A Distributed Algorithm for Multi-constrained Anypath Routing in Wireless Mesh Networks

Transcription

1 A Distributed Algorithm for Multi-constrained Anypath Routing in Wireless Mesh Networks Xi Fang, Dejun Yang, and Guoliang Xue Arizona State University Abstract Anypath routing, a new routing paradigm, has been proposed to improve the performance of wireless networks by exploiting the spatial diversity and broadcast nature of the wireless medium. In this paper, we study the problem of finding an anypath subject to multiple (K) constraints, which has been proved to be NP-hard. We present a polynomial time distributed K-approximation routing algorithm. Our algorithm is as simple as Bellman-Ford s shortest path algorithm. Extensive experiments show that our algorithm is very efficient and its result is as good as that obtained by the best centralized algorithm which however requires global information. 1. Introduction For unreliable wireless networks, due to the spatial diversity and broadcast nature of wireless medium, it is usually less costly to transmit a packet to any node in a set of neighbors than to one specific neighbor [10]. This observation motivates the emergence of a novel routing technique, known as opportunistic routing. It has been shown that opportunistic routing can significantly improve the performance of wireless networks [1][3]. Biswas and Morris [1] designed and implemented ExOR, an opportunistic routing protocol for wireless mesh networks. Chachulski et al. [3] introduced MORE by combining opportunistic routing and network coding together. Based on the concept of opportunistic routing, Dubois-Ferrière [10] introduced a new routing paradigm, called anypath routing, which was subsequently studied in [11][13][19][25]. In anypath routing, each packet is broadcast to a forwarding set composed of several neighbors (called forwarders), and the packet is retransmitted only if none of the forwarders receives it. As long as one of them receives this packet, it can be forwarded on. One of the key research issues in anypath routing is how to find an optimal anypath with respect to delay, cost, energy consumption, etc. This problem often reduces to finding an optimal forwarding set for each node. On one hand, a node with more forwarders can have less forwarding cost or delay to any of its forwarders. On the other hand, each neighbor does not make as much progress as the next hop in the optimal single path. Therefore, having too many forwarders may lead to a counterproductive backfire, such as increasing the likelihood of a packet veering away from the optimal single path, which ultimately even results in loops in the routing topology [11]. Previous works mainly study how to compute a shortest delay or least cost anypath, with only one QoS (quality of service) metric taken into account [10][11][19]. Dubois- Ferrière [10] addressed the shortest anypath problem, Dubois- Ferrière et al. [11] studied the least-cost anypath routing Fang, Yang, and Xue are all affiliated with Arizona State University, Tempe, AZ. This research was supported in part by NSF grants , , and ARO grant W911NF The information reported here does not reflect the position or the policy of the federal government. problem, and Laufer et al. [19] focused on the problem of computing a shortest anypath if different nodes are allowed to use different transmission rates. A closely related problem is the multi-constrained path problem (MCP), which seeks a path connecting a source node to a destination node that satisfies multiple QoS constraints, such as cost, delay and reliability [4][15][22]. When the number of constraints is greater than one, MCP is a well-known NP-hard problem and it has been studied by many researchers[12][17][20][23][24]. In this paper, we study the problem of computing an anypath subject to K constraints, called multi-constrained anypath routing problem (MCAP). Xi et al. [13] proved that this problem is NP-hard, and presented a polynomial time K- approximation algorithm, called Greedy. However, Greedy is a centralized algorithm, in which the global information of the entire network is needed to compute an anypath for a sourcedestination pair. This requirement restricts the application of Greedy for actual wireless networks. In order to overcome this restriction, we design a distributed algorithm for solving MCAP in this paper. We propose a distributed routing algorithm with provably good performance guarantee, called DMART. Each node just needs the local information from one-hop neighbor. DMART computes a K- approximation after at most n iterations and each iteration can be done in O( (log + K)) time by each node, where is the max out-degree of network graph G, and n is the number of nodes in this network. When K=1, our algorithm is the optimal algorithm for the corresponding problem. The shortest anypath algorithm proposed in [10] is a special case of DMART when K = 1 and all the vertex weights are equal to 1. When K 2, our result is a distributed O(1)-approximation algorithm for the NP-hard problem. Extensive simulations show that our algorithm converges very fast (although the theoretical upper bound on the number of iterations is n) and its result is as good as that obtained by the best centralized counterpart Greedy. We must emphasize that DMART just needs local information from one-hop neighbors, while Greedy requires the global information of the entire network. The rest of this paper is organized as follows. In Section 2, we review anypath routing, describe our network model, and define the problem to be studied. In Section 3, we present our approximation algorithm, analyze its properties, and describe its distributed implementation. Section 4 reveals our numerical results. We conclude this paper in Section System Model and Problem Formulation In this section, we first review the anypath routing introduced in [10], then describe the network model, and finally formulate the problem studied in this paper.

