WIRELESS ad-hoc networks are emerging day by day

Transcription

1 Load and Energy Aware Topology Control in Wireless Ad-hoc Networks Tandra Chakraborty, Fazlay Rabbi, Aungon Nag Radon, Student Member, IEEE and Ashikur Rahman Department of Computer Science, University of Calgary, Calgary, AB, Canada, T2N 1N4 Department of Computer Science & Engg. (CSE), BUET, Dhaka 1000, Bangladesh School of Computer Science & Engg. (CSE), UITS, Dhaka 1212, Bangladesh {tandra050387, rabbi cse05 Abstract Most of the topology control protocols for wireless ad hoc networks focus on minimizing power consumptions while maintaining connectivity. Those protocols are static in nature due to the fact that all the nodes use only a fixed set of (energy efficient) relays. However, using the same set of relays all the time causes some nodes to become heavily loaded creating congestion, unbalanced traffic and high latency in the network. Instead of generating a single topology with fixed relay sets, we propose a distributed algorithm that generates a family of connected topology with (possibly) disjoint relay sets. Thus, by creating different energy aware relay sets of each node the proposed algorithm performs a time-delayed rotation of data forwarding duty among those sets in order to evenly distribute loads in the network. We illustrate the efficiency and efficacy of the proposed algorithm through simulation experiments. I. INTRODUCTION WIRELESS ad-hoc networks are emerging day by day due to its nature of being infrastructure independent. These networks [1] dominates other networking technology in applications like battlefields, natural disasters, classroom environment etc. Often an ad-hoc wireless network consists of mobile nodes equipped with volatile power sources such as batteries or solar cells. Thus energy consumption control is a prime concern in designing such networks. The transmission power between a pair of communicating nodes is the predominant factor in the overall energy consumption. As this transmission power grows at least quadratically with the distance between the communicating parties, using one or more neighbors for relaying data packets saves energy. Thus any energy saving topology control algorithm replaces longer links by one or more shorter links. In general, a topology control algorithm works as follows: Each node in the network classifies all the neighbors within its maximum transmission range as relays and non-relays. This classification is shaped by an objective function (often dubbed as a rule in the protocol) suitably chosen by the algorithm based on certain goals to achieve. Thus, if the goal is to conserve energy then a neighbor requiring higher transmission power for direct communication is an obvious choice of being classified as a non-relay. After the classification step each node reduces its transmission power to the minimum required to communicate only to all of its relay neighbors. Authors are listed in the alphabetic order of their last names. Fig. 1. An example topology Fig. 2. Energy Aware Topology Thereby a node abandons direct communication to all its nonrelay neighbors and uses one or more relays to reach those neighbors. Although many solutions are proposed in the literature [2], [3] and [4] in determining appropriate relay sets to produce very sparse topologies, those protocols choose a fixed set of relays all the time. This type of static strategy causes some nodes, chosen as relays, to become heavily loaded and creates congestion, unbalanced traffic and high latency in the network. To further illustrate the problem, let us consider the example topology of Fig. 1. We assume that node a is the source node and node j is the destination node. The circle denotes the maximum communication range of node a. From this figure it is obvious that node a can not reach node j directly and must use either node d or h for relaying to j. Moreover a can reach any node in this topology using nodes b, c, d and e as relays. It also costs minimum energy if a relay is used instead of direct transmission. The traditional minimum energy protocols [2], [3] suggest to construct the set S 1 = {b, c, d, e} as a fixed relay set for node a. By eliminating the links to non relays of node a we obtain the topology of Fig. 2, which can lead to minimum energy consumptions. However, if we closely observe Fig. 2, we can observe that a must always use d in order to reach j, because node h is not included in its relay set. Eventually d will become heavily loaded if it is frequently used for relaying to j. On the contrary, if we construct a second relay set comprising S 2 = {b, c, h, e} and perform a timedelayed rotation between S 1 and S 2, then the load could be evenly distributed between d and h for communicating to j. In this paper we propose a distributed algorithm to evenly distribute load while maintaining connectivity. Our approach is free from any communication overhead, i.e., does not require any load feedback from a node s neighborhood. The proposed

