57 in Low-Duty-Cycle Sensor Networks

57 in Low-Duty-Cycle Sensor Networks

LIJIE XU, Nanjing University of Posts and Telecommunications

GUIHAI CHEN, Nanjing University

JIANNONG CAO, The Hong Kong Polytechnic University

SHAN LIN, Stony Brook University

HAIPENG DAI and XIAOBING WU, Nanjing University

FAN WU, Shanghai Jiao Tong University

Multihop broadcasting in low-duty-cycle Wireless Sensor Networks (WSNs) is a very challenging problem, since every node has its own working schedule. Existing solutions usually use unicast instead of broadcast to forward packets from a node to its neighbors according to their working schedules, which is, however, not energy efficient. In this article, we propose to exploit the broadcast nature of wireless media to further save energy for low-duty-cycle networks, by adopting a novel broadcasting communication model. The key idea is to let some early wake-up nodes postpone their wake-up slots to overhear broadcasting messages from its neighbors. This model utilizes the spatiotemporal locality of broadcast to reduce the total energy consump- tion, which can be essentially characterized by the total number of broadcasting message transmissions.

Based on such model, we aim at minimizing the total number of broadcasting message transmissions of a broadcast for low-duty-cycle WSNs, subject to the constraint that the broadcasting latency is optimal. We prove that it is NP-hard to find the optimal solution, and design an approximation algorithm that can achieve a polylogarithmic approximation ratio. Extensive simulation results show that our algorithm outperforms the traditional solutions in terms of energy efficiency.

Categories and Subject Descriptors: C.2.2 [Computer-Communication Networks]: Network Protocols General Terms: Algorithms, Design

A preliminary version of this article was presented in Proceedings of the 10th IEEE International Conference on Mobile Ad-hoc and Sensor Systems (IEEE MASS 2013) [Xu et al. 2013].

This work was partly supported by the State Key Development Program for Basic Research of China (973 Program) (Grant No. 2012CB316201), National Natural Science Foundation of China (Grants No. 61472252, No. 61321491, No. 61133006, No. 61373130, and No. 61422208), NUPTSF (Grant No. NY214169), NSF of Jiangsu Province (Grant No. BK20141319), EU FP7 CROWN project under Grant No. PIRSES-GA-2013- 610524, Funding for Central Institutions of China (No. 20620140515), CCF-Tencent Open Fund, ANR/RGC Joint Research Scheme (RGC No. A-PolyU505/12), NSF CNS-1239108, CNS-1218718, and IIS-1231680.

Authors’ addresses: L. Xu, School of Computer Science and Technology, Nanjing University of Posts and Telecommunications, 9 Wenyuan Road, Qixia District, Nanjing 210023, China; email: [email protected]; G.

Chen (corresponding author), State Key Laboratory for Novel Software Technology, Nanjing University, 163 Xianlin Avenue, Qixia District, Nanjing 210023, China, he is also with Shanghai Key Laboratory of Scalable Computing and Systems, Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang District, Shanghai 200240, China; email: [email protected]; J. Cao, Internet and Mobile Computing Lab, The Hong Kong Poly- technic University, Hung Hom, Kowloon, Hong Kong; email: [email protected]; S. Lin, Department of Electrical and Computer Engineering, Stony Brook University, Stony Brook, NY 11794-2350, USA; email:

[email protected]; H. Dai and X. Wu, State Key Laboratory for Novel Software Technology, Nan- jing University, 163 Xianlin Avenue, Qixia District, Nanjing 210023, China; emails: [email protected], [email protected]; F. Wu, Shanghai Key Laboratory of Scalable Computing and Systems, Shanghai Jiao Tong University, 800 Dongchuan Road, Minhang District, Shanghai 200240, China; email: [email protected].

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 2015 ACM 1550-4859/2015/07-ART57 $15.00c DOI: http://dx.doi.org/10.1145/2753763

1. INTRODUCTION

Wireless Sensor Networks (WSNs) have been widely used for various applications, such as environmental monitoring [Liu et al. 2013a, 2013b; Li and Liu 2009], scientific exploration [Li et al. 2013], and navigation systems [Wang et al. 2013]. Many of these applications require broadcasting to frequently disseminate system configurations and code updates to the whole network. The total energy consumption and the broadcasting latency are the main performance metrics for evaluation of broadcasting algorithms.

It is important and very challenging to minimize the energy consumption of broad- casting for low-duty-cycle WSNs, in which every sensor node has its own working schedule to wake up periodically to perform sensing and communication tasks. Ex- isting solutions for broadcasting in low-duty-cycle WSNs (such as Guo et al. [2009], Hong et al. [2010], Wang and Liu [2009], Sun et al. [2009], Niu et al. [2013], Su et al.

[2009], Jiao et al. [2010], and Li et al. [2011]) usually implement one-hop broadcast with multiple unicasts, which is energy inefficient especially for applications of large message broadcasting, such as code update. Actually, the broadcast nature of wire- less media offers opportunities to reduce the total number of broadcasting message transmissions, even for duty-cycled networks where every node has its own schedule.

To improve the energy efficiency of broadcasting, nodes should adjust their working schedules to maximize the number of receivers for each forwarding message.

