1. Introduction. Abstract

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1 Determining a equence of table Multicast teiner Trees in Mobile Ad Hoc Networks Dr. Natarajan Meghanathan nmeghanathan@jsums.edu Jackson tate University, Jackson, M 3927 Abstract We show that given the complete knowledge of future topology changes, it is possible to determine the sequence of stable multicast teiner trees (called the stable mobile multicast teiner tree) in a multicast session such that the number of tree transitions is minimal. The algorithm to determine the stable mobile multicast teiner tree and the minimum number of tree transitions is called OptTreeTrans. It is based on the following greedy approach: whenever a multicast teiner tree is required, choose the teiner tree that will exist for the longest time. The above strategy is repeated over the duration of the multicast session. We prove that OptTreeTrans gives the optimal number of tree transitions and simultaneously yields the stable mobile multicast teiner tree. We study the performance of OptTreeTrans in terms of the number of tree transitions and the number of links constituting the teiner tree for different conditions of node mobility and network density.. Introduction Wireless networks provide ubiquitous communication capability and location-independent information access to mobile users [2]. A mobile ad hoc network (MANET) is a dynamically reconfigurable self-organizing wireless network with no fixed infrastructure or a centralized administration [8]. MANETs function in a peer-to-peer mode of operation. Routes in MANETs are often multi-hop due to the limited radio propagation range of wireless devices. Each mobile node in MANET also functions as a router to establish endto-end connections between any two nodes [7]. Multicasting in wired and wireless networks has emerged as a desirable and essential technology in many applications such as audio/ video conferencing, corporate communications, collaborative and groupware applications, distance learning, stock quotes, distribution of software, news etc [9]. Using multicast, a single stream of data can be distributed to multiple recipients, with data duplicated only when necessary [9]. This is more advantageous than multiple unicast transmissions of the same data independently to each and every receiver, which also leads to network clogging. MANETs are deployed in applications such as disaster recovery, rescue missions, military operations in a battlefield, conferences, crowd control, outdoor entertainment activities, etc. One common feature among all these applications is oneto-many and many-to-many communications among the participants [6]. Hence, to support these applications, efficient ad hoc multicast routing protocols are required. tability of paths and trees is an important design criterion to be considered in the development of multi-hop unicast and multicast routing protocols for resource-constrained environments like that of MANETs. Wireless links are shared and are limited in their bandwidth. Frequent route discovery attempts could congest the network and also knock out the battery power at critical nodes. The battery reserves at the nodes are to be treated precious as they may be deployed in environments where recharging is next to impossible. Also, nodes in energy-constrained environments cannot afford to lose their battery power quickly. For multi-media applications that require the packets to be delivered in-order with negligible variance in the inter-packet delay, frequent changes in the routes traversed by the packets could result in out-of-order packet delivery with high jitter. In the case of reliable data transfer applications, failure to receive an acknowledgement packet within a particular timeout interval can also trigger retransmissions at the source side. As a result, the application layer at the receiver side might be overloaded in handling out-of-order, lost and duplicate packets. Thus, stability of the communication structures such as unicast paths and multicast trees is important from the point of view of Quality of ervice too. This forms of our motivation for this paper. We model an ad hoc network as a unit disk graph [5], in which vertices represent wireless nodes and a bi-directional edge exists between two vertices if the corresponding nodes are within the transmission range of each other. In general, given a weighted graph G = (V, E), and a subset of vertices (called the multicast group or teiner points) V, the teiner tree is the minimum-weight tree of G connecting all the vertices of. In the unit disk graph considered in this paper, the weight of each edge is unity, and all the edges in the teiner tree are contained in the edge set of the graph. Accordingly, we define the minimum teiner tree as the tree with the least number of edges required to connect all the vertices in the multicast group (i.e., the set of teiner points). Unfortunately, the problem of determining a minimum teiner tree in an undirected graph like that of the unit disk graph is NP-complete. Efficient heuristics (e.g., [4]) have been proposed in the literature to approximate a minimum teiner tree. In this paper, we show that aiming for the minimum teiner tree in MANETs, results in multicast trees that are highly unstable. The multicast tree has to be frequently rediscovered, and this adds considerable overhead to the resource-constrained network. By adding a few more links and nodes to the tree, it is possible to increase its stability. We define stability of a multicast teiner tree in terms of the number of times the tree has to change for the duration of a multicast session. We propose an efficient, polynomial-time greedy algorithm called OptTreeTrans, to determine the minimum number of tree transitions during the period of a multicast session from a source node to a set of nodes constituting the multicast group in the network. Given the complete knowledge of future topology changes, the