2 al. [9] address the fault tolerance issue by constructing several subgraphs that preserve minimum energy paths and maintain bi-connectivity. However the effect of load is safely ignored. Rahman et al. [10] introduce LMECN algorithm that uses a cross layer approach and some assistance from the MAC layer to construct a subgraph of the initial topology. However LMECN algorithm is not load aware. Fig. 3. Relay region R u v Fig. 4. Relay graph G a R algorithm creates different sets of relays from each node s neighbor set and rotates data forwarding duty among those relay sets. The decision for choosing a relay set is taken at the routing layer. Because of this dynamic rotation of relay sets, nodes in a particular set can remain inactive for a certain period of time and some other nodes in other sets perform data forwarding. The relay sets are constructed in such a way that the network remains connected all the time. Rest of the paper is organized as follows. Section II describes related research works. Section III illustrates the notations used in our algorithm. Section IV presents the basic idea to architect load and energy aware topology control. Section V gives the proof of correctness of the proposed algorithm where we prove that the connectivity of the network is preserved all the time. Simulation models and various performance aspects of the protocol are presented in Section VI. Finally, Section VII concludes the paper. II. RELATED WORKS A significant amount of research has been directed to topology control problems for wireless ad-hoc networks but very few of them address the issue of load awareness. Ramanathan et al. [5] consider the problem of adjusting the transmission powers of nodes and present two centralized algorithms CONNECT and BICONN-AUGMENT. They introduce two heuristics which do not preserve connectivity. Cone- Based Topology Control (CBTC), proposed by Li et al. [6], generates a connected graph structure. A serious drawback of the algorithm is the need to decide on the suitable initial power level and the increment at each step. Bahramgiri et al. [4] augment the CBTC algorithm [6] later on to provide fault tolerance. However there is no guarantee that the proposed modification preserves minimum-energy paths. Rodoplu and Meng [3] introduce the notion of relay region and provide a distributed position-based network protocol optimized for minimum-energy consumption in mobile wireless networks. However, the algorithm does not consider load effects in the network. Later on, Rodoplu s [3] algorithm is modified by Li [2] which trims more unnecessary links. But the algorithm has the problem of partially enclosed nodes. Shen et al. [7] propose a distributed topology control algorithm, which preserves minimum-energy property and bi-connectivity at the same time. But the algorithm uses another algorithm presented in [8] to identify cut vertices which can not be achieved using only local information. Acquiring global topology information is expensive and wastes more energy. Harichandan Roy et A. Power Model III. DEFINITIONS We use the the well known, generic, two-ray, channel path loss model [1]. In this model, the minimum transmission power to send a message from node s to node t at distance d is approximated by P s t = Kd α where α [2, 4] is the path loss factor, and K is a global constant. B. Relay Region First defined in [3], relay region of a node v with respect to another neighbor u, denoted by R u v, is the collection of points x, y such that relaying through v to any point in R u v consumes less energy than direct transmission to that point. Fig. 3 shows the relay region of v with respect to u. Mathematically: R u v = { x, y P u v x,y < P u x,y } (1) C. Relay Graph of a Node Relay graph of a node s, denoted by G s R = (V R, E R ), includes all its neighbors in its vertex set V R. The edge set E R contains all the direct edges to its neighbors and all edges between two neighbors (u, v) where u can be used as energy efficient relay to reach v from s. For a formal definition, let us assume that N(s) is the set of neighbors of node s. Thus, V R = N(s) and E R is defined as follows: E R = {(u, v) u, v V u, v s Loc(v) R s u } {(s, v) v N(s) v s} (2) where Loc(v) is the geographic location of node v. In order to construct relay graph of a node s, at first we determine the relay region of each node u N(s). Then for each node v N(s), if node v lies in the relay region of node u, we draw an edge from u to v. Finally, we draw edges from the source