Sleeping Schedule-Aware Local Broadcast in Wireless Sensor Networks

Broadcast is widely used in applications in wireless sensor networks (WSNs). In the last decade, the broadcast problem in WSNs has been well studied. However, few of existing broadcasting strategies have considered the scenarios with sleeping schedules, which have been emerging as a prevalent energy-saving method for WSNs. In WSNs with sleeping schedule, each node switches on and off periodically, rendering the broadcast problem more difficult. To handle the periodical sleep issue, we focus on designing effective sleeping schedule-aware broadcast algorithms. We practically propose SALB, a sleeping schedule-aware local broadcast algorithm. In SALB, a typical local algorithm for constructing connected dominating set is employed to form the broadcast backbone. To guarantee proper transmission of broadcast messages, a sleep-aware forwarding mechanism is implemented. Moreover, heuristic strategies are used to decrease the number of transmissions and the broadcast latency. Theoretical analysis shows that the number of transmissions for SALB is within 4(min(Δ, | T | ) + c ) (c is constant) is constant) times of the optimal value. And the broadcast latency of SALB is within 4 | T | + 1 times of the optimal value (Δ is the maximum degree in the network, | T | is the scheduling period length). The performance of SALB is evaluated via simulations.


Introduction
Energy is regarded as scarce resource in wireless sensor networks (WSNs).It is shown that most energy is wasted in sensor node's idle listening.To handle this issue, sleeping schedule has been proposed to preserve energy in WSNs.With sleeping schedule, each node is switched on and off periodically.Nodes turn on to detect object and receive and forward data and then turn to sleep to save energy.As a simple yet efficient method, sleeping scheduling has been widely used in WSN applications like environment monitoring and object tracking applications [1].
Broadcast is a fundamental operation in WSNs for routing discovery, information dissemination, and so on [2].Naive broadcast methods such as flooding always lead to massive redundancy and intolerable latency, wasting the energy dramatically [2].In order to design energy-efficient broadcast algorithms, a lot of effort has been devoted to reduce data transmission and broadcast latency.For instance, many MCDS-based approaches are proposed to minimize transmission redundancy [3][4][5][6][7].And lots of coloring-based collision-avoiding approaches are used to reduce latency [8][9][10].
In this paper, we focus on the sleeping schedule-ware broadcast problem in WSNs, which is quite different from that in traditional WSNs.In traditional WSNs, each node is assumed to be nonsleeping.Based on the broadcast nature of wireless medium, a node can deliver one broadcast message to all its neighbors by one transmission.While in the WSNs with sleeping schedule, each node can only receive messages International Journal of Distributed Sensor Networks when it is active, so not all neighbors of a senor node can receive the broadcast message by one transmission.This difference renders the broadcast problem more difficult in such situations.Existing broadcast algorithms did not consider the sleeping schedule issues and thus are not suitable.
To solve the sleeping schedule-aware broadcast problem, we practically propose a local algorithm in this paper, namely, the sleeping schedule-aware local broadcast (SALB) algorithm.In SALB, we first use a typical local MCDS construction algorithm to form a virtual broadcast backbone.With this virtual backbone, we design an active slot-based forwarding mechanism for each node, which guarantees the success of broadcast and can help reduce transmission redundancy and latency.The numbers of transmissions for SALB are within 4(min(Δ, ||) + ) (c is constant) times of the optimal value.And the broadcast latency of SALB is within 4|| + 1 times of the minimum value.(Δ is the maximum degree in the network; || is the scheduling period length).
The rest of this paper is organized as follows.We review the related work in Section 2 and present the models and assumption in Section 3. SALB is proposed in Section 4. We evaluate its performance in Section 5 and discuss the tradeoff between the number of transmissions and the broadcast latency in Section 6.In Section 7, we conclude this paper.

