MULTI-HOP wireless networks consist of nodes with a

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1 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS 1 Minimum Latency Broadcast Scheduling in Duty-Cycled Multi-Hop Wireless Networks Xianlong Jiao, Student Member, IEEE, Wei Lou, Member, IEEE, Junchao Ma, Student Member, IEEE, Jiannong Cao, Senior Member, IEEE, Xiaodong Wang, Member, IEEE, and Xingming Zhou Abstract Broadcast is an essential and widely-used operation in multi-hop wireless networks. Minimum latency broadcast scheduling (MLBS) aims to find a collision-free scheduling for broadcast with the minimum latency. Previous work on MLBS mostly assumes that nodes are always active, and thus is not suitable for duty-cycled scenarios. In this paper, we investigate the MLBS problem in dutycycled multi-hop wireless networks (MLBSDC problem). We prove both the one-to-all and the all-to-all MLBSDC problems to be NPhard. We propose a novel approximation algorithm called for the one-to-all MLBSDC problem, and two approximation algorithms called and for the all-to-all MLBSDC problem under the unit-size and the unbounded-size message models respectively. The approximation ratios of the, and algorithms are at most 17 T, 17 T + 2 and ( + 22) T respectively, where T denotes the number of time slots in a scheduling period, and denotes the maximum node degree of the network. The overhead of our algorithms is at most constant times as large as the minimum overhead in terms of the total number of transmissions. We also devise a method called Prune to further reduce the overhead of our algorithms. Extensive simulations are conducted to evaluate the performance of our algorithms. Index Terms Broadcast scheduling, duty cycle, multi-hop wireless networks, approximation algorithms. 1 INTRODUCTION MULTI-HOP wireless networks consist of nodes with a limited transmission range. Broadcast is one of the most essential operations in multi-hop wireless networks, and is widely used for routing discovery, data collection and code update, etc. The two most commonly used broadcast tasks are the one-to-all broadcast and the all-to-all broadcast (also called gossiping in [1]). The one-to-all broadcast aims to disseminate a message from one source node to all the other nodes, while the all-to-all broadcast aims to distribute the messages from all the nodes to all the other nodes. In multi-hop wireless networks, although the packets transmitted by a node can be received by all the nodes within its communication range, two parallel transmissions to one common node can cause signal collision, and the common node will receive neither of the two messages. Minimum latency broadcast scheduling (MLBS) aims to minimize the broadcast latency while ensuring the transmissions are collision-free. The MLBS problem in multi-hop wireless networks has been Xianlong Jiao is with the School of Computer, National University of Defense and Technology, Changsha, China, and the Department of Computing, The Hong Kong Polytechnic University, Kowloon, Hong Kong. xljiao@nudt.edu.cn Wei Lou, Junchao Ma and Jiannong Cao are with the Department of Computing, The Hong Kong Polytechnic University, Kowloon, Hong Kong. {csweilou, csjma, csjcao}@comp.polyu.edu.hk Xiaodong Wang and Xingming Zhou are with the School of Computer, National University of Defense and Technology, Changsha, China, {xdwang, xmzhou}@nudt.edu.cn This work is supported in part by Hong Kong GRF grants (PolyU 5236/6E, PolyU 512/8E), PolyU ICRG grant 1-ZV5N, National Basic Research Program of China grant 26CB33, Natural Science Foundation of China grants (61723, , 61334), Hunan Provincial Natural Science Foundation of China grant 9ZZ434. proved to be NP-hard in [2], and several approximation algorithms [2] [9], which follow the assumption that all the nodes are always active, have been proposed for this problem. However, it is well known that nodes in multi-hop wireless networks often switch between the active state and the sleep state to save the energy powered by batteries [1] [12]. Unlike in conventional scenarios without sleep cycles, a node in duty-cycled scenarios with active/sleep cycles may require transmitting several times to inform all its neighboring nodes with different active time. Therefore, most of the previously proposed broadcast scheduling algorithms are not suitable for duty-cycled scenarios. In this paper, we investigate the MLBS problem in dutycycled multi-hop wireless networks (MLBSDC problem). We prove that both the one-to-all and the all-to-all MLBSDC problems are NP-hard. We first present one novel approximation algorithm for the one-to-all MLBSDC problem. This algorithm provides a broadcast scheduling with a constant approximation ratio of 17 T, which greatly improves the previously known guarantee of 24 T + 1 in [13], where T denotes the number of time slots in a scheduling period. We then propose two algorithms called and for the allto-all MLBSDC problem under the unit-size message model and the unbounded-size message model respectively. Under the unit-size message model, the messages can be sent only one by one without combination. Under the unbounded-size message model, the messages can be combined as one message. To the best of our knowledge, this is the first work to investigate the all-to-all MLBSDC problem. The and algorithms achieve the approximation ratios of 17 T + 2 and ( + 22) T respectively, where denotes the maximum node degree of the network. We prove that the total numbers of transmissions scheduled