node s to all other nodes in N(s). Relay graph of the example topology of Fig. 1 is shown in Fig. 4. From Fig. 4, it is clear that node g lies in the relay region of node e. Similarly node i and node h lie in the relay region of node d, and node f lies in the relay region of node b. D. Level Graph Level graph G s L of a node s is a labeled relay graph where each vertex is associated with a positive integer representing its hop distance on the longest path from the node s within its relay graph. G s L can be constructed from Gs R using the following procedure. Let us denote the level of each node v in G s L by level(v). Clearly, level(s) = 0. For all other neighbors, at first we set the level of each node to zero. Then we construct

3 Fig. 5. Level graph G a L Fig. 6. Relay set R 1 a Algorithm 1 Procedure Find-Level(s) 1: level(s) 0 2: for each v N(s) do 3: level(v) 0 4: end for 5: Let E(s) be the set of edges of node s 6: Sort the edges of E(s) in ascending order of their lengths 7: for each edge (s, x) E(s) in sorted order do 8: Let E(x) be the set of outgoing edges from x 9: for each edge (x, y) E(x) do 10: if level(y) < level(x) + 1 then 11: level(y) level(x) : end if 13: end for 14: end for a set E(s) with all outgoing edges of s and sort them in ascending order of length. Next, for each edge (s, x) E(s), we determine the set of all outgoing edges from x. Let us denote this set by E(x). Then, for each edge (x, y) E(x), if level(y) < level(x) + 1, we assign level(y) level(x) + 1. This process continues until no more new edges could be added. The complexity of this algorithm is O(n). Algorithm 1 summarizes this procedure. Fig. 5 shows the level graph of node a which is obtained from the relay graph of Fig. 4. From Fig. 5, we observe that level(h) > level(d) which is obvious because the length of ah is greater than the length of ad. E. Level Sets In a level graph, all the neighbors that have the same level comprise a level set. Thus, the level sets of a node are disjoint partitions of its neighbor set where the partitions are formed based on their levels. Intuitively, a node has n level sets where n is the maximum level of any neighbor within its level graph. Formally, the i th level set of a node s, denoted by L i s, is: L i s = {v v N(s) v s level(v) = i} (3) In Fig. 5, node a has at most two level sets as maximum level of its neighbors is two. These two level sets are L 1 a = {b, c, d, e} and L 2 a = {f, g, h, i} respectively. The algorithm for determining level sets of a node s is shown in Algorithm 2. The algorithm begins with assigning maximum level of s s neighbors to variable M axlevel. Then each neighbor v is included in the level set L j s based on its level j. Algorithm 2 Procedure Level-Set(s) 1: MaxLevel max {level(v) : v N(s)} 2: for i = 1 to MaxLevel do 3: L i s 4: end for 5: for all v N(s) {s} do 6: j level(v) 7: L j s L j s {v} 8: end for IV. THE PROPOSED ALGORITHM In this section we propose a distributed algorithm that constructs several relay sets of each node in the network. We refer to our algorithm as Load and Energy Aware Topology Control or LEATC in brief. The main idea of LEATC is described next. A. The Main Idea The overall objective is to construct multiple relay sets of each node in the network and then periodically rotate each node s relay sets for forwarding data packets to the destinations. Thus, periodic rotation of relay sets ensures that a particular relay node is not overwhelmed with too many data transmissions. When constructing multiple relay sets, we make them as disjoint as possible. The overall procedure is described next. Relay sets of a node are formed only from its 1-hop neighbors in such way so that the node can reach to any other nodes in the network using the nodes in any relay set (i.e. the connectivity is guaranteed). The relay sets of a node are constructed from its level sets as follows. Let us denote the i th relay set of a node s by Rs. i The first relay set of a node is composed of all 1-hop neighbors that are at level 1 in the level graph, i.e.: Rs 1 = { v v L 1 } s (4) The first relay set Ra 1 for node a of the example topology of Fig. 1 is shown in Fig. 6. The red boundary in the figure contains the nodes which are in Ra, 1 consequently, Ra 1 = {b, c, d, e}. It is easy to see that all other neighbors within maximum communication range of a are reachable via one of the relays in Ra. 1 The other relay sets of a node are formed from its remaining nodes within maximum transmission range. Let us see how a node s determines its i-th relay set where 1 < i MaxLevel. When forming i-th relay set, the connectivity is guaranteed by covering all two hop neighbors of s that are reachable via some nodes in first relay set Rs. 1 Let N1 2 (s) be the set of neighbors reachable via one of the relays in Rs, 1 i.e. N1 2 (s) = N(v) (5) for all v R 1 s Also let R be the set of nodes that are members of prior relay sets, Rs, 1 Rs 2... Rs i 1, i.e., R = i 1 k=1 R k s (6)