Related Work
Data-transmission issues in duty-cycled WANETs have recently attracted much of researchers' attention.Dousse et al. [11] established a bound on the transmission latency of sensor networks with uncoordinated schedule.Lu et al. proved the NP-hardness of minimizing end-to-end communication delay in low-duty-cycle sensor networks in [12].Cao et al. proposed the pipeline forwarding pattern for sensor networks with a sleeping schedule to decrease the delay [13].Keshavarzian et al. analyzed the delay of several known wakeup patterns and proposed a new multiparent scheduling pattern for sensor networks in [14].Gu and He proposed a dynamic data forwarding scheme for extremely low-dutycycle sensor networks based on the expected latency and reliability model [15].However, these works mainly focus on the latency issue caused by the sleep schedule, and none of them refer to the broadcast problem.
Since broadcast plays an important role in WSNs, a lot of research work has been done on this area.The simplest approach for broadcast is blind flooding, where each node is obligated to forward a packet upon receiving it for the first time.However, it has been shown that blind flooding can lead to serious redundancy and collisions, a situation known as broadcast storm [2].Therefore, great effort has been made by improving the flooding approach to reduce broadcast redundancy and collisions, as well as broadcast latency.For example, [2,16] proposed probabilistic forwarding methods to avoid massive redundant transmissions.Based on the construction of virtual broadcast backbone and broadcasting along the backbone, [3][4][5][6][7] proposed different technologies to reduce broadcast redundancy.Specifically, much recent research has been done to minimize the broadcast transmissions and minimize the broadcast latency based on different

𝑛
The number of nodes in the network The period of sleeping schedule Δ The maximum degree of nodes in the network  (V) The set of neighbors of node V  (V) The set of 1-hop neighbor dominators of node V  2 (V) The set of 2-hop neighbor dominators of node V SL  (V) The active time-slot of node V network models.Generally, both the minimum transmission broadcast problem and the minimum latency broadcast problem are NP-hard, and a lot of efficient algorithms have been proposed [3][4][5][6][7][8][9][10].
Recently, some researches focusing on the sleeping schedule-aware broadcast problem have emerged.Wang et al. discussed the broadcast problem in duty-cycle sensor networks in [17].Hong et al. proposed a series of work on the minimum-redundancy broadcast problem for dutycycle wireless ad hoc networks [18,19].The other topic of minimum-latency broadcast problem has been investigated in [20,21].However, none of them proposes a local algorithm for sensor network considering both transmission and latency issues of broadcast.

Model and Assumption
We assume that  sensors node are deployed in a twodimensional plane with equal maximum transmitting range of one unit.The network can be modeled as a connected UDG (, ), where  is the set of nodes and  is the edge set.An edge {, V} ∈  if and only if the distance between nodes u and v is within each other's communication range.Unlike the literature focusing on tuning sleeping schedule [12][13][14], we follow the assumption in [11], where each node determines its sleeping schedule completely and uncoordinatedly.We assume that the scheduling period  is divided into||timeslots with fixed and equal length, denoted by 1, 2, . . ., || accordingly.We also assume that each node v randomly chooses an active time-slot SL  (V) ∈  independently.Each time-slot is assumed to be long enough for sending or receiving a data packet [12].The contention and collision issues of wireless channel are assumed to process by the MAC protocols, for example, S-MAC [22].Therefore we will not need to consider the impact of factors like channel conflict.Following the models in [12,15], we assume that a node can wake up to transmit at any time, but can receive only in its active time-slot.The set of sending time-slots of node V is SL  (V).The global time synchronization is guaranteed by protocols like flooding time synchronization protocol (FTSP) [23].
The notations used in this paper are listed in Table 1.