2 2 2.1 Anypath Routing In anypath routing, each node broadcasts a packet to a set of its neighbors. As long as one of them receives this packet, this packet can be forwarded on. This set of neighbors is called a forwarding set, which is similar to the next hop for each node in classic routing. Node v is called a forwarder of node u if v is in u s forwarding set. Since more than one forwarder may receive the same packet, unnecessary redundant forwarding should be suppressed. Therefore, the nodes in a forwarding set are each given a priority in relaying the received packet. A higher priority is assigned to the node with shorter distance or less cost to the destination. A node forwards a received packet only if all higher priority nodes fail to receive it. As a result, this node will forward the received packet to the destination while other lower priority nodes suppress their forwarding. For each packet, the source keeps rebroadcasting it until some forwarder receives it or a threshold is reached. Once a forwarder receives this packet, it repeats the same procedure until this packet is delivered to the destination. Mathematically, an anypath from a source to a destination is a directed graph where every node (but the source) is a successor of the source, and every node (but the destination) is a predecessor of the destination. We use Fig. 1 to illustrate the concept of the anypath. An anypath is shown in bold dashed arrows. For instance, s, v 1 and v 2 have forwarders {v 1, v 2, v 3 }, {v 3 } and {v 3, v 6 } respectively. Since at every hop only one forwarder forwards the packet, every packet from s traverses only one of the available paths to reach the destination t. The dashed red line in Fig. 1 shows a possible path of a particular packet. Different packets could traverse different paths, therefore paths are determined on-the-fly, depending on which nodes in the forwarding sets successfully receive the packet at each hop. The anypath shown in Fig. 1 is composed of the union of 7 different paths between s and t. As discussed in [10][11][13], we study the case that anypaths are acyclic. Intuitively, if an anypath is acyclic, all of the potential paths are simple without repeated vertices. This can guarantee that no packet will traverse a forwarder more than once. Fig. 1. Illustration of anypaths. The link labels show link delivery probabilities; the vertex labels show (w 1, w 2 ) pairs (explained in Section 2.2). For instance, w 1 and w 2 may represent the average transmission time and the average energy consumption of the node for each transmission, respectively. 2.2 System Model A wireless mesh network is modeled as a directed graph G = (V, E), where E is the set of m edges and V is the set of n vertices. In this paper, the following terms are used interchangeably: edge and link, vertex and node. Each edge (v, u) E is associated with a packet delivery probability p(v, u). We define the hyperlink delivery probability p(v, J) as the probability that a packet transmitted by node v is received by at least one of the nodes in set J. As demonstrated in [19][21], the loss of a packet at different receivers occurs independently in practice. Thus, the hyperlink delivery probability can be computed as p(v, J) = 1 j J (1 p(v, j)). In addition, each node v V is also associated with K vertex weights w k (v), 1 k K. There are many possible explanations for vertex weights. For instance, when K = 1, w 1 (v) may represent average transmission time for each transmission. When K = 2, w 1 (v) may represent average transmission time and w 2 (v) may represent average energy consumption for each transmission. Now we introduce a metric called expected