2 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS 2 by the, and algorithms are at most 15, 17 and 4 times as large as the minimum total numbers of transmissions respectively. Moreover, we devise a method called Prune to further reduce the overhead of our algorithms in terms of the total number of transmissions. We prove that the total numbers of transmissions scheduled by our algorithms together with the Prune method, called -P, -P and -P, are at most 1, 12 and 3 times as large as the minimum total numbers of transmissions respectively. Finally, we evaluate the performance of our algorithms by conducting extensive simulations under different network configurations. The remainder of this paper is organized as follows. Section 2 details the related work. Section 3 introduces the network model, the problem formulation and some graph-theoretic definitions. Section 4 details the algorithm, and gives the basic processes of the and algorithms as well as the Prune method, which are detailed in the supplemental material. Section 5 and Section 6 analyze and evaluate the performance of our algorithms respectively. Section 7 concludes this paper and discusses our future work. 2 RELATED WORK Lots of studies have been done on the broadcast problem [14] [16]. The simplest implementation of broadcast is flooding, which may cause a large amount of contention and collision [17]. Therefore, lots of efforts [17], [18] have been made to improve the efficiency of the broadcast. The multi-hop wireless network is often modeled as a unit disk graph (UDG) [19] when the nodes have the same transmission radius. The MLBS problem aims to find a collision-free scheduling for broadcast with the minimum latency, which has been proven to be NPhard in both general graphs [14] and unit disk graphs [2]. There are many algorithms presented for the one-to-all MLBS problem [2] [7]. Gandhi et al. [2] present an approximation algorithm with a constant ratio of more than 4. This ratio is improved to 16 by Huang et al. [3], and is recently improved to 12 by Gandhi et al. [4]. For multi-hop wireless networks where the interference radius is α times as large as the transmission radius, Chen et al. [5] give an efficient algorithm with a ratio of O(α 2 ). The ratio turns to be 26 if α equals to 2. Mahjourian et al. [6] further consider the carrier sensing range when designing efficient algorithms. Huang et al. [7] give an efficient solution for the one-to-all MLBS problem under physical interference model. Much work has also been done for the all-to-all MLBS problem [1], [2], [4], [8], [9], [2], [21]. Recently, Gandhi et al. [2] investigate this problem under the unit-size message model, and propose an approximation algorithm of a constant ratio more than 1. The ratio is improved to 27 by Huang et al. [8], [9], and is recently improved to 2 by Gandhi et al. [4]. The all-to-all MLBS problem under the unboundedsize message model for radio networks has been studied in [1], [2], [21]. Gasieniec et al. [1] give two gossiping algorithms which work in O(n d log 2 n) and O(d 3/2 log 3 n) time respectively, where d is the network graph-theoretic diameter and n is the network size. An O(n 3/2 log 2 n)-time gossiping algorithm is developed in [2]. Gasieniec et al. [21] propose an O(d)-time gossiping algorithm for known radio networks with a certain maximum node degree. Nevertheless, none of the work mentioned above has taken the active/sleep cycles into consideration. The broadcast problem in duty-cycled scenarios has been studied in [22] [24]. To the best of our knowledge, the only work so far to study the MLBSDC problem is done by Hong et al. [13]. Their algorithm called is proposed for the one-to-all MLBSDC problem. first constructs a shortest path tree, which is defined with the latency function on the network and contains the information of network topology and each node s sleeping schedule. then constructs the maximal independent set (MIS) at each layer, and uses a layered D2- coloring approach to color the nodes in the MIS. Finally, schedules the transmissions based on the colors of the nodes in the MIS to avoid the collision. However, simply uses two times of the number of colors at one layer to divide the transmissions between two layers, so it incurs high broadcast latency and a large approximation ratio of 24 T + 1. In this paper, we further investigate the one-to-all and all-to-all MLBSDC problems and give efficient solutions for these problems. Unlike, our proposed algorithm