2 algorithm operates on the following principle: Whenever a multicast tree connecting a given source node to all the members of a multicast group is required, choose the multicast tree that will keep the source connected to the multicast group members for the longest time. The above strategy is repeated over the duration of the multicast session and the sequence of stable multicast teiner trees obtained by running this algorithm is called the stable mobile multicast teiner tree. The complexity of algorithm OptTreeTrans depends on the complexity of the underlying heuristic used to determine the longest existing multicast teiner tree during each tree transition (also called here after as tree repair). We use the Kou. et. al s [4] well-known O( V 2 ) heuristic, as the underlying heuristic to determine the longest existing multicast tree. The complexity of OptTreeTrans can then be written as O( V 2 T 2 ), where V is the number of nodes in the network, is the multicast group size and T is the duration of the multicast session. We also define the minimum mobile teiner tree as the sequence of approximations to the minimum teiner tree obtained by directly using Kou s heuristic whenever required. We compare the performance of the stable mobile multicast teiner tree with that of the minimum mobile multicast teiner tree in terms of the number of tree repairs and the average number of links in the multicast tree (averaged over time) at different conditions of node density and node mobility. The rest of the paper is organized as follows: In ection 2, we discuss the Kou s heuristic to approximate the minimum teiner tree in wireless ad hoc networks. In ection 3, we introduce the algorithm to find the stable mobile multicast teiner tree and the optimal number of tree transitions; give its proof of correctness, and analyze its complexity. In ection 4, we present our simulation environment and discuss the simulation results with respect to the stable mobile multicast teiner tree and the minimum mobile multicast teiner tree. In ection 5, we conclude the paper. 2. Heuristic to Approximate Minimum teiner Tree Determining a minimum teiner tree is a well-known NPcomplete problem. everal heuristics have been proposed in the literature to solve the minimum teiner tree problem. We use the Kou et. al s [4] well-known O( V 2 ) heuristic to approximate the minimum teiner tree in graphs representing snapshots of the network topology. In this section, we give the outline of the heuristic in Figure, followed by its complexity analysis and approximation ratio. Input: An undirected graph G = (V, E) Multicast group V Output: A tree T H for the set in G tep : Construct a complete undirected weighted graph G C = (, E C ) from G and where (v i, v j ) E C, v i and v j are in, and the weight of edge (v i, v j ) is the length of the shortest path from v i to v j in G. tep 2: Find the minimal spanning tree T C in G C (If more than one minimal spanning tree exists, pick an arbitrary one). tep 3: Construct the sub graph G of G, by replacing each edge in T C with the corresponding shortest path from G (If there is more than one shortest path between two given vertices, pick an arbitrary one). tep 4: Find the minimal spanning tree T in G (If more than one minimal spanning tree exists, pick an arbitrary one). tep 5: Construct the minimum teiner tree T H, from T by deleting edges in T, if necessary, such that all the leaves in T H are members of. Figure : Kou et. al s Heuristic to find an Approximate Minimum teiner Tree tep is the most time consuming step of the heuristic. The shortest path between any two vertices on G = (V, E) can be found in O( V 2 ) time. There are /2 pairs of vertices that form the edges of G C. Hence, the graph G C can be constructed at time complexity O( V 2 ). With the Kruskal s minimal spanning tree algorithm run on G C = (, E C ), step 2 can be completed in O( 2 ) time. The sub graph G can be constructed in step 3 in O( V ) time as all the shortest paths have been already calculated in step. tep 4 can be completed in O( V 2 ) time by running Kruskal s algorithm on G. tep 5 visits at most V nodes to construct the minimum teiner tree T H. In our case, since all the edges in the teiner tree are contained in the input edge set E, there is no need to execute tep 5. The minimal spanning tree T determined in step 4 is also the teiner tree T H. umming up the complexities of steps through 5, the overall time complexity of the heuristic is O( V 2 ). Let D H be the total number of edges in the teiner tree T H, produced by the above heuristic. Let D MIN be the total length on the edges of the minimal teiner tree T MIN. Let m be the total number of leaves in T MIN. Then, according to [4], D H / D MIN 2(-/m). 3. table Mobile Multicast teiner Tree In a dynamically changing topology, the minimum multicast teiner tree approximated using Kou s heuristic has to be frequently updated as it will have the minimal nodes and links. The source might be disconnected from one or more members of the multicast group. 3.. Mobile Graph The algorithm to determine the stable mobile multicast teiner tree and hence the optimal number of tree transitions uses the notion of a mobile graph [3] to represent the sequence of network topology changes. A mobile graph is defined as the sequence G M = G G 2 G T of static graphs that represents the network topology changes over some time scale T, the duration of the multicast session.

