IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 31, NO. 4, APRIL

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1 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 1, NO. 4, APRIL Towar Efficient Distribute Algorithms for In-Network Binary Operator Tree Placement in Wireless Sensor Networks Zongqing Lu, Stuent Member, IEEE, Yonggang Wen, Member, IEEE, Rui Fan, Member, IEEE, Su-Lim Tan, Member, IEEE, an Jit Biswas, Member, IEEE Abstract In-network processing is toute as a key technology to eliminate ata reunancy an minimize ata transmission, which are crucial to saving energy in wireless sensor networks (WSNs). Specifically, operators participating in in-network processing are mappe to noes in a sensor network. They receive ata from ownstream operators, process them an route the output to either the upstream operator or the sink noe. The objective of operator tree placement is to minimize the total energy consume in performing in-network processing. Two types of placement algorithms, centralize an istribute, have been propose. A problem with the centralize algorithm is that it oes not scale to large WSN s, because each sensor noe is require to know the complete topology of the network. A problem with the istribute algorithm is their high message complexity. In this paper, we propose a heuristic algorithm to place a treestructure operator graph, an present a istribute implementation to optimize in-network processing cost an reuce the communication overhea. We prove a tight upper boun on the minimum in-network processing cost, an show that the heuristic algorithm has better performance than a canonical greey algorithm. Simulation-base evaluations emonstrate the superior performance of our heuristic algorithm. We also give an improve istribute implementation of our algorithm that has a message overhea of O(M) per noe, which is much less than the O( NM log M) an O( NM) complexities for two previously propose algorithms, Sync an MCFA, respectively. Here, N is the number of network noes an M is the size of the operator tree. Inex Terms Sensor networks, operator tree placement, heuristic algorithm. I. INTRODUCTION WIRELESS sensor networks (WSNs), owing to recent avances in wireless communication an microsystem technologies, have enable a school of novel applications [1], [], such as isaster management, factory automation, environment monitoring, offshore exploration, to name a few. In these applications, a large number of sensor noes, each of which is capable of sensing, computing an communicating, are eploye to operate autonomously or collaboratively in remote environments. However, each sensor noe has only a Manuscript receive April 1, 01; revise December 6, 01 Z. Lu, Y.G. Wen an R. Fan are with the School of Computer Engineering, Nanyang Technological University, Singapore ( {luzo000, ygwen, fanrui}@ntu.eu.sg). S.L. Tan is with Singapore Institute of Technology, Singapore ( forest.tan@singaporetech.eu.sg). J. Biswas is with Institute for Infocomm Research, 186 Singapore ( biswas@ir.a-star.eu.sg). Digital Object Ientifier /JSAC /1/$1.00 c 01 IEEE limite amount of system resource at its isposal. In particular, the limite energy resource for the sensor noes is a major factor that etermines the lifetime of WSNs. Therefore, energy optimization stans out as an imperative step in the esign, operation an application of WSNs [], [4]. One of the most prominent technologies for conserving energy in WSNs is in-network processing [], [6]. Specifically, in-network processing leverages the computational capacity of sensor noes to perform aggregation (or fusion) operations en route, eliminating ata reunancy an minimizing ata transmission to save energy for sensor noes. Normally, the operators participating in in-network processing are place in existing sensor noes, which receive the ata from ownstream operators, process them an route the output along a particular path to either upstream operators or the sink noe. The energy consumption in this scheme epens on the ata transmission from one operator to its upstream counterpart, which in turn epens on the placement of operators over the sensor topology. It has been shown [7] that the placement of operators can greatly affect the transmission energy cost of in-network processing. In this research, we focus on eveloping algorithms for in-network operator placement in wireless sensor networks, with the objective of minimizing total in-networking processing cost an ata transmission energy. Previously propose algorithms for optimal