Problem Formulation.
We consider the multisource broadcast in this paper, in which each node in the network is possible to broadcast the data packets to all the other nodes.
In the general WSN, with the benefit of broadcasting nature of wireless media, each node can broadcast the data packets to all neighbor nodes with only one transmission.As a result, the objective of traditional broadcasting algorithm is to determine the forwarding nodes in broadcasting.Each forwarding node retransmits the data packets once after receiving them to complete the broadcasting process.However, considering the effect of sleeping schedule, the active time-slots of nodes are always different.A node cannot guarantee that all its neighbors are able to receive the data package successfully with one transmission.As a result, the broadcasting problem becomes different from the traditional WSN and thus is needed to be redefined.We define a broadcast backbone () on (, ) as a subset of V, where the broadcast backbone is the set of forwarding nodes.The data package will be forwarded in the network on the broadcasting backbone.Also we define a broadcast schedule BS(B) as the set of SL  (V) in which v is the node in broadcasting backbone B. We also define Cov  (V) as the set of neighbor nodes in which node v covers at time-slot i.The sleeping schedule-aware broadcast problem can be described as follows.
Definition 1 (the sleeping schedule-aware broadcast (SAbroadcast)).Given a WSN with sleeping schedule which is modeled as a UDG (, ), find a connected backbone () for broadcast and a corresponding broadcast schedule BS() so that Different broadcasting algorithms have different broadcast backbones and broadcast schedules, which determine the number of transmissions and the broadcast latency.Finally we present a useful definition.
When node  receives a data package at time-slot SL  () and sends data to node c through node b, if there is no inversion, node c can receive the data package within the minimum latency: SL  () − SL  () time-slots.If an inversion exists, for example, SL  () ≥ SL  (), the latency of node c receiving the data package will be increased by || timeslots.If the inversion appears k times, the time-slots will be increased by ||.Therefore, to decrease the latency, we usually try to avoid the appearance of inversions in the sequence of forwarding nodes from the source node to the destination node.

Local Algorithm for SA-Broadcast Problem
In this section, we introduce the details of the proposed SALB algorithm.The design of SALB algorithm consists of two parts: construction of a broadcast backbone and the active time-slot-oriented forwarding mechanism.

Construction of Broadcast
Backbone.Among existing broadcast backbone constructing algorithms, the MCDS is proved to have the minimum forwarding nodes, performing well in reducing both transmission and latency [24].Therefore, we would like to use a MCDS as the broadcast backbone.Many MCDS constructing algorithms have been proposed so far, like [25][26][27][28].Among them, a widely used local algorithm for mobile ad hoc networks proposed by Alzoubi et al. in [25] has constant message complexity and constant approximation ratio.Therefore, we employ this algorithm with some modification here to construct the broadcast backbone of SALB.
As described in [7,25], the construction of broadcast backbone can consist of two phases: dominator election and dominator connection.The elected dominators and connectors form a connected dominating set (CDS), acting as the broadcast backbone.In the construction phase of the broadcast backbone, we assume that all nodes are in the active state.After the construction, each node will become a dominator, a dominate, or a connector.According to [25], there must be a dominator within three hop distance of any dominator.During the broadcast, each dominator delivers message to its neighbors, and connectors relay messages among dominators.Each node maintains a forwarding node list FWD LIST, containing the IDs of destination nodes and their active time-slots.The FWD LIST will be used in designing the forwarding mechanism.

Electing Dominators.
We assume that each node in the network obtains all its 1-hop neighbors' ID and active timeslot by exchanging beacon messages.To reduce inversions in the broadcast process, we would like to make the active timeslot of each dominator smaller than its neighbors' .Let (V) be the set of node V's 1-hop neighbors.We define a metric  to represent the possibility of no inversion happening while a node  transmits to its neighbors: In electing the dominators, nodes with larger value of  will be more likely to win.Initially, all nodes are in the "Blank" state.Then the modified election procedure based on that in [25] with metric  is as follows.
(i) A Blank node becomes a dominator if it has the largest  among all its Blank 1-hop neighbors and then broadcast a message IamDominator (ID, i) with its ID and active time-lot i (ID is used to break the tie).
(ii) A Blank node becomes a dominator if there are no Blank nodes nor dominators in its 1-hop neighbors (knowledge from the received IamDominatee message) and then broadcast the IamDominator(ID, i) message.
(iii) A Blank node becomes a dominatee if it receives a IamDominator message and then broadcast message IamDominatee(ID, i).
After election, if a dominator  has the largest ID among all its dominatee V's neighbor dominators, it stores the ID and the schedule of active time-slots of V in its FWD LIST.Each // Choosing 1-hop connectors for dominator (1) If  == 0 then stop, else execute the following steps; (2) For each dominatee in HOP1 LIST, compute the number  of dominators in ; (3) Choose the dominatee with largest  as 1-hop connector; (4) Remove the item of V from the HOP1 LIST, and remove all dominators connected by V from , go to Step 1.

