improved algorithm for minimum data aggregation time ... - Springer Link

Jrl Syst Sci & Complexity (2008) 21: 626–636

IMPROVED ALGORITHM FOR MINIMUM DATA AGGREGATION TIME PROBLEM IN WIRELESS SENSOR NETWORKS∗ Jianming ZHU · Xiaodong HU

Received: 10 March 2008 / Revised: 8 September 2008 c °2008 Springer Science + Business Media, LLC Abstract Wireless sensor networks promise a new paradigm for gathering data via collaboration among sensors spreading over a large geometrical region. Many applications impose delay requirements for data gathering and ask for time-efficient schedules for aggregating sensed data and sending to the data sink. In this paper, the authors study the minimum data aggregation time problem under collision-free transmission model. In each time round, data sent by a sensor reaches all sensors within its transmission range, but a sensor can receive data only when it is the only data that reaches the sensor. The goal is to find the method that schedules data transmission and aggregation at sensors so that the time for all requested data to be sent to the data sink is minimal. The authors propose a new approximation algorithm for this NP-hard problem with guaranteed performance ratio log7∆|S| + c, 2 which significantly reduces the current best ratio of ∆ − 1, where S is the set of sensors containing source data, ∆ is the maximal number of sensors within the transmission range of any sensor, and c is a constant. The authors also conduct extensive simulation, the obtained results justify the improvement of proposed algorithm over the existing one. Key words Approximation algorithm, data aggregation, wireless sensor network.

1 Introduction Due to existing and emerging applications in various situations, wireless sensor networks (WSNs) have recently emerged as a premier research topic. A WSN consists of a sink node and a number of small-sized sensor nodes spreading over a geographical area where the end user can access data from the sink. All nodes are equipped with capabilities of sensing, data processing, and communicating with each other by means of a wireless ad hoc network. A wide range of tasks can be performed through WSNs, such as condition-based maintenance and the monitoring of a large area with respect to some given physical quantities. In contrast to traditional communication networks (e.g., the Internet) which are address-centric, WSNs are intrinsically data-centric. For instance, the user may be just interested in the highest temperature in some specified areas, which could indicate danger of fire. Jianming ZHU Graduate School of Chinese Academy of Sciences, Beijing 100049, China. Email: [email protected]. Xiaodong HU Institute of Applied Mathematics, Chinese Academy of Sciences, Beijing 100190, China. Email: [email protected]. ∗ This work was supported in part by the National Natural Science Foundation of China under Grant No. 70221001, 10531070, 10771209, 10721101, and Chinese Academy of Sciences under Grant No. kjcx-yw-s7.

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In many applications of WSNs, hundreds of tiny low-cost sensor devices are implanted inside or beside the phenomenon, and linked by a wireless medium. The stringent resource constraint and the sheer number of sensors pose unique challenges on data transmission. In particular, the sensors operate on batteries, which may be irreplaceable or rechargeable. Moreover, with lowpower radio transceivers, a senor (sender) could not send data to other sensors located outside its transmission range. To send the data to those sensors, it has to use some intermediate nodes to relay the data. Second, collision resulting from a large number of simultaneous data transmission creates response implosion[1] : when two or more sensors send data to a common neighbor at the same time, the data collide at the common neighbor which will not receive any of these data. Data aggregation[2−3] is proposed to achieve energy-efficient data transmission in WSNs. When the end user needs to extract a specified information from the sensing field, data from different sensors are aggregated in intermediate nodes on the way when they are sent to the data sink. Data aggregation is the combination of data according to a certain aggregation function (e.g., logical and/or, minima and maxima). The data sent by a sender far from the data sink is received by any of its neighbor (receiver) closer to the sink; the receiver fuses the data received with its own data (possibly null), stores the fused data as its new data and then sends it towards the sink. To employ data aggregation and guarantee collision free transmission, a proper schedule for data transmission at sensors is needed so that all sensors just need only one data transmission through integration of sending their own data with relaying data from other sensors. In addition, some applications also demand time-efficient data transmission where data should be sent to the data sink within a certain period of time from the moment they are requested, otherwise, the data will be useless. In this case, parallel sending-receiving at different sensors are desirable for reducing the transmission delay. Thus, using data aggregation with a proper transmission schedule, lots of energy can be saved since each node needs to send (aggregated) data at most once. Motivated by various applications of time-efficient data aggregation[4] , in this paper we study the Minimum Data Aggregation Time (MDAT) problem: Given a WSN and a distinguished data sink d that is interested in data on a subset S of sensors, the goal is to find a sendingreceiving schedule such that all data on S are aggregated to d in a minimum time. This problem is NP-hard and has a (∆ − 1)-approximation algorithm[5] , where ∆ is the maximal number of sensors within the transmission range of any sensor. In this paper, we propose a novel technique that reduces the approximation ratio to log7∆|S| + c, where c is a constant. Our simulation study 2 shows that the proposed method is significantly better than the existing algorithm in terms of not only theoretical but also practical performances[5] . The remainder of this paper is organized as follows. In Section 2, we introduce the model, formalize the MDAT problem, and then present some related results. In Section 3, we propose our algorithm, then give the theoretical proof of its performance guarantee. In Section 4, we evaluate the performance of the proposed algorithm through simulation and compare it with that of the existing algorithm. The paper is concluded in Section 5.