3 3.2. Algorithm for the Optimal Number of Tree Transitions Algorithm OptTreeTrans operates on the following simple greedy strategy: whenever a multicast teiner tree, here after denoted as (s-)-tree, connecting given a source node s to all the members of the multicast group, i.e., set of teiner points, is required, select a multicast teiner tree that will exist for the longest time. Let G M = G G 2 G T be the mobile graph generated by sampling the network topology at regular instants t, t 2,, t T of a multicast session. When an (s-)-tree is required at sampling time instant t i, the strategy is to find a mobile sub graph G(i, j) = G i G i+ G j such that there exists at least one multicast (s-)-tree in G(i, j) and none exists in G(i, j+). A multicast (s-)-tree in G(i, j) is selected using Kou s heuristic. uch a tree exists in each of the static graphs G i, G i+,, G j. If there is a tie, the (s-)-tree with the smallest number of constituent links is chosen. If sampling instant t j+ t T, the above procedure is repeated by finding the (s-)-tree that can survive for the maximum amount of time since t j+. A sequence of such maximum lifetime multicast teiner (s-) trees over the timescale of G M forms the stabile mobile multicast teiner tree in G M. The pseudo code is given in Figure 2. Input: G M = G G 2 G T, source s, multicast group Output: (s-) Mobiletabletree // stable mobile multicast teiner tree Auxiliary Variables: i, j Initialization: i=; j= Begin OptTreeTrans while (i T) do 2 Find a mobile graph G(i, j) = G i G i+ G j such that there exists at least one (s-)-tree in G(i, j) and {no (s-)-tree exists in G(i, j+) or j = T} 3 (s-) Mobiletabletree = (s-) Mobiletabletree U {Minimum teiner (s-)-tree in G(i, j) } 4 i = j + 5 end while 6 return (s-) Mobiletabletree End OptTreeTrans Figure 2: Pseudo code for algorithm OptTreeTrans In a mobile graph G M = G G 2 G T, the number of tree transitions can be at most T. The minimum teiner tree Kou s heuristic will have to be run at most T 2 times, each time on a graph of V nodes. As Kou s heuristic is of O( V 2 ) complexity, the complexity of OptTreeTrans is O( V 2 T 2 ) Proof of Correctness Given a mobile graph G M = G G 2 G T, the source node s and the multicast group, let the number of tree transitions in the mobile multicast teiner tree, (s-) Mobiletabletree, generated by OptTreeTrans be m. To prove m is the optimal number of tree transitions, assume the contrary: there exists another mobile multicast teiner tree (s-) Mobiletabletree in G M and the number of tree transitions in (s-) Mobiletabletree is m < m. Let epoch, 2 epoch,, m+ epoch be the set of sampling time instants in each of which the mobile multicast teiner tree (s-) Mobiletabletree suffers no tree transitions. Let epoch, 2 ' epoch,, m ' + ' epoch be the set of sampling time ' instants in each of which the mobile multicast teiner tree init, j (s-) Mobiletabletree suffers no tree transitions. Let t( s ) and end, k t( s ) be the initial and final sampling time instants of j init, j epoch where j m+. Let t ( s )' end, k and t( s )' be the initial and final sampling time instants of k epoch where ' k m +. Note that init, t = init, t and end, m+ t = end, m ' + ' t to ' indicate (s-) Mobiletabletree and (s-) Mobiletabletree span over the same sequence of sampling time instants. Now, we claim (j, k; j m+ and k m +) such that j epoch k epoch, i.e., init, k init, j end, j end, k ' t' < t < t < t and ' at least one (s-)-tree existed in [ init, k t,, end, k ' t ]; ' otherwise, m m. But OptTreeTrans made a tree transition after end, j t since there was no common (s-)-tree outside [ init, j t,, end, j t ]. This implies there is no common (s-)-tree in [ init, k t,, end, k ' t ].=> Contradiction. Thus, m ' m. 4. imulations 4.. imulation Conditions The node mobility model used in all of our simulations is the Random waypoint model [], a widely used mobility model in MANET studies. Each node starts moving from an arbitrary location to a randomly selected destination with a randomly chosen speed in the range [0.. v max ]. Once the destination is reached, the node selects another destination and continues to move with a different speed. The v max values used are: 0 and 50 m/s. The pause time between two consecutive direction changes is 0 seconds. We obtain a centralized view of the network topology by generating mobility trace files for 000 seconds. The network size is 500 m x 300 m. The density of the network is varied by performing simulations with 25 and 75 nodes. The wireless transmission range of nodes in all three cases is 250m. imulation time is