Algorithm 1
// Choosing 1-hop connectors for dominatee (1) Find V  from the SPCON LIST which connects to most isolated dominators in ; (2) Mark V  as an 1-hop connector, and removes all nodes it connects to from ; (3) Store V  's ID and active time-slot in FWD LIST; (4) Go to Step 1. until  == 0.
Algorithm 2 dominatee then records all of its dominator's IDs and active time-slots in its FWD LIST.

Connecting Dominators.
Here we present the modified dominator connecting phase based on the algorithm from [25].For each dominator, we call the connecting nodes adjacent to its 2-hop dominators the 1-hop connectors and the connecting nodes 2-hop away from its 3-hop dominators the 2-hop connectors.
First we connect dominators with its 2-hop neighbor dominators.After the election phase, each dominatee V broadcasts an ANNOUNCE message containing the IDs of nodes in (V).Hence, each dominator  is able to obtain the set of all its 2-hop dominators  2 (), and the dominatees through which the nodes in  2 () can be reached.Dominator  keeps this information in a list HOP1 LIST{V, (V) ∪  2 ()} and uses a working set  =  2 () for the 1-hop connector selection.Dominator  then broadcasts an ANNOUNCE message containing the IDs of all nodes in  2 (), so that all dominatees in () can obtain the information of 's 2-hop neighbor dominators.If dominator  does not have the largest ID among its dominatee V's 1hop dominators, V will not be chosen as 's 1-hop connectors.And the item of V will be removed from 's HOP1 LIST, and all V's 1-hop dominators will be removed from .After the removal, dominator  chooses 1-hop connectors for its 2hop dominators using the procedure shown in Algorithm 1.
After choosing its 1-hop connector, each dominator puts the ID of its 1-hop connectors and the connected dominators in  in a HOP1 CONN message and then broadcast.If a dominatee receives a HOP1 CONN message and finds its ID included, it marks itself as a connector and adds the ID and active time-slots of nodes attached in the HOP1 CONN message into its FWD LIST.
Next we connect the dominator and its 3-hop neighbor dominators.After a dominator  broadcasts the IDs of all nodes in  2 () with the ANNOUNCE message, its dominatee V is able to gather the information of 2-hop neighbor dominators of .Then V puts its 1-hop and 2-hop dominators' information in an ANNOUNCE message and broadcast.When dominator  receives all the ANNOUNCE messages broadcasted by its dominatees, it is able to know the 2-hop neighbor dominators  2 () for each dominator  in  2 (), which will then be stored in a list.When a dominatee V receives the ANNOUNCE message from other dominatees, it checks their dominators.If there is some dominatee  which does not share 1-hop dominators with V, V will mark  as a special dominatee and mark its corresponding dominators as special dominators.Dominatee V maintains a list SPCON LIST to store the special dominators for each special dominatees and then broadcast them with an ANNOUNCE message.
When dominator  receives the ANNOUNCE from dominatee V, it checks the special dominators inside the message.If there are nodes in the message which are neither 2-hop neighbor dominators of  nor the special dominators for the 2-hop neighbor dominators of any node in  2 (),  will mark them as isolated dominators.If a dominatee V satisfies the following conditions: (1) there are isolated dominators marked by  in V's dominators and (2)  owns the largest ID among all isolated dominators and V's dominators [25],  will choose V as a 2-hop connector and notify V with the isolated dominators it needs to connect to.Receiving the message from , dominatee V fetches the isolated dominators it needs to connect to and store them in .After that, it finds the corresponding 1-hop connectors using the procedure shown in Algorithm 2.
After that, each 2-hop connector stores the information of its isolated dominators in HOP1 CONN message and delivers the message to its 1-hop connectors.When these 1hop connectors receive the message, they store the ID, as well as the active time-slots of the source 2-hop connectors and the isolated dominators they needs to connect to in FWD LIST.

