Edge-Based Beaconing Schedule in Duty-Cycled Multihop Wireless Networks

Transcription

1 Edge-Based Beaconing Schedule in Duty-Cycled Multihop Wireless Networks Quan Chen, Hong Gao, Yingshu Li, Siyao Cheng, Jianzhong Li Harbin Institute of Technology, Georgia State University Abstract Beaconing is a fundamental networking service where each node broadcasts a packet to all its neighbors locally. Unfortunately, the problem Minimum Latency Beaconing Schedule (MLBS) in duty-cycled scenarios is not well studied. Existing works always have rigid assumption that each node is only active once per working cycle. Aiming at making the work more practical and general, MLBS problem in duty-cycled network where each node is allowed to active multiple times in each working cycle (MLBSDCA for short) is investigated in this paper. Firstly, a modified first-fit coloring based algorithm is proposed for MLBSDCA under protocol interference model. After that, a (ρ + 1) 2 W -approximation algorithm is proposed to further reduce the beaconing latency, where ρ denotes the interference radius, and W is the maximum number of active time slots per working cycle. When ρ and W is equal to 1, the approximation ratio is only 4, which is better than the one (i.e., 10) in existing works. Furthermore, two approximation algorithms for MLBSDCA under physical interference model are also investigated. The theoretical analysis and experimental results demonstrate the efficiency of the proposed algorithms in term of latency. I. Introduction In multihop wireless networks, beaconing is an essential operation for many networking protocols in which each node broadcasts a packet to all of its one-hop neighbors [1]. For example, in the algorithms of discovering network topology, constructing routing protocols, triggering data collections, etc. each node needs to collect their neighbors information at first [2] [5]. Assume all the nodes are synchronous and the transmission time is divided into time slots allowed for one packet. Then the beaconing latency is defined as the number of time slots for all the nodes to broadcast the packet to their neighbors conflict-free. Obviously, the smaller the beaconing latency is, the better the network performance achieves. Thus, the problem of finding a conflict-free schedule to minimize the beaconing latency, named as Minimum Latency Beaconing Scheduling problem (MLBS), attracts many attentions of researchers. Many scheduling algorithms [10] [18] have been proposed for the networks that nodes always keep awake. According to [15], the MLBS problem is NP-hard even when each node has uniform beaconing radius and interference radius ρ = 1, so that the aforementioned algorithms [10] [18] are approximation ones, and they achieve high performance in term of efficiency. However, for many wireless networks, such as wireless sensor networks, the energy is quite limit. In order to save energy, the energy harvesting technology and the duty-cycled working mode is often adopted, in which each node has two states, i.e., the active state and sleep state, and all the functional modules are turned off in sleep state to save energy. Since working mode of duty-cycled network is completely different from the traditional network, the MLBS problem should be reconsidered and the following two challenges problems should be solved. 1) Problem 1. For a node which wants to broadcast a packet to all of its neighbors, it must take the active time slots of its neighbors under consideration. Sometimes, multiple transmissions are required in order to guarantee all of its neighbors to receive the packet. 2) Problem 2. For a node which wants to receive the packet from its neighbors, it must make use of all of its active time slots during a working cycle, and try to find a best schedule to minimize the beaconing latency. Currently, the MLBS problem in duty-cycled networks is rarely studied. To the best of our knowledge, such problem is only studied by Wang et al. [1] recently. Based on graph coloring techniques, Wang proposed two constant-approximation algorithms to solve such problem under protocol interference model, furthermore, Wang proposed another efficient algorithm for the physical interference model. All these algorithms are efficient and effective. However, they assume that all the nodes have only one active time slot per working cycle. So they only provided a solution for the aforementioned challenge problem 1. When each node has arbitrary active time slots in a working cycle [6] [9], the challenge problem 2 becomes very important for reducing the beaconing latency. In this case, the algorithms in [1] which are node-based scheduling algorithms may waste many active time slots for transmission and result in a large beaconing latency. In order to overcome the above shortcomings and provide a more general solution, this paper studies the MLBS problem in Duty-Cycled networks with Arbitrary active time slots allowed in a working cycle(mlbsdca). Since node-based scheduling algorithm is no long suitable in this scenario, an edge-based scheduling framework is designed and several approximation algorithms are proposed for the MLBSDCA problem under both protocol and physical interference model. In summary, the main contributions of this paper are given as follows. 1) In this paper, the MLBSDCA problem is defined under both protocol and physical interference model. And an edge-based scheduling framework is proposed to solve such