4 000 seconds. The network topology is sampled for every second. The multicast group size is varied from 2 through 24. Each data point in Figures 3., 3.2, 3.3, 3.4, 4. and 4.2, is averaged over 5 randomly generated mobility trace files Figure 3.: 25 Nodes, vmax = 0 m/s Figure 3.3: 25 Nodes, vmax = 50 m/s tested on 5 randomly picked multicast groups for each value of the group size. Figure 3.2: 75 Nodes, vmax = 0 m/s Figure 3.4: 75 Nodes, vmax = 50 m/s Figure 3: Minimum Mobile Multicast teiner Tree Vs table Mobile Multicast teiner Tree Figure 4.: vmax = 0 m/s Figure 4.2: vmax = 50 m/s Figure 4: Availability Lifetime Tradeoff in Minimum Mobile Multicast teiner Tree When an (s-) teiner tree is required at sampling time instant ti and stability is not to be considered, then Kou s heuristic is run on static graph Gi and the (s-) tree obtained is used as long as it exists. The procedure is repeated till the last sampling time instant tt is reached. We refer to the sequence of multicast teiner trees generated by the above strategy as minimum mobile multicast teiner tree. We compare the performance of the stable mobile multicast teiner tree with that of minimum mobile teiner tree in terms of number of tree transitions and the average number of edges in the mobile teiner trees, which is the number of links in the constituent (s-) teiner trees, averaged over time imulation Results The stability of the minimum mobile multicast teiner tree decreases exponentially as the multicast group size is increased. Aiming for the minimum number of edges, it is tough to find a teiner tree that would keep together all the multicast group members for a longer time. On the other hand, by accommodating 20-50% more edges, the stable mobile teiner tree becomes insensitive to the multicast group size. The decrease in the number of tree transitions is by a factor of 2 to 7 (with 25 nodes, 0 m/s) and 6 to 5 (for 75 nodes). For a given node mobility and multicast group size, the number of tree transitions in the case of a stable mobile multicast teiner tree is reduced by half. This could be attributed to the tendency of algorithm OptTreeTrans to make use of the available nodes and links as much as possible in order to maximize the stability of the trees. On the contrary,

5 in the case of a minimum mobile teiner tree, the average number of links in the constituent (s-) trees remains the same with increase in node density, while the stability of the minimum mobile teiner trees decreases with increase in node density. This shows that when we aim for a minimum teiner (s-) tree, we do not make use of the increased number of nodes and links to improve the lifetime of the tree. As the network density increases, the physical distance of a link (or the hop) is equal to 70-80% of the transmission range of the nodes. Hence, the links in a minimum mobile multicast teiner tree are highly vulnerable to failure as the constituent nodes of the link are likely to move away from the reach of one another anytime. On the other hand, in the case of the stable mobile teiner trees, the physical distance of a link is only 50-60% of the transmission range of the nodes. This justifies the stability of the trees and also the increase in the number of links in the constituent (s-) trees of a stable mobile multicast teiner tree, compared to that of the minimum mobile multicast teiner tree. The availability-lifetime tradeoff in the case of the minimum mobile teiner trees is well illustrated in Figures 4. and 4.2. With 25 nodes in the network, it is tough to find a tree that connects the source node of the multicast session to all the members of the multicast group for most