Active Time-Slot-Oriented Forwarding Mechanism.
In a conventional WSN, broadcast can be finished if each node // The forwarding mechanism for node v when receiving a broadcast packet (1) If the packet is not received for the first time, then it is just dropped and the following steps are skipped.// According to the ID and SEQ of the packet.

Algorithm 3
in the broadcast backbone forwards the packet to all its neighbor nodes only once receiving a packet.However, in a network with sleeping schedule, nodes have to forward the packet according to the schedule of active time-slots of all its receivers.Therefore, to execute the broadcast operation correctly, we have to design the forwarding mechanism for nodes in the broadcast backbone, that is, broadcast schedule.To distinguish the packets from the same source node or different source nodes, each broadcast packet includes the ID of its source node, as well as an increasing sequence number SEQ which is maintained by the source node.Based on the FWD LIST list kept by each node, the forwarding mechanism for a node when receiving a broadcast packet is shown in Algorithm 3.
When a connector or dominator node  starts a broadcast operation, it keeps  as the set of all nodes in the FWD LIST and starts the forwarding process as in step (3).If  is a dominatee, it forwards the packet to some adjacent dominator directly.From the above forwarding mechanism, we can see that nodes do not send the packet to the nodes in the FWD LIST one by one, but forward according to their active time-slots.Therefore, the numbers of transmissions are greatly reduced because an effective transmission can cover all active neighbors in the corresponding time-slot.Meanwhile, a node starts the forwarding process once it receives a packet such that following up active neighbor nodes can receive the packet as soon as possible, reducing the broadcast latency.

Performance Evaluation
In this section, we first present theoretical analysis of the transmission number, broadcast latency, and the complexity of SALB.Then, we conduct simulations to evaluate the performance of SALB.

Theoretical Analysis.
Since the broadcast virtual backbone of SALB is a CDS, if each node in the virtual backbone succeeds in delivering broadcast messages to its neighbor nodes, all nodes in the network are guaranteed to receive the broadcast message.Based on this fact, with the active slot-based forwarding mechanism, SALB obviously provides correct broadcasting operation.Next we give two theorems on broadcast transmission and latency of SALB.
Assuming the minimum number of transmission to complete a broadcast in the network is  min , we have the following.
Proof.Assume that the WSN is modeled as a UDG (, ).According Lemma 1 in [7], the dominating set  elected in the virtual backbone's constructing phase of SALB is a maximal independent set of .During the broadcasting of SALB, each dominator in  will transmit the message to nodes in its FWD LIST.For each dominator , the nodes in FWD LIST are a subset of ().Since the transmission of node  is only according to the schedule of the active slots of nodes in (), the number of necessary transmissions is at most min(Δ, ||).To cover its 2-hop neighbor dominators, the 1-hop connectors of node  need at most  2 transmission totally, where  2 is the number of node 's 2-hop neighbor dominators.Also, the 2-hop connector of node  will need to transmit message to node 's 3-hop neighbor dominators through their 1-hop connectors, respectively.Denoting the number of node 's 3hop neighbor dominators as  3 , the numbers of transmissions to cover all 3-hop neighbor dominators of node  are at most 2 3 because each 3-hop dominator connects to only one 1hop connector in the worst case.Therefore, the numbers of transmissions  to finish the broadcast equal ||(min(Δ, ||+  2 + 2 3 )).
Let the size of MCDS of  be #MCDS, according to Lemma 2 in [25]; we have || ≤ 4#MCDS + 1.And, according to Lemma 2 in [28], both  2 and  3 are bounded by constants in a UDG.Hence we use a constant  to denote the upper bound of  2 + 2 3 .On the other hand, in any WSN which can be modeled as a UDG, the minimum transmission  min of broadcast is obviously has lower bound #MCDS.Summarizing all above, we have (2) The theorem holds.Theorem 3 shows that in situations with sparse node density or short sleeping scheduling period, the broadcast transmission of SALB will be closer to the optimal value.
Next we analyze the broadcast latency of SALB.Denoting the minimum broadcast latency in a WSN with sleeping schedule by  min , we have the following.