weight of anypath transmissions (EWATX), which is a generalization of EATX [3][10][19][25] and defined as ζ k (v, J) = w k(v) p(v,j), 1 k K. We take the case of K = 2 above as an example. ζ 1 (v, J) (ζ 2 (v, J), respectively) represents the expected transmission time (the expected energy consumption, respectively) for a packet sent by v to be successfully received by at least one node in J. We now define the k th anypath weight from v to t along an anypath P as W k (v, P ) = w k(v) p(v,j(v, P )) + W k(j(v, P ), P ), (2.1) where J(v, P ) is v s forwarding set for anypath P and W k (J(v, P ), P ) is the k th anypath weight of the set J(v, P ) along anypath P. It is intuitively defined as a weighed average of the k th anypath weights from the nodes in J(v, P ) to the destination along P : W k (J(v, P ), P ) = j β J(v,P ) α(j β, P )W k (j β, P ), (2.2) with j β J(v,P ) α(j β, P ) = 1(β [1, J(v, P ) ]), where the coefficient α(j β, P ) denotes the probability of a node with priority β (i.e. node j β ) forwarding a received packet. Obviously, node j β forwards a packet only when it receives this packet and none of the higher priority nodes receives it, which happens with the probability (1 p(v, j 1 )) (1 p(v, j β 1 ))p(v, j β ). Thus the coefficient α(j β, P ) can be computed as follows: α(j β, P ) = p(v,j β ) β 1 q=1 (1 p(v, j q )), (2.3) p(v,j(v, P )) with the denominator being the normalizing constant. Note that p(v,j(v, P )) = 1 j β J(v,P ) (1 p(v, j β )) = j β J(v,P ) p(v,j β ) β 1 q=1 (1 p(v, j q )). Consider the example of K = 2 discussed before. W 1 (v, P ) (W 2 (v, P ), respectively) represents the expected total transmission time (the expected total energy consumption, respectively) necessary for a packet sent by v to be successfully received by the destination node t along anypath P. We now use the network depicted in Fig. 1 as an example to illustrate how to compute anypath weights, and focus on the transmission time (k = 1). The expected transmission time of a packet sent from v 4 to t via the forwarding set J = {t, v 6 } can be calculated as W 1 (v 4, P ) w 1 (v 4 ) = 1 (1 p(v 4, t))(1 p(v 4, v 6 )) + p(v 4, t)w 1 (t, P ) + (1 p(v 4, t))p(v 4, v 6 )W 1 (v 6, P ) 1 (1 p(v 4, t))(1 p(v 4, v 6 )) (1 0.5) 1 2 = + = 3. 1 (1 0.5)(1 1) 1 (1 0.5)(1 1)

3 3 Note that t is actually a forwarder with the first anypath weight W 1 (t, P ) = 0, and that v 6 s first anypath weight can be calculated as W 1 (v 6, P ) = w1(v6) p(v + W 6,t) 1(t, P ) = = 2. Adding an extra forwarder, however, is not always beneficial although it reduces the forwarding delay or cost from a node to any of its forwarders. For example, the expected transmission time of a packet sent from v 4 to t via the forwarding set {t} is = 4, while the expected transmission time via {t, v 5} is (1 0.5) 0.2 ( ) 1 (1 0.5)(1 0.2) = 6.7. Since the concept of relay priority is introduced for forwarders, MAC protocols for anypath routing must be designed to guarantee this mechanism. Some MAC protocols have been proposed to provide the priority for the forwarders [5][16][18]. In addition, the MAC periodically measures average transmission time, energy consumption, delivery probability and other information, and notifies the network layer to update this information. The MAC also needs to handle interference, schedule links, and recover from packet loss. However, the details of the MAC are abstracted from the routing layer. As in [8][9][13][19], practical routing protocols only incorporate the information reported by the MAC into the routing metrics in order to abstract from the MAC details, and we take the same approach. Please refer to [5][16][18] for the MAC designs which support the implementation of anypath routing. 