of the duration of the multicast session. The connectivity of the nodes in the network is limited. On the other hand, with 75 nodes, it is possible to keep the source connected to all the members of the multicast group for most of the duration of the multicast session. The tradeoff is that we will be making more tree transitions with 75 nodes compared to that with 25 nodes. This availability-lifetime tradeoff is more dominant at smaller values of the multicast group size and fades off as the multicast group size approaches 25 nodes. Note that the number of edges in a mobile sub graph is inversely proportional to the number of constituent static graphs. When an algorithm is applied over a mobile sub graph spanning multiple static graphs, the algorithm is offered only limited choices in selecting a multicast teiner tree and is forced to lose sight over its design objective. Thus, Kou s heuristic is merely used as a tool to find the stable mobile multicast teiner tree. The optimal number of tree reconstructions does not depend on the underlying algorithm or heuristic used as we always try to find the longest living multicast teiner tree in the mobile graph. Nevertheless, the run-time complexity of OptTreeTrans depends on the underlying algorithm or heuristic used to determine the constituents of the stable mobile multicast teiner tree. 5. Conclusions The high-level contribution of our paper is a polynomialtime algorithm OptTreeTrans to determine the sequence of stable multicast teiner trees in a MANET environment. The algorithm can determine the longest-living multicast teiner tree over a given span of time using any heuristic developed (for the NP-complete problem) to approximate the minimum teiner tree. When used with the well-known Kou s heuristic, whose complexity is O( V 2 ), the complexity of algorithm OptTreeTrans is O( V 2 T 2 ), where V is the number of nodes in the network, is the multicast group size, and T is the duration of the multicast session during which a source node is sending data to all the members of the multicast group. imulation results indicate that for a given node density, the number of tree transitions in a stable mobile multicast teiner tree increases linearly with node mobility while the number of tree transitions in a minimum mobile teiner tree increases exponentially. Also, for a given node mobility, the stability of minimum mobile multicast teiner tree decreases with increase in node density; while the stability of stable mobile multicast teiner tree increases with increase in node density. The average number of links in a minimum mobile multicast teiner tree remains the same irrespective of the increase in node density; where as algorithm OptTreeTrans chooses a teiner tree that has more constituent links and can cover the multicast group members for a longer time. As part of future work, we are working on developing a practical stable multicast tree-discovery and maintenance protocol that can make use of the looking ahead technique of algorithm OptTreeTrans. References [] C. Bettstetter, H. Hartenstein and X. Perez-Costa, tochastic Properties of the Random-Way Point Mobility Model, Wireless Networks, pp , Vol. 0, No. 5, eptember [2] C. C. Chiang, M. Gerla and L. Zhang, Adaptive shared tree multicast in mobile wireless networks, GLOBECOM 98, pp , 998. [3] A. Farago and V. R. yrotiuk, MERIT: A calable Approach for Protocol Assessment, Mobile Networks and Applications, Vol. 8, No. 5, pp , October [4] L. Kou, G. Markowsky and L. Berman, A Fast Algorithm for teiner Trees, Vol 5, pp. 4-45, Acta Informatica, pringer-v erlag, 98. [5] F. Kuhn, T. Moscibroda and R. Wattenhofer, Unit Disk Graph Approximation, Proceedings of DIALM-POMC, pp. 7 23, October [6] M. Lee and Y. Kim, PatchODMRP: an ad hoc multicast routing protocol, 5 th IEEE International Conference on Information Networking, pp , 200. [7]. Lee and C. Kim, Neighbor supporting ad hoc multicast routing protocol, First Annual Workshop on Mobile and Ad hoc Networking and Computing, MobiHOC, pp.37-44, [8]. J. Lee, W. u, J. Hsu, M. Gerla and R. 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