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ALGORITHM OptPathTrans OPTIMIZED FOR USE WITH LOCATION UPDATE VECTORS AND ITS COMPARISON TO MINIMUM HOP ROUTING IN MOBILE AD HOC NETWORKS Alexander Roy Geoghegan, Natarajan Meghanathan Jackson State University, Jackson MS, USA roygeoghegan@gmail.com, natarajan.meghanathan@jsums.edu ABSTRACT In an earlier work, we proposed the OptPathTrans algorithm to minimize the number of path transitions during a source-destination (s-d) session in a mobile ad hoc network. The sequence of longest-living stable paths determined over the duration of the s-d session is called the Stable Mobile Path (SMP). But, the average hop count per static path for SMP is significantly larger than the minimum required hop count for a path between the source and destination. Also, algorithm OptPathTrans requires complete knowledge about future topology changes over the duration of the s-d session. In this paper, we illustrate the effectiveness of predicting the future topology changes using the location and mobility information of the nodes in the form of Location Update Vectors (LUVs) learnt at the time of determining a static stable path of the SMP. The modified algorithm is referred to as OptPathTrans-LUV and the sequence of predicted static stable paths (that also actually exists) is referred to as SMP-LUV. Simulation results illustrate that the average lifetime per static path of SMP-LUV can be as large as 88% of the lifetime per static path obtained for SMP. On the other hand, the average hop count per static path of SMP-LUV can be as low as 80% of the hop count per static path for SMP. Keywords: Stability, Hop Count, Path Lifetime, Simulations, Location Prediction 1 INTRODUCTION topology changes, algorithm OptPathTrans operates using the following greedy principle: Whenever a A Mobile Ad hoc Network (MANET) is a path between the source node s and destination node dynamic distributed system of arbitrarily moving d is required at time t, choose the s-d path that exists resource-constrained wireless nodes. Due to the for the longest time since t. The above strategy is limited battery charge per node, each node operates applied over the duration of the s-d session and the with a reduced transmission range, thus necessitating sequence of stable paths obtained is called the Stable the need for multi-hop routes. In the presence of Mobile Path (SMP). SMP yields the longest-living node mobility, routes are susceptible to break over stable paths, but the average hop count per static path time and have to be frequently discovered. Route of the SMP is also significantly higher than the discoveries are normally accomplished through a minimum hop count. flooding-based Route-Request-Reply cycle during Algorithm OptPathTrans requires complete which the source node broadcasts Route Request knowledge about the future topology changes for the (RREQ) messages targeted towards the destination. entire duration of the s-d session. This may not be The destination learns of the best route to the source easy to obtain in real-implementation scenarios. In through the RREQ messages collected and notifies this paper, we illustrate the effectiveness of the source about the selected route through a Route predicting the future topology changes using the Reply (RREP) packet [1]. Frequent flooding of the location and mobility information of the nodes learnt network can significantly exhaust the battery at the time of determining a stable static path of the resources of the nodes as well as the network SMP. The location and mobility information of the bandwidth [2]. It is highly important that the routes nodes at the time of determining a static stable path discovered by a routing protocol are stable and be are stored in the form of a Location Update Vector able to sustain node mobility as long as possible. (LUV) and these LUVs are used to predict the future In [3], Meghanathan and Farago proposed a changes in network topology. The modified polynomial-time algorithm called OptPathTrans to algorithm is referred to as OptPathTrans-LUV and determine the sequence of longest-living stable paths the sequence of predicted static stable paths (that over the duration of a