Theorem 4. The broadcast latency of SALB is bounded by
Proof.Denote the virtual broadcast backbone constructed in SALB by . is a connected dominating set of the UDG  corresponding to the WSN.By adding edges connecting each dominator and its dominatees in , we obtain a new graph   .According to Lemma 5 in [28], the hop-distance between any two nodes  and V in   is less or equal to three times of the minimum distance between them in .As shown in Figure 1, assume that the path with minimum distance in  between nodes  and V is   ( 0 ,   ) =  0  1 ⋅ ⋅ ⋅   , where  =  0 , V =   .If   is a dominator, let   be its dominatee.If   is a dominatee, let   =   .It is obvious that there exists a path      +1  +1 in .According to the connecting phase in SALB, at most two nodes are needed to connect two nodes   and  +1 .Therefore, nodes  0 and   can be connected with a path    ( 0 ,   ) =  0  0  0  1  1  2  3  2 ⋅ ⋅ ⋅     in graph   , where  0 ,  1 , . . .are connecting nodes in .If the minimum hopdistance between  and V is  in graph , then the maximum distance between them in graph  is bounded by 3 + 2.
The minimum broadcast latency  min in the network is actually the maximum-minimum transmission latency from the broadcast source  to each node in the network along the paths in graph .Assume that the path in  with the latency  min is   (,   ).While using SALB, according to (3), the transmission latency of forwarding the broadcast message along the path    (,   ) in graph   is less or equal to (4|| + 1) min .This theorem holds.
The construction of virtual broadcast backbone dominates the time and message complexities of SALB algorithm.According to [7,25], the time and message complexities of the CDS constructing algorithm we used in forming the virtual backbone of SALB are both ().Therefore, the time and message complexities of SALB are both (), where  is the size of the WSN.

