases they hold. The location information helps localizing the routing procedure and managing nodal mobility. Since the location information should be kept up-to-date, the virtual backbone nodes need to maintain the inter-connection among themselves. Among many characteristics of wireless networks, nodal mobility is one of the important features. When nodes move, any sparse structure is subject to lose the connectivity very soon. In the construction of the virtual backbones for wireless networks, the nodal mobility should be considered so that the resulting structure, the virtual backbone, becomes more robust and at the same time requires less maintenance cost. [26, 43] propose a virtual backbone scheme which is more reliable and hence requires less maintenance need for mobile ad-hoc networks. The authors considered two factors: the stability and the coverage of nodes. Their scheme also generates a constant approximation ratio (at most 8) virtual backbone. Moreover, by constructing the backbone using nodes with more coverage and more stability, the lifetime of the resulting backbone increases significantly. The message and time complexities are O(∆n) and O(n), respectively. The extensive experimental results show that the resulting backbone is more robust than the backbones generated from other existing backbone schemes. Against the smallest size backbones, the lifetime of the connectivity and the coverage increased by 148% and 490% on average under various mobility scenarios . Even against the equivalent size backbones, the lifetime increased by 88% and 32% on average. In terms of maintenance cost, the authors of [26, 43] measured potential maintenance need which essentially counts the number of the disconnected backbone and non-backbone nodes. For a backbone node, when it loses a link to its neighbor that was originally in the backbone, maintenance may be required to repair the damaged connectivity. For a non-backbone

node, when it loses all the links to its neighbors that were originally in the backbone, maintenance may be required to repair the damaged coverage. In this way they measured the potential maintenance needs for several virtual backbone schemes. The experimental results show significant improvements for the potential maintenance needs. One more interesting point is that the conventional two phases (MIS construction and interconnection) are interleaved so that after the new MIS nodes are selected, those nodes select intermediate nodes which interconnect them to existing MIS nodes. This integration of phases leads to a faster construction of the virtual backbone, which is preferable in highly dynamic environments.

3.2 Balancing power consumption While the most research on energy-efficient broadcasting focuses on optimizing the total power in the broadcast tree, [9, 10] present a multicast problem aiming at balancing the power consumption in the multicast tree. Wireless ad-hoc networks require every host’s cooperation, for example without any help of routers each host has to exchange messages to find out the routes. In this situation, if some hosts are completely drained of their power, the whole network may be partitioned or may not operate normally. Hence the authors identified a slightly different energy-efficient broadcast tree problem where the objective is to balance the power consumption. In the paper, the network lifetime is defined as the first time that any host is completely drained of power. And the problem is to find a broadcast tree which minimizes the maximum power consumption of hosts. [9, 10] find an interesting property that an MST has the minimum longest edge among all spanning trees and the MLE (Minimum Longest Edge) algorithm was proposed based on the property. MLE consists of two phases: the MST construction phase and the redundancy elimination phase. Every relaying (transmitting) host is represented as a nonleaf node in the MST. And the transmission range of a relaying host is represented as the length of the longest edge between the node and a leaf node. Among the nonleaf nodes, and hence their corresponding transmissions, there may be unnecessary transmissions; for example, if a node A’s children are all within the transmission range of A’s parent, then A does not need to transmit. In this way the MST from the first phase is reformed into another spanning tree after the second phase. Note that the reformed tree preserves the

10

Proceedings of the 10th WSEAS International Conference on COMPUTERS, Vouliagmeni, Athens, Greece, July 13-15, 2006 (pp469-481)

minimum longest edge property. The second phase of MLE also implicitly makes use of WMA property. The experimental results show that the maximum transmitting power of MLE is decreased by 15% against BIP on average.

3.3

of partially received signals can further improve the energy efficiency.

References

Hitch-hiking

Recently an interesting concept of hitch-hiking has been used in the study of energy efficient multicast problem [1, 37, 38]. The key idea of hitch-hiking [1] is to use not only the fully received signal but also the partially received signal. For every signal reception, SNR (Signal-to-Noise Ratio) is computed and the received signal is determined to be whether a full reception or a partial reception based on the SNR. [1] provides two thresholds γp and γacq . γp is the threshold for successful decoding of the packet and γacq is the threshold for successful time acquiring and hence successful decoding of header. If the received signal has the SNR γ in between γacq and γp , the signal is said to be a partially received signal. Partially received signals can be combined so that together they can be used to decode the complete signal. This combining process is called hitchhiking. The hitch-hiking approach requires the assumption that the nodes can buffer the partially received packet so that they can be used to decode the complete packet later. Aggressive use of hitch-hiking is expected to result in reducing the number of relaying nodes as well as reducing the transmission energy. [37, 38] present the combination of WMA property with hitch-hiking concept and the experimental results show significant improvement on the total transmission energy consumption.

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4 Conclusions

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