placement of in-network operator tree can be classifie into two categories, centralize an istribute. In centralize algorithms, each noe is require to know the topology of the entire network, limiting its scalability in large WSNs. Existing istribute algorithms can eliminate this constraint. However, they often incur a substantial message overhea: (O( NM log M) for Sync [8] an O( NM) for MCFA [9], where N is the number of network noes an M is the size of the operator tree). Other istribute algorithms, e.g. the greey algorithm of [10], has a high cost to search for an optimal operator placement. These extra communication costs increase the energy cost of network operations, possibly negating the energy savings of in-network processing. In this paper, we propose a istribute heuristic algorithm to place a tree-structure binary operator graph over the sensor topology, with an objective to reuce the message overhea an save energy. Inspire by the three-factory problem [11], we evelop a heuristic that places each operator iniviually to minimize the total cost of routing ata from its ownstream

2 744 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 1, NO. 4, APRIL 01 operators to the sink noe. We then evelop an area-restricte flooing mechanism to solve the local minimization problem, which has a low message overhea. Distribute message passing protocols, base on the area-restricte flooing mechanism, are presente for numerical analysis. Our theoretical an numerical analyses show the performance avantage of our propose algorithm in reucing communication an energy cost. Our technical contributions inclue the following: We propose a geometric heuristic, base on the threefactory problem, as a mathematical basis to place each operator iniviually. We establish a tight upper boun on the minimum energy cost for in-network operator tree placement an prove that our performance ratio is significantly better than that of the previously propose greey algorithms. We introuce a novel area-restricte flooing mechanism to reuce the message overhear of placing iniviual in-network operators. We evelop a parameter-free istribute algorithm to place a tree of operators without requiring knowlege of the network topology. The message overhea of the algorithm is O(M) per noe. We provie both mathematical analyses an simulationbase evaluations that show our approach has better performance than the canonical greey algorithm an has much smaller message overhea than the Sync an MCFA algorithms. The rest of this paper is organize as follows. Section II reviews the existing work. Section III formulates the optimization problem for operator tree placement in a sensor network. We then characterize the optimal solution in Section IV. Section V presents our propose heuristic algorithm an proves its approximation ratio. An improve istribute implementation of our heuristic algorithm is presente in Section VI. In Section VII, we give a simulation-base evaluation of our heuristic algorithm an compare it to other algorithms. We conclue the paper in Section VIII. II. RELATED WORK The problem of optimal operator placement has been an important research subject. One school of prior works optimizes the throughput as in [1], [1], [14]. The other school of prior works optimizes the energy cost, incluing centralize algorithms [1], [16], [17], [18] that requires the global topology information, an istribute algorithms [19], [0], [1], [], [8], [10]. Bonfils et. al. [19] propose a ecentralize algorithm for operator placement, which progressively refines the placement of operators by neighbor exploration an placement aaptation. The approach that an operator is graually move towars optimal placement is calle in-network relaxation or operator migration. Other works in the same category inclue [0], [1]. As these algorithms of operator migration are base only on local information (information from neighbors), they suffer from oscillating change, which might force the placement of an operator to a ifferent irection before reaching the optimal placement. They are also prone to local minima an they cannot guarantee the optimality of operator placement base on local information only. In [], 1-meian point is consiere as the optimal placement in network an a istribute search algorithm was propose to fin the optimal operator placement. However, this algorithm is esigne to hanle only one operator placement. Abrams et. al. [10] propose a greey algorithm, which places each operator on the noe with minimize input ata transmission cost. Obviously, the