2 problem. 2) Under the protocol interference model, a modified firstfit coloring based algorithm is firstly given to solve the MLBSDCA problem. To further reduce the beaconing latency, a dynamic scheduling algorithm with approximation ratio of (ρ + 1) 2 W is proposed. When ρ and W is equal to 1, the approximation ratio is only 4, which is better than the one (i.e., 10) in [1]. In addition, we prove the lower bound of MLBSDCA problem when W = 1 in [1] is incorrect. 3) Two approximation algorithms are also proposed for the MLBSDCA problem under physical interference model. 4) The extensive simulations are carried out to evaluate the performance of our proposed algorithms. The proposed algorithms can significantly reduce the beaconing latency comparing with the ones in [1]. The rest of this work is organized as follows. Section II surveys the related works. Section III presents the problem definition. Section IV and V explain the proposed algorithms for MLBSDCA under protocol and physical interference model respectively. The simulation results are shown in Sections VI. Section VII concludes this paper. II. Related Works MLBS and its variants have attracted extensive attentions from researchers. In node always-awake networks, the prior works [10] [15] adopt a simplest setting that all the nodes have uniform beaconing radius and interference radius ρ = 1. After then, Sen et al. [16] proposed a first-fit coloring based algorithm for MLBS, of which the approximation bound was proved to be at most 7 by Wan et al. [19]. Besides the simplest setting, Wan et al. [17] assume all the nodes have uniform interference radius ρ 1, and they present a stripcoloring based algorithm with an approximation bound at most 5 when ρ = 1. For arbitrary ρ 1, the approximation ratio is proved between 3 and 6. A more general model that each node u has arbitrary interference radius ρ(u) 1 is exploited in [18]. Based on first-fit coloring technique, the authors in [18] proposed an approximation algorithm with approximation bound at most 61 for the MLBS problem under protocol interference model. An efficient algorithm for the MLBS problem under the physical interference model is also proposed in [18]. However, all these methods are not suitable for duty-cycled WSNs where node can only receive packet when it is in active state. In duty-cycled networks, few papers have addressed this issue. To the best of our knowledge, the MLBS problem in duty-cycled networks is only studied by Wang et al. [1] recently. Based on first-fit coloring, they first propose a 74-approximation algorithm for such problem under protocol interference model. Then they propose a strip-coloring based algorithm which can achieve approximation bound at most 10 when ρ = 1, and between 6 and 12 in general. However, they have a strong assumption that all the networking nodes have only one active time slot per working cycle. When there are arbitrary active time slots in a working cycle [6] [9], these algorithms may result in a very large beaconing latency. Besides the beaconing problem, the minimum latency broadcasting problem in duty-cycled networks under protocol interference model is studied by [21] [24] and several approximation algorithms have been proposed. In addition, the delay efficient flooding problem in duty-cycled networks is studied by Guo et al. [6]. However, these methods are not suitable for beaconing in duty-cycled networks. III. Problem Definition Assuming G = (V, E) denote a multihop duty-cycled network, where V = {1, 2,..., N} is the set of nodes and E = {e uv 1 v u N} denotes the neighborhood relationship among nodes. That is, node u and v can communicated with each other by one-hop messages if e uv E. Therefore, NB(u) = {v e uv E} is the set of all neighbors of u. According to the above discussion, each node has two states, i.e., the active state and sleep state, and switches between them periodically. Let T = ({0, 1, 2,..., T 1}) be a working cycle that contains T time slots. Then the working plan of node u (denoted by W(u)) can be defined as the active time slots in a working cycle, i.e., W(u) = {t 1, t 2,..., t k } {0, 1, 2,..., T 1}. For any node u, it can only receive data at time slots t (t W(u)) in every working cycle. If it wants to send a message, it can switch to active state at the time slot when the receiving node is awake. The duty cycle is defined as the ratio of the length of u s working plan over a whole working cycle, i.e., W(u) / T. For example, Fig.1 gives an example of a dutycycled network where the working plan of each node is given in the bracket. Since a node may have multiple active time slots in its working plan, the node-based scheduling algorithm in which each node is allowed only one active time slot per working cycle cannot be used. Thus, in this paper, the edge-based scheduling framework is proposed. Let e uv denote the transmission from node u to node v. To exploit the edge-based scheduling, we assume both e uv and e vu are in E in this paper and their corresponding transmissions are e uv and e vu respectively. Then the beaconing problem in duty-cycled networks is to obtain a conflict-free schedule for all the edges in E which considers the working plan of the receiving node. It is called Edge-based Schedule in this paper. To get an Edge-based Schedule of network G, the interference model is quite important. This paper first discusses the protocol interference model, and then the physical interference model. Under protocol interference model, the interference range of node u is a disk with radius ρ(u) centered at u, where ρ(u) 1. Therefore, the conflicted nodes by u (i.e., the nodes that cannot receive messages when u is transmitting), denoted by INF(u) satisfies that INF(u) = {v dist(v, u) ρ(u)}, where dist(v, u) denotes the distance between u and v. Based on these symbols, the definition of Edge-based Schedule under the protocol interference model is given as follows. Definition 1 (Edge-based Schedule): Given the network G = (V, E), the Edge-based Schedule of G, denoted by S, is a