constructs several MIS es according to the active time slots of the nodes, determines every node s parent based on the MIS es, applies two proper D2- coloring methods by different orderings to color the parent nodes and schedules the broadcast based on the colors. These D2-coloring methods utilize the special geometric properties of the MIS es and require smaller numbers of colors than that used in, and thus achieves lower broadcast latency and a smaller approximation ratio than. 3 PRELIMINARIES 3.1 Network Model We model the multi-hop wireless network as a UDG G = (V, E), where V contains all the nodes in the network, and E is the set of edges, which exist between any two nodes if their Euclidean distance is no larger than the transmission radius. A node cannot send or receive the message at the same time. We denote by n the number of nodes in the network. Like [11] and [12], we assume that nodes determine the active/sleep time without coordination in advance, and thus do not require additional communication overhead. The duty cycle is defined as the ratio of the active time to the whole scheduling time. The whole scheduling time is divided into multiple scheduling periods of the same length. One scheduling period T is further divided into unchanged T unit time slots, i.e., T = {, 1,..., T 1}. Every node v chooses one active time slot A(v) in T randomly and independently, and wakes up at this time slot in every scheduling period to receive the message. If node v wants to send a message as required, it can wake up at any time slot to transmit the message as long as the receiver node is awake and there is no collision for this transmission. 3.2 Problem Formulation This paper studies the one-to-all and the all-to-all broadcast problems in duty-cycled scenarios. In the one-to-all broadcast

3 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS 3 problem, one distinguished node disseminates its message to all the other nodes. The one-to-all broadcast completes when every node receives the message. In the all-to-all broadcast problem, every node has a message to send to all the other nodes. The broadcast task completes when every node receives the messages from all the other nodes. We model the broadcast scheduling as assigning the transmitting time slots for all the nodes, i.e., assigning a function TTS : V 2 N, where N denotes the natural number set. The objective of broadcast scheduling is to minimize the largest transmitting time slot. Furthermore, the broadcast scheduling in duty-cycled scenarios requires taking the following two constraints into account. First, a transmitter node can transmit messages to a receiver node only when the receiver node is awake. Second, the transmissions should be carefully scheduled to avoid the collision. We have the following lemma about the hardness of the MLBSDC problem. The proof of this lemma is given in Appendix B of the supplemental material. Lemma 1. Both the one-to-all and the all-to-all MLBSDC problems are NP-hard. 3.3 Graph-Theoretic Definitions We denote by G[U] the subgraph of G induced by a subset U of V. We add edges between two nodes in G[U] if their hop distance is 2, and denote by G 2 [U] the resulting graph. If there is no edge between any two nodes in G[U], we call the subset U an Independent Set (IS) of G. An MIS of G is not a subset of any other IS of G. It is known that a node can be adjacent to at most five nodes in an IS of a UDG [25]. U 1 is a cover of U 2, if, for any node in U 2, there exists at least one node in U 1 which is adjacent to this node. A minimal cover of U is a cover of U in which the removal of any node destroys the covering property. A proper coloring of G is to assign natural numbers representing the colors to all the nodes, while ensuring adjacent nodes achieve varied numbers. A proper D2-coloring of G is to color all the nodes so that the nodes within two-hop neighborhood achieve different colors. The minimum node degree and the maximum node degree of G are denoted by δ(g) and (G) respectively. We denote by δ (G) = max U V δ(g[u]) the inductivity of G. According to [26], at most 1 + δ (G) colors suffice to color the nodes in G by a proper smallest-degreelast ordering. For any IS U of a UDG G, the inductivity of G 2 [U] is no larger than eleven [3]. 