Compared with always-awake networks, low-duty-cycle sensor networks usually yield a notable increase on communication latency due to periodic sleeping [Gu and He 2007], and thus latency is always taken as the first consideration for such networks. In this article, we mainly focus on the problem of how to achieve energy-efficient broadcast with minimum latency for low-duty-cycle WSNs. To achieve optimal latency and high energy efficiency of broadcasting, we come up with a novel broadcasting communica- tion model, which fully exploits the spatiotemporal locality of broadcasting to reduce the total number of broadcasting message transmissions. The basic idea is to allow nodes to adjust their wake-up schedules to overhear forwarding messages sent by their neighbors. Some nodes may postpone their wake-up slots to receive the broadcasting message, increasing their latency. But these nodes can be carefully selected so that they are not on latency-critical paths, which indicates their schedule changes do not af- fect the minimum broadcasting latency. Based on such a broadcasting communication model, we find that the total energy consumption for broadcasting can be essentially characterized by the total number of broadcasting message transmissions, and thus our objective is to design a broadcast with minimum total number of broadcasting message transmissions for low-duty-cycle WSNs, subject to the constraint that the broadcast- ing latency is optimal, which we call the Latency-optimal Minimum Energy Broadcast Problem (LMEB).

The main contributions of this work are as follows:

—To the best of our knowledge, this is the first work that both utilizes the spatiotem- poral locality of broadcasting and proposes a solution with a provable approximation

ratio, for energy-efficient broadcast problem with minimum latency constraint in low-duty-cycle WSNs.

—We prove that the LMEB problem is NP-hard. Then, we model the LMEB problem as the Directed Latency-optimal Group Steiner Tree Problem (DLGST) by capturing the spatiotemporal characteristic of multihop broadcasting, and propose an efficient solution for this problem.

—Based on the solution to the DLGST problem, we further devise a novel Broadcasting Schedule Construction Algorithm to derive the solution to the LMEB problem, which essentially avoids the redundant transmissions and reduces the collision probability as much as possible.

—We show that the approximation ratio of our solution is O(log N· log dmax), where N and d_max denote the number of sensor nodes and the maximum node degree, respectively.

—Extensive simulation results show that our solution makes a significant improve- ment over the traditional solutions in terms of energy efficiency.

The rest of the article is organized as follows: Section 2 summarizes the related work. Section 3 illustrates the network model and formally states the problem. A Detailed descriptions of our proposed scheme and performance analysis are presented in Section 4, followed by the simulation results and the discussions about practical issues in Sections 6 and 5, respectively. We conclude the article in Section 7.

2. RELATED WORK

The broadcast problem in low-duty-cycle WSNs has received a lot of attention from the research community in the past few years [Guo et al. 2009; Hong et al. 2010; Wang and Liu 2009; Sun et al. 2009; Niu et al. 2013; Su et al. 2009; Jiao et al. 2010; Li et al. 2011;

Zhu et al. 2010; Guo et al. 2011; Lai and Ravindran 2010b; Han et al. 2013a, 2013b, 2013c; Cheng et al. 2013; Xu and Chen 2013; Kyasanur et al. 2006].

Guo et al. [2009] proposed Opportunistic Flooding to make probabilistic forwarding decisions at the sender based on the delay distribution of next-hop nodes. Hong et al.

[2010] studied the Minimum-Transmission Broadcast Problem in uncoordinated duty- cycled networks and proved its NP-hardness. They proposed a centralized approxima- tion algorithm with a logarithmic approximation ratio and a distributed approximation algorithm with a constant approximation ratio for this problem. Wang and Liu [2009]

proposed a broadcasting scheme to achieve the controllable tradeoff between energy and latency by using a dynamic-programming approach. Another solution ADB [Sun et al. 2009], which is designed to be integrated with the receiver-initiated MAC proto- col [Sun et al. 2008], was proposed to reduce both redundant transmissions and delivery latency of broadcasting by avoiding collisions and transmissions over poor links. In Niu et al. [2013], the authors investigated the energy-efficient broadcast problem with min- imum latency constraint in low-duty-cycle WSNs with unreliable links, and proposed a distributed heuristic solution to tackle this problem. In Han et al. [2013a], the authors studied the duty-cycle-aware Minimum-Energy Multicasting problem in WSNs both for one-to-many multicasting and for all-to-all multicasting. Han et al. [2013c] studied the problem of minimizing the expected total transmission power for reliable data dissemination in duty-cycled WSNs. Due to the NP-hardness of the problem, they designed efficient approximation algorithms with provable performance bounds for it. Cheng et al. [2013] proposed a novel Dynamic Switching-based Reliable Flooding (DSRF) framework, which is designed as an enhancement layer to provide efficient and reliable delivery for a variety of existing flooding tree structures in low duty-cycle WSNs. However, all of these works inefficiently implement one-hop broadcast with multiple unicasts, which do not fully utilize the spatiotemporal locality of broadcasting.

Actually, the broadcast nature of wireless media offers opportunities to reduce the total number of broadcasting message transmissions, even for low-duty-cycle networks.

To achieve higher energy efficiency of broadcasting, a few works that make the best of the spatiotemporal locality of broadcasting were proposed recently. In Guo et al.

[2011], the authors considered link correlation and devised a novel flooding scheme to reduce energy consumption of broadcasting by making nodes with high correlation be assigned to a common sender. Lai et al. [2010b] proposed a Hybrid-cast protocol that adopts opportunistic forwarding with delivery deferring to shorten broadcasting latency and transmission number. However, all of these existing solutions are heuristic and fail to provide a provable approximation ratio. Moreover, all of them mainly focus on energy efficiency optimization but do not take latency constraint into account.