source-destination (s-d) also actually exists) by this algorithm is referred to as session. Given the complete knowledge of future SMP-LUV. Such a strategy can be implemented in practice by letting each node to include its LUV in successive flooding-based broadcast route the RREQ message being broadcast during the route discoveries initiated by LPBR. discovery process. The rest of the paper is organized as follows: Our recently proposed Location Prediction Section 2 describes the original algorithm Based Routing (LPBR) protocol [4] also utilizes the OptPathTrans. Section 3 describes the proposed LUVs to gather the location and mobility OptPathTrans-LUV. Section 4 illustrates the path information of the nodes. LPBR works as follows: lifetime and hop count obtained for algorithms During a flooding-based route discovery, all the OptPathTrans and OptPathTrans-LUV and compares nodes in the network, except the destination, include them with respect to the lifetime and hop count of their latest location and mobility information in the minimum hop paths over the same period of form a LUV record appended to the RREQ message. simulation time. Section 4 also illustrates the tradeoff The destination gathers all the RREQ messages and between path lifetime and hop count for stability- sends back a Route-Reply (RREP) message on the based and minimum-hop based routing. Section 5 minimum hop path to the source. If the discovered concludes the paper. path fails, the source does not opt for another route discovery; instead, waits for the destination to 2 ALGORITHM TO DETERMINE THE predict a new path using the LUV information. The OPTIMAL NUMBER OF PATH destination locally constructs the global topology TRANSITIONS (OptPathTrans) based on the predicted locations for the nodes and sends a LPBR-RREP message to the source on the 2.1 Terminology minimum hop predicted path. If the predicted path exists in reality, the LPBR-RREP message reaches A mobile graph [6] is defined as the sequence the source and an expensive flooding-based route GM = G1G2 … GT of static graphs that represents the discovery is avoided. This procedure is repeated network topology changes over some time scale T. In every time the path used for communication fails. this research work, we sample the network topology However, if the predicted path does not exist in periodically for every 0.25 seconds (i.e., 4 network reality, an intermediate node (that could not further snapshots per second), which could in reality be the forward the LPBR-RREP message to the source) instants of data packet origination at the source. sends back a LPBR-RREP-ERROR message to the A mobile path [6], defined for a source- destination. The destination clears the database of destination (s-d) pair, in a mobile graph GM = G1G2 LUVs and waits for the source to launch a new … GT is the sequence of paths PM = P1P2 … PT, flooding-based route discovery. The LPBR protocol where Pi is a static path between the same s-d pair in has been observed to significantly reduce the number Gi = (Vi, Ei), Vi is the set of vertices and Ei is the set of times the flooding-based route discoveries are of edges connecting these vertices at time instant ti. initiated. As LPBR discovers minimum hop paths That is, each static path Pi can be represented as the both in the actual as well as the predicted graphs, the sequence of vertices v0v1 … vl, such that v0 = s and vl hop count of LPBR is only at most 8% more than = d and (vj-1,vj) ∈ Ei for j = 1,2, …, l. The timescale that obtained for the minimum-hop based Dynamic of tT normally corresponds to the duration of a Source Routing (DSR) protocol [5]. session between s and d. LPBR and OptPathTrans-LUV are different in The Stable Mobile Path (SMP) for a given the following aspects: LPBR is a distributed routing mobile graph and s-d pair is the sequence of static s- protocol to discover the sequence of minimum hop d paths such that the number of route transitions is as paths such that the number of flooding-based minimum as possible. A Minimum Hop Mobile Path broadcast route discoveries is as low as possible. (MHMP) for a given mobile graph and s-d pair is the OptPathTrans-LUV is a centralized algorithm to sequence of