3 Fig. 1. The example of a duty-cycled network. set of tuples with size E, i.e., S = {( e uv, t uv ) e uv E} and S satisfies the following two conditions: 1) For any transmission schedule ( e uv, t uv ), t uv = C T + t, where C 0 is any non-negative integer and t W(v). 2) For any ( e uv, t uv ) and ( e u v, t u v )(u u ), t uv = t u v if and only if v INF(u ) and v INF(u). Based on Definition 1, a qualified edge-based schedule first requires the receiving nodes of each transmission to be in active state (Condition 1) and then all transmissions at the same time slot are collision free (Condition 2). For a qualified edge-based schedule S, the beaconing latency [1], i.e., BL(S), is defined as the number of working cycles required by S. Thus, the MLBSDCA problem can be formalized as follows. Given a duty-cycled network G = (V, E) and the working plans for all nodes {W(v) v V}, the output is an Edge-based Schedule S min = { (( e uv, t uv ) e uv E} that satisfies Definition 1 and has the smallest beaconing latency. Theorem 1. The MLBSDCA problem under protocol interference model is NP-hard. Proof. According to [1], it is NP-hard to minimize the beaconing latency when each node in the network has only one active time slot per working cycle. Apparently, the above problem is a special case of the MLBSDCA problem, and thus, the MLBSDCA problem is also NP-hard. IV. Beacon Scheduling Algorithms under Protocol Interference Model To obtain Edge-based Schedule, a graph, named as Super Conflict Graph, must be constructed first to describe the conflict relationship among edges. Based on Super Conflict Graph, two algorithms are proposed then for solving the MLBSDCA problem under protocol interference model. A. Super Conflict Graph Construction Before introducing the concept of Super Conflict Graph, the induced subgraph of G at time slot i (0 i T 1) should be given firstly. The induced subgraph of G at time slot i, denoted by G i, is formed by the nodes that are active at time slot i and their neighbors. Specifically, G i = ( ) V i ( u V i NB(u)), E i, where V i = {u i W(u)} contains all active nodes at time slot i, NB(u) denotes the neighbor set of u, and E i = u V i {e vu v NB(u)} consists of all edges from u s neighbor to u( u V i ). It can be found that V i indicates the set of nodes that can receive packet at time slot i, and E i presents the available transmissions that can be scheduled at time slot i. However, we cannot evoke all edges in E i for transmitting due to the Fig. 2. The Super Conflict Graph (SCG) of the graph in Fig.1. confliction. Therefore, a conflict graph at time slot i (0 i T 1) should be constructed according to G i. Definition 2. (Conflict Graph) Let G i denote the induced subgraph of G at time slot i, then H i is called as the Conflict Graph of G i, which can be constructed as: 1) For e uv E i, a vertex labeled as e uv is created in H i, which is denoted by H i.e uv for distinguish. 2) For any two vertexes H i.e uv and H i.e u v (u u ), they are connected in H i if v INF(u ) or v INF(u), which means the transmission of e uv and e u v are conflicted. Then, the Super Conflict Graph can be defined as: Definition 3 (Supper Conflict Graph (SCG)) Given a duty-cycled network G = (V, E), the Super Conflict Graph (SCG) of G, denoted by H S, is actually a series of conflict graphs, that is H S = {H i 0 i T 1}, where H i is the conflict graph at time slot i for any 0 i T 1. The definition of Supper Conflict Graph also provides a way for constructing it. Based on such definition, the SCG of Fig.1 is presented in Fig.2, where H 0, H 1, H 2 and H 3 are the conflict graphs at each time slot respectively. Taken H 0 as an example, since only nodes n 0 and n 2 are awake at time slot 0, G 0 only contains two nodes and four edges, i.e., e 10, e 12, e 32, and e 42. So there exist four corresponding vertexes in H 0, and their conflict relationship can be constructed as in Fig.2. Notice that, since W(n 0 ) = {0, 1} 1, edge e 10 is existed in both induced subgraphs G 0 and G 1. Accordingly, there is a vertex labeled as e 10 in both H 0 and H 1. In addition, by exploiting broadcasting nature, e 10 and e 12 are not conflicted in H 0. The following theorem guarantees that the size of the SCG is bounded, which can be easily verified. Theorem 2. The number of vertexes and edges in the Super Conflict Graph H S are at most E W and E W (δ (G)+ (G)/2 1) respectively, where W = max u V ( W(u) ), (G) = max u V NB(u) and δ = max u V INF(u). According to the construction of H S, the MLBSDCA problem can be converted to the one of coloring all labels (not all vertexes) in H S if we use different colors to denote different working cycles and ( e uv, C T +i) S when H i.e uv is colored C. Notice that, some vertexes in different conflict graphs may share the same label in H S. For such vertexes, only one color is required, that is, the vertexes with label e 10 (in Fig.2) can be deleted from H 1 if the one with same label have been already assigned a color in H 0. Therefore, to minimize the number of