3. MOTIVATION

3.1. Network Model and Assumptions

Without loss of generality, we assume that N sensor nodes are uniformly and densely deployed in a circular sensory field with a fixed radius of R and the sink node is located at the center of the sensory field. Each node has the same communication range rc. Also, it is assumed that time is divided into a number of equal time slots and each time slot is set long enough so that it can accommodate the transmission of the potential large broadcasting message. Each time slot is either in sleep state, where each node will turn its radio off, or in active state, where each node will keep awake for a short duration of listening interval to make the event sensing and channel listening at the beginning.

In our model, we assume all the sensor nodes are operated at low-duty-cycle mode, where each sensor node determines its own working schedule depending on a partic- ular power management protocol (e.g., Cao et al. [2005]) immediately after deploy- ment. For simplicity, we assume the working schedule of each node is periodic and alternates between one active state and L− 1 sleep states. Here, we use Ts( j) to rep- resent the scheduled active time slot in each period of working schedule for any node j. Figure 1 explicitly illustrates an example of the periodic working schedule where L = 5 and Ts(·) = 3. Further, we use the undirected spatiotemporal topology graph G = (V, E, W, L) to represent the network topology and nodes’ working schedules, where V represents the set of N nodes including the sink nodev0and all sensing nodes {v1, . . . , vN−1}, E represents the set of all communication links, W denotes the set of working schedules for all nodes, and L denotes the schedule period length of each node.

We denote by d(vi, vj) the point-to-point transmission latency from nodevi to nodevj

for any edge (vi, vj)∈ E, and d(vi, vj) can be determined as follows:

Ifvi = v0,

d(vi, vj)=

T_s(vj)− Ts(vi)+ 1, i f T_s(vj)≥ Ts(vi);

T_s(vj)− Ts(vi)+ L + 1, otherwise, (1)

and ifvi = v0,

d(vi, vj)=

T_s(vj)− Ts(vi), i f T_s(vj)> Ts(vi);

T_s(vj)− Ts(vi)+ L, otherwise. (2) The same as with most literature for low-duty-cycle WSNs (e.g., Guo et al. [2009], Hong et al. [2010], Wang and Liu [2009], Niu et al. [2013], Su et al. [2009], Jiao et al.

[2010], Li et al. [2011], Gu and He [2007], Zhu et al. [2010], Guo et al. [2011], Han et al. [2013a], and Cheng et al. [2013]), we assume time synchronization is achieved, and each node can transmit its packets at any time, while it can only receive the pack- ets from its neighbors in active states. Specifically, each nodevi will wake up at the beginning of the active state and keep listening for a period of listening interval; if any broadcasting packet in which the target receiver ID isvi is received, it will keep receiving until all packets of the broadcasting message are received and then go to sleep immediately; otherwise, it will go to sleep immediately. If any sender wants to send the broadcasting message to its receiver, it will set a timer to wake up itself at the beginning of the receiver’s next active state to finish the transmission, and then go to sleep.

Besides this, we also have the following basic assumptions:

(1) Each node cannot do sending and receiving simultaneously.

(2) Each node is aware of the working schedules of all its neighboring nodes within two hops; this can be realized via local information exchange between neighboring nodes initially after the network is deployed.

(3) For simplicity, we do not consider the packet collision problem due to the fact that the low-duty-cycle operation inherently reduces the probability of collision to a great extent, which has been experimentally verified in Wang and Liu [2009].

(4) The working schedules of any node and its neighbors are different from each other.

It is usually true for low-duty-cycle WSNs, since we usually improve the network performance (e.g., to minimize average detection delay) by carefully designing the working schedules of all nodes (e.g., Cao et al. [2005]) to make the neighboring nodes rotate the sensory coverage. Further, this assumption will be relaxed in Section 4.5.

3.2. Problem Statement

In traditional solutions for broadcasting, all nodes will receive the broadcasting mes- sage at their scheduled wake-up time slots, which could lead to the minimum broad- casting latency but, however, draw much more energy consumption since any one-hop broadcast is actually realized by a number of unicasts. To achieve higher energy effi- ciency of broadcasting, we come up with a novel broadcasting communication model that is based on the spatiotemporal locality of broadcasting. This model defines two kinds of receivers, that is, DelayedReceivers and InstantReceivers, for any sender. The sender will send the broadcasting message to each InstantReceiver, and also it will send a short beacon packet that only contains the ID of some InstantReceivervj, say Beacon(vj), to each DelayedReceiver. Upon receiving the Beacon(vj) from the sender, any DelayedReceiver will go to sleep immediately and defer its message receiving time by setting a timer to wake up itself at the next active state of the InstantReceivervj. Note that, the DelayedReceiver can be aware of the working schedule of the InstantRe- ceivervjdue to the assumption that each node is aware of the working schedules of all its neighboring nodes within two hops.

Figure 2 illustrates a simple example for the one-hop broadcast case, where the number labeled within each pair of brackets denotes the scheduled wake-up time slot (e.g.,v0(3) represents T_s(v0) = 3) and the schedule period length L is set as 10.

Fig. 2. (a) Broadcast without deferring. (b) Broadcast with one DelayedReceiver. (c) Broadcast with two DelayedReceivers. (d) The optimal broadcast.