greey placement is not optimal an it can be much worse when the greey placement is backwar to the sink. Further, the istribute implementation of this greey algorithm consiere only the placement aaptation an the authors i not elaborate how to fin the initial placement for each operator in a istribute manner. In [8], Sync was propose to achieve the optimal placement for a tree-structure operator graph with optimal in-network processing cost. However, Sync requires that the network shoul have full time synchronization, an all the noes shoul know when other noes finish information upating an when they finish broacasting upate information. Furthermore, it incurs a huge message overhea for searching the optimal placement. To aress this issue, Lu et. al. [9] propose a minimum-cost forwaring base istribute algorithm (MCFA) to optimize the message overhea. Although MCFA has less message overhea than Sync, the message overhea is still very heavy. III. SYSTEM MODEL AND PROBLEM FORMULATION In this section, we first present moel assumptions for the in-network processing in WSNs, an then formulate the operator tree placement as a joint placement-routing optimization problem. A. Moel Assumptions In this research, we assume that a WSN is eploye in a two-imensional region, where each noe has capabilities of sensing, communication an computing. For sensing, each noe can observe a local variable an encapsulate the result as a ata object. For communication, each noe can communicate with other noes that fall in with a raius (r c )anthenetwork is fully connecte (no isolate noes). For computing, each noe can take all the ata objects from ownstream noes an compute a summary of these objects into another ate object, which will be forware to noe in the upstream. Uner these assumptions, we moel the WSN as an unirecte graph G =(V,E), wherev enotes the set of vertices representing sensor noes an E enotes the set of eges representing communication links between noes. The carinality of the vertex set is N. For two noes p, q V within the communication raius, we enote the ege as (p, q) E. For any ege (p, q) E, we enote C(p, q) as a chosen istance metric between the two incient noes of p an q. Examples of the istance metric inclue hop counts, square of Eucliean istance between two noes, an transmission energy, etc. Moreover, we assume the communication link is symmetric C(p, q) =C(q, p). Figure 1(a) illustrates a typical wireless sensor network, where each noe can play three roles, incluing sensing noe, relay noe an sink noe. The sensing noe collects ata object, the replay forwars information along a chosen routing

3 LU et al.: EFFICIENT DISTRIBUTED ALGORITHMS FOR IN-NETWORK BINARY OPERATOR TREE PLACEMENT IN WIRELESS SENSOR NETWORKS 74 sink sensor relay s 1 s n 1 n B n 8 n n 6 n 7 (a) s s 4 n n 4 sink sensor operator s 1 s n 1 n operator sensor ata object B n 8 n o 1 n o o n 7 6 (c) s s 4 n n 4 o root o 1 o s 1 s s s 4 Fig. 1. Sensor network (a), operator tree (b) an placement (c). Note that operator can be place on any network noe an (c) is an example of placement that might not be optimum. (b) path, an the sink noe collects all the ata objects. We assume that a set X V of noes are ata sources of interests (sensors) an the sense ata nees to be gathere at sink noe or base station B V as shown in Fig. 1(a). Moreover, in orer to reuce transmitte ata size to save energy, in-network ata aggregations or computations (efine as operators) are esire to process the sense ata along the routing paths. We assume that these sensors an operators formulate a routing tree, through which ata objects are processe an route towar the sink noe. We call it an operator tree, which is a irecte graph, enote as T = (X, O, A), where X enotes the set of sensors (ata sources), O enotes the set of operators, an A enotes the set of irecte eges. The carinality of O is t an the size of operator tree (the number of sensors an operators) is M. We assume the operator tree is a complete binary tree as illustrate in Fig. 1(b). In the rest of paper, the complete binary operator tree is simplifie as operator tree unless otherwise specifie. For each sensor x X, we enote x the size of ata object generate by sensor x. For each operator o O, we enote Δ o as the set of chilren of operator o, o the size of ata object generate by operator o, r o the ata reuction ratio of operator o, which is efine as the ratio between the size of output ata object an the sum size of input ata objects. We assume r o 1/, which is reasonable because output ata of