4 Algorithm 1 Group First-Fit Coloring Algorithm Input: The network graph G and its SCG H S ; Output: The Edge-based schedule for G; 1: for i = 0 to T 1 do 2: Ordering the nodes in H i with smallest-last ordering; 3: while there exist uncolored nodes do 4: for i = 0 to T 1 do 5: e uv the first uncolored node in H i ; 6: x the smallest color that has not been used by any preceding neighbor of H i.e uv in the conflict graph H i ; 7: S S ( e uv, x T + i); 8: Remove vertexes with label e uv from other conflict graph H j ( j i); 9: return The Edge-base schedule S for G; colors, finding a proper conflict graph to begin the coloring process for each vertex is quite challengeable, and we will propose two algorithms to solve it in the following sections. B. Group First-Fit Coloring based Beaconing Schedule Firstly, we propose a solution for the MLBSDCA problem based on a first-fit coloring algorithm. First-fit coloring is a classic greedy algorithm, which requires each vertex is colored by order and assigned the first color it fits. Let {1, 2,...k} denote the available colors, and < v 1, v 2,...v n > be a vertex ordering of V. The first-fit coloring mainly works as follows: For i = 1 up to n, it assigns v i the smallest number that has not been used by any preceding neighbors of v i. If B i denotes the set of neighbors of v i which precedes v i, then = 1 + max 1<i n B i is the upper bound of the number of used colors by first-fit coloring under the ordering < v 1, v 2,...v n >. The value is called the inductivity of the ordering < v 1, v 2,...v n >. According to [20], under the entire vertex ordering of V, the smallest-last ordering can achieve the smallest inductivity. The smallest-last ordering is produced as follows: 1)let v n denote the vertex with smallest degree in G, then delete v n from G; 2) For i = n 1 to 1, v i denotes the vertex with smallest degree in the left graph G and then delete v i from G. In this paper, we exploit the first-fit coloring under smallestlast ordering for the MLBSDCA problem. Since coloring the SCG H S is much different, a Group First Fit Coloring(GFFC) algorithm is proposed, in which we color all the conflict graphs from H 0 to H T 1 simultaneously. It mainly works as follows. Initially, each conflict graph H i is ordered by the smallest-last ordering. For i = 0 to T 1, we color the vertex with smallest order in H i by first-fit coloring and then remove all vertexes with same label from other conflict graphs H j ( j i). This process repeats until there exists no uncolored vertexes in H S. The detail is shown in Algorithm 1. Notice that, GFFC depends on a fixed ordering for each conflict graph H i. However, H S is a highly dynamic graph. Once a vertex is colored, the vertexes with same label in other conflict graphs are removed so that the degree and ordering for the left vertexes is changed. This results in the coloring method based on fixed ordering is not suitable for H S. C. Dynamic Largest First Beaconing Scheduling Algorithm In this subsection, we introduce a more efficient algorithm which assumes non-ordering for MLBSDCA under protocol interference model, known as the Dynamic Largest First Beaconing Scheduling(DLFBS) algorithm. Different from GFFC, DLFBS is scheduled by working cycles. Initially, a vertex merging process is conducted on SCG H S to exploit the broadcasting nature in duty-cycled networks. Let the new SCG be H S. For each working cycle C = 0 to K (K denotes the last working cycle when all vertexes in H S are scheduled), it executes the following two steps: 1)Searching and Scheduling the isolated vertexes in the SCG H S ; 2)Exploiting a dynamic largest degree first scheduling algorithm to schedule the nonisolated vertexes in working cycle C. 1) Vertex Merging in SCG: In duty-cycled networks, the broadcasting nature can be exploited when several neighbors share a common active time slot. This can greatly increase the throughput in a time slot and then reduce the beaconing latency. Therefore, for each conflict graph, we merge the vertexes with same senders into a Super Vertex in DLFBS. That is, in conflict graph H i, the set of vertexes containing sender u, i.e., the vertexes {H i.e uv v NB(u)&i W(v)}, are merged into a Super Vertex H i.e u. And the vertex H i.e uv is called the Child Vertex of H i.e u. Obviously, H i.e u will inherit the conflict relationship of all its child vertexes. That is, for any other vertexes in H i, if it conflicts with any child vertex of H i.e u, we create an edge between it and H i.e u. For super vertex H i.e u, it will be seen as a whole in scheduling. For example in Fig.2, there are two vertexes H 0.e 10 and H 0.e 12 in conflict graph H 0 share the same sender n 1. Thus, we can merge H 0.e 10, H 0.e 12 into a super vertex H 0.e 1. If H 0.e 1 is scheduled in Cth working cycle, the transmissions e 10, e 12 can be both transmitted at 0th time slot in Cth working cycle. 2) Searching and Scheduling Isolated Vertexes: In SCG H S, if a vertex in H i, i.e., H i.e x, is an isolated vertex, then it s conflicted with no vertexes in H i. And if H i.e x is scheduled at time slot C T + i, it won t incurs collisions to any other vertexes scheduled in Cth working cycle. Furthermore, once an isolated vertex is scheduled in some conflict graph H i (0 i T 1) in Cth working cycle, the vertexes with same label in the whole H S can be removed, which is different from nonisolated vertexes. As a result, some vertex, i.e., H j.e y ( j i), may become the isolated vertex again in conflict graph H j. This process is called the Isolated Vertexes Spreading process. In DLFBS, we exploit an iterative scheduling and removing algorithm to schedule the isolated vertexes as many as possible at the beginning of each working cycle. The detailed algorithm is shown in Algorithm 2. In addition, when we schedule an non-isolated vertex in H i and remove the vertexes with same label in H j ( j i), we can also obtain new isolated vertexes. For example, in Fig.2, if we schedule H 3.e 23 in Cth working cycle and remove the vertexes with label e 23 in H j ( j 3), H 2.e 43 would become a new isolated vertex in conflict graph H 2. For this case, we would handle it in next subsection. Notice that, after us merging the super vertexes in step 1, it becomes a little trivial to remove a vertex in SCG. If e x is a normal vertex, we not only need to remove the normal vertex with same label in other conflict graphs, but also the child vertex with same label from some super vertexes. If e x is a