4 MINIMUM LATENCY BROADCAST SCHEDUL- ING 4.1 One-To-All Broadcast Algorithm In this subsection we detail the approximation algorithm for the one-to-all MLBSDC problem. The pseudocode of algorithm is given in Appendix A of the supplemental material. To achieve the lower bound of this problem and to efficiently solve this problem, algorithm first constructs a shortest path tree T S PT rooted at source node s as follows. We assume that source node s starts the broadcast operation at time slot. For every edge (u, v) E, the latency Lat(u, v) of this edge is determined as follows: A(v) + 1, if u = s; Lat(u, v) = A(v) A(u), if u s and A(v) A(u) > ; (1) A(v) A(u) + T, else Specifically, for each edge (u, v), the latency of this edge contains the waiting latency and the transmitting latency of the transmission from node u to node v. The shortest path tree T S PT can be achieved by applying Dijkstra s algorithm [27] with the latency. Note that, the latency of the shortest paths from node s to the other nodes in this tree is the minimum latency the other nodes can receive the message from node s. So the trivial lower bound of the one-to-all MLBSDC problem is the maximum latency of these shortest paths, denoted by D. Unfortunately, to avoid the collisions among the transmissions, we cannot directly schedule the broadcast based on T S PT. Instead, we construct a broadcast tree T B based on T S PT, and then schedule the broadcast according to the broadcast tree. We denote by P(v) the parent node of node v in T B. After constructing T S PT, we divide all the nodes into different layers L, L 1,..., L D according to the latency of the shortest paths from node s to all the nodes in T S PT. This layering operation is used to construct the broadcast tree and to schedule the broadcast. According to the definition of Eq. (1), it is easy to find that nodes at the same layer L i have the same active time slot j, where j equals to (i 1) mod T. To construct T B, we require to determine every node s parent node. Since the MIS of a UDG has some special geometric properties as discussed in Section 3.3, we determine every node s parent node based on the MIS. To efficiently use the special geometric properties of the MIS, we construct several MIS es as follows. All the nodes except node s are partitioned into different subsets U, U 1, U 2,..., U T 1 according to their active time slots. For each subset U j ( j T 1), an MIS Q j of G[U j ] is constructed layer by layer. Therefore, Q j includes several IS es at different layers. Then all the nodes at each layer L i ( i T 1) are divided into two subsets: an IS M i and a subset L i \M i, where M i belongs to Q j and j equals to (i 1) mod T. Each subset U j is divided into two sets Q j and U j \Q j. The broadcast tree is constructed layer by layer. At each layer L i, the parent nodes of nodes in M i are chosen from some nodes at higher layers, and some nodes in M i are chosen as the parent nodes of nodes in L i \M i and nodes at lower layers with the same active time slot. The choosing process is based on the rule that if a node covers most nodes that do not have a parent yet, this node is first chosen as the parent node. To facilitate coloring the parent nodes, we collect the parent nodes of nodes in Q j and U j \Q j into two sets W 1 j and W 2 j respectively based on the chosen order of these parent nodes. Note that, a parent node in W 1 j has at most five children nodes in Q j and W 2 j is an IS of G. To use as few colors as possible, we take advantage of these parent nodes different geometric properties concerning the IS and color them using two proper D2-coloring methods f 1 j and f 2 j by two different orderings: the front-to-back ordering (i.e., from the first node to the last node in W 1 j ) and the smallest-degree-last ordering. We denote

4 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS 4 v 9 v 6 v 7 v1 v 11 v 1 v 2 s v 5 v 3 (a) The topology of G v 9 v 6 v 1 v 4 v 8 v 2 s v1 v 7 v 11 2 v 9 3 v 2 v 5 3 v v v 1 3 v 3 v 4 s v 7 v 1 v 11 v 5 (b) Shortest path tree T S PT v 3 (c) Broadcast tree T B Fig. 1. An example to illustrate the algorithm by m 1 j and m 2 j the numbers of colors required by f 1 j and f 2 j respectively. Finally we schedule the transmissions from the parent nodes to their children nodes based on the colors of the parent nodes. This scheduling starts at time slot, and works layer by layer. At each layer L i, the transmission from the parent node P(u) to each node u in M i is scheduled at time slot t + f 1 j (P(u)) T + j, where t is the current time slot, f 1 j (P(u)) is the color of node P(u), and j equals to (i 1) mod T. The current time slot t increases by m 1 j T so that all these transmissions can finish. The transmission from the parent node P(w) to each node w in L i \M i is scheduled at time slot t + ( f 2 j (P(w)) + m 1 j ) T + j. The current time slot t increases by m 2 j T so that these transmissions can finish. Then the scheduling handles the next layer in a similar manner until all the layers are handled. Example 1. We take Fig. 1 as an example to illustrate the algorithm. The network consists of twelve nodes as shown in Fig. 1(a). Node s is the source node. The scheduling period T is {,1,2,3}. Table 1 lists the active time slots of all the nodes. According to algorithm, we first define the latency of each edge and construct the shortest path tree T S PT rooted at node s as shown in Fig. 1(b). In Fig. 1(b), the number labeled on each edge represents the latency of that edge. All the nodes are then divided into different layers L, L 1,..., L 1 according to the latency of the shortest paths from node s to these nodes in T S PT. Table 1 lists the layers of all the nodes. TABLE 1 Active time slots, layers