minimum hop static s-d paths. discover stable paths on a sequence of network graphs predicted using the LUVs. The hop count of 2.2 Algorithm OptPathTrans SMP-LUV (Stable Mobile Path determined by OptPathTrans-LUV) would be far larger than that of Algorithm OptPathTrans operates on the the minimum hop count, but smaller than that of the following greedy strategy: Whenever a path is SMP, as also observed in the simulations in this required, select a path that will exist for the longest paper. In between two successive route discoveries, time. Let GM = G1G2 … GT be the mobile graph LPBR uses several minimum hop paths, whereas, generated by sampling the network topology at OptPathTrans-LUV uses only one stable path. regular time instants t1, t2, …, tT of an s-d session. However, as both LPBR and OptPathTrans-LUV are When an s-d path is required at sampling time instant based on the LUVs of the nodes gathered at the time ti, the strategy is to find a mobile sub graph G(i, j) = of route discovery, the lifetime of the stable paths Gi ∩ Gi+1 ∩ … ∩ Gj such that there exists at least determined using OptPathTrans-LUV can be one s-d path in G(i, j) and no s-d path exists in G(i, considered as an upper bound on the time between j+1). A minimum hop s-d path in G(i, j) is selected. Such a path exists in each of the static graphs Gi, obtained from taking intersections of the static Gi+1, …, Gj. If time instant tj+1 ≤ tT, the above graphs generated based on the actual locations of the procedure is repeated by finding the s-d path that can nodes at time instant ti and the predicted locations of survive for the maximum amount of time since tj+1. the nodes at time instants ti +1, ti +2, ..., tj. A sequence of such maximum lifetime static s-d paths over the timescale of a mobile graph GM forms 3.2 Location Update Vector (LUV) and Location the Stabile Mobile s-d path in GM. The pseudo code Prediction of the algorithm is given in Figure 1. The Location Update Vector (LUV) for a node Input: GM = G1G2 … GT, source s, destination d collected at time instant ti contains the following Output: PS // Stable Mobile Path information: ID of the node, X and Y co-ordinates of Auxiliary Variables: i, j - representing sampling the node at ti, the velocity and direction of movement time instants of the node with respect to the X-axis at ti. We Initialization: i=1; j=1; PS = Φ assume the LUV of a node is known at every sampling time instant ti at which we need to Begin OptPathTrans determine a new stable path as part of the algorithm OptPathTrans-LUV. Using the LUVs of every node 1 while (i ≤ T) do in the network at time instant ti, the predicted mobile graph Gpred(i, j) = Gactual(i) n Gpred(i+1) n Gpred(i+2) 2 Find a mobile graph G(i, j) = Gi ∩ Gi+1 ∩ … n ….. Gpred(j) is constructed. ∩ Gj such that there exists at least one s-d To construct a predicted static graph at time path in G(i, j) and {no s-d path exists in G(i, instant ti+k (where ti ≤ ti+k ≤ tj), we need to predict the j+1) or j = T+1} location (i.e., the X and Y coordinates) of every node in the network at time ti+k. This is done using the 3 PS = PS U {minimum hop s-d path in G(i, j) } LUVs of the nodes at time instant ti. We now explain how to predict the location of a node (say node u) at 4 i=j+1 a time instant ti+k based on the LUV gathered from u at time ti. Let (Xui, Yui) be the X and Y co-ordinates 5 end while of node u at time instant ti. Let Angleui and Velocityui represent the angle of movement with respect to the 6 return PS X-axis and the velocity at which u is moving at time instant ti. The distance traveled by node u from time End OptPathTrans instant ti to ti+k would be: Distanceu(ti to ti+k) = (ti+k – ti)* Velocityui. We assume each node is initially Figure 1: Pseudo code for algorithm OptPathTrans configured with information regarding the network boundaries: [0, 0], [Xmax, 0], [Xmax, Ymax] and [0, Ymax]. 