2.3 Problem Definition As defined in [13], the decision version of multi-constrained anypath problem (denoted by DMCAP) is to find a feasible anypath P from source s to destination t such that W k (s, P ) W k, 1 k K, where positive constant W k defines the k th QoS constraint. However, since there might exist more than one solution to DMCAP, we define the following metric, called anypath length, to compare multiple feasible solutions to DMCAP. The anypath length from a node v to the destination t along an anypath P is defined as l(v, P ) = max 1 k K W k (v, P ) W k. (2.4) This motivates us to study OMCAP, an optimization version of MCAP, which is formally defined as follows. Definition 2.1: OMCAP(G, s, t, W, w, p, K): Instance: A directed graph G=(V, E, w, p), with a delivery probability p(e) (0, 1] associated with each edge e E and K positive vertex weights w k (v)(1 k K) associated with each vertex v V ; a constraint vector W=(W 1,W 2,...,W K ) where each element is a positive constant; and a source-destination pair (s, t). Problem: Find an s-t anypath P such that l(s, P ) is minimized. Xi et al. [13] proved the NP-hardness of this problem, and presented a centralized K-approximation algorithm Greedy. However, Greedy needs the global information of the entire network, which restricts its practical implementation. In this paper, we study the distributed algorithm and implementation for solving OMCAP. 3. An Efficient Distributed K-Approximation Algorithm In this section we first present a distributed algorithm DMART to compute anypaths from all the nodes to a destination t, then analyze its performance, and finally describe its distributed implementation. 3.1 Distributed Routing Algorithm First we need to define an auxiliary vertex weight ω m (v) w for each node v V as follows: ω m (v) = max k (v) 1 k K W k. Likewise we define the auxiliary anypath weight (AAW) from v to the destination t along an anypath P as: W m (v, P ) = ω m(v) p(v,j(v, P )) + W m(j(v, P ), P ), wherew m (J(v, P ), P ) = α(j β, P )W m (j β, P ). j β J(v,P ) We call W m (J(v, P ), P ) the auxiliary anypath weight of the set J(v, P ) along anypath P. DMART s initialization phase for node v: w 1: ω m (v) max k (v) 1 k K W k, Q(v), J(v, P ) 2: if v = t then 3: W m (0) (v, P ) 0 4: for each k do 5: W k (v, P ) 0 6: else 7: W m (0) (v, P ) 8: for each k do 9: W k (v, P ) DMART s i th iteration for node v: 1: Update Q(v) such that all the elements in it (denoted by j) are sorted in the increasing order of W m (i 1) (j, P ). W m (i 1) (j, P ) is obtained in the path vector updating operation (see Section 3.3). 2: F, W m (i) (v, P ) W m (i 1) (v, P ) 3: while Q(v) do 4: j Extract-Min(Q(v)), F F {j} 5: W m(v, P ) ωm(v) p(v,f ) + j β F α(j β, P )W m (i 1) (j β, P ) 6: if W m (i) (v, P ) > W m(v, P ) then 7: J(v, P ) F, W m (i) (v, P ) W m(v, P ) 8: for each k do 9: W k (v, P ) w k(v) p(v,j(v,p )) + W k(j(v, P ), P ) DMART requires each node to first initialize the data structures and then execute a sequence of iterations. For every node v V we use a variable W m (i) (v, P ) to denote the AAW from v to t along anypath P computed in the i th iteration. In addition, each node v keeps a data structure Q(v) to store all its outgoing neighbors. At the beginning of the i th iteration, each node v sorts the nodes in Q(v) in the increasing order of their AAWs which are obtained in the (i 1) th iteration. We now illustrate how DMART works using a simple example, shown in Fig. 2. In this example, K = 2 and W 1 = W 2 = 1. In the initialization phase, each node calculates its auxiliary vertex weight and initializes the necessary data structures.