and transmitting time slots of all the nodes ID s v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 1 v 11 A Layer L L 1 L 1 L 3 L 5 L 5 L 2 L 8 L 9 L 5 L 8 L 1 TTS v 4 v Next we construct the MIS es. All the nodes except node s are first partitioned into four subsets, where U = {v 1, v 2, v 4, v 5, v 8, v 9 }, U 1 = {v 6, v 11 }, U 2 = {v 3 }, and U 3 = {v 7, v 1 }. Then we can achieve all the MIS es constructed layer by layer, e.g., Q = {v 1, v 4, v 9 }, and Q contains two IS es M 1 and M 5, where M 1 = {v 1 } and M 5 = {v 4, v 9 }. Afterward, we construct the broadcast tree T B layer by layer. At layer L 1, M 1 = {v 1 } and L 1 \M 1 = {v 2 }. Node v 1 s parent is chosen from layer L, and thus is node s. Node v 1 in M 1 is chosen as the parent of node v 2 in L 1 \M 1. At layer L 2, the unique node v 6 belongs to M 2, and its parent is node v 1 chosen from layer L 1. The other layers are handled in a similar manner, and the final broadcast tree is shown in Fig. 1(c). Then we color the parent nodes in the broadcast tree. For instance, we collect the parent nodes of nodes in Q and U \Q = {v 2, v 5, v 8 } into two sets W 1 and W 2 respectively, i.e., W 1 = {s, v 3, v 6 } and W 2 = {v 1, v 4 }. We color the parent nodes in W 1 and W 2 using two D2-coloring methods f 1 and f 2 by the front-to-back ordering and the smallest-degree-last ordering respectively. Nodes s and v 3 are both colored with, because they are not adjacent to their respective children nodes v 1 and v 4, i.e., f 1 (s) = f 1 (v 3 ) =. Node v 6 is adjacent to node s s child node v 1, and thus should be colored with 1, i.e., f 1 (v 6 ) = 1. f 1 requires two colors to color the parent nodes, and thus m 1 = 2. By the smallest-degree-last ordering, both the nodes v 1 and v 4 are colored with, i.e., f 2 (v 4 ) = f 2 (v 1 ) =. f 2 requires only one color to color the parent nodes, and thus m 2 = 1. Finally, we schedule the transmissions based on the colors of the parent nodes. The scheduling proceeds layer by layer, and the current time slot t is. At layer L 1, node v 1 in M 1 is informed by its parent node s at time slot t + f 1 (s) T + =. Then t increases by m 1 T = 2 4 = 8. Node v 1 informs its child node v 2 in L 1 \M 1 at time slot t + f 2 (v 1 ) T + = 8. Then t increases by m 2 T = 4. The transmissions at other layers are scheduled using the similar method. Table 1 lists the transmitting time slots of all the nodes scheduled by. 4.2 All-To-All Broadcast Algorithms All-To-All Broadcast Algorithm To solve the all-to-all MLBSDC problem under the unitsize message model, we propose the algorithm. This algorithm contains two processes. During the first process, all the messages are gathered to one special node. During the second process, these messages are broadcast one by one from this special node to all the other nodes. The details and pseudocode of algorithm are given in Appendix A of the supplemental material All-To-All Broadcast Algorithm algorithm is proposed for the all-to-all MLBSDC problem under the unbounded-size message model. algorithm also contains two processes: data gathering and broadcasting. Note that, under this message model, the special node only needs to broadcast one combined message after receiving all the other messages. Moreover, different from the data gathering process in algorithm, the data gathering process

5 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS 5 in algorithm is similar to a data aggregation process [28], [29]. The details and pseudocode of algorithm are given in Appendix A of the supplemental material. 4.3 Prune Method We propose the Prune method to reduce the overhead of our proposed algorithms in terms of the total number of transmissions. The Prune method is based on the observation that a node can inform the nodes at the same layer and a lower layer with the same active time slot by one transmission. The details of the Prune method are given in Appendix A of the supplemental material. We call our three algorithms together with the Prune method as -P, -P and - P respectively. 5 PERFORMANCE ANALYSIS In this section we analyze the correctness, approximation ratios, overhead and time complexity of our algorithms. The proofs of Theorems 2, 3, 5, 6, 7, 8 and Lemma 2 are given in Appendix B of the supplemental material. 