4 Algorithm 3 LEATC(s) 1: j 1, i 1, MaxLevel 0, R 2: Broadcast NDM, Collect responses and Construct Neighbor Set, N(s) 3: Construct Relay Graph, G s R = (V R, E R ) 4: Find-Level(s) 5: Level-Set(s) 6: MaxLevel max {level(v) : v N(s)} 7: while j MaxLevel do 8: T = { v v L j s} R 9: if T then 10: R i s T 11: R R Rs i 12: N1 2 (s) = v R N(v) s 1 13: U s = N1 2 (s) R, 14: if U s then 15: OSLA(U s, s, j, Rs, i R) 16: OSLB(U s, s, i, Rs, i R) 17: end if 18: Ps i max { } Kd α (s, v) : v Rs i 19: j j + 1, i i : else 21: j j : end if 23: end while In order to construct R i s, at first we include all the neighbors at level i in the level graph of s. We also exclude all the nodes that are already included in any prior relay sets by subtracting the set R from the set R i s. These two steps are combined in the following mathematical step: R i s = { v v L i s} R (7) Then, we add some additional nodes to the set Rs i for ensuring connectivity by selecting nodes from the level i + 1 first and then from the level i 1 as needed. We form a set U s = N1 2 (s) R, which is the set of two hop neighbors excluding members of prior relay sets. If U s = then no additional relays are necessary. Otherwise, in each iteration, s selects a neighbor v Rs i+1, such that v is not in R and the list of neighbors of v covers the maximum number of nodes in U s, i.e., N(v) U s is maximized. Next, s includes v in Rs i and sets U s = U s N(v). The iterations continue until U s becomes empty or there is no way to further improve the coverage. We call this process one step look ahead (OSLA) as it requires a search for additional relays from the immediate next level. After one step look ahead if the set U s still remains nonempty then additional relays are added from the immediate previous level i 1. This time we iteratively select a neighbor v Rs i 1 under the same criteria, i.e., N(v) U s is maximized. The iteration continues until U s becomes empty. This process is called one step look back (OSLB) as we search from the immediate previous level. The overall procedure just described is implemented by LEATC(s) in Algorithm 3. LEATC(s) begins with initializing necessary variables. Variable j is the marker of level and Algorithm 4 Procedure OSLA(U s, s, j, R i s, R) 1: while U s do 2: select a v L j+1 s v / R such that N(v) U s is maximum 3: U s U s N(v) 4: Rs i Rs i {v} 5: R R {v} 6: if there is no such v L j+1 s then 7: break 8: end if 9: end while Algorithm 5 Procedure OSLB(U s, s, i, Rs, i R) 1: while U s do 2: select a v Rs i 1 such that N(v) U s is maximum 3: U s U s N(v) 4: Rs i Rs i {v} 5: end while i is the marker of relay set. Variable M axlevel stores the maximum level of nodes in N(s). Variable Ps i denotes minimum necessary power needed to reach all nodes in Rs. i After initialization, node s sends Neighbour Discovery Message (NDM) at maximum power P max and constructs neighbor set N(s) when it gets back the reply messages. Then, s constructs relay graph G s R as described in Section III- C. Next, Find-Level(s) procedure is called which is defined in Algorithm 1. Find-Level(s) procedure assigns levels of the neighbors of s (See Section III-D). Then, Level-Set(s) procedure is called to form different level set of s which is defined in Algorithm 2 (see Section III-E). After that, we iteratively generate different relay sets of s. Iteration starts from the first level and terminates when M axlevel is reached. On each iteration the OSLA and OSLB is invoked as needed. OSLA and OSLB are implemented separately in Algorithm 4 and Algorithm 5 respectively. Finally, the power for each relay set Ps i is determined based on the maximum distance from s to relays in Rs. i It is easy to see that the way relay sets are constructed the following relation always holds. B. Load Aware Graph G i P 1 s P 2 s P 3 s... P MaxLevel s (8) Algorithm LEATC generates several subgraphs of the initial graph G. Load aware graph G i is constructed by drawing directed edges from every node in the network to its relay nodes in i th relay set. Mathematically, we denote G i as G i = (V, E i ), whose vertex set V consists of all the nodes in the network and whose edge set E i is defined as follows: E i = for all s V for all v Rs i where R i s is the i th relay set of node s. {(s, v)} (9)