Figure 2(a) shows a traditional solution, in which the sink nodev0delivers the message to its neighbors one by one to realize the broadcasting (i.e., to set nodesv1,v2,v3, v4

as the InstantReceivers). It requires total energy consumption of E_total= 4 × k × e^d_s + 4 × k × e^d_r, where k denotes the number of data packets contained in a broadcasting message, and e_s^dand e^d_r denote the energy consumption when sending and receiving a data packet, respectively. As shown in Figure 2(b), if the sink nodev0delivers the beacon packet Beacon(v2) to the DelayedReceiverv1and delivers the broadcasting message to the InstantReceivers {v2,v3,v4}, nodev1will defer its message receiving time by setting a timer to wake up itself at the next scheduled active time slot of the InstantReceiver v2(i.e., time slot 5) and the total energy consumption for broadcasting will be E_total^ = e^b_s+ 3 × k × es^d+ er^b+ 4 × k × e^dr, where e_s^band e^b_r denote the energy consumption when sending and receiving a beacon packet, respectively. As shown in Wang et al. [2006], it is usual that a data packet has a length of 133 bytes and a beacon packet has only a length of 19 bytes, which indicates that e^b_s+ e_r^bis far less than e_s^din practice. Thus, total energy benefit of deferring the message receiving time of any receiver, that is,

 = Etotal− E_total^ = k × e^d_s − (e^b_s + e^b_r), must be greater than zero. For applications with a large broadcasting message that contains a large number of data packets (i.e., code update), especially, this benefit will be significant as k  1. Moreover, we can easily find that based on such a broadcasting communication model, the total energy benefit will increase as the number of InstantReceivers decreases, which implies that total energy consumption for broadcasting can be essentially characterized by total number of broadcasting message transmissions under this model. Figure 2(c) shows an example of broadcast with two DelayedReceivers, that is, the sink nodev0delivers the beacon packet Beacon(v2) to the DelayedReceiver v1, delivers the beacon packet Beacon(v4) to the DelayedReceiver v3, and delivers the broadcasting message to the InstantReceivers {v2,v4}. According to the previous conclusion, we can find that it must have a higher energy efficiency than the case in Figure 2(b). Obviously, the schedule in Figure 2(d) must be the optimal solution, where the sink nodev0delivers the beacon packet Beacon(v4) to the DelayedReceivers {v1, v2, v3} and delivers the broadcasting message to the InstantReceiverv4.

According to the previous example, we can find that total energy consumption for broadcasting will benefit from receive deferring. Based on such a broadcasting commu- nication model, we present the definitions of Forwarding Sequence and Broadcasting Schedule in low-duty-cycle WSNs as follows.

Definition 3.1 (Forwarding Sequence). For any forwarder vi of the broadcasting message, its Forwarding Sequence S_f(vi) is defined as a sequence of its receivers sorted based on the scheduled wake-up time, namely,

S_f(vi) = <[r₁¹, . . . , r₁^k1], r1, . . . , [r¹_j, . . . , r^k_j^j], rj>, (3) where r^k_j(k= 1, . . . , kj) and the underlined r_j, respectively, denote the DelayedReceivers and InstantReceivers of nodevi. Specifically, the forwarderviwill send the short control packet Beacon(r_j) to each DelayedReceiver r^k_jand send the broadcasting message to each InstantReceiver rj. Here, [ ] denotes an optional item.

Definition 3.2 (Broadcasting Schedule). Given a spatiotemporal topology graph G= (V, E, W, L), the schedule strategy of any node vi ∈ V , say M(vi), can be defined as follows:

M(vi)= (α, β), (4)

where

α ∈ {0, . . . , N − 1}, β =

Sf(vi), α > 0;

NULL, α = 0.

In Equation (4), the variableα denotes node vi’s total forwarding number of the broad- casting message, and ifviis the forwarder (i.e., M(vi)·α > 0), β will denote the Forward- ing Sequence S_f(vi), which represents that once receiving the broadcasting message, nodevi will send the short beacon packet or the broadcasting message to each node in Sf(vi) in sequence. Obviously, M(vi)·α must be equal to the number of InstantReceivers in S_f(vi). Here, NULL denotes the omitted item and it is obvious that M(vi)·β = NULL for any nodevi with M(vi)· α = 0. Specifically, it must have M(v0)· α > 0 for the sink nodev0.

Here, a broadcasting schedule M in the network can be defined as the set of all nodes’

schedule strategies:

M= {M(vi)|vi ∈ V }, (5)

such that I_α= {vi|vi ∈ V and M(vi)· α > 0} subjects to

(1) connectivity, that is, there must exist a subtree T = (Iα, E^T), where E^T ⊆ E and for any edge (vi, vj)∈ E^T, it must havevj ∈ M(vi)· β if vi is the parent ofvj; (2) coverage, that is,

vi∈Iα M(vi)· β = V − {v0};

(3) nonredundancy, that is, M(vi)· β

M(vj)· β = ∅ for any vi, vj ∈ Iα(i= j).

In the preceding definition, note that, we assume each node cannot send the beacon packets until the broadcasting message is received in order to avoid potential simul- taneous sending and receiving, as well as to simplify the problem. As stated before, we will utilize total number of broadcasting message transmissions to characterize total energy consumption for broadcasting. Here, we take two broadcasting schedules shown in Figure 3 as an example to illustrate our problem. There is no deferring for each node (i.e., no DelayedReceiver but only InstantReceivers exist in the network) when adopting Schedule 1, which achieves the minimum broadcasting latency 17 but the maximum number of broadcasting message transmissions 5. For Schedule 2, the

Fig. 3. (a) The original topology graph with two defined broadcasting schedules. (b) Illustration of Schedule 1.

