# On Performance Evaluation of Reliable Topology Control Algorithms in Mobile Ad Hoc Networks (Invited Paper)

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2 TABLE I COMPUTATION TIME OF THE CONSIDERED ALGORITHMS. O(m + n log n) O(n 2 ) O(m + n log n) FLSS O(m(m + n)) LTRT O(k(m + n log n)) Fig. 1. v u w The local view of node u at time t and (t +Δt). a rather small average node degree. The node degree of a node is the number of edges that connect to it. By using Prim s algorithm, the time complexity of is computed to be O(n 2 ) when using simple searching and O(m+n log n) when using Fibonacci heap. Here, m is the number of edges and n denotes the number of vertices. also has the lowest complexity among the five algorithms that are evaluated in this work, as shown in Table I. Toussaint [4] proposed Relative Neighborhood Graph () that attempts to remove redundant edges while maintaining the connectivity. An edge (u, v) is redundant if there is a node w satisfying d(w, u) <d(u, v) and d(w, v) <d(u, v), where d(u, v) is the distance between u and v. The constructed topology is the remained graph after removing all redundant edges. is also carried out in the local graph of each node to become a localized topology control algorithm. Cartigny et al. [9] prove that with the same graph, the topology generated by LTRT is a sub-graph of s topology. Therefore, the average node degree and total transmission power of are higher than those of. can be implemented with a computational cost of O(n 2 ). By using the simple idea that an edge will be removed if there is a 2-hop path between the two nodes consuming a lower transmission power than the directed way, Rodoplu and Meng [1] proposed another topology control algorithm. Li and Halpern [5] then extended the algorithm by using k-hop paths instead of 2-hop paths. Accordingly, the algorithm can be explained as carrying out Dijkstra s algorithm [11] for a graph with the weight function E = d α where E and d are the weight and length of the edge, respectively. Localized version of this algorithm is called Local Shortest Path Tree () and can run in O(m + n log n) time when using Fibonacci heap. Although,, and can preserve a connected topology, the network is still 1-edge connected. Consequently, the connection can easily be dropped when one link is broken. In order to come up with a fault-tolerant solution, Li and Hou [6] proposed Fault-tolerant Local Spanning Subgraph (FLSS) algorithm that guarantees k-edge connectivity if the original network is k-edge connected. In terms of maximum transmission power, FLSS is proven to be a min-max optimal algorithm. However, the complexity of FLSS is O(m(m+n)), which is not applicable for dense networks. Recently, Miyao et al. [1] proposed Local Tree-based Reliable Topology (LTRT), a reliable topology control algorithm that can preserve k-edge connectivity. While FLSS is a minmax optimal algorithm, LTRT is a near-optimal one, but it has much lower complexity, O(k(m + n log n)). The initial proposal of tree-based reliable topology (TRT) was made by Ansari et al. [12] and LTRT can be considered as a localized version of TRT. III. MOBILITY EFFECTS ON TOPOLOGY CONTROL A. Mobility effects In static networks, to reduce energy consumption, topology control is used to help each node decide its own transmission power while also guaranteeing the network connectivity. After constructing the topology, every node only needs to transfer information in its range. However, in MANETs, the connectivity might be lost as a consequence of node movement. Fig. 1 is a circumstance of local graph in MANETs. The figure demonstrates a local view of node u and its local graph after an interval Δt. From the local view of u, there are two types of neighbor movements that can cause the loss of connectivity: moving out of the local area, like v, and moving into the local area, like w. If node u does not have the updated information of its neighbors, it will lose the connection with v and will not be able to use w to calculate the new topology. The two situations imply the two methods of improvement, namely frequently updating the topology and using redundancy of edges. These two fault-tolerant strategies are described next. B. Fault-tolerant strategies The first strategy that can be applied to maintain the network connectivity is to frequently update the topology. This is obvious because whenever the nodes know the updated information of their neighbor, they can decide the appropriate transmission power to keep their connections. However, the size of the refreshing interval needs to be considered. If the movement is slow while the interval is too short, the cost of topology construction will be needlessly