5 V. PROOF OF CORRECTNESS OF LEATC In this section we prove that LEATC generates connected topologies. Consider an n-node, multi-hop, wireless ad-hoc network deployed on a two-dimensional plane. Suppose, each node can adjust its transmission power up to a maximum value P max. Such a network can be modeled as a graph G = (V, E), with the vertex set V representing the nodes, and the edge set defined as follows: E = {(x, y) (x, y) V V d(x, y) R max } (10) where d(x, y) is the distance between nodes x and y, and R max is the distance reachable by using P max. The graph G defined in this way is called initial network. Lemma 5.1: G 1 is a minimum energy path preserving connected subgraph of G. Proof: At first, we show that G 1 = (V, E 1 ) preserves all minimum energy paths of G between every pair of nodes. Suppose, the cost of an edge s, t is γ( s, t ) = K d α (s, t). Since a path is defined by consecutive edges, the cost of a path π = s 0, s 1,..., s n+1 becomes γ(π) = n i=0 γ( s i, s i+1 ). Suppose, by way of contradiction, there exist nodes s, d and a simple path δ in G such that γ(δ) < γ(δ ) for any simple path δ from s to t in G 1. Let δ = s 0, s 1,..., s n+1 with s 0 = s and s n+1 = t. Then for all i = 0, 1,..., n, it must be true that s i, s i+1 E 1. Otherwise, level(s i+1 ) > 1 in the level graph of s i. Which also implies that in the relay graph of s i there exists a node r between the node pair s i, s i+1 such that Loc(s i+1 ) R si r. But then γ( s i, r, s i+1 ) < γ( s i, s i+1 ) and the path δ can be replaced by a least cost path by replacing the edge s i, s i+1 with s i, r, s i+1, which is a contradiction. So G 1 preserves all minimum energy paths in G between all pairs of nodes. Thus it immediately follows that G 1 is connected providing at least the minimum energy paths between all pairs of nodes. Lemma 5.2: For any node s, let N(N(s)) be the set of nodes that are within two hops of s in G. Then all the members of N(N(s)) are always reachable from s in any load aware subgraph G i of G. Proof: We prove this lemma by induction. Basis: According to Lemma 5.1, G 1 is a connected topology. So the nodes in N(N(s)) are always reachable from s in G 1. Induction Hypothesis: Let us assume that, s can reach all nodes of N(N(s)) in G i. Then it suffices to show that s can also reach all nodes of N(N(s)) in G i+1. Induction Step: Let, w be a node which is outside of the maximum power range of node s in G. Suppose s can reach node w in G i through node v where v N(s) for some v Rs. i So, w N(N(s)). Two cases may arise in G i+1. case 1: Let, v is the only node to reach node w. So, v is retained in Rs i+1 according to LEATC. case 2: There exists other node say x to reach w. According to LEATC v is replaced by x in Rs i+1. So, from the way LEATC algorithm constructs the relay sets, we can easily conclude that node s can reach node w R i+1 s using some relay in Rs i+1. Therefore, nodes in N(N(s)) are always reachable in G i+1. Theorem 5.3: For any integer i 1, G i is a connected subgraph of G. Proof: Let, π =< u 0, u 1,...u i, u i+1,...u k > be a path between node u and node v in the initial graph G = (V, E) where u = u 0 and v = u k. Let us assume that a link between u i and u i+1 does not exist in the graph G i. But u i+1 N(N(u i )) in G for all 0 i < k. Therefore according to Lemma 5.2, all nodes in N(N(u i )) are always reachable from u i in all G i. So if the edge u i, u i+1 is not a direct edge in any of G i, the edge can be replaced by a path between u i u i+1 and a longer path between u and v always exists in G i for all i. As u and v were arbitrarily chosen, the result can be generalized for any node pair in G. VI. SIMULATION RESULTS To evaluate the performance of the proposed algorithm we use a custom written simulator. In order to nullify the effect of MAC layer on the performance of the proposed algorithm, we use an ideal MAC layer for all the simulation experiments. A. Scenario Generation We simulate our algorithm using four rectangular grids. The size of the grids are 600m 600m, 800m 800m, 1000m 1000m, and 1m 1m. Different grid sizes control the density of the deployment. We uniformly distribute 50 to 250 nodes over these deployment areas. Each node has a maximum transmission range of 250m. Fig. 7 shows a random deployment of 50 nodes over a 600m 600m grid area. This is the initial graph G for our algorithm. After running LEAT C, we get several connected subgraphs G i of G. G 1 and G 2 are shown in Fig. 8 and Fig. 9 respectively. Both G 1 and G 2 reduce a significant amount of edges from the initial graph G. B. Performance Metrics We observe some statistical properties of the load aware graphs. The description and significance of each performance metric is given below. Here we assume that the set of nodes in G is V = v 1, v 2,..., v N and the maximum levels in the level graph of each node are m 1, m 2..., m N. Average Number of Relay Sets (ANRS): ANRS is measured by dividing the total number of relay sets of each node by the total number of nodes. A high value of ANRS is desirable because if a large number of relay sets is available for rotation then it is possible to distribute load more evenly. Average Size of Relay Sets (ASRS): ASRS is measured by dividing the total cardinality of each relay sets of all nodes by the total number of nodes. Mathematically, N mi i=1 j=1 ASRS = Rj v i (11) N The higher value of ASRS indicates that more nodes are participating in the formation of relay sets and consequently the chances of even load distribution is higher. Power Consumptions: For power requirements we measure Average Power of all Relay Sets (APRS) which is the total