4 4 (a) Initialization (b) Iteration 1 (c) Iteration 2 (d) Iteration 3 (e) Iteration 4 (f) Iteration 5 (g) Iteration 6 (s-t anypath) (h) Optimal s t anypath Fig. 2. Execution of DMART from every node to t. The link labels show link delivery probabilities; the vertex labels show (w 1, w 2 ) pairs of the vertices; the labels next to each vertex show the currently computed anypath weights (W 1, W 2 ) (the first parenthesis in the first bracket), the AAW (W m) (the second element in the first bracket), and the forwarding set (the second bracket). The original network graph is shown in Fig.1. Fig. 2(b)-2(g) show the situations after each iteration. Fig. 2(h) shows an optimal s-t anypath. W m (0) (t, P ) is set to 0, and W m (0) (v, P ) = for all v V \{t} since so far no anypath has been found for them. At the beginning of the first iteration, v 4, v 5 and v 6 know that W m (0) (t, P ) = 0. Now we take v 4 as an example. Q(v 4 ) has three elements t, v 5 and v 6. Note that W m (0) (v 5, P ) = W m (0) (v 6, P ) =. Thus in the first loop (Lines 3-7), v 4 extracts t (Line 4), and computes W m(v, P ) = 8 (Line 5). Since W m (1) (v 4, P ) = > W m(v, P ) = 8, Line 7 updates the AAW for v 4. In the second loop, F is updated to {t, v 5 }. Obviously, W m(v, P ) =, and thus Lines 6-7 will be skipped. In the third loop, F is updated to {t, v 5, v 6 }. Likewise W m(v, P ) =, and Lines 6-7 will be skipped. The result of the first iteration is shown in Fig.2(b). At the beginning of the second iteration, for example, v 1 knows W m (1) (v 3, P ) = and W m (1) (v 5, P ) = 20. Similar to the procedure above, after two loops (Lines 3-7) v 1 will select v 5 as its forwarder. The same procedure repeats until some stopping criterion is satisfied. We will discuss the stopping criterion in Section 3.3. We note that the s t anypath obtained by DMART has a length of 13.1 = max{ 6.1 }, while as shown in Fig. 2(h) 1, the optimal one has a length of 13 = max{ 6.2 1, 13 1 }. Although the anypath so computed is not optimal, its length is within a factor of 2 of that of the optimal one. 3.2 Algorithm Analysis We now prove that the anypath computed by DMART is a K-approximation to the OMCAP problem. We first prove that for any node its anypath so computed is actually its shortest AAW anypath. Then we use the result in [13] to show the relationship between the anypath so computed and the optimal solution to the original OMCAP problem. Let θ(v) denote v s optimal AAW. We use W m [J] (v) to denote v s AAW via the forwarding set J if all the nodes in J use their shortest AAW anypaths. Here we introduce two lemmas Lemmas 3.1 and 3.2 which were proved in [13]. Lemma 3.1: For all of node v s neighbors j 1, j 2,..., j z, where z is the number of v s outdegree, if θ(j 1 ) θ(j 2 ) θ(j z ), then there must exist an AAW optimal forwarding set (AOFS) of the form {j 1, j 2,..., j b } for some b {1, 2,..., z}, called a full AAW optimal forwarding set (FAOFS). Lemma 3.1 implies that if we want to find an AOFS, we only need to check forwarding sets {j 1 }, {j 1, j 2 },... The complexity of the algorithm can therefore be reduced to polynomial time from exponential time. Note that the proof of this lemma in [13] constructs a full AAW optimal forwarding set from a particular forwarding set. This constructed forwarding set is called a minimum full AAW optimal forwarding set (MFAOFS) in [13]. We still use this name in this paper. When all the nodes on a shortest AAW anypath from v to t are using their MFAOFSs, we call it a full shortest AAW anypath. Lemma 3.2: The MFAOFS of v is {j 1, j 2,..., j b } with optimal AAWs θ(j 1 ) θ(j 2 ) θ(j b ). We use S µ to denote the forwarding set {j 1, j 2,..., j µ }, 1 µ b. Then we have W m [S1] (v) > W m [S2] (v) > W [S b] m (v) = θ(v). Now we define a distance index for each node v. Since the full shortest AAW anypath from v to the destination t is an acyclic directed graph (recall that we only consider the case in which anypaths are acyclic), we can use the topological order [7] to index all the nodes on this anypath. In other words, t has a distance index of 0, and v has the largest index. Since there may exist more than one shortest AAW for node v, node v could have different topological orders in different shortest anypaths. We select the smallest one as v s distance index (denoted by D(v)). Let D denote max v V D(v). Obviously, D V. Lemma 3.3: In iteration i, each node with distance index D(v) = i computes its shortest AAW anypath to the destination t. Proof. We prove this lemma by using mathematical induction. Obviously, only the destination node t has a distance index of 0. In the zeroth iteration (i.e. the initialization phase), t can find its own shortest AAW anypath (although it is trivial). Now we assume that at iteration i 0 each node with the distance index D(v) [0, i] finds or has found its shortest AAW anypath to the destination t. We will prove the claim that in iteration (i+1) each node v with the distance index D(v) = i + 1 computes its shortest AAW anypath to the destination t.