5.1 Correctness Analysis Theorem 1. Both the and -P algorithms provide correct and collision-free broadcast schedulings. Proof: We first prove the correctness of the algorithm. Since the broadcast scheduling provided by this algorithm works layer by layer, we only need to prove that all the nodes at each layer can be informed. We prove the correctness by induction. At the first layer, L = {s}. Since node s initiates the message, it is true for the first layer. We assume that all the nodes at the layers before L i (1 i D) have been informed. The nodes at layer L i are partitioned into two subsets: M i and L i \M i. The parent nodes of nodes in M i are chosen from nodes at higher layers. Clearly these parent nodes can cover the nodes in M i and have been informed according to the inductive assumption. Hence the nodes in M i can be informed by their parent nodes. For each node w in L i \M i, its parent node P(w) is picked from M i or M i, where i < i, and (i 1) mod T and (i 1) mod T are both equal to j. According to the construction method of Q j, node P(w) must exist because otherwise node w can be added into Q j. Moreover, we consider two cases. In the first case, node P(w) M i. According to algorithm, node P(w) will be informed before node w. In the second case, node P(w) M i. According to the inductive assumption, node P(w) have been informed. Therefore, node w can be informed by its parent node P(w). The difference between the and -P algorithms is the second phase of the broadcast process. So, to prove the correctness of the -P algorithm, we only require to prove that nodes in L i \M i can be informed during the second phase. Recall that the parent nodes of nodes in L i \M i are picked from M i or M i. According to the Prune method, all the nodes in M i and M i have transmitted the message after the second phase of the broadcast process at layer L i finishes. Hence nodes in L i \M i can be informed by their parent nodes. Next, we prove that the transmissions scheduled by the algorithm from the parent nodes to their children nodes are collision-free. Since the transmissions are scheduled strictly layer by layer, we only consider the transmissions at each layer. At each layer L i, the messages are first delivered to the nodes in M i and then to the nodes in L i \M i. So we distinguish between two cases. Case 1: The messages are delivered to the nodes in M i without collision. We prove it by contradiction. Suppose there are two nodes u 1 and u 2 in M i. Signal collision occurs when their parent nodes P(u 1 ) and P(u 2 ) deliver messages to them. That is, these two parent nodes are scheduled to deliver messages to their respective children nodes at the same time slot, and one child node is covered by both two parent nodes. Without loss of generality, we assume node u 1 is covered by both two nodes P(u 1 ) and P(u 2 ). According to the rule of choosing parent nodes and the coloring method used in algorithm, node P(u 1 ) must be picked as the parent node earlier than node P(u 2 ) (otherwise node u 1 will become the child node of node P(u 2 )), and these two parent nodes must have different colors. Therefore, the transmissions from these two parent nodes to nodes u 1 and u 2 are separated, which contradicts the assumption. Case 2: The messages are delivered to the nodes in L i \M i without collision. Since both the rule of choosing parent nodes and the coloring method in this case are similar to those used in the previous case, we can prove it by using the similar proof to that for Case 1. The transmissions scheduled by the -P algorithm are also based on the colors of the parent nodes. It is easy to prove that these transmissions are collision-free by using the similar proof to that for Case 1. Theorem 2. Both the and -P algorithms provide correct and collision-free broadcast schedulings. Theorem 3. Both the and -P algorithms provide correct and collision-free broadcast schedulings. 5.2 Approximation Ratio Analysis Lemma 2. To color the parent nodes, the required numbers of colors m 1 j 5 and m 2 j 12, where j T 1. Theorem 4. The approximation ratios of both the and -P algorithms are at most 17 T. Proof: The lower bound for the one-to-all MLBSDC problem has been proved to be D in [13]. It is easy to find that and -P have the same worst-case broadcast latency, so we only require to prove the approximation ratio of algorithm. We denote by D OT AB the broadcast latency of. Since the transmissions are scheduled strictly layer by layer, the broadcast finishes when all the nodes at the last layer receive the messages. In the worst case, there are D + 1 layers in T S PT. According to Lemma 2, it follows that D OT AB = 1 i D(m 1 j + m 2 j ) T (5 + 12) T D = 17 T D

6 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS 6 Theorem 5. The approximation ratios of both the and -P algorithms are at most 17 T + 2. Theorem 6. The approximation ratios of both the and -P algorithms are at most ( + 22) T. 5.3 Overhead Analysis Theorem 7. The total numbers of transmissions scheduled by the, -P,, -P, and -P algorithms are at most 15, 1, 17, 12, 4 and 3 times as large as the minimum total numbers of transmissions respectively. 5.4 Time Complexity Analysis Theorem 8. The time complexity of all the, - P,, -P, and -P algorithms is at most O( T 2 n 2 + n 3 ). 