2.3 Complexity of Algorithm OptPathTrans Let (Xui+k, Yui+k) be the predicted location of node u at time instant ti+k. The values of Xui+k and In a mobile graph GM = G1G2 … GT, the number Yui+k are given by Xui+Offset-Xui+k and Yui+Offset- of route transitions can be at most T. A path-finding Yui+k respectively. The offsets in the X and Y-axes algorithm will have to be run T times, each time on a depend on the angle of movement and the distance graph of n nodes. If we use O(n2) Dijkstra algorithm traveled by node u from time ti to ti+k. These are [7], where n is the number of nodes in the network, calculated as follows: the worst-case run-time complexity of OptPathTrans Offset-Xui+k = Distanceu(ti to ti+k) * cos(Angleui) is O(n2T). Offset-Yui+k = Distanceu(ti to ti+k) * sin(Angleui) 3 ALGORITHM OptPathTrans OPTIMIZED If (Xui+k < 0), then Xui+k = 0 FOR USE WITH LOCATION UPDATE If (Xui+k > Xmax), then Xui+k = Xmax VECTORS (OptPathTrans-LUV) If (Yui+k < 0), then Yui+k = 0 If (Yui+k > Ymax), then Yui+k = Ymax 3.1 Actual and Predicted Mobile Sub Graph As stated above, when a node is predicted to Let Gactual(i, j) = Gactual(i) n Gactual(i+1) n cross over the boundary of the network, we just Gactual(i+2) n ….. Gactual(j) denote the actual mobile locate the node to be at the network boundary. sub graph obtained from taking intersections of the static graphs generated based on the actual locations 3.3 Algorithm OptPathTrans-LUV of the nodes at time instants ti, ti +1, ti +2, ..., tj. Let Gpred(i, j) = Gactual(i) n Gpred(i+1) n Gpred(i+2) n ….. Let t1, t2, …, tT denote the regular sampling time Gpred(j) denote the predicted mobile sub graph instants of an s-d session When an s-d path is required at sampling time instant ti, the strategy is to 4 SIMULATIONS find the actual mobile sub graph Gactual(i, j) = Gactual(i) n Gactual(i+1) n Gactual(i+2) n ….. Gactual(j) We conduct our simulations in both square and and the predicted mobile sub graph Gpred(i, j) = circular network topologies for different conditions Gactual(i) n Gpred(i+1) n Gpred(i+2) n ….. Gpred(j) such of network density and node mobility. The area of that there exists at least one common s-d path in both both the square and circular network topologies Gactual(i, j) and Gpred(i, j) and no such common s-d considered is 1000,000 m2. This corresponds to a path exists in both Gactual(i, j+1) and Gpred(i, j+1). square network of side 1000m and a circular network The minimum hop s-d path that exists in both of radius 564m. The transmission range of a node in Gactual(i, j) and Gpred(i, j) is selected. Such a path both these network topologies is 250m. The network exists in each of the actual static graphs Gactual(i), density is varied by conducting the simulations with Gactual(i+1), Gactual(i+2), …, Gactual(j) and Gpred(i+1), 25 nodes (low), 75 nodes (moderate) and 125 nodes Gpred(i+2), …, Gpred(j). If tj+1 ≤ tT, the above (high) – which corresponds to an average procedure is repeated by finding the s-d path that can neighborhood density (the number of neighbors per survive for the maximum amount of time since tj+1 node) of 5, 15 and 25 respectively. We observed a satisfying the above constraint. The sequence of such probability of network connectivity of 0.3-0.4, 1 and predicted static s-d paths over the time scale T is 1 in the low, moderate and high density scenarios called the Stable Mobile Path – predicted using LUV respectively. (represented as SMP-LUV). At each time instant, the The node mobility model used is the Random path-finding algorithm has to be run twice, once on Waypoint model [8], one of the most widely used the actual mobile sub graph and another time on the models for simulating mobility in MANETs. predicted mobile sub graph. If we use O(n2) Dijkstra According to this model, each node starts moving algorithm, where n is the number of nodes in the from an arbitrary location to a randomly selected network, the worst-case run-time complexity of destination with a randomly chosen speed in the OptPathTrans-LUV is O(2n2T) = O(n2T). The pseudo range [vmin .. vmax]. Once the destination is reached, code of OptPathTrans-LUV is given below: the node stays there for a pause time and then continues to move to another randomly selected destination with a different speed. We use vmin = 0 Input: GM = G1G2 … GT, source s, destination d and pause time of a node is 0. The values of vmax Output: PS-LUV // Stable Mobile Path – LUV used are 10, 30 and 50 m/s representing scenarios of Auxiliary Variables: i, j - representing