5 5 Since all the nodes in v s MFAOFS have distance indices less than i+1 (recall the definition of the distance index), we know that all the nodes in v s MFAOFS have found their shortest AAW anypaths before the (i+1) th iteration. We are now ready to prove that v can find its MFAOFS in the (i + 1) th iteration. By Lemma 3.1, we know there exists an MFAOFS of the form {j 1, j 2,..., j b } for v. The while loop (Lines 3-7) will check {j 1 }, {j 1, j 2 }... By Lemma 3.2, if j 1 and j 2 are in v s MFAOFS, using {j 1, j 2 } always provides a smaller AAW than just using {j 1 }. Thus the condition in Line 6 will be true and the forwarding set is updated to {j 1, j 2 }. The same procedure is repeated until the MFAOFS is found. The claim is therefore proved. The proof is complete. Theorem 3.1: DMART computes a K-approximation to the OMCAP(G, s, t, W, w, p, K) problem after at most D V iterations and each iteration can be done in O( (log + K)) time by each node, where is the max out-degree of graph G. Proof. Lemma 3.3 indicates that a shortest AAW anypath for each node to the destination t can be found by DMART. According to [13], we know that a shortest AAW anypath is actually a K-approximation to the OMCAP(G, s, t, W, w, p, K) problem. Therefore DMART can find a K-approximation anypath for each node after at most D V iterations. We now analyze the running time of each iteration. For each node, Line 1 takes O( log ) time. If we store some status variables as in [13][19], Line 5 takes O( ) in total. Obviously, Lines 4, 6 and 7 take O(1) time, with O( ) in total. Lines 8-9 take O(K ) time. Thus each node needs O( (log + K)) time to finish one iteration. Although the theoretical upper bound on the number of iterations is V, our numerical results in Section 4 show that the actual number of iterations is far smaller than V. In addition, obviously each node can finish the initialization phase in O(K) time. Remark. When K=1, our algorithm is the optimal algorithm for the corresponding problem. The shortest anypath algorithm proposed in [10] is a special case of DMART when K = 1 and all the vertex weights are equal to 1. When K 2, our result is the first distributed O(1)-approximation algorithm for the NP-hard OMCAP problem. 3.3 Distributed Implementation Inspired by the Distributed Bellman-Ford protocol proposed in [2], we present the following synchronous proactive protocol based on our DMART algorithm. Each node maintains a routing table entry for each destination <destination, K anypath weights, auxiliary anypath weight, forwarding set>. The timeline is divided into a sequence of time intervals of a constant length, each of which is used for one iteration in DMART. In each time interval, each node runs DMART to update its anypath to each destination. If the entries in the routing table change, this node sends path vector tuples <destination, K anypath weights, auxiliary anypath weight> to all its immediate neighbors. This is called path vector updating. In the next iteration, its immediate neighbors can use these new path vectors to update their routing tables. Since our synchronous proactive protocol requires a rough time synchronization, we also propose an asynchronous proactive table-driven protocol. Every node periodically sends path vector tuples <destination, K anypath weights, auxiliary anypath weight> to all its immediate neighbors. The updating operation frequency depends on the size and the dynamic of the network. Whenever the entries in the routing table change, this node also triggers the path vector updating. Once a node receives the path vector updates, it uses DMART to update the anypath to the destination. If this computation leads to a routing table change, it triggers a path vector updating. Now we discuss the stopping criteria for the iterations in our synchronous proactive protocol. Obviously, for a dynamic network this protocol should periodically update the path vector. However, for a static network, we may ask when we can terminate the algorithm iteration. Although we know we can terminate the algorithm after the upper bound of the number of iterations, the tight upper bound D cannot be obtained easily and the loose bound V sometimes is too large. We propose the following guessing strategy. An obvious stopping criterion is W (i 1) m (v, P ) = W m (i) (v, P ), v V. We need the following simple global status checking as a building block: every node v sends one bit to a leader (elected using well-known leader-election algorithms) to notify whether W m (i 1) (v, P ) = W m (i) (v, P ). If this leader finds W m (i 1) (v, P ) = W m (i) (v, P ), v V, it sends back one bit iteration termination signaling. Although this global status checking involves global information exchanges, obviously the overhead is very small especially if the data fusion technique is used. In order to reduce the number of global status checking times, the guessing strategy works as follows: after b 0, b 1,...b h,... iterations (b is an integer chosen by network administrators), we do global status checking. Obviously, although we do not know the value of D, after at most log b D global status checking operations, the algorithm will terminate. 