operator is much less than input ata like query processing, stream processing, etc [], [4]. Moreover, α o is efine as the ratio of the larger input ata object size to the smaller input ata object size of operator o. Note that the final ata object (the ata object from root operator) will be transmitte to sink noe, as shown in Fig. 1(b). In this research, we focus on overlaying the operator tree atop of the WSN topology as shown in Fig. 1(c). Specifically, we ientify a subgraph (i.e., a complete tree) in the WSN topology as the operator tree an route the ata objects via the subgraph to the sink noe. In this case, we efine the in-network processing cost as the routing cost for the set of ata objects forware to the sink noe. Specifically, for a ata object i of size i route from noe p to noe q, its innetwork processing cost is i Cq p,wherecp q = C(p, i) + C(i, )+C(, j)+c(j, q), (p, i,, j, p) is routing path from noe p to q. As a result, for a given operator tree T,the total in-network processing cost can be erive as f(g, T )= a A a C a h a t, (1) where a is the size of the ata object route on arc a A, a h an a t enote the hea an the tail noe of arc a, respectively. B. Problem Statement In this research, we consier the problem of the operator tree placement. The operator tree placement problem consists of two steps. First, a subset of noes in the WSN are chosen as the set of operators. Secon, a set of paths are chosen to route the set of ata objects from their sources to the sink noe, via the set of chosen operators. Note that these two steps are not separate. The esign objective for the operator tree placement is to minimize the total in-network processing cost. Mathematically, the optimal operator tree placement problem can be state as follows. Let P enotes the operator placement, such as P (o, p) = 1 if operator o is place at noe p, otherwise P (o, p) = 0. R enotes the routing scheme that routes ata from the noes generating the ata to the noe requiring the ata. The problem is to ientify an optimal scheme of (P,R ) that minimizes the total innetwork processing cost, enote as the following: (P,R )=argminf(g, T ; P, R), () (P,R) where f(g, T ; P, R) enotes the total in-network processing cost uner a chosen placement scheme of (P, R). In this paper, we focus on eveloping efficient istribute algorithms to solve this joint placement-routing problem. In aition, we also aim to reuce the message overhea, efine as the number of messages exchange in network for istribute operator placement. IV. DYNAMIC PROGRAMMING APPROACH FOR OPTIMAL SOLUTION The optimal placement of an operator tree, mathematically, can be forme as a ynamic programming problem, which can be solve in a polynomial time. Uner the ynamic programming framework, the iterative equation for the optimal innetwork operator tree placement problem is given as follows, min ν V o c Δ o ( oc C p oc ν cost o p = +cost oc p ) if o R oc min ν V ( o Cν sink + () o c Δ o ( oc C p oc ν + cost oc p )) if o = R oc where R enotes the root operator, p enotes the optimal placement of o, cost o p enotes the optimal cost of routing all ata objects in subtree of operator o to noe p,ancost oc p oc enotes the minimum cost of routing all the ata to the subtree root noe at operator o c. This ynamic programming problem can be solve via a bottom-up approach to calculate the minimum cost to route all

4 746 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 1, NO. 4, APRIL 01 B sink s 1 s o h o g o o o 1 o Fig.. Comparison between heuristic an greey algorithm, o h chosen by heuristic algorithm an o g chosen by greey algorithm, respectively. o 1 o the ata to the root noe from a subtree. Define cost o p operator o O an p V to be the minimum possible communication cost of routing all ata objects in subtree of o to noe p (for o is root operator, cost o p is the minimum in-network processing cost of operator tree). For x X anchore at noe q, efine cost x q =0;ancost x p = if p q. The cost of cost o p can be calculate recursively, using the following equation, cost o o c Δ o min ν V ( oc Cν p + costoc ν ) if o R p = o c Δ o min ν V ( oc Cν p + cost oc ν ) + o C p sink if o = R After computing cost o p recursively, o O, p V,mapP with the set of pairs o O an p V is use in recursive unfoling of cost R p R,whereRisanchore at noe p R.Then P is the optimal placement of operator tree. This off-line iterative algorithm is centralize an its computational complexity can be calculate as O(N M). Although the algorithm can fin an optimal placement in a polynomial time, it