6 PERFORMANCE EVALUATION In this section, we implement our algorithms using Matlab, and evaluate the broadcast latency and the total numbers of transmissions of our algorithms by extensive simulations. The broadcast latency is the total time slots required by all the nodes to receive the broadcast messages. The performance of the and -P algorithms is compared with that of the algorithm [13]. Since the broadcast latency of our algorithms together with the Prune method is the same as that of our algorithms without this method, we only give the broadcast latency of our algorithms without this method. All the nodes are randomly deployed in a rectangle area of 2m 2m. We study the effect of different network configurations including the network size, the transmission radius, and the duty cycle on the performance of the algorithms. The network size ranges from 2 to 1 with an interval of 2. We vary the range of the transmission radius from 2m to 6m. The duty cycle which equals to 1/ T varies from.1 to.2, where T ranges from 1 to 5 with an interval of 1. A typical application scenario to choose these sets of parameters is wireless sensor networks, e.g., a number of sensor nodes are randomly deployed in a farmland to monitor the growing environment of the crops. We conduct the experiments with one configuration changed and the other two fixed. Every experiment is run on 2 randomly generated graph topologies. Moreover, for the oneto-all broadcast simulations, we carry out the experiments 1 times for each graph topology, and randomly choose one node as the source node in each experiment. The average performance of these experiments is reported. 6.1 Impact of Network Size First, we evaluate the impact of network size on the performance of the algorithms. The transmission radius is fixed to 3m, and the duty cycle is set as.5 in these experiments. As shown in Fig. 2(a) and Fig. 2(b), the broadcast latency of all the algorithms grows with the increase of the network size because more nodes require to be informed. performs better than especially when the network size grows, which validates the performance analysis in previous section. outperforms because only requires the special node to broadcast once. From Fig. 2(c), Fig. 2(d) and Fig. 2(e), we can observe that the total numbers of transmissions of all the algorithms increase when the network size increases, because the forwarding nodes have to transmit more times when they have more neighboring nodes to inform. Moreover, and -P outperform, and Prune method reduces the total number of transmissions, which validates our analysis in previous section. 6.2 Impact of Transmission Radius Next, we study the performance variation of the algorithms under different transmission radiuses. These experiments test 4 nodes with the duty cycle of.5. From Fig. 2(f) and Fig. 2(g), we can see that, with the increase of the transmission radius, the broadcast latency of, and decreases while the broadcast latency of grows slowly. This is because a node may have more neighbors with the increase of the transmission radius, which makes the number of layers in the shortest path tree rooted at the source node (or the special node) decreases., and all use layer-by-layer scheduling methods, and thus their broadcast latency decreases. Although the number of layers decreases, the required number of colors to color the parent nodes may increase. Since has to broadcast 4 messages, its broadcast latency increases slowly due to the increase of the number of colors. When the transmission radius increases, a forwarding node can inform more nodes, and thus the total numbers of transmissions of all the algorithms decrease as shown in Fig. 2(h), Fig. 2(i) and Fig. 2(j). Moreover, performs better than, and Prune method brings benefits to our algorithms. 6.3 Impact of Finally, we evaluate the impact of the duty cycle on the performance of the algorithms. These experiments are run with the network size of 4 and the transmission radius of 2m. As shown in Fig. 2(k) and Fig. 2(l), the broadcast latency of all the algorithms grows with the decrease of the duty cycle. When the duty cycle decreases, the scheduling period contains more time slots, and there are more layers in the shortest path tree. Therefore the latency of the broadcast scheduled layer by layer will increase. performs better than, and has lower broadcast latency than. Fig. 2(m), Fig. 2(n) and Fig. 2(o) show that the total numbers of transmissions of all the algorithms grow with the decrease of the duty cycle. This is because when the duty cycle decreases, the number of time slots in a scheduling period increases, and thus a forwarding node may require more transmissions to inform all its neighboring nodes with different active time slots. Moreover, schedules fewer transmissions than, and the Prune method reduces the total numbers of transmissions of our algorithms. 7 CONCLUSION AND FUTURE WORK This paper studies the MLBSDC problem. We prove that both the one-to-all and the all-to-all MLBSDC problems are NPhard, and propose efficient approximation algorithms for these