4. Performance Evaluation In this section, we present some numerical results to evaluate the performance of DMART. We implemented DMART for the case of K = 2 in a practical setting using our synchronous proactive protocol. The two vertex weights represent average transmission time and power consumption for each transmission, respectively. We compared our implementation with the SAF algorithm in [19] and the Greedy algorithm in [13]. As in [13], we extended the SAF algorithm to support EWATX. Since SAF can only deal with one metric each time, we used SAF to compute the anypath lengths, with respect to the metrics transmission delay and power consumption, respectively (denoted by SAF-D and SAF-C). We assume that each node was equipped with an b/g wireless LAN client adapter. Each node can choose 10, 20, 30, 50, or 63mW as its transmit power level. In addition, each node randomly selected 1, 6, 11, or 18M bps as its transmission rate [6]. According to [6], the corresponding maximum transmission ranges R max are 213, 198, 149, and 122m when the adapter is being used at the maximum transmit power. Since wireless cards may operate on different power levels, the actual transmission range of each transmission rate configuration must satisfy R max ( Pt P max ) 1 γ [14], where the path loss exponent γ = 2, P max = 63mW and P t is the

6 6 (a) Comparison of average anypath lengths (b) Average number of iterations Fig. 3. (c) Average number of path vector updating operations for each node Numerical Results (d) Average running time for each iteration actual transmit power (10, 20, 30, 50, and 63mV ) of a wireless card. The link delivery probability was inversely proportional to the distance with a random gaussian deviation of 0.1. The two constraints were set to 50ms and 625mW. We uniformly distributed nodes in a 1000m 1000m square region, with the number of nodes chosen to be 150, 200, For each network size, we randomly generated 1000 test cases by randomly choosing the transmission ranges and coordinates for all the nodes. We performed all tests on a 1.8GHz Linux PC with 2G bytes of memory. Fig. 3(a) shows the lengths of the anypaths computed by DMART, Greedy, SAF-D and SAF-C for a random sourcedestination pair. We observe that DMART and Greedy always outperform SAF-D and SAF-C. This is as expected, because DMART and Greedy combine two metrics and thus use more information in decision making, while with only one metric taken into account each time SAF-D and SAF-C obtain biased results. In addition, the average anypath lengths obtained by DMART and Greedy are almost the same. This is because both of DMART and Greedy approximate OMCAP by looking for the shortest AAW anypath. According to our experiment data, in 97% test cases the anypaths computed by DMART and Greedy are the same. However, we must emphasize that DMART is a distributed algorithm which just needs local information from one-hop neighbor, while Greedy is a centralized algorithm and needs the global information of the entire network. Fig. 3(b) shows the average number of iterations required by DMART to compute anypaths from all nodes to a destination. We observe that the average number of iterations is around 20. Fig. 3(c) shows how many path vector updating operations a node needs. In all cases studied, the number of updating operations is no more than 6. Fig. 3(d) shows that the average running time which each node needs to finish one iteration is also very small. Thus DMART is very efficient. 5. Conclusion In this paper, we have studied the problem of finding an anypath subject to K constraints, and presented a Bellman-Fordlike distributed K-approximation algorithm. When K = 1, our algorithm is the optimal algorithm for the corresponding problem. When K 2, our result is a distributed O(1)- approximation algorithm for the NP-hard OMCAP problem. Numerical results show that our algorithm is very fast and its result is as good as the anypath obtained by the best centralized algorithm. REFERENCES [1] S. Biswas and R. Morris, ExOR: Opportunistic Multi-Hop Routing for Wireless Networks, ACM SIGCOMM 05, Philadelphia, PA, USA, Aug.2005, pp [2] Dimitri P. Bertsekas, and Robert G. Gallager, Distributed Asynchronous Bellman-Ford Algorithm, Data Networks, [3] S. 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