(c) Illustration of Schedule 2.

number of broadcasting message transmissions can be reduced to three without in- creasing the broadcasting latency as nodesv1andv4defer their receiving time to the scheduled wake-up time slots of v2 and v5, respectively. From the preceding exam- ple, we can find that there could exist multiple broadcasting schedules that have the same minimum broadcasting latency but different numbers of broadcasting message transmissions. Accordingly, our objective is to address the following LMEB.

PROBLEM 1 (LMEB). Given an undirected spatiotemporal topology graph G =

(V, E, W, L), find an efficient broadcasting schedule M to optimize the total number of broadcasting message transmissions, that is, to minimizeN−1

i=0 M(vi)· α, subject to the constraint that the broadcasting latency is minimized.

THEOREM3.3. The LMEB problem is NP-hard.

Note that the proofs of all the theorems in this article will be included in th Appendix.

4. APPROXIMATION ALGORITHM

In order to solve the LMEB problem, in this section, we propose an efficient approxi- mation solution.

4.1. Overview

To better capture the spatiotemporal characteristic of multihop broadcasting, we first transform the original topology graph into a directed Spatiotemporal Relationship Graph (SRG). Then, we prove that the LMEB problem on the original topology graph is equivalent to the DLGST on its corresponding SRG, and solve it by adopting a

Fig. 4. Overview of solution to the LMEB problem.

deterministic randomized-rounding based approach. Based on the solution to the DL- GST problem, finally, we devise a novel BSC-A to derive the solution of the LMEB problem, which essentially avoids the redundant transmissions and reduces the colli- sion probability as much as possible. Figure 4 explicitly illustrates the general process of our solution.

4.2. Graph Transformation

Definition 4.1 (Coverage Set). Given a Sender-InstantReceiver pair (vs, vr) and a time slot t (t∈ {0, . . . , L − 1}), the coverage set CS(vs, vr, t) is defined as follows:

(1) if t< Ts(vr), CS(vs, vr, t) = {x|x ∈ N(vs)− {v0} and Ts(x)∈ {t + 1, . . . , Ts(vr)}}, (2) if t> Ts(vr), CS(vs, vr, t) = {x|x ∈ N(vs)− {v0} and Ts(x) /∈ {Ts(vr)+ 1, . . . , t}}, (3) if t= Ts(vr), CS(vs, vr, t) = {x|x ∈ N(vs)− {v0}},

in whichv0denotes the sink node, and N(vi) denotes the neighbors set of nodevi. OBSERVATION 1. Given a spatiotemporal topology graph G = (V, E, W, L) and any edge (vs, vr) ∈ E, if it requires that node vsbe the sender (i.e., forwarder) and node vr

be the InstantReceiver of nodevs, then an efficient broadcasting schedule must make sure that when being received byvr, the broadcasting message also has been received by all the nodes in the coverage set CS(vs, vr, Tc(vs))− {vr}, where Tc(vs) denotes the time slot that the uncovered nodevsreceives the broadcasting message.

As an example, in Figure 5(a), the senderv0is assumed to receive the broadcasting message at its scheduled wake-up time slot, namely, T_c(v0)= Ts(v0)= 3. If we let node v3be the InstantReceiver of the senderv0, according to Observation 1, all the nodes in CS(v0, v3, Tc(v0)) = {v1, v2, v3} must be ensured to have been covered at the coverage time ofv3(i.e., the time slot 7 when the uncovered nodev3 receives the broadcasting message); this is because any schedule that makes the coverage time ofv1 or v2 be preceded by that ofv3will never benefit from both broadcasting latency and number of broadcasting message transmissions. In other words,v1as well asv2must be covered by one of the following three ways:

(1) covered by the senderv0at time slot Ts(v3) as the DelayedReceiver;

(2) covered by any other sender before time slot T_s(v3) as the DelayedReceiver; or

Fig. 5. (a) An example of one-hop topology. (b) Illustration of SRG.

(3) covered by the sender v0 or any other sender before time slot T_s(v3) as the InstantReceiver.

In order to better exhibit the spatiotemporal characteristic of broadcasting, any one-hop broadcast (e.g., Figure 5(a)) can be characterized by a directed SRG (e.g., Figure 5(b)) where each edge represents one broadcasting message transmission and each vertex represents a coverage set. For any vertexvi^in SRG, we let S(vi^) denote the coverage set that represented byv^_iand IR(v_i^) denote the InstantReceiver in S(v_i^). Also, we let Ts(v_i^) denote the coverage time slot of vertexv_i^and set Ts(v_i^)= Ts(IR(v^_i)). Specif- ically, any directed edge (v_i^,v^_j) in SRG represents one broadcasting message transmis- sion from a sender s∈ S(v_i^) to the InstantReceiver IR(v^_j) at time slot T_s(IR(v^_j)), and vertexv^_j represents the resulting coverage set CS(s, IR(v^_j), Tc(s)) after this transmis- sion where Tc(s)= Ts(v^i). Specifically, we set S(v^₀)= {v0} and Ts(v₀^)= Ts(v0) for the root vertexv₀^ in SRG. For each directed edge (v^_i, v^_j) in SRG, we use a Sender-InstantReceiver pair, that is, P(v^_i, v^_j)= <Sender, InstantReceiver>, to mark it.