sampling low, moderate and high node mobility respectively. time instants Note that, two nodes a, b are assumed to have a Initialization: i=1; j=1; PS-LUV = Φ bidirectional link at time t if the Euclidean distance between them at time t (derived using the locations Begin OptPathTrans-LUV of the nodes from the mobility trace file) is less than or equal to the wireless transmission range of the 1 while (i ≤ T) do nodes. We obtain a centralized view of the network topology by generating mobility trace files for 1000 2 Find a predicted mobile sub graph Gpred(i, j) seconds. Each data point in Figures 3-6 is an average and an actual mobile sub graph Gactual(i, j) computed over 5 mobility trace files and 5 randomly such that there exists at least one common s-d selected s-d pairs from each of the mobility trace path in both Gpred(i, j) and Gactual(i, j) and no files. The starting time of each s-d session is such common s-d path exists in both Gpred(i, uniformly randomly distributed between 1 to 20 j+1) and Gactual(i, j+1) or j + 1 = T seconds. In Figures 3-6, the Minimum Hop Mobile Path, the Stable Mobile Path determined by 3 PS-LUV = PS-LUV U {Minimum hop predicted s- algorithm OptPathTrans and the Stable Mobile Path d path that exists in both Gpred(i, j) and determined by algorithm OptPathTrans-LUV are Gaactual(i, j)} identified as MHMP, SMP-Opt and SMP-LUV respectively. 4 i=j+1 The performance metrics evaluated are the average lifetime per static path for a mobile path and 5 end while the time averaged hop count of the mobile path under the conditions described above. The time averaged 6 return PS-LUV hop count of a mobile path is the sum of the products of the number of hops per static path and the number End OptPathTrans-LUV of seconds each static path exists divided by the number of static graphs in the mobile graph. For example, if a mobile path spanning over 10 static Figure 2: Pseudo code for OptPathTrans-LUV graphs comprises of a 2-hop static path p1, a 3-hop Figure 3.1: Average Lifetime per Figure 3.2: Average Lifetime per Figure 3.3: Average Lifetime per Static Path (25 Nodes) Static Path (75 Nodes) Static Path (125 Nodes) Figure 3: Average Lifetime per Static Path in the Mobile Path (Square Network Topology) Figure 4.1: Average Lifetime per Figure 4.2: Average Lifetime per Figure 4.3: Average Lifetime per Static Path (25 Nodes) Static Path (75 Nodes) Static Path (125 Nodes) Figure 4: Average Lifetime per Static Path in the Mobile Path (Circular Network Topology) Figure 5.1: Average Hop Count Figure 5.2: Average Hop Count Figure 5.3: Average Hop Count per Static Path (25 Nodes) per Static Path (75 Nodes) per Static Path (125 Nodes) Figure 5: Average Hop Count per Static Path in the Mobile Path (Square Network Topology) Figure 6.1: Average Hop Count Figure 6.2: Average Hop Count Figure 6.3: Average Hop Count per Static Path (25 Nodes) per Static Path (75 Nodes) per Static Path (125 Nodes) Figure 6: Average Hop Count per Static Path in the Mobile Path (Circular Network Topology) static path p2, and a 2-hop static path p3, with each approach and OptPathTrans-LUV cannot effectively existing for 2, 3 and 5 seconds respectively, then the make use of the increase in network density and time-averaged hop count of the mobile path would be determine paths with relatively larger lifetime than (2*2 + 3*3 + 2*5)/10 = 2.3. those determined in low-density networks. The average lifetime per static path determined 4.1 Average Path Lifetime by the Minimum hop path algorithm is only 10%- 40% (i.e., 10 times smaller at the worst case) of the For a given level of node mobility, as we average lifetime per static path for an SMP. On the increase the network density, the difference in the other hand, the average lifetime per static path lifetimes of the paths discovered by algorithms determined for SMP-LUV is 30%-88% (i.e., 3.5 OptPathTrans and OptPathTrans-LUV increases. times smaller at the worst case) of the average OptPathTrans effectively makes use of the increased lifetime per static path obtained for an SMP. The availability of the nodes and the knowledge of the difference in the lifetime per static path for Minimum locations of the nodes over the entire simulation time