7 IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS 7 Broadcast Latency (time-slot) (a) Broadcast latency Broadcast Latency (1 4 time-slots) (b) Broadcast latency P (c) Total number of transmissions (1 4 ) P (d) Total number of transmissions P Broadcast Latency (time-slot) Broadcast Latency (1 4 time-slots) P (e) Total number of transmissions (f) Broadcast latency (g) Broadcast latency (h) Total number of transmissions (1 4 ) P P Broadcast Latency (1 3 time-slots) Broadcast Latency (1 4 time-slots) (i) Total number of transmissions (j) Total number of transmissions (k) Broadcast latency (l) Broadcast latency P (m) Total number of transmissions (1 4 ) P (n) Total number of transmissions P (o) Total number of transmissions Fig. 2. Performance variation under different network configurations problems.we prove that our algorithms provide correct and collision-free broadcast schedulings, achieve small approximation ratios and have low overhead in terms of the total number of transmissions. The Prune method is proposed to reduce the overhead of our algorithms. Extensive simulation results verify the efficiency of our algorithms and the Prune method. As our future work, we plan to solve the MLBSDC problem under physical interference model. Under this model, we require to consider the impact of both the interference and noise on the transmissions. Since the solution in [7] for the MLBS problem under physical interference model also uses the geometric properties of the MIS, we believe that our algorithms can be extended to solve the MLBSDC problem under physical interference model. The approximation ratios still exist, but may be affected by some other factors, such as the ratio of the interference radius to the transmission radius. REFERENCES [1] L. Gasieniec and A. Lingas, On adaptive deterministic gossiping in ad hoc radio networks, Information Processing Letters, vol. 83, no. 2, pp , 22.

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Dijkstra, A note on two problems in connexion with graphs, Numerische Mathematik, vol. 1, no. 1, pp , [28] B. Yu, J. Li, and Y. Li, Distributed data aggregation scheduling in wireless sensor networks, in Proc. of IEEE INFOCOM, 29. [29] P.-J. Wan, S. C.-H. Huang, and L. Wang, Minimum-latency aggregation scheduling in multihop wireless networks, in Proc. of ACM MobiHoc, 29. Xianlong Jiao is a Ph.D. student in the School of Computer, National University of Defense Technology, Changsha, China. He is currently an exchanging student in the Department of Computing, The Hong Kong Polytechnic University, HKSAR, China. His research interests are in the areas of wireless ad hoc and sensor networks, and mobile computing. He received the B.E. and M.E. degrees in the School of Computer, National University of Defense Technology, Changsha, China in 23 and 26 respectively. Wei Lou is an assistant professor in the Department of Computing, The Hong Kong Polytechnic University, Hong Kong, China. His research interests are in the areas of mobile ad hoc and sensor networks, peer-to-peer networks, mobile computing, and computer networks. He has worked intensively on designing, analyzing and evaluating practical algorithms with the theoretical basis, as well as building prototype systems. His research work is supported by several Hong Kong GRF grants and Hong Kong Polytechnic University ICRG grants. Junchao Ma is a PhD student in Department of Computing at The Hong Kong Polytechnic University. He received the B.E. degree in Automation from Zhejiang University, China in 26 and the M.E. degree in Department of Electronic and Information Engineering from The Hong Kong Polytechnic University, Hong Kong in 27. His research interests are broadly in the fields of wireless ad hoc and sensor networks, such as sleep scheduling, routing, topology control and privacy. He is a student member of the IEEE. Jiannong Cao is a full professor and acting head of the Department of Computing at Hong Kong Polytechnic University, and the director of the Internet and Mobile Computing Lab in the department. He is an adjunct professor of National University of Defense Technology, Northeastern University, and Northwest Polytechnic University in China. He also held a visiting scholar at the Institute of Software, Chinese Academy of Science. He received the B.Sc. degree in computer science from Nanjing University, China, in 1982, and the M.Sc. and Ph.D. degrees from Washington State University, USA, in 1986 and 199, all in computer science. Xingming Zhou has been with National University of Defence Technology since 1978, where he is currently the head of scholar committee of National Key Laboratory on Parallel and Distributed Processing. He has been the Academician of Chinese Academy of Science (CAS) since He has been the cochair of many international conferences and workshops on parallel computing, network computing, and mobile networks. His current research interests include high performance computing and network computing. Xiaodong Wang received the B.E., M.E. and Ph.D. degrees in computer science in 1994, 1998 and 21 respectively, all from the National University of Defence Technology, Changsha, China. He is a member of IEEE Communication Society and China Computer Federation. His main research interests are wireless networks and mobile computing, including mobile ad hoc networks and wireless sensor networks.