high. The second strategy is preserving k-edge connectivity. As the previous example, a connection between two nodes can be easily broken in mobile networks. As a result, the algorithms that only guarantee 1-edge connectivity, such as, and, may not be suitable for MANETs because the loss of only one edge may lead to the loss of the whole network connectivity. Although there is a trade-off between maintaining connectivity and minimizing energy consumption, FLSS and LTRT should be more suitable for MANETs. Because they can generate k-edge connected topologies, the network will be more fault-tolerant. Moreover, these two algorithms are also more scalable than the others because the value of k can easily be changed. We can achieve more fault-tolerant topologies by using bigger values of k. However, the cost of topology construction and the total transmission power will be

3 Average node degree Average transmission range [m] Fig. 2. Number of nodes Average node degree. higher in such a case. The trade-off between connectivity and energy consumption can be seen clearly by evaluating LTRT and FLSS for different values of k. A. Simulation setups IV. PERFORMANCE EVALUATION We evaluate the performance of the five algorithms, i.e.,,,, FLSS, and LTRT, by using Network Simulator version 2 (NS2). To simulate the algorithms in MANETs, we use random waypoint mobility model that was first proposed by Johnson and Maltz [13] and has been considered as a typical mobility model to evaluate routing protocols in MANETs [7] [8]. The model gives mobile nodes pause times between changes of moving speed or direction. In each process, after a pause time in current position, a mobile node moves to a random destination with speed in specified range. Until arriving to the chosen destination, the node keeps its direction and speed. In our simulation, the scenarios are generated on an area fixed to 1m 1m with the maximum transmission range set to 25m for each node. The number of nodes, moving speeds and topology update intervals are changed in conducted experiments. Corresponding to each different number of nodes or moving speed, one hundred scenarios are generated to find the average results for each experiment. With FLSS and LTRT, the value of k is also individually assigned in each experiment. We introduce a metric named connectivity ratio to evaluate the performance of the considered algorithms in MANETs. Connectivity ratio measures the network connectivity at a certain snapshot in time. Every node always keeps its neighbor list obtained by the latest topology construction. However, at an arbitrary moment, the positions of all nodes may change and a node may be too far from another to keep the link. Connectivity ratio demonstrates the connectivity status of the whole graph constructed by all nodes and existing links. It is the percentage of connected node pairs out of all possible node pairs in the network. A node pair (u, v) is called connected if there is a path from u to v and vice versa. If n is the number of nodes and V is the set of nodes, connectivity ratio, which is Fig. 3. Number of nodes Average transmission range. indicated by C, can be calculated by the following equation. C = u,v V c uv n(n 1) where { 1, if u v and (u, v) is connected, c uv =, otherwise. Connectivity ratio is calculated without data transmitting. In each simulation, a snapshot is taken right before an arbitrary topology update. For example, if t is the time of a topology update, the time of taking snapshot will be (t ε), where ε is the assigned error. B. Simulation results 1) Node degree and transmission range: We first make a comparison between the five algorithms in terms of average node degree and transmission range. The algorithms are evaluated with different scenarios with the number of nodes varying from 5 to 15. In this evaluation, LTRT and FLSS are executed with k =2. Because this evaluation can be done without mobility, the scenarios are generated with moving speed set to zero. Fig. 2 demonstrates average node degree of the topologies constructed by the five algorithms, respectively. As topology control algorithms attempt to preserve network connectivity with minimal energy consumption, they tend to find the fewest and nearest 1-hop neighbors for each node. Therefore, the average node degree is not affected by the number of nodes in the network. However, the network density has a strong relationship with the needed transmission power. In sparse networks, because every node is far from each other, a node may have to use higher transmission power to connect enough neighbors. Fig. 3 also demonstrates that the topologies generated by all algorithms require a higher average transmission range when the number of nodes is smaller. As demonstrated in the two figures, LTRT needs significantly higher node degree than FLSS, but LTRT needs slightly higher transmission range.,, and preserve only 1-edge connectivity, and thus, evidently need lower node degree and transmission range. (1)