hop routing and OptPathTrans-LUV decreases with period and determines stable paths with the longest increase in network density and node mobility. The lifetime. Both the Minimum-hop based routing lifetimes per static path obtained for SMP-LUV can be as large as, three times the lifetime per static path two metrics. If a routing algorithm has a lower Path obtained using Minimum hop based routing. lifetime ratio, then it implies the routes determined In general, at low and high mobility conditions, by that algorithm are more stable. Similarly, if a for a given routing algorithm and network density, routing algorithm has a lower Hop count ratio, then it the average lifetime per static path incurred in a implies the routes determined by that algorithm have circular topology is higher than that incurred in a a hop count that is closer to that of the minimum. square topology. On the other hand, in moderate mobility conditions, the average lifetime per static path incurred in a square topology is higher than that incurred in a circular topology. 4.2 Average Hop Count The average hop count per static path determined by algorithms OptPathTrans and OptPathTrans-LUV is respectively 20%-88% and 11%-66% more than the minimum hop count for square network topology. On circular network topologies, the two stable mobile path algorithms incur relatively higher hop count. The average hop count per static path determined by algorithms OptPathTrans and OptPathTrans-LUV is respectively 20%-150% and 16%-82% more than the minimum hop count for circular network topology. OptPathTrans determines static stable paths whose hop count could be as large as 24% and 37% more than that determined by OptPathTrans-LUV in square and circular network topologies respectively. In general, the hop count of a minimum hop path is higher for square network topologies than circular topologies. On the other hand, the hop count of a stable path is higher for circular network topologies than square topologies. With increase in network density, the average hop count per minimum hop path decreases. For the two stability-based algorithms (especially OptPathTrans), the average hop count per stable path increases as the network density is increased. This is attributed to the nature of these two algorithms to determine long-living stable paths by accommodating few more nodes on the path that would increase the average lifetime of the constituent links of the path. Figure 7: Path Lifetime – Hop Count Tradeoff 4.3 Path Lifetime – Hop Count Tradeoff (Square Network Topology) Figures 7 and 8 capture the tradeoff between We observe that OptPathTrans has the highest path lifetime and hop count for each of the three Hop count ratio and MHMP has the highest Path algorithms (Minimum hop based routing, lifetime ratio. This implies that the minimum hop OptPathTrans, OptPathTrans-LUV) for square and count paths cannot have the longest lifetime – what circular network topologies respectively. The Path we call as the lifetime-hop count tradeoff. On the lifetime ratio is defined as the ratio of the average other hand, we observe OptPathTrans-LUV to have a path lifetime per static path for the SMP determined relatively lower Path lifetime ratio compared to by algorithm OptPathTrans to that of the average MHMP and a relatively lower Hop count ratio path lifetime per static path for a MHMP or the compared to OptPathTrans. Thus, OptPathTrans- SMP-LUV. The Hop count ratio is defined as the LUV effectively balances the tradeoff between path ratio of the average hop count of either the SMP or lifetime and hop count tradeoff as much as possible. the SMP-LUV to that of the average hop count per We observe the Path lifetime ratios of SMP- static path for a MHMP. Note that the ratios are LUV are relatively lower for square networks. This formulated using the optimum values for each of the implies for square networks, OptPathTrans-LUV determines paths with lifetime close to that determined by OptPathTrans. The Hop count ratios compared to Minimum hop routing, and also obtained for SMP-LUV in circular networks are a generally provides a lower time-averaged hop count little higher, but are still contained within 1.8. We than