2 Preliminaries 2.1 Network Model In view of miniature design of sensor devices, this paper considers WSNs in which all nodes are fixed and homogeneous. It is assumed that the underlying WSN consists of stationary nodes (sensor nodes and a sink node) distributed in Euclidean plane. The transmission range of any

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sensor node is a unit disk (circular region with unit radius) centered at the sensor. As usual, a WSN is modelled as a unit disk graph (UDG) G = (V, E), where there is an edge (u, v) ∈ E joining u and v if and only if the Euclidean distance ||u − v|| between u and v is at most one. It is assumed that G is connected. Denote by ∆ the maximum degree of G. It is further assumed that communication in WSNs is deterministic and proceeds in synchronous rounds controlled by a global clock. In each time round, • any node can send data or receive data but cannot do both at the same time; • data sent by any sender reaches simultaneously all its neighbors; • a node receives data only if the data is the only data that reaches it within this round; • each receiver updates its data as the combination of its old data and the data received; • the time consumed by a single sending-receiving-fusing-storing is normalized to one. An instance of MDAT problem is denoted by (G, S, d), the underlying WSN G = (V, E), the sink node d(∈ V ), and the set S(⊆ V ) consisting of nodes possessing data requested by d. The solution of (G, S, d) is a schedule {(S1 , R1 ), (S2 , R2 ), · · · , (Ss , Rs )} such that Sr (resp. Rr ) is the set of senders (resp. receivers) in round r, r = 1, 2, · · · , s, and all data on S must be aggregated to d within s rounds. As usual (e.g., in [6]), it is assumed that each sensor knows its geometric position, which is considered the unique ID of the sensor (the data aggregated may include some of these IDs). It is further assumed that the sink has global knowledge of all IDs in the WSN. When it needs data of particular interests at some sensor nodes, it sends the request to S together with the schedule {(S1 , R1 ), (S2 , R2 ), · · · , (Ss , Rs )}. Upon receiving the request, sensor nodes will send their data or receive data from others as specified in the schedule towards the sink d. For energy-efficient data transmission, it is assumed that each node sends data at most once and all data is received by exactly one receiver (i.e., one of neighbors of the node). Hence, every (Sr , Rr ) gives implicitly the 1-1 correspondence between Sr and Rr in a way that v ∈ Sr corresponds to its receiver in Rr which is the only neighbor of v in Rr . The value s is called the data aggregation time of solution {(S1 , R1 ), (S2 , R2 ), · · · , (Ss , Rs )}. The MDAT problem is to find the schedule with minimum data aggregation time tOPT (G, S, d). This problem has been proved to be NP-hard[5] . 2.2 Related Work Most of works on data aggregation focus on energy efficiency. There are two recent works on time efficiency[7−8] . They studied a special cast of MDAT problem, called convergecasting problem, where data on all sensors in the network are required to be sent and aggregated to the data sink, Annamalai et al.[7] proposed a centralized heuristic that constructs a tree rooted at the sink node according to proximity criterion (a node is assigned as a child to the closest possible parent node) and to assign each node a code and a time slot to communicate with its patent node. However, the miniature hardware design of nodes in WSNs may not permit employing complex radio transceivers required for spread spectrum codes or frequency bands systems. Additionally, the heuristic is evaluated only through simulations. More recently, Kesselman and Kowalski[8] devised a randomized distributed algorithm for convergecasting in n-node WSNs that has the expected running time O(log n). An assumption central to their model is that sensor nodes have the capability of detecting collisions and adjusting transmission ranges, and that the maximum transmission range could be large enough to cover all the nodes in the network. This complicates the sensor hardware design and poses challenge to lowtransmission-range constraint on sensors. MDAT problem is first formulated and studied in a more recent work[5] . It is proved that this problem is NP-hard and has an approximation algorithm with guaranteed performance