5 Algorithm 2 Searching and Scheduling the Isolated Vertexes Input: The SCG H S and the scheduling working cycle C; Output: The Part of the Edge-based Schedule for G; 1: e x NULL; 2: while There exist isolated vertexes in H S do 3: for i = 0 to T 1 do 4: e x the isolated vertex in H i ; 5: S S ( e x, C T + i); 6: Remove e x from the whole SCG H S ; 7: return The Part of the Edge-based Schedule for G; super vertex, we need to conduct the same process as above for all its child vertexes. Lemma 1 guarantees the correctness of Algorithm 2. We omit the proof here for space limitation. Lemma 1. The generated transmission schedules by Algorithm 2 are conflict-free. 3) Dynamic Largest Degree First Scheduling: In the following, we introduce the method to handle the non-isolated vertexes in H S. According to the aforementioned discussion, label e uv may appear multiple times in SCG H S. To decide which vertex with label e uv should be scheduled, we exploit the dynamic largest first scheduling method, in which the vertex with maximum degree in the H S will be prior scheduled. In addition, different from the GFFC, we didn t schedule the conflict graphs from H 0 to H T 1 sequentially. The conflict graph with maximal (H i )(the maximum degree in H i ) will be first considered. In order to schedule the non-isolated vertexes conflict free, each vertex in H S has three states, i.e., the White State, Black State and Red State, to indicate whether a vertex can be scheduled in current working cycle. Initially, all the vertexes are marked White. If a vertex in H i is marked Black, which means it is a scheduled vertex, then all vertexes connected to it in H i cannot be scheduled in current working cycle. We mark these vertexes Red. The algorithm mainly works as follows. First, the White vertex with maximum degree over H S, i.e., H k.e max, is found and scheduled (S S ( emax, C T + k)). Second, the vertex H k.e max will be marked Black, and its neighbors in H k will be marked Red. After that, the vertexes with label e max are removed from other conflict graphs, i.e., H j (0 j T 1& j k). Third, since there may exist new isolated vertexes in H S, we call Algorithm 2 again to schedule such isolated vertexes. The above process repeats until there is no White vertexes in SCG H S. Finally, the left vertexes are either Black or Red in H S, which means either they have been scheduled or cannot be scheduled in current working cycle. At this point, all the Black vertexes are then deleted from H S. As a result, some vertexes connected to the Black vertexes may become isolated vertexes. Notice that these isolated vertexes cannot be scheduled in current working cycle(incurs collision to the Black vertexes), and will be scheduled at the beginning of the next working cycle. The detail is shown in Algorithm 3. Now, the DLFBS algorithm is completely introduced. As for the example in Fig.1, the edge-based schedule generated Algorithm 3 Dynamic Largest Degree First Scheduling Input: The SCG H S and the scheduling working cycle C; Output: The Part of the Edge-based Schedule for G; 1: Mark all the nodes in H S White; 2: e max 0, k 0; 3: while There existed White vertexes in H S do 4: for i = 0 to T 1 do 5: e i the White vertex with maximum degree in H i ; 6: if Degree(e i ) > Degree(e max ) then 7: e max e i, k i ; 8: S S ( emax, C T + k); 9: Mark the vertex e max in H k Black; 10: Mark all the neighboring vertexes of e max in H k Red; 11: Remove e max from other conflict graphs H j ( j k); 12: if There exist isolated vertexes then 13: Call Algorithm 2 to schedule the isolated vertexes; 14: Remove all the Black vertexes from H S ; 15: return The Part of the Edge-based Schedule for G; by DLFBS is given in Fig.3. Theorem 3 guarantees the correctness of DLFBS. Fig. 3. The results of the beaconing schedule for Fig.1. Theorem 3. DLFBS generates a conflict-free edge-based schedule for MLBSDCA under the protocol interference model. Proof. First, according to H S, for e uv E, there exists at least one vertex with label e uv in H S. Obviously, for e uv E, the transmission e uv is scheduled by DLFBS. Otherwise, the vertexes with label e uv won t be deleted from H S and the DLFBS algorithm cannot stop. Second, by Lemma 1, all the isolated vertexes in each working cycle are scheduled conflict-free in Algorithm 2. In addition, these isolated vertexes won t be conflicted by the vertexes scheduled later. Thus, all the isolated vertexes scheduled by Algorithm 2 are conflict-free. Third, we will prove the vertexes scheduled in Algorithm 3 are conflict-free. We prove it by contradiction. Assume there exists a confliction, then there are at least two vertexes, i.e., H i.e x and H i.e y, connected and scheduled in the same conflict graph H i (0 i T 1) at working cycle C. W.l.o.g., let H i.e x be scheduled first. According to Algorithm 3, H i.e x will be marked Black. Since H i.e x and H i.e y are connected in H i, then H i.e y will be marked Red and won t be scheduled in working cycle C, which is a contradiction. Therefore, the edge-based schedule generated by DLFBS is conflict-free. D. Performance Analysis of DLFBS algorithm Theorem 4 gives the lower bound for MLBSDCA under protocol interference model. Theorem 4. The lower bound on the latency of any optimal beaconing schedule for MLBSDCA under the protocol interference model is at least (G)+1 W T, where (G) denotes the