6 Y Coordinate (m) Y Coordinate (m) Y Coordinate (m) X Coordinate (m) X Coordinate (m) X Coordinate (m) Fig. 7. Initial graph, G (no. of nodes 50) Fig. 8. G 1 (no. of nodes 50) Fig. 9. G 2 (no. of nodes 50) Average Number of Relay Sets (ANRS) Grid size 600 X 600 Grid size 800 X Grid size 1000 X 1000 Grid size 1 X Number of Nodes Average Size of Relay Sets (ASRS) Grid size 600 X Grid size 800 X 800 Grid size 1000 X 1000 Grid size 1 X Number of Nodes Average Power of Relay Sets (APRS) mw Grid Size 600 X 600 Grid Size 800 X 800 Grid Size 1000 X Number of Nodes Fig. 10. Average number of relay sets (ANRS) Fig. 11. Average size of relay sets (ASRS) Fig. 12. Power consumptions power consumed to reach all relay sets from each node divided by the total number of nodes. Mathematically N mi i=1 j=1 AP RS = P v j i (12) N C. Result Analysis 1) Number of relay sets: Average number of relay sets (ANRS) for different number of nodes is shown in the Fig 10. ANRS increases with the increase in node density. Consequently, in denser networks each node has more relay sets available for rotation which provides more opportunity for load distribution. 2) Size of relay sets: Average size of relay sets (ASRS) is shown in the Fig 11. For a particular deployment area, ASRS increases with the increase in number of nodes. Also ASRS increases in denser networks. Therefore a relay set may contain more number of nodes in a dense network than in a sparse network. Having more number of nodes in a set provides more load balancing option. 3) Power consumptions: Average power consumption to reach all relay sets (APRS) is shown in Fig 12. Sparser networks consume more power than denser networks. We observe a trade off between power consumption and load distribution. When the network density increases, a node gets more relay sets for even load distribution but the overall power consumptions to reach all those relay sets also increase. VII. CONCLUSION Topology control problem is a fundamental research problem for wireless ad-hoc networks. While most of the existing research efforts focus on reducing either energy consumption or interference, the load balancing issue is often overlooked by the research community. Nevertheless, the load balancing is an important issue that needs to be separately addressed for power constrained wireless devices deployed under ad-hoc network settings. This paper addresses joint load balancing and energy-aware topology control by generating a family of topologies in a distributed manner where load balancing is achieved by periodically rotating relay sets of each node. Simulation results show that the proposed algorithm performs better in denser networks than sparser ones and more load balancing is achieved at the cost of higher power consumption. REFERENCES [1] T. Rappaport, Wireless communications: principles and practice. IEEE, [2] L. Li and J. Halpern, Minimum-energy mobile wireless networks revisited, in ICC, pp , June 1. [3] V. Rodoplu and T. H. Meng, Minimum energy mobile wireless networks, JSAC, vol. 17, no. 8, pp , [4] M. Bahramgiri, M. Hajiaghayi, and V. S. Mirrokni, Fault-tolerant and 3 dimensional distributed topology control algorithms in wireless multihop networks, in ICCCN, pp , 2. [5] R. Ramanathan and R. Rosales-Hain, Topology control of multihop wireless networks using transmit power adjustment, in INFOCOM, 0. [6] L. Li, J. Y. Hlapern, P. Bahl, Y. Wang, and W. R., Analysis of a con-based topology control algorithm for wireless multi-hop networks, PODC, 1. [7] Z. Shen, Y. Chang, C. Cui, and X. Zhang, A fault-tolerant and minimum-energy path-preserving topology control algorithm for wireless multi-hop networks, in CIS, pp , 5. [8] A. Kazmierczak and S. Radhakrishnan, An optimal distributed ear decomposition algorithm with applications of biconnectivity and outerplanarity testing, TPDS, vol. 11, no. 2, pp , 0. [9] H. Roy, S. K. De, M. Maniruzzaman, and A. Rahman, Fault-tolerant power-aware topology control for ad-hoc wireless networks. IFIP Networking, [10] S. Z. M. Ahmed, M. Shariar and A. Rahman, Analysis of minimumenergy path-preserving graphs for ad-hoc wireless networks, in SPECTS, 8.