IMPROVED ALGORITHM FOR MINIMUM DATA AGGREGATION TIME PROBLEM 629

ratio of ∆ − 1. Given an MDAT instance (G, S, d), a Shortest Path Tree (SPT) T of (G, S, d) is a tree in G consisting of the shortest paths from d to nodes in S. The hight of T , denoted by h(G, S, d), equals to the length of the longest path in T form d to leaf nodes of T . The following lower bound can be easily obtained be reversing the argument in the estimation of multicasting time in a telephone network[9] . Lemma 1 tOPT (G, S, d) ≥ max{h(G, S, d), log2 |S|} for any MDAT instance (G, S, d). However, data collection through aggregation in WSNs is not simply the reverse of multicast or broadcast in traditional telephone networks. For example, when the underlying topology of WSN is n-node complete graph, that is, every node can reach every other node in the network, the data aggregation time equals n while the broadcasting time and the height of SPT both equal to 1. 2.3 Our Contribution The main contribution is a new approach for dealing with the MDAT problem in WSNs. It first embeds WSNs in the plane, and then uses a novel partition technique to divide the plane into many small cells. Each cell is a hexagon and considered as a node of an auxiliary graph. Since degrees of all nodes in the auxiliary graph are upper bounded, we are able to design an approximation algorithm with performance ratio log7∆|S| + c, which is significantly better than 2 ∆ − 1.

3 Approximation Algorithm In this section we present an approximation algorithm that has two stages: construct an aggregation tree and then schedule data aggregation/transimssion of the tree. Before we propose our algorithm, we describe the algorithm, ASDA (Shortest Data Aggregation) introduced in [5], which will be used in our method for scheduling data aggregation/transimission. The pseudocode is enclosed in Appendix for easy reference. Algorithm ASDA proceeds by incrementally constructing smaller and smaller shortest path trees rooted at d that span nodes possessing all data of interests. ASDA initially sets T1 to an SPT of (G, S, d), then implements a number of iterations. Each iteration produces a schedule of a round. In the r-th iteration, Tr is a shortest path tree rooted at d spanning a set of nodes that possess all data aggregated from S till round r − 1. ASDA picks from the leaves of Tr the senders for round r, and then initially sets Z r to be the set of leaves of Tr excluding d, and it remains the property that every non-leaf neighbors of a leaf in Tr other than d has a neighbor on Z r . The leaves of Tr other than d are examined in the decreasing order of the number of their neighbors in G that are non-leaf node in Tr . A leaf is eliminated from Z r if and only if the elimination does not destroy the property of Z r . When all leaves of Tr other than d are examined, the remaining nodes in Z r form the set Sr of the senders in round r. Subsequently, ASDA eliminates Sr from its consideration by setting Tr+1 = Tr \Sr and enders the (r + 1)-th iteration. For the simplicity of presentation, denote by NG (U ) the set {v|(u, v) ∈ E, u ∈ U, v ∈ V \ U } for U ⊆ V , where V is the node-set of G. The following theorem gives the guaranteed performance of Algorithm ASDA . Theorem 2[5] Given any MDAT instance (G, S, d), Algorithm ASDA produces a solution with data aggregation time tSDA (G, S, d) ≤ (∆ − 1)h(G, S, d) + 1. 3.1 Partitioning the Plane into Cells In this subsection, we describe how the plane is partitioned into small cells of hexagons and

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the auxiliary graph G0 is constructed. Given an MDAT instance (G, S, d), we obtain the unit disc graph G = (V, E), the geometric representation of underlying WSN. We partition the plane into regular hexagons of size 0.5 in such a way that every node in V is in exactly one of the hexagons. Refer to Fig. 1, where Sij denotes the small hexagon for i = 1, 2, · · · , m and j = 1, 2, · · · , n.