6 Fig. 4. The possible conflicted transmissions with edge e uv. maximum degree in G, and W denotes the maximum number of active time slots in a working cycle. Proof. Assume the node with the maximum degree in G is node u. To receive the beacon messages from all of its neighbors, it has to be scheduled at least (G) times. Since node u can wake up at most W time slots per working cycle, then the required number of working cycles is at least (G) W. In addition, node u must broadcast its packet to its neighbors, this takes at least one transmission. Therefore, the beaconing latency is at least (G)+1 W T. To obtain the approximation ratio of DLFBS, we need to analyze the maximum degree in H S. For simplicity, we assume the interference radius is equal to ρ for each node. When each node has arbitrary interference radius ρ(u), the following lemma can still hold by setting ρ = max{ρ(u), u V}. Lemma 2. The maximum degree in the super conflict graph H S, i.e., (H S ), is at most (ρ + 1)2 ( (G) + 1) 1, where ρ is the interference radius and the beaconing radius r = 1. Proof. To derive the upper bound of the maximum degree in H S, we consider a vertex in H S by the following two cases: 1) It is a normal vertex, i.e., H i.e uv. Consider the example in Fig.4, when edge e uv is transmitting, the nodes in u s interference range cannot receive any messages, i.e., the nodes lies in the disk centered at u of radius ρ. Obviously, to transmit messages to these nodes, the senders must be in the disk centered at the u of radius ρ + 1. Since we merge the vertexes with same sender to a super vertex in H i, the upper bound of the number of conflicted vertexes by H i.e uv is equal to the number of the senders lie in the disk centered at the u of radius ρ + 1 except u. Let φ denote the average number of nodes per unit area, then the number of conflicted vertexes is 2π (ρ + 1) 2 φ 1. Also, we can have 2π r 2 φ (G) + 1 (the largest number of degree in G). Therefore, the number of conflicted vertexes is at most (ρ + 1) 2 ( (G) + 1) 1. 2) It is a super vertex, which means it has multiple receivers. Since the range of the disk centered at receiver v of radius ρ is in the disk centered at node u of radius ρ + 1 (as in Fig.4), then the conflicted senders by the receiver v is still in the disk centered at node u of radius ρ + 1. As a result, the number of conflicted edges is also (ρ + 1) 2 ( (G) + 1) 1. Therefore, the maximum number of conflicted vertexes by a vertex in H i is at most (ρ+1) 2 ( (G)+1) 1, which means the maximum degree in H S is (ρ + 1)2 ( (G) + 1) 1. Fig. 5. The example of the analysis for Wang s lower Bound. Theorem 5. The approximation ratio of the DLFBS algorithm is at most (ρ + 1) 2 W. Proof. Let H i.e x be the last vertex scheduled by the DLFBS algorithm. It is easy to verify that H i.e x is scheduled at most in (H i ) + 1 working cycles when all of its neighbors are both scheduled in H i. According to Lemma 2, we have (H i ) (ρ+1) 2 ( (G)+1) 1. Thus, the required number of working cycles is at most (ρ+1) 2 ( (G)+1). In addition, according to Theorem 4, the optimal beaconing latency is at least (G)+1 W T. Therefore, the approximation ratio of DLFBS algorithm is at most (ρ + 1) 2 W. According to Theorem 5, when W = 1, the approximation ratio is equal to (ρ + 1) 2. And when ρ = 1, the approximation ratio is only 4, which is better than the one in [1] (the best approximation ratio is 10 when ρ = 1). E. Discussion of Wang s Lower Bound Analysis In [1], Wang et al. derive a lower bound of beaconing latency of any optimal schedule for MLBCDA when W = 1, which is (ω(h) 1) T (see Lemma 1 in [1]), where H is the conflicted graph over all nodes in original graph G and ω(h) is the clique number of H. However, we find their derived bound is incorrect. In their analysis, the conflicted graph H is constructed as follows: for any pair of nodes, i.e., u and v, there exists an edge between u and v if only if they cannot transmit simultaneously (if they share a same neighbor or they are in each other s beaconing range). Consider a special graph in Fig.5(a), where each node wakes up at different active time slots. Its conflicted graph H is constructed and shown in Fig.5(b). As we can see, the conflict graph H is a clique itself, and then we have w(h) = 5. According to [1], the beaconing latency is at least 4 T. However, we can see that node a can receive packets from its neighbors b, c and d at the 1th time slot per working cycle conflict-free. This would consume 3 working cycles. Similarly, node b, c, d and e can receive their neighbors packets within 2, 3, 2 and 2 working cycles respectively. The beaconing latency is 3 T, which is lower than their derived lower bound 4 T. The reason is that they construct the conflicted graph based on nodes, not on edges, which may result in inaccuracy. For example, although node b and e are conflicted (they share the same neighbors c), but e ba and e ed can be transmitted simultaneously.