4 Connectivity ratio (C) [%] 1 Fig Average moving speed [m/s] Effect of moving speed on network connectivity. Connectivity ratio (C) [%] Fig Topology update interval [s] Effect of topology update interval on network connectivity. Reachability ratio [%] Fig Average moving speed [m/s] Reachability ratio of the packets being transmitted. 2) Effects of moving speed on network connectivity: We then evaluate the performance of all algorithms in case of mobile nodes. In the same area as the previous experiment, the number of nodes is fixed to 1. In order to evaluate the effect of moving speed on network connectivity, the scenarios are generated with an average moving speed, which varies from 1m/s to 25m/s. LTRT and FLSS are evaluated with k =2. In the experiment measuring connectivity ratio, the topology update interval is set to five seconds (i.e., δt =5s). Fig. 4 shows that the moving speeds have a strong effect on connectivity ratio. All the evaluated algorithms, especially,, and, are considerably sensitive to mobility. When the average moving speed is larger than 15m/s, no algorithm can generate a topology that has connectivity ratio higher than 1%. As mentioned before, connectivity ratio is calculated without data transmitting. Therefore, we attempt to conduct another experiment to measure the success of transmitting data packets under the effect of moving speed. In this experiment, δt =1s and data packets are randomly transmitted every second by a flooding method. We calculate the reachability ratio denoting the percentage of nodes, which received at least one packet, out of the all nodes. Because this experiment is conducted with δt =1s and data packets are randomly transmitted during simulation time, the calculated reachability ratio proximately implies the state of network connectivity at δt/2 seconds after a topology update (i.e., five seconds). Therefore, it can be compared with connectivity ratio that is calculated with δt =5s in the previous experiment. Note that all the snapshots used to calculate connectivity ratio are taken right before the next topology update. Fig. 5 demonstrates that the moving speeds also have a considerable effect on reachability ratio. LTRT and FLSS send data packets more successfully than others when k = 2. On the other hand, the calculated reachability ratio is lower than connectivity ratio corresponding to each moving speed. This is because it depends on the type of routing protocol used in the simulation. In our simulation, we only evaluate reachability ratio by using the Ad hoc On-Demand Distance Vector (AODV) routing protocol. Further evaluation with other routing protocols will be addressed in our future work. 3) Effects of topology update interval on network connectivity: Besides moving speed, the effect of topology update interval, δt, on connectivity ratio is also evaluated by running simulations with different intervals, from one to ten seconds. In this experiment, the average moving speed is fixed to 1m/s and the number of nodes in all scenarios is set to 1. As evident from Fig. 6, frequent topology updating can help preserve network connectivity. The shorter the topology update interval is, the higher the connectivity ratio will be. The result also demonstrates that LTRT is more reliable than FLSS. 4) Comparison between LTRT and FLSS: We then evaluate the two reliable topology control algorithms, LTRT and FLSS, with different values of k which varies from one to seven. We conduct two experiments to compare LTRT and FLSS in terms of average transmission range and connectivity ratio. The number of nodes is fixed to 1. The experiment measuring average transmission range is conducted without mobility. On the other hand, in the experiment evaluating the effect of the change of k on connectivity ratio, the average moving speed is 2m/s and topology update interval, δt, is set to five or ten seconds. Fig. 7 shows the average transmission range that is needed for the two algorithms corresponding to each value of k. The result shows that when k is bigger, not only the computation time becomes longer, but the needed transmission range also becomes higher. Although the topology control algorithms, certainly, have no meaning if all the nodes in the constructed topology still need a transmission range close to the maximum (i.e., 25m), as shown in the figure, LTRT needs a slightly

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