The following Spatiotemporal Relationship Graph Construction Algorithm (SRGC-A) will introduce how to efficiently construct a directed SRG G^ = (V^, E^, W^, L) from a undirected spatiotemporal topology graph G = (V, E, W, L) in detail: Initially, SRG only contains a root vertex v₀^ where S(v₀^) = {v0} and Ts(v₀^) = Ts(v0). Starting with considering the sink nodev0as the sender, we respectively regard each neighborvi of the sink as the InstantReceiver, then insert a directed edge from the vertexv₀^ to the newly added vertexv^_newand set S(v_new^ )= CS(v0, vi, Ts(v0)), IR(v_new^ )= vi, P(v^₀, v_new^ )=

<v0,vi>. For any newly added vertex v^new, we in turn select each nodevi ∈ S(vne^ w) as the sender and select each nodevj ∈ N(vi)− {v0} as the InstantReceiver, then search all the vertices in SRG to check whether there exists a vertexv^with S(v^)= CS(vi, vj, Ts(vne^ w)) and IR(v^)= vj. If so, we just insert a directed edge fromv^newtov^with P(vne^ w, v^)= <vi, vj>; otherwise, we add a new vertex v^as well as the directed edge (v_ne^ _w, v^) into SRG and then set S(v^) = CS(vi, vj, Ts(vne^ w)), IR(v^) = vj, T_s(v^) = Ts(vj), P(vne^ w, v^) = <vi, vj>. The preceding process repeats until no new vertex addition to SRG is possible.

Algorithm 1 shows the detailed process of SRGC-A.

THEOREM 4.2. The worst-case time complexity of SRGC-A is O(N²d_max⁶ ), where dmax

denotes the maximum node degree in the network.

It is noteworthy to mention that SRGC-A could be more efficient and offer better time performance guarantee than that shown in Theorem 4.2, if a high-efficient search algorithm, such as the hash-based search algorithm, is adopted in practice.

For convenience of description, as shown in Figure 5(b), the root vertexv₀^ in SRG is directed denoted by the coverage set{v0} and any nonroot vertex v^_i in SRG is directly

ALGORITHM 1: Spatiotemporal Relationship Graph Construction Input: The undirected spatiotemporal topology graph G= (V, E, W, L).

Output: The directed spatiotemporal relationship graph G^= (V^, E^, W^, L).

V^= {v0^};

S(v₀^)= {v0}; Ts(v₀^)= Ts(v0); f lag(v^₀)= 1; //v0∈ V is the sink node while{v^|v^∈ V^and f lag(v^)== 1} = ∅ do

select any vertexvne^ w∈ {v^|v^∈ V^and f lag(v^)== 1};

for each nodevi∈ S(v^new) do

for each nodevj∈ N(vi)− {v0} do is f ound= 0;

for each vertexv^∈ V^do

if S(v^)== CS(vi, vj, Ts(vne^ w)) and IR(v^)== vjthen add a directed edge (v^new, v^) into E^;

P(v^new, v^)=<vi,vj>;

is f ound= 1; break;

end end

if is f ound== 0 then

add a new vertexv^into V^;

add a directed edge (vne^ w, v^) into E^; S(v^)= CS(vi, vj, Ts(v^new)); IR(v^)= vj; Ts(v^)= Ts(vj);

f lag(v^)= 1;

P(vne^ w, v^)=<vi,vj>;

end end endf lag(vne^ w)= 0;

end

denoted by the coverage set S(v^_i), in which the underlined node denotes the InstantRe- ceiver IR(v_i^), and the number labeled within any vertexv^_irepresents its coverage time slot Ts(vi^). We can find that SRG well captures the spatiotemporal characteristic of broadcasting and one broadcasting schedule can be implicitly represented by a subtree of SRG that is rooted from the vertex{v0} and consists of vertices that collectively cover all the nodes in the original topology graph. As an example of multihop broadcasting, Figure 6(a) shows the resulting SRG by performing SRGC-A on the original topology graph in Figure 3(a).

Next, we first define the DLGST and then show that our target problem can be transformed into the DLGST problem. Essentially, the DLGST problem is a variant of the classic Group Steiner Tree Problem (GST) [Reich and Widmayer 1990]. Given a weighted graph G = (V, E), a root r ∈ V , and a set of groups where each group is defined as a subset S ⊆ V , the classic GST problem is to find a minimum weight r-rooted subtree containing at least one vertex from each group.

Definition 4.3 (Latency of Tree). Given a spatiotemporal tree T = (V, E, W, L), the latency of T , say D(T ), can be defined as follows:

D(T )= max

v∈V −{v0}{DT(v0, v)}, (6)

wherev0denotes the root of the tree, and D_T(i, j) denotes the End-to-End (E2E) latency from vertex i to vertex j on T .

Fig. 6. (a) SRG of the original topology graph in Figure 3(a) (underlined letters denote the InstantReceivers).

(b) The simplified SRG (dashed edges constitute LGT).

PROBLEM2 (DLGST). Given a directed spatiotemporal graph G= (V, E, W, L) with weight we = 1 for each directed edge e ∈ E and a family of groups (i.e., subsets of vertices) f = {g1, g2, . . . , gk}(gi ⊆ V ), find a directed minimum weight subtree T^∗ = (V^∗ ⊆ V, E^∗ ⊆ E, W^∗ ⊆ W, L) rooted from the root vertex v0, subject to the constraints that

(1) V^∗∩ gi= φ for any i ∈ {1, . . . , k};

(2) D(T^∗) is minimized.