7 V. Beacon Scheduling Under Physical Interference Model Under Physical Interference Model, the collision happens when the value of the signal to interference plus noise ratio (SINR) is less than a certain threshold β. Assumes all the nodes transmit at a fixed power P. To receive the messages from node u successfully, the SINR value at the receiver v is required to satisfy S INR(u) = ηpd(u,v) α ηpd(w,v) α ξ+ w u β, where α is the path-loss exponent which usually belongs to [2, 4], η is a positive reference loss parameter of power, ξ is the background noise, and node w denotes any other transmitting nodes at the same time slot. Then the MLBSDCA problem under physical interference model is then to construct an edge-based schedule with minimum latency under condition that the SINR value of all receivers is no less than β. Assume the beaconing radius is one, and λ = ( ) ηp 1/α βξ denote the ratio of the beaconing radius and the maximum transmission radius, and ζ(x) denotes the Riemann zeta function in the form ζ(x) = 1 j=1 j. Let x ρ = 1 + ( ) β(16ζ(α 1)+8ζ(α) 6) 1/α. 1 λ Then the following lemma can α provide a sufficient condition for all the nodes to receive the message successfully under physical interference model. Lemma 3. [18] Assume E = { e uv e uv E} is a set of transmissions. If for any e uv and emn in E (u m), d(u, m) > ρ, then all transmissions in E can transmit successfully under the physical interference model. Notice that the interference range ρ above is a relative loose condition. If we use ρ to construct the SCG Ĥ S according to Lemma 3, and then merge the super vertexes in Ĥ S and the new SCG is called Ĥ S, and finally schedule the vertexes in Ĥ S as in GFFC and DLFBS, we can still obtain a solution for MLBSDCA. These two methods are named GFFC-ph and DLFBS-ph respectively. However, these methods may produce a much higher beaconing latency. Fortunately, we found that the DLFBS algorithm can be well suit for the physical interference model by making a little modification. To distinguish, the new algorithm is just called DLF. According to DLFBS, two vertexes cannot be scheduled simultaneously if there exists an edge between them in Ĥ S. However, under physical interference model, two connected vertexes in Ĥ S can be still scheduled if the SINR value is lower than β. Since ρ is a much higher bound, there would exist many such vertexes. Based on this, we design the DLF algorithm by make a little modification in Algorithm 3. Similar as in Algorithm 3, the White vertex with maximum degree, i.e., H k.e max, is first considered. Different from Algorithm 3, H k.e max cannot just be scheduled here. This is because it may cause collisions to the existed scheduled vertexes in H k (including the isolated and Black vertexes scheduled at current working cycle) under physical interference model. If it is, we mark H k.e max Red. Otherwise, we just do as in DLFBS, i.e., schedule H k.e max at Cth working cycle, mark it Black and remove the vertex with same label from H j ( j k). When H k.e max is scheduled, the isolated vertexes scheduling process is also invoked when there exists isolated vertexes. The following two theorems prove the correctness and the approximation ratio of the DLFBS-ph and DLF algorithm, and the proof are omitted due to space limitation. Theorem 6. The edge-based schedule generated by the DLFBS-ph and DLF algorithm is conflict-free. Theorem 7. The approximation ratio of the DLFBS-ph and DLF algorithm is at most ( ρ) 2 W. VI. Simulation Results In this section, the performance of proposed algorithms is evaluated through extensive simulations, which are mainly divided into two scenarios, i.e., the simulation under both protocol interference model and physical interference model. Under protocol interference model, there are two existing algorithms proposed for MLBS problem in duty-cycled networks, i.e., FFBSD [1] and SCBSD [1], which assume single active time slot per working cycle. In the experiments, these two algorithms are both evaluated to demonstrate that our proposed algorithms, especially the DLFBS algorithm, can benefit a lot from scheduling over all active time slots. Under physical interference model, Wang et al. [1] proposed an efficient algorithm for the MLBS problem in duty-cycled networks, i.e., FFBSD-ph. It is implemented and compared in the simulations. In the simulations, we mainly focus on the performance of beaconing latency of the proposed algorithms under various network topologies. Firstly, we randomly deployed 100 nodes in a 200m 200m field and test their performance under different transmission range and interference range under the protocol interference model. Secondly, as in [26], to test more network topologies, Networkx [25] was used to generate different network topologies with number of nodes from 100 to 600. Thirdly, under the physical interference model, the parameters are set as, β = 1, ξ = 0.1, α = 2.5, η = 1 and P = 300 respectively. In all simulations, the duty cycle is set from 10% to 35% and the working plan of each node is generated randomly to test a wide range of configurations. In the simulation results, each plotted point represents the average of 100 executions. A. Beaconing Latency in Random Deployed Network First, the beaconing latency of the proposed algorithms under protocol interference model when nodes are randomly deployed is evaluated. In this simulation, different T, Duty Cycle and interference range ρ are tested. Figure 6 shows the performance of beaconing latency when the interference range is equal to the beaconing radius r. The first observation from Fig.6(a) and Fig.6(b) is that the beaconing latency of DLFBS is much less than the one of other algorithms in all scenarios, in which the beaconing latency is decreased by 200% on average. The effectiveness of DLFBS is mainly attributed to it can make full use of the whole active time slots of each node and maximize the throughput per working cycle (which are shown in VI.C), i.e., by iterative