S

S

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Sn

S S

S

S n

S

S

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Sm

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Figure 1



Partition the plane into cells of hexagons

Sm Sn Let Vij denote the set of nodes in Sij , which may be empty. Then we have V = i=1 j=1 Vij . 0 0 0 Now we construct an auxiliary graph G = (V , E ) of G as follows: We associate every nonempty subset Vij with a node vij in V 0 . Thus, ¯ V 0 = {vij ¯ Vij 6= ∅, i = 1, 2, · · · , m; j = 1, 2, · · · , n}. There is an edge between two different nodes vij and vkl , if there exist u ∈ Vij and v ∈ Vkl such that (u, v) ∈ E, that is, ¯ E 0 = {(vij , vkl ) ¯ vij 6= vkl , ∃u ∈ Vij , v ∈ Vkl , s.t. (u, v) ∈ E}. With the auxiliary graph G0 produced, we define the new data sink d0 ∈ V 0 in G0 to be the node vij with d ∈ Vij , and the new source field S0 = {vij ∈ V 0 | Vij ∩ ({S} ∪ d) 6= ∅}. In the end we can obtain a new MDAT instance (G0 , S0 , d0 ), and find the shortest path tree T 0 of (G0 , S0 , d0 ). 3.2 Construct Aggregation Tree In this subsection we will show how to construct a data aggregation tree Td for the original problem (G, S, d) with the help T 0 . It is done in two steps. Refer to the pseudocode in the following. In the first step, given T 0 in G0 , Let V 0 (T 0 ) (resp. V (T )) and E 0 (T 0 ) (resp. E(T )) be the node-set and edge-set of T 0 (resp. T ), respectively. For each edge (vij , vkl ) ∈ E 0 (T 0 ), by the definition of G0 , there must exist u ∈ Vij and v ∈ Vkl such that ||u, v|| ≤ 1. There may be many such pairs of u and v. We just take any of them for each edge of E 0 (T 0 ), and put nodes u, v into V (T ), and edge (u, v) to E(T ). At the same time, we construct two node-sets Vijh and V c which stand for the head set in Vij and set of clusters of all Vij , respectively. That means each Vij may have many heads, but has at most one cluster. Suppose now that vij is the parent of vkl in T 0 , we put u into Vijh and v into V c . Notice that each hexagon has at most one cluster. When all edge (vij , vkl ) ∈ E 0 (T 0 ) have been considered, we put d into both V (T ) and V c . Now every hexagon has exactly one cluster. Next we connect these components in the current graph