requires the precise knowlege of the shortest path between all pairs of noes, an any change in network topology or communication link cost will incur re-execution of the searching algorithm. Moreover, although polynomial, the cost of running the algorithm may be prohibitively large for wireless sensor network of sizable imension. In fact, the ynamic programming algorithm for the optimal placement can be also solve in istribute way, such as the Sync algorithm propose in [8] an the MCFA algorithm propose in [9], respectively. Sync has a message overhea per noe of O( NM log M), an MCFA has messages overhea per noe of O( NM). Although Sync an MCFA woul fin the optimal placement in a istribute manner, their message overhea at each noe is very high. Therefore, the cost of running these istribute algorithms woul be also extremely large in WSNs. V. HEURISTIC PLACEMENT ALGORITHM In this section, we will present our propose heuristic operator placement algorithm, investigate its approximate ratio to the optimal solution an compare the performance of our heuristic algorithm an a canonical greey algorithm [10]. A. Heuristic Algorithm Given an operator tree as shown in Fig.1(b), the operators will be assigne to network noes hierarchically from bottom to top. In our research, we propose a heuristic algorithm that assigns each operator o to the noe accoring to (4): p =argmin{ o (q, sink) + i (i, q) }, (4) q i Δ o Fig.. Operator Placement of Heuristic an Optimal Algorithm where (q, sink) enotes the minimum cost between two noes. The crucial part of (4) is o (q, sink), which enables the heuristic algorithm to be equippe with the sense of irection. In this case, the heuristic algorithm will place each operator towar the sink as shown in Fig.. The key observation is that the heuristic placement is nearer to the optimal placement than the greey placement, where the operator will be hoste by the noe with minimize i Δ o i (i, q). As the final ata object of the operator tree will be transmitte to the sink, the ifference between the optimum solution an the greey algorithm is the orientation of each operator placement. Our heuristic solution graually places the operators towars the sink, while the greey algorithm constantly chooses the noe with minimize ata input communication cost. That is the reason why we a o (q, sink) rather than just i Δ o i (i, q) in (4), which actually benefits the latter operator placement an communication cost, an makes a further step of approximation to the optimal solution. o (q, sink) + i Δ o i (i, q) is calle placement cost for operator o an noe q, an enote by cost q o. As iscusse in Section IV, the placement of operators is only etermine after fining minimum in-network processing cost. As such, the placement of each operator is correlate an globally etermine. Unlike the optimal solution, our heuristic algorithm separates each operator placement an makes it locally etermine so as to reuce the computational complexity an message overhea of fining an efficient solution. The placement of operators in our heuristic algorithm epens only on the placement of chilren of the operator an the sink position. It follows that it is relatively easier to fin the placement for each operator an to implement the heuristic algorithm in a istribute manner. When the operator tree has only one operator, the heuristic algorithm fins the same placement for operator as the optimal solution. When there are more than one operator, our heuristic algorithm may incur slightly more in-network processing cost than the optimum solution. The penalty will be investigate in next subsection. B. Optimality Performance An immeiate task woul be to characterize the performance of our propose heuristic algorithm. In this section, we aopt an Eucliean cost metric to show that our heuristic algorithm is boune an its approximate ratio is lower than that of the greey algorithm.