8 (a) T =20 (b) Duty Cycle=20% Fig. 6. The Beaconing Latency When ρ is Equal to r. (a) T =20 (b) Duty Cycle=20% Fig. 7. The Beaconing Latency When ρ is Equal to 1.5r. scheduling the isolated vertexes and merging super vertex. One can also observe that, although considering multiple active time slots in coloring, GFFC reduces the beaconing latency by only 13% comparing with FFBSD. This is because that GFFC is also a coloring based method with a fixed ordering as in FFBSD, which cannot maximize the throughput per working cycle to further reduce the beaconing latency. Another finding is that, SCBSD performs much worse than other methods. This is because, in SCBSD all the nodes are partitioned based on strips and two strips apart from a certain distance can be scheduled simultaneously. However, the distance is set much rigorous. Many transmissions which are not that far away can still be scheduled simultaneously. Figure 7 displays the beaconing latency when the interference radius is equal to 1.5r. Similar as in Fig.6, GFFC and DLFBS both perform better than the existing methods, i.e., SCBSD and FFBSD, in term of beaconing latency. This demonstrates that two methods can both benefit from scheduling over the multiple active time slots, especially the DLFBS algorithm. Compared to the scenario when ρ = r in Fig.6, the beaconing latency of all methods are increased about 40% due to the large interference range. B. Beaconing Latency in Networkx Generated Graph Second, we exploit Networkx to investigate the beaconing latency under different network topologies, i.e., network size (i.e., N) and average number of neighbors(i.e., ψ). The interference range is set to ρ = r in this scenario. Since SCBSD assumes each node knows its location information, then it cannot be used and is omitted in this group of experiments. Figure 8 presents the beaconing latency of three algorithms under different N and ψ. Firstly, we can see that if we set ψ = 10, the beaconing latency of all methods almost stay unchanged when the number of nodes N increase from 100 to 600. On the other hand, the beaconing latency grows fast when we fixed N and increase ψ. This demonstrates the beaconing latency is highly related to the average degree in the network. Under all these network topologies, the beaconing latency of DLFBS is far lower than the one of FFBSD and GFFC, which demonstrates the efficiency of DLFBS in beaconing. Figure 9 compares the beaconing latency when we set N = 200 and ψ = 10, and vary the Duty Cycle and the length of working cycle T. In this experiment, we see the same trend of each method as in Fig.6 and Fig.7, where the beaconing latency of DLFBS increases a little when we increase T and fix Duty Cycle, while the latency of other methods grow fast. This is because when T is increased, the active time slots per working cycle ( T Duty Cycle) is also increased. And DLFBS can make better use of the increased number of active time slots to reduce the beaconing latency. C. Throughput Analysis In this group of experiments, we analyze the average throughput per working cycle of four algorithms, in which we verify the effectiveness of DLFBS from another perspective. Figure 10 presents the average throughput per working cycle of each method when nodes are randomly deployed in the field and ρ is set to r and 1.5r respectively. In this experiment, we set T = 20 and vary the duty cycle from 10% to 35%. Comparing with SCBSD and FFBSD, the throughput of DLFBS is increased by 4.5 and 2.7 times respectively when duty cycle is 35% in both two scenarios. This demonstrates the great efficiency of DLFBS in beaconing from another perspective. Figure 11 shows the average throughput per working cycle of each method under the Networkx scenario, where we set ψ = 10, and N = 100 and 200 respectively. Similarly, the throughput of DLFBS is still far higher than the one of other methods in this scenario, and DLFBS can performs even better when duty cycle grows. In addition, comparing Fig.11(b) with Fig.11(a), the throughput of each method grows at least 100% when the number of nodes in the network is more. This is because more transmissions need to be scheduled in this case. D. Performance Under Physical Interference Model Finally, we evaluate the performance of proposed algorithms under physical interference model, where N = 100 and the parameters are set as introduced before. The result is shown in Fig.12. Comparing with FFBSD-ph, the beaconing latency of DLF is decreased by 8.7 times on average in Fig.12(a). This is mainly due to DLF cannot only exploit the multiple active time slots of each node to reduce beaconing latency, but also make full use of the SINR value at each receiving node when scheduling under physical interference model. Comparing with GFFC-ph and DLFBS-ph, DLF reduce the beaconing latency by 4.3 times and 2.4 times on average respectively. Additionally, when we fix Duty Cycle and increase T, the beaconing latency of FFBSD-ph also grows fast in Fig.12(b) under physical interference model. This is because it exploits only one active time slot per working cycle for each node. VII. Conclusion In this work, several efficient algorithms are developed for MLBSDCA. First, a modified first-fit coloring based algorithm

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