THEOREM 4.4. The LMEB problem on the original topology graph is equivalent to the DLGST problem on its corresponding SRG, where the vertices whose coverage sets contains a common sensor node belong to one group.

4.3. Solution to the DLGST Problem

According to Theorem 4.4, our objective thus turns to solve the DLGST problem on SRG.

To this end, we come up with an efficient solution. Overall, we first find a Latency- optimality Guaranteed Tree (LGT) from SRG and approximate our problem as the GST problem on LGT; a deterministic randomized-rounding based algorithm is then proposed to solve this problem.

4.3.1. LGT Construction.Here, we define the Minimum Latency Path Tree (MLPT) in any graph G as a spanning subtree of G where the E2E delay from the root to each vertex is minimal. We can easily have the following conclusion.

THEOREM4.5. Under our proposed broadcasting communication model, the minimum broadcasting latency must be equal to D(T_min), where T_min denotes the MLPT in the original topology graph.

According to Theorem 4.5, we can further simplify SRG by removing all the vertices whose minimum root-to-vertex latencies are more than D(T_min) and the associated edges from SRG. This is because our expected subtree of SRG, which represents the latency-optimal broadcasting schedule, will absolutely not include any vertex whose minimum root-to-vertex latency in SRG is more than the optimal broadcasting la- tency. Thus, our target problem can be further reduced to the DLGST problem on the simplified SRG.

We use OPTGST(T ) and OPTDLGST(G) to denote the cost of the optimal solution for the GST on any tree T and that for the DLGST problem on any directed graph G, respectively, and the following conclusion holds.

THEOREM 4.6. Given a simplified SRG G^ where the vertices whose coverage sets contains a common sensor node belongs to a group, we must have

OPTGST(T^)≤ h(T^)· OPTDLGST(G^), (7) where T^ denotes any latency-optimal spanning subtree of G^ and h(T^) denotes the height of tree T^. Suppose that the parameters R, L, and r_care fixed, then h(T^) must be bounded by a constant.

For any latency-optimal spanning subtree of the simplified SRG, we call it the LGT.

According to Theorem 4.6, obviously, we are expected to find a LGT with lower height to achieve a better performance guarantee. Here, we adopt the following approach, which is similar to the Bellman-Ford Algorithm, to construct the LGT.

—Initialization: Given a simplified SRG G^, we let D_min(v^₀, v^_i) and hopcount(v^₀, v^_i) denote the minimum E2E latency and the hop count from rootv^₀ to vertex v_i^, re- spectively. Initially, we set Dmin(v₀^, v₀^)= hopcount(v₀^, v₀^)= 0, and set Dmin(v^₀, vi^)= hopcount(v^₀, v^_i)= ∞ and p(v^_i)= null for any v_i^= v₀^, where p(v) denotes the parent of vertexv.

—Iteration: For each edge (v^_i, v^_j) in G^, if D_min(v₀^, v_i^)+ d(v_i^, v^_j) < Dmin(v^₀, v^_j), we will update D_min(v^₀, v^_j)= Dmin(v^₀, v_i^)+ d(v_i^, v^_j), hopcount(v^₀, v^_j)= hopcount(v₀^, v^_i)+ 1 and set p(v^j) = vi^. If Dmin(v^₀, vi^) + d(vi^, v^j) = Dmin(v^₀, v^j), we will check whether hopcount(v^₀, v^_i)+ 1 < hopcount(v₀^, v^_j); if so, we update hopcount(v^₀, v^_j) = hopcount(v^₀, v^_i)+ 1 and set p(v^_j)= v^_i. The preceding process is repeated until there is no update in G^.

For the original topology graph with D(T_min) = 17 (i.e., Figure 3(a)), we can de- rive the LGT (i.e., Figure 6(b)) by adopting the preceding approach on its SRG (i.e., Figure 6(a)).

4.3.2. Edge Selection on LGT.As seen previously, accordingly, we can approximate our problem as the GST problem on LGT, which has guaranteed the optimality of broad- casting latency. In Garg et al. [1998], the authors proposed an efficient method to address the GST Problem on tree. However, Garg et al. [1998] required that the input tree should be a binary one where each group is a subset of its leaves and groups are pairwise disjoint, and also it only gives a probabilistic solution. Based on the solution in Garg et al. [1998], we devise a deterministic method, which consists of three steps:

(1) Tree Transformation

Given a LGT T^ = {V^, E^, W^, L}, we first convert T^ into a binary tree in which each group is a subset of its leaves and groups are pairwise disjoint via the following operations:

—For each internal (i.e., nonroot and nonleaf) vertexv_i^in LGT, we insert an edge from vertexvi^ to a newly added vertexvne^ w sharing the same S(·) and IR(·) with vi^ (i.e., S(v^new)= S(v_i^) and IR(vne^ w)= IR(v_i^)).

—For each leaf vertexv_i^in LGT, if|S(v_i^)| > 1, we insert |S(v_i^)| edges from v_i^to|S(v^_i)|

newly added vertices sharing the same S(·) and IR(·) with v_i^.

—For each nonleaf vertexv_i^ with more than two children, we first add a new vertex v^newsharing the same S(·) and IR(·) with vi^into LGT; specifically, ifv^iis not the root, we replacev_i^withv_new^ to be the child of p(v_i^). Then, we delete an edge fromv_i^to any

Fig. 7. (a) Illustration of LGT (i.e., T^). (b) Illustration of the transformed binary LGT (i.e., T^∗).

childv^j of vi^, and insert the edges (v^new, v^i) and (v^new, v^j). This process is repeated until a binary tree is fully built.