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to form the tree T . For every 1 ≤ k, l ≤ m, let v be the unique cluster in Vkl , then we connect v with all nodes in Vklh whenever Vklh 6= ∅. In particular, the cluster d is connected with all nodes in the Vijh with d ∈ Vij . It is easy to see that T is a tree. In the second step, let Veij = (S \ V (T )) ∩ Vij , then Veij is the set of nodes in the intersection of S and the hexagon Sij but outside T . Notice that for each nonempty Veij , T contains at least one node in the hexagon Sij , say u. According to the way of partitioning the plane, {u} ∪ Veij induces a complete subgraph. Now connect u with every node in Veij , and in the end delete all leaf nodes outside S recursively in the current tree constructed. The final tree is the data aggregation tree Td as desired. Algorithm CAT (Construct Aggregation Tree) Input MDAT instance (G, S, d). Output An aggregation tree Td rooted at d. 1 Embed G = (V, E) into the plane partition it into m × n hexagons Sij of size 0.5 for i = 1, 2, · · · , m, j = 1, 2, · · · , n. Vij ← V ∩ Sij ; 2 Generate auxiliary graph G0 = (V 0 , E 0 ) and MDAT instance (G0 , S0 , d0 ), T 0 ← a shortest path tree of (G0 , S0 , d0 ) and E0 ← E 0 (T 0 ); 3 Vijh ← ∅, i = 1, 2, · · · , m, j = 1, 2, · · · , n; V c ← {d}, V (T ) ← {d}, and E(T ) ← ∅, 4 while E0 6= ∅ do begin 5 Take (vij , vkl ) ∈ E0 and (u, v) ∈ E such that vij is the parent of vkl in T 0 and u ∈ Vij and v ∈ Vkl ; 6 V (T ) ← V (T ) ∪ {u, v} and E(T ) ← E(T ) ∪ {(u, v)}; 7 Vijh ← Vijh ∪ {u} and V c ← V c ∪ {v}; 8 E0 ← E0 \ {(vij , vkl )}; 9 end-while 10 for every v ∈ V c do begin 11 Vij ← the set containing v; 12 Put an edge (u, v) into E(T ) for every u ∈ Vijh ; 13 end-for 14 Td ← T ; e ij ← (S \ V (Ts )) ∩ Vij , i = 1, 2, · · · , m, j = 1, 2, · · · , n; 15 V e ij 6= ∅ do begin 16 for every i = 1, 2, · · · , m, j = 1, 2, · · · , n with V 17 Take a node u ∈ Vij ∩ V (Td ); e ij ; 18 Connect u with every node in V 19 end-for 20 Td ← a shortest path tree of (Td , S, d); 21 Output Td . 3.3 Schedule Aggregation Time Our algorithm SAT for Scheduling Aggregation Time also consists of two steps as follows. e ij as follows: consider every seven hexagons as a In the first step, schedule all nodes in V group (see Fig.2). In each group, the nodes coming from different hexagons are set to aggregate and transmit data to their parents. In the second step, schedule all nodes in T obtained at the end of Step 13 in Algorithm CAT by running Algorithm ASDA with T as an input.

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Figure 2





 



 

 







 



 









 

















 





 











 











 

 



 



Considering every seven hexagons into a group

Algorithm SAT Schedule Aggregation Time on Td Input: Td . Output: A data aggregation schedule for instance (G, S, d) of MDAT problem. 1 Merge all hexagons into groups of seven hexagons e ij to their parents in Td 2 From labelled hexagons 1 to 7, schedule all nodes in V with aggregation time t1 . 3 Use T obtained at the end of Step 13 in Algorithm CAT as the input in Step 1 of Algorithm ASDA , and then get the schedule ASDA (T ) with aggregation time t2 . 4 Output the combined schedule in above Steps 2 and 3 with final aggregation time t = t1 + t2 . 3.4 Performance Analysis of Algorithm In this subsection, we will prove that integrating Algorithms CAT with SAT yields a new algorithm, denoted by AMDAT , for MDAT problem that has guaranteed approximation performance ratio log7∆|S| + c. For this purpose, we first show some lemmas in the following. 2 Lemma 3 Each Vij induces a complete subgraph of G. Proof This is true since each hexagon Sij has size of length 0.5 and there is an edge between two nodes in G if and only if the distance between them at most 1. Lemma 4 Let t1 be the data aggregation time required for all nodes in Veij . Then t1 ≤ 7∆ log2 |S| tOPT (G, S, d). ¯ Proof Let δ = max{|Veij |¯ i = 1, 2, · · · , m; j = 1, 2, · · · , n}. Notice that in each group, the nodes located in different hexagons can send data to their parents at the same time without causing transmission conflict since the distance between these nodes are bigger than 1. Thus, we have t1 ≤ 7δ ≤ 7∆. This, together with Lemma 1, leads to t1 ≤ log7∆|S| tOPT (G, S, d). 2 Lemma 5 Let ∆0 be the maximum degree of nodes in auxiliary graph G0 . Then ∆0 ≤ 18. Proof Recall that G is a unit disc graph and all hexagons vij have the same size of length as 0.5. Since every node in V that is located in Vij can reach only nodes located in those 18 hexagons as shown in Fig. 3. Hence, each vij ∈ V 0 are adjacent to at most 18 other vij ’s.