OptPathTrans. OptPathTrans-LUV is able to also observe that in low density networks, MHMP accomplish this through effective prediction of the has almost the same Path lifetime ratio in both square locations of each node using the Location Update and circular network topologies. As we increase the Vectors (LUVs) available for the nodes at the time of network density, the Path lifetime ratio for MHMP in determining a new stable path. The effectiveness of circular networks decreases – implies, the minimum location prediction is that even if the predicted hop paths in circular networks are more stable than locations of the two nodes are different from the that of square networks in moderate and high density actual locations of the nodes, there exists a link networks. Another interesting observation is that the between two nodes in the predicted graph if they are hop count of a SMP determined by algorithm within their transmission range. This helps to OptPathTrans has a relatively higher Hop count ratio significantly increase the lifetime of the paths in circular networks (than square networks), determined by OptPathTrans-LUV compared to especially at moderate and high network density. those of the minimum hop paths, and at the same time, the increase in the hop count is well restricted. The mobile sub graphs of algorithm OptPathTrans span over a larger period of time, and the number of links in such mobile sub graphs are relatively lower. The stable path (which is a minimum hop path in the mobile sub graph) determined between a source and destination would have to go through several intermediate nodes. On the other hand, the mobile sub graphs of algorithm OptPathTrans-LUV have relatively more links and the stable path is basically a minimum hop path in such a mobile sub graph with more links. Thus, OptPathTrans-LUV effectively reduces the tradeoff between path lifetime and hop count. As future work, we will work on extending OptPathTrans-LUV to a distributed routing protocol for MANETs. 6 ACKNOWLEDGMENTS The research is supported through the National Science Foundation grant (CNS-0851646) entitled: REU Site: Undergraduate Research Program in Wireless Ad hoc Networks and Sensor Networks,” hosted by the Department of Computer Science at Jackson State University, USA. The authors also acknowledge Dr. Loretta Moore, Dr. Xuejun Liang and Mrs. Brenda Johnson (all at Jackson State University) for their services to this program. 7 REFERENCES [1] C. S. R. Murthy and B. S. Manoj, “Ad Hoc Wireless Networks: Architectures and Protocols,” Prentice Hall, June 3, 2004. [2] S. Ni, Y. Tseng, Y. Chen and J. Sheu, “The Broadcast Storm Problem in a Mobile Ad Hoc Figure 8: Path Lifetime – Hop Count Tradeoff Network,” Proceedings of ACM International (Circular Network Topology) Conference on Mobile Computing and Networking, pp.151-162, 1999. 5 CONCLUSIONS [3] N. Meghanathan and A. Farago, “On the Stability of Paths, Steiner Trees and Connected As hypothesized, algorithm OptPathTrans-LUV Dominating Sets in Mobile Ad Hoc Networks,” does in fact offers an alternative to both Minimum Elsevier Ad Hoc Networks, Vol. 6, No. 5, pp. hop routing as well as OptPathTrans. OptPathTrans- 744 – 769, July 2008. LUV is able to yield higher average path lifetime [4] N. Meghanathan, “A Location Prediction Based Reactive Routing Protocol to Minimize the Number of Route Discoveries and Hop Count per Path in Mobile Ad hoc Networks,” The Computer Journal, Vol. 52, No. 4, pp. 461-482, Oxford Press, British Computer Society, July 2009. [5] D. B. Johnson, D. A. Maltz and J. Broch, “DSR: The Dynamic Source Routing Protocol for Multi-hop Wireless Ad hoc Networks,” in Ad hoc Networking, Chapter 5, C. E. Perkins, Eds. Addison Wesley, pp. 139 – 172, 2000. [6] A. Farago and V. R. Syrotiuk, “MERIT: A Scalable Approach for Protocol Assessment,” Mobile Networks and Applications, Vol. 8, No. 5, pp. 567 – 577, October 2003. [7] T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, “Introduction to Algorithms,” 2nd Edition, MIT Press/ McGraw Hill, Sept. 2001 [8] C. Bettstetter, H. Hartenstein and X. Perez-Costa, “Stochastic Properties of the Random-Waypoint Mobility Model,” Wireless Networks, vol. 10, no. 5, pp. 555-567, 2004.