5 LU et al.: EFFICIENT DISTRIBUTED ALGORITHMS FOR IN-NETWORK BINARY OPERATOR TREE PLACEMENT IN WIRELESS SENSOR NETWORKS 747 As shown in Fig., o 1, o an o are the operator placements of optimal solution, meanwhile o 1, o an o are the operator placements of heuristic algorithm. The ifference between the optimal placement an the heuristic placement varies accoring to ifferent ata object size an ata reuction ratio at each operator. In orer to evaluate our heuristic algorithm, we give its approximate ratio to the optimal solution. It is known that (4) will be the three factory problem 1 [] when the operator has two chilren as shown in Fig.. Note that the three-factory problem is applicable for any istance metric. Before proving the approximate ratio, first we introuce two lemmas about three factory problem, which help us conclue the further theorems. Lemma 1. When r 1, the solution point of the three-factory problem is locate in area BOC enclose by arc BOC an line BC as shown in Fig. 4, where O is the Fermat point of triangle ABC. Proof. To locate the Fermat point: construct two equilateral triangles on any of the three sies of the given triangle; construct the circumscribe circle for each equilateral triangle; the two circles intersect at the Fermat point. As shown in Fig. 4, arc BOC an arc ÂOB are part of circumscribe circle of constructe equilateral triangle for BC an AB, respectively. For point O an O arc BOC, BOC = 10 o an BO C = 10 o. We use a, b an c to enote the weight at point A, B an C as shown in Fig. 4, an r = a b + c (note that in our real scenario, a is the size of ata object generate by operator an r is the ata reuction ratio of operator). For the three factory problem, we have the following [6]: cos( B BC) = a b c. b c To solve the three-factory problem, we can construct a circle by using point B, C an BB as a tangent line. Similarly, we can construct another circle for A, B an A A. Then the intersection of two circles insie the triangle ABC is the solution [6], as O in Fig. 4. Note that when the weight of one point is no less than the sum of other two, the solution is locate at the point with the largest weight [6]. When a = b = c, B BC = 10 o, the solution is point O as shown in Fig. 4, which is also known as the Fermat point. We call the area inclose by arc BOC an line BC Fermat area. When r 1/, wehave cos( B BC) ( b+ c ) b c 1 b c. As B BC 10 o thus BO C 10 o ( B BC = BO C), arc BC, which is a part of circle constructe using B, C an BB as a tangent line, will be in the Fermat area. So the three factory solution point O BC is also in the 1 Three factory problem can be state as follows: Given three points, a, b, c. Each one is connecte with a fourth point, u. Suppose the length of ua = r 1, ub = r,anuc = r. Choose the point u so that k 1 r 1 + k r + k r is minimize. k 1, k, k, are arbitrary positive weighing factors. Fermat point is a point such that the total istance from the three vertices of the triangle to the point is the minimum possible. Fig. 4. The Three Factory Problem Fermat area. For example O in Fig. 4 is the solution point for a =1, b =1. an c =0.8. One implication of Lemma 1 is that the optimal operator placement always locates insie the Fermat area when r 1/. Lemma. As shown in Fig. 4, the point O insie the Fermat area, which maximizes b BO + c CO, is locate on arc BOC an eterministic, where α = b / c (assume b c ). Proof. As b BO + c CO inclues istances BO an CO, we can quickly infer O must be on arc BOC. As O arc BOC an θ enotes O BC, wehave sin10 o BC = sinθ O C = sin(60o θ) O. () B From (), we get b BO + c CO = BC [ b (cos θ 1 sin θ) + c sin θ]. To maximizes b BO + c CO, we nee to maximize: max { b(cos θ 1 sin θ)+ c sin θ}, (6) θ an max {α cos θ α sin θ + sin θ}, (7) θ max{sin(θ + φ)}, tan φ = θ α α. (8) As 0 θ<60 o,when1 α, from (8) we have θ = π φ. So point O can be etermine by α. For example, when α =1, θ =0 o,anwhenα =, θ =0, in other wors point O locates at point C. Whenα>, tan φ = α α >,so φ< 60 o.as0 θ<60 o, b BO + c CO will be a negative value. So (8) is not applicable for α>. However, since o is locate at C when α =, we can conclue that O is always locate at C when α>. The term of b BO + c CO can be seen as the communication cost in the operator placement context. It follows that the point in the Fermat area that maximizes the communication cost can be erive as:

6 748 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 1, NO. 4, APRIL 01 It locates at the mipoint of the arc of the Fermat area when α =1; It locates on the arc between the mipoint an the point with the smaller ata output when 1 <α<; It locates at the point with the smaller ata output when α. We enote C (T ), C(T ) an Ĉ(T ) as the in-network processing cost of the optimal solution, the heuristic algorithm an the greey algorithm, respectively, for the operator tree T, where each operator is compatible with the constraint that r 1/. The following theorem summarize the approximate ratio for our heuristic algorithm. Theorem 1. For an operator tree T an an ata reuction ratio at each operator is no more than 1/, the performance of our propose heuristic algorithm is boune above: C(T ) α α +1C (T ) if 1 α C(T ) αc (T ) if <α< C(T )=C (T ) if α Proof. Case 1: if 1 α From Lemma 1, it is known that for an operator with r 1/, the optimal operator placement (the solution of the three-factory problem) is locate insie the Fermat area. The operator placement with the minimum communication cost for the input ata is B, where the communication cost C β = c BC. At the same time, when 1 α, from Lemma, we have the worst case with the maximum communication cost of input ata C γ = b BO + c CO, where O BC = π α arctan α. (9) It follows that C γ C β = b BO + c CO c BC From () an (9), we can obtain C γ C β = α BO + CO. BC = α sin(60 θ)+ sin θ = ( cosθ sin θ)α +sinθ = α α +1. Assume that optimal solution always has the best case of communication cost of input ata for each operator an the heuristic solution always has the worst case. From (), we can obtain C(T ) C (T ) α α +1. Case : if <α< When α>, from Lemma we have O, which maximizes b BO + c CO, is always locate at point C. Anthe best case for the communication cost of input ata is still B. So we have C γ = b BC c BC = α. C β In network processing cost, (C*(T)) Greey Heuristic r Fig.. Comparison of in-network processing cost between heuristic algorithm an greey algorithm for T, where all the operators have r 1/, when α =1. C(T ) Similarly, C (T ) α. Case : if α As r = a b + c 1 an α, by summing up we can obtain b a + c. In the three-factory problem, when b a + c, the solution is always locate at point B, no matter where point A is locate [] as shown in Fig. 4. Although our heuristic algorithm uses the sink noe as the parent for each operator placement ecision, in this case, the resulte operator placement is always the chil that has larger ata output, which is the optimal placement for the operator. If the placement of an operator tree is globally optimal, the placement of each operator is also locally optimal with respect to its chilren an the parent operator [16]. In this case, the optimal solution an the heuristic algorithm have the same placement for each operator, C(T )=C (T ). From Theorem 1, we can see the in-network processing cost of the heuristic algorithm is at worst three times of that of the optimum solution (C(T ) < C (T )). For the greey algorithm, Ĉ(T ) 1 1 r C (T ), whenr<1/an α =1(in [10], only the results about α =1are provie). As shown in Fig., the heuristic algorithm performs much better than the greey algorithm for α =1. The processing cost resulte from the greey algorithm increases ramatically with the increase of r. It follows, from the mathematical analysis above, that our heuristic algorithm has much better approximate ratio to the optimal solution than the placement resulte from the greey algorithm. VI. LOW-OVERHEAD DISTRIBUTED IMPLEMENTATION As shown in Section V, when the ata reuction ratio of an operator is no more than 1/, the solution of the threefactory problem is locate in the Fermat area, no matter where the parent of the operator is. It follows that we only nee to fin the heuristic operator placement within the Fermat area. In this section, we escribes an efficient way to fin the placement for each operator in the network, using our

7 LU et al.: EFFICIENT DISTRIBUTED ALGORITHMS FOR IN-NETWORK BINARY OPERATOR TREE PLACEMENT IN WIRELESS SENSOR NETWORKS 749 C receiver = C s s C s sener + C sener / C s C s sener + C sener / C s C s receiver = Cs Fig. 6. An illustration of the search region involve in the area-restricte Flooing heuristic algorithm in a istribute manner. This is achieve by proposing an area-restricte flooing mechanism to locate the operator placement within a esire area, minimizing the number of noes involve in flooing an reucing the message overhea. A. Area-restricte Flooing In orer to minimize the messages overhea of the istribute implementation of our heuristic algorithm, we propose an area-restricte flooing approach to locate each operator placement. As shown in Fig. 6, s an enote the two chilren of an operator that nees to be place, the area enclose by the ashe line is calle the search region, two times of the Fermat area. Weusethesearch region instea of the Fermat area, because the noes o not have the location information of the sink an the esire placement can be at either sie of line s. Thus, using the search region can guarantee to fin the esire heuristic operator placement. Notice that the flooing area is set up base on the cost between noes, instea of geographic information. For example, Fig. 1 can be seen as a coorinate, where the location of a noe is base on its minimum costs to noe s