IMPROVED ALGORITHM FOR MINIMUM DATA AGGREGATION TIME PROBLEM 633





 

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Figure 3





The degree of vij is at most 18

Lemma 6 Let ∆(T ) be the maximum degree of T generated at the end of Step 13 in Algorithm CAT, ∆(G(T )) the maximum degree of induced graph G(T ), and T 0 the shortest path tree of the auxiliary graph G0 . Then, i) T is a tree rooted at d with ∆(T ) ≤ 18 and ∆(G(T )) ≤ 361. ii) Suppose h0 is the height of T 0 , and h is the height of T . Then h ≤ 2h0 ≤ 2h(G, S, d). Proof i) Recall that T is constructed in two steps. At first, all edges in T 0 are replaced by the corresponding edges in G. These edges may be separated since one node in T 0 may correspond to two or more different nodes in G. In the second step, we join every cluster with all heads in the same hexagon. Since T 0 is a tree, so is T . Notice that the maximum degree of T is no more than that of T 0 . Thus, from Lemma 5 we deduce ∆(T ) ≤ ∆0 ≤ 18. By the way of constructing T , the number of nodes in each hexagon is at most 19. ii) By Lemma 3, in each hexagon the cluster can be joined with nodes in the head set directly. Then the length of each path in T is at most two times the length of the corresponding path in T 0 . Hence, we have h ≤ 2h0 . By the way of constructing (G0 , S0 , d0 ), every path in G corresponds to a path in G0 with equal or shorter length. Thus, h0 ≤ h(G, S, d). Theorem 7 Given an MDAT instance (G, S, d), Algorithm AMDAT produces an approximation solution whose data aggregation time is at most log7∆|S| + c times that of the optimal 2 solution, where c is a constant. Proof Let t2 be the data aggregation time of schedule ASDA (T ) obtained in Step 3 of Algorithm SAT. Then from Theorem 2 and Lemmas 5–6, we deduce t2 ≤ ∆h ≤ 361h ≤ 722h(G, S, d). Thus by Lemma 4, we have the data aggregation time of schedule returned by AMDAT is t = t1 + t2 ≤ ( log7∆|S| + 722)tOPT (G, S, d). 2

4

Simulation

In this section, we will compare the performance of Algorithm AMDAT with that of existing algorithm ASDA through simulation. The obtained numerical results show that AMDAT has not only theoretically (as proved in the last section) but also practically better guarantee than ASDA . Let tMDAT (resp. tSDA ) be the aggregation time of Algorithm AMDAT (resp. Algorithm ASDA ). In our simulation we randomly place some nodes in a 100 × 100 square, and choose one of them as the sink d. We consider the broadcast case where S contains all given nodes except for d. In order to simulate various network environment, in the first set of simulations, WSNs are

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constructed with different transmission ranges of sensors while the number of sensors is fixed. In the second set of simulations, WSNs are constructed with different numbers of sensors while the transmission range is fixed. At each simulation point, the simulation is done 20 times, and we take the average of obtained results. In order to evaluate the effectiveness of Algorithms AMDAT and ASDA , we use the lower bound of the minimum data aggregation time, max{h(G, S, d), log2 |S|} (refer to Lemma 1), as the performance benchmark. Figs. 4–6 show the curves of two algorithms’ aggregation times versus the transmission ranges, where the numbers of sensor nodes are set at 60, 100, and 150, respectively. Note that bigger transmission ranges make graphs denser (i.e., more nodes and more edges between nodes). Observe that the lower bound is almost a constant, which equals log2 |S| since h(G, S, d) will become small as the transmission range becomes large. In addition, the obtained results for different number of sensor nodes have the same features. From the numerical results we can draw the following conclusions. 

lowerbound T(SDA) T(MDA)





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lowerbound T(SDA) T(MDA)

     



 

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The aggregation time versus the transmission range with 60 sensor nodes 

 



Figure 5



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lowerbound T(SDA) T(MDA)

    

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The aggregation time versus the transmission range with 100 sensor nodes

 aggregation time

 