—We check each edge (vi^, v^j) in the preceding binary tree; if S(vi^)= S(v^j) and IR(vi^)= IR(v^_j), we set the weightw(vi^,v^j)= 0; otherwise, w(v^i,v^j)= 1.

Here, we partition all the nonroot vertices in LGT T^ into N − 1 groups, that is, {g1, . . . , gN−1}, where any vertex v^ ∈ V^ belongs to group g_i if and only if vi ∈ S(v^) (i∈ {1, . . . , N −1}). Correspondingly, we also partition all the leaves in the transformed binary LGT into N− 1 pairwise disjoint groups, which respectively correspond to {g1, . . . , gN−1}. Figure 7 illustrates an example of tree transformation in which the members in one group are marked with the same color. Apparently, we can safely draw the conclusion that the GST Problem on LGT is equivalent to the minimum weight GST Problem on the transformed binary LGT in which each group g_i (i∈ {1, . . . , N − 1}) is a subset of its leaves and all groups are pairwise disjoint.

(2) Randomized Rounding

Let T^∗ = (V^∗, E^∗) be the transformed binary LGT; as shown in Garg et al. [1998], the minimum weight GST Problem on T^∗ can be formulated as the following 0-1 Integer Programming:

(IP) min 

e^∗∈E^∗

we^∗xe^∗

s.t. 

e^∗∈∂ S

xe^∗≥ 1, ∀S ⊂ V^∗so that r∈ S

and S∩ gi = φ f or some i ∈ {1, . . . , N − 1}

we^∗ ∈ {0, 1}, xe^∗ ∈ {0, 1}, ∀e^∗ ∈ E^∗,

(8)

where r denotes the root vertex of T^∗, and∂ S denotes the set of edges with exactly one end point in S.

In the preceding formulation, the binary variable xe^∗indicates whether to select the edge e^∗ or not. Given a group g_i, apparently, it requires that at least one edge with exactly one end point in S should be selected for any vertex set S that separates the root from g_i. Here, x_e∗can be relaxed to the range of [0,1] and regarded as the capacity of edge e^∗, which implies that any cut that separates the root from all the vertices in a given group has capacity of at least 1. According to the Max-flow Min-cut Theorem, the maximum flow from the root to any group must be at least 1. In other words, there

must exist a flow whose value is exactly 1 from the root to any group. Thus, we can relax the preceding Integer Programming to the following Linear Programming.

(LP) min 

(u,v)∈E^∗

w(u,v)x_(u,v) s.t. 

u∈g



v∈Vg

fg(v, u) = 1



(u,v)∈Eg

fg(u, v) = 

(v,w)∈Eg

fg(v, w), ∀v ∈ Vg− g − {r}

0≤ fg(u, v) ≤ x(u,v)≤ 1, ∀(u, v) ∈ Eg

w(u,v)∈ {0, 1}, ∀(u, v) ∈ Eg

∀g ∈ {g1, . . . , gN−1},

(9)

where f_gdenotes the flow from the root to group g and T_g= (Vg, Eg) denotes the subtree of T^∗, which consists of the paths from the root r to each leaf vertex in group g.

Similar to Garg et al. [1998], we adopt the following approach, which is called the Randomized-rounding based Edge Selection Algorithm (RES-A), to make the edge selection.

—We define a Selected Edge Set, which is initially set as empty.

—We make the following random selection operation: Each edge e^∗in T^∗is marked with probability of_x^xe∗

p(e∗ ), in which xe^∗can be figured out from Equation (9) and p(e^∗) denotes

the parent edge of e^∗. For any edge e^∗ with one end point is the root; specifically, it is marked with probability of x_e∗. An edge is added into the Selected Edge Set if and only if the edges including itself and all its ancestors are marked.

—We check whether the GST is generated by combining all the edges in the Selected Edge Set and the zero-weight edges in T^∗; if yes, the edge selection process is ter- minated; otherwise, we repeat the preceding random selection operation until the edge selection is terminated or the random selection operation has been repeated for

η · log(N − 1) · log max1≤i≤N−1|gi| rounds, where η is a constant.

The following lemma, which has been proven in Garg et al. [1998], explicitly shows the performance of the aforementioned randomized-rounding based approach.

LEMMA4.7 [GARG ET AL. 1998]. For a binary tree in which each group is a subset of its leaves and groups are pairwise disjoint, the probability that its root fails to reach any group g after one round random selection operation is at most about 1−_{64 log max}¹_{1≤i≤N−1|gi|}.

(3) Edge Compensation and Reduction

Different from Garg et al. [1998], which only gives a probabilistic solution, we will make sure our solution is deterministic by edge compensation. If the root is not con- nected to some group g after executing RES-A, specifically, we will establish the min- imum weight path from the root to group g and then add the edges on this path that have not been selected by RES-A into the Selected Edge Set. Finally, we further reduce the solution on the transformed binary LGT to that on the original LGT by removing all the zero-weight edges from the Selected Edge Set.

4.4. Broadcasting Schedule Construction

By adopting the previously mentioned solution, we can approximately obtain the min- imum Group Steiner Tree on LGT that consists of the edges in Selected Edge Set, called T^G = (V^G, E^G, W^G, L), which implicitly represents a latency-optimal broad- casting schedule that the total number of broadcasting message transmissions is at most|E^G|. We can easily find that the broadcasting schedule represented by T^Gmust