Simulation Results.
We conduct simulations to evaluate the performance of SALB on a costumed simulator developed using PARSEC [29], which is a C-based distributed discreteevent simulation language.In simulations, the network is randomly deployed in a 200 m * 200 m dimension area.To maintain reasonable network connectivity, the radio radius of each node is set to 35 m, resulting in at least 0.5 nodes/100 m 2 and a node degree of at least 19 in the following experiments.Each node randomly chooses an active time-slot from .All results are average of ten runs.In each run, the broadcast source node is chosen randomly.
We first observe the broadcast transmission of SALB.In this simulation, we compared SALB with the modified classical tree-based broadcast scheme [30], namely the Treealgorithm.The Tree-algorithm can be stated as follows: generate a spanning tree of the network  rooted in the source node  and the broadcast finishes when each node on this tree sends message to all its children according to their active time-slots.Obviously total transmission of Tree-algorithm is exactly  − 1.We let || = 20 to observe the impact of network size on the broadcast transmission.As the network size is scaling up, the transmission of both Tree-algorithm and SALB increases (Figure 2).When the network size is relatively small (e.g.,  < 300), the transmission of SALB is a bit more than that of Tree-algorithm.That is because when || is fixed and the network size is small, each node will choose a different active time-slot with a high probability.In this case, each dominator had to transmit to its dominatees one by one, which is similar to the unicast scenario.On the other hand, there exist redundant paths between two dominators in the virtual backbone of SALB, which also causes the result.However, when the network size is large enough, for example,  > 300, more nodes will have identical active time-slots, and thus the active time-slot-oriented forwarding mechanism of SALB can save mode transmission.When  > 900, the transmission of SALB is only 50% of the Tree-algorithm.Then we fix the network size to 500 nodes to observe the impact of || on the transmission.For the Tree-algorithm, the transmission remains unchanged since it is determined by the network size.For SALB, the transmission increases as || becomes larger (Figure 3).When || is small, nodes share identical active time-slots with a high probability, and thus SALB can save more transmission (e.g., || < 60 in  Figure 3).As || increases, nodes have different active timeslots gradually, together with the redundant paths among dominators, leading to a bit more transmission than the Treealgorithm.
We then evaluate the broadcast latency of SALB.Without considering the collision of wireless channel, if each node retransmits the broadcast message as long as receiving it, the broadcast will finish within the minimum latency, namely, OPT.We compare the broadcast latency of SALB with OPT in the following simulations.First we fix || = 100 to observe the impact of network size on latency.As the network is scaling up, both the latency of SALB and OPT decrease (Figure 4).The reason behind is when || is fixed, the increase of node will make more of them share identical active time-slots.As a result, in SALB, one transmission of a dominator at some time-slot can cover mode neighbor nodes, accelerating the broadcast process.We also find that the broadcast latency of SALB never exceeds 3.5 times of the OPT (Figure 5).Then International Journal of Distributed Sensor Networks we fix network size to 500 nodes to see the impact of ||.In accordance with theoretical analysis, both the latency of SALB and OPT increase linearly (Figure 6).That is because in the cases with fixed network size and increasing ||, few of nodes share identical active time-slots.Broadcast process had to borrow the unicast method, leading to the increase of latency.We also find that as || is increasing, the latency of SALB remains within 3 times of OPT (Figure 7).

Discussion
In the design of SALB algorithm, we apply a heuristic strategy to decrease the number of transmissions and the broadcast latency.However, the two objectives always conflict with each other.Next we will illustrate an example.We consider a network shown as in Figures 8 and 9.The period of sleeping schedule  = {1, 2, 3, 4, 5}.Node  is the source node of broadcasting.The number in the circle of a node means the active time-slot of the node and the underlined number means the time of package arrives.To minimize the broadcast latency, the broadcasting schedule will be  → , → ,  → , and  → {, } (Figure 8).The latency is 8 timeslots and the numbers of transmissions are 4.To minimize the numbers of transmissions, the broadcasting schedule will be  → , → {,,} and  →  (Figure 9).The numbers of transmissions are 3, and the broadcast latency is the time of node  receiving the data package, 11 timeslots.We can observe from this example that minimizing the broadcast latency may cause the number of transmission to be increasing and vice versa.Therefore, in the real scenarios, a tradeoff between two objectives is required.

Conclusion
In this paper, we studied the broadcasting problem while considering sleeping schedule in WSN.First we formulated the sleeping schedule-aware broadcast algorithm.Then we   proposed a local broadcast algorithm SALB.In SALB, we modified a classical local algorithm for constructing connected dominating set to form the broadcast backbone and designed a forwarding mechanism to handle the periodically sleeping issue of nodes.We proved that the number of transmission of SALB is within 4(min(Δ, ||) + ) (c is constant) times of the optimal value, and the latency is within 4|| + 1 times of the optimal value.Moreover, simulations results showed that the performance of SALB is better than the tree-based broadcast algorithm.In the best case, the SLAB saved 50% transmission of the Tree algorithm.As the network is scaling up and the period of sleeping schedule is increasing, the latency of SALB remains within the constant times of the optimal value.

Figure 2 :Figure 3 :
Figure 2: Impact of network size on transmission.

Figure 4 :
Figure 4: Impact of network size on latency.

Figure 5 :
Figure 5: Impact of network size on ratio to OPT.

Figure 7 :Figure 8 :
Figure 7: Impact of |T| on latency ratio to OPT.