The aggregation time versus the transmission range with 150 sensor nodes

IMPROVED ALGORITHM FOR MINIMUM DATA AGGREGATION TIME PROBLEM 635

1) Algorithm AMDAT performs better than Algorithm ASDA in all cases, particularly when the transmission range is larger. The reason is that AMDAT constructs the aggregation tree according to clusters formed, which enables more sensors to transmit data at the same time than shortest path tree based algorithm ASDA . 2) The aggregation time for both algorithms monotonically increases as the transmission range increases; In other words, both algorithms perform effectively when the transmission range is small (the data aggregation time will be shorter when the graph becomes sparse). The reason is that collision will more (less) likely occur when transmission ranges become larger (smaller), thus more (less) times are needed for large (small) transmission ranges. Figs. 7–9 show the curves of two algorithms’ aggregation times versus the number of sensor nodes where the transmission range is set at 30, 50, and 80, respectively. Note that bigger transmission ranges make graphs denser (more edges between nodes). Fig. 7–9 are very similar in features to Figs. 4–6, from which we can draw similar conclusions; In particular, the data aggregation time increases monotonically as the number of sensor nodes increases. 

lowerbound T(SDA) T(CDA)





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Figure 7



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The aggregation time versus the number of sensors with transmission range of 30   aggregation time

 

 



Figure 8





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The aggregation time versus the number of sensors with transmission range of 50

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Figure 9





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The aggregation time versus the number of sensors with transmission range of 80

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5 Conclusions In this paper we have studied MDAT problem aiming for time-efficient data aggregation in WSNs. We propose an algorithm with performance ratio of log7∆|S| + c, which significantly 2 improves the current best ratio. We notice that if, instead of applying Algorithm ASDA , we just schedule data aggregation tree Td in a greedy way, we can reduce c to a small constant. We have also made extensive simulation study on the algorithms proposed in this paper and in the literature. The obtained results show that the aggregation task can be finished quickly if we set the transmission range of sensors small (but keep the graph connected). This does not mean, however, the smaller the transmission range is, the more time-efficient the schedule will be, because for collecting data, we should also consider the time required for broadcasting/multicasting enquiries from the data sink to specified sensors, and in this direction of transmission, the larger the transmission range is, the shorter the broadcast/multicast time will be. Thus, it is worth studying how to find a suitable value of transmission range that tradeoffs times required for data aggregation and enquiry broadcast/multicast. References [1] T. Imielinski and S. Goel, DataSpace: querying and monitoring deeply networked collections in physical space, IEEE Personal Communication, 2000, 7(5): 4–9. [2] H. Chen, H. Mineno, and T. Mizuno, A meta-data-based data aggregation scheme in clustering wireless sensor networks, in Proceedings of the 7th International Conference on Mobile Data Management (MDM), 2006, 154. [3] F. Hu, C. May, and X. Cao, Data aggregation in distributed sensor networks: towards an adaptive timing control, in Proceedings of the 3rd International Conference on Information Technology: New Generations (ITNG), 2006, 256–261. [4] C. Y. Lu, B. M. Blum, T. F. Abdelzaher, J. A. Stankovic, and T. He, RAP: a real-time communication architecture for large-scale wireless sensor networks, in Proceedings of the 8th IEEE Real-Time and Embedded Technology and Application Symposium (RTAS), 2002, 55–66. [5] X. Chen, X. Hu, and J. Zhu, Minimum data aggregation time problem in wireless sensor networks, Lecture Notes in Computer Sciences, 2005, 3794: 133–142. [6] N. Bulusu, J. Heidemann, and D. Estrin, GPS-less low cost outdoor localization for very small devices, Technical Report 00–729, Computer Science Department, University of Sourthern California, April, 2000. [7] V. Annamalai, S. K. S. Gupta, and L. Schwiebert, On tree-based convergecasting in wireless sensor networks, in Proceedings of the 1st IEEE Wireless Communication and Networking Conference (WCNC), 2003, 4(1): 1942–1947. [8] A. Kesselman and D. Kowalski, Fast distributed algorithm for convergecast in ad hoc geometric radio networks, in Proceedings of the 2nd Annual Conference on Wireless on Demand Network Systems and Services (WONS), 2005, 119–124. [9] A. Bar-Noy, S. Guha, J. Naor, and B. Schieber, Message multicasting in heterogeneous networks, SIAM Journal on Computing, 2000, 30(2): 347–358.