an to noe. In this case, our propose area-restricte flooing mechanism oes not require any location information. Assume that every noe knows the minimum communication costs to both the sink an noe (we will show how to obtain these information in next subsection), noe s will initialize an area-restricte flooing algorithm to fin a noe as the operator placement within the search region, which minimizes (4). We call the noe that starts the flooing an initiator, like noe s in Fig. 6, an call the noe that finishes the flooing a terminator, like noe in Fig. 6. To fin the heuristic operator placement an minimize the message overheas, we nee to aress several issues, incluing 1) How to control the flooing within the search region? ) In orer to fin the operator placement accurately, the minimum cost fiel of an initiator must be set up for the noes in the search region. In other wor, every noe in the search region nees to know the minimum cost to the initiator. Then the noes can properly calculate the placement cost accoring to (4). ) In orer to get the minimum placement cost of noes in the search region, a straightforwar way is to collect all the placement costs at one noe. However, the process for each noe to sens its to one particular noe will incur huge amount of message overhea, one nees to evelop an intelligent strategy to efficiently collect all the costs an fin out the noe with the minimum placement cost in the search region? 4) How to combine the proceures of setting up the minimum cost fiel of the initiator an fining operator placement into one flooing roun? Area-restricte floo for searching each heuristic operator placement is initiate by broacasting an avertisement (ADV) packet at the initiator. It floos from the initiator to the terminator. ADV is transmitte uring area-restricte flooing, which carries the following information: C s : the minimum cost between the initiator an the terminator. Csener s : the cost between the initiator an the current sener that broacasts the packet. cost o : the current minimum placement cost obtaine at the sener. In orer to restrict the flooing area, when the noe receives an ADV packet, it will re-broacast the packet only if the following two conitions are satisfie: Csener s +C sener / Cs : From Lemma, we know that all the points in the search region satisfy this conition as shown in Fig. 1. Although some points outsie the search region may also satisfy this conition, we inten to use it to simplify the computation of guaranteeing the flooing covering the search region. Csener s <Cs receiver Cs an C receiver <C sener Cs :Tomaketheflooing forware from the initiator to the terminator an avoi evious path an loop that the packets may travel, these constraints will straighten the traveling paths from the initiator to the terminator. In other wor, the receivers are suppose to have a smaller minimum cost to the terminator than the sener, an a higher minimum cost to the initiator than the sener. The aforementione proceure aresses the first issue, which restricts the flooing in the area enclose by the soli line as shown in Fig. 6 in a simple an efficient way. To tackle the secon issue, although a straightforwar flooing will eventually fin the minimum cost path to the initiator for each noe, it will incur significant amount of message overhea. Here we aapt the cost fiel establishment algorithm [7] to set up the cost fiel for the initiator. A backoff timer is enable after the noe receives ab ADV packet, the expire time of which is set to be proportional to the cost between the sener an the receiver (time coefficient enote by λ) as follows: T elay = λc(p, q). During T elay, the noe might receive more than one ADV packet. However, the noe will only broacast the ADV with the least cost to the initiator an iscar all other ADV packets receive after the timer expires. We use Fig. 7 as an example to illustrate how the cost fiel establishment algorithm works to reuce the ADV broacast, explaine as follows: At time t, noe a broacasts an ADV packet that inclues Cs a. After noe b, c an receive the ADV packet from noe a, theysetc(s, b) =Cs a +4, C(s, c) =Cs a +,an

8 70 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 1, NO. 4, APRIL 01 C(s, ) =Cs a +, λ. C(s, c) =Cs a +, λ c C(s, ) =Cs a +4, 4λ b 4 a Time t after noe a broacaste ADV (a) C(s, ) =Cs a +4., 0.λ. Cs c = Cs a + c Cs b = Cs a +4 b 4 a C a s C a s Time t +4λ after noe b broacaste ADV (c) C(s, ) =Cs a +4.,.λ. Cs c = Cs a + c C(s, b) =C a +4, λ b 4 a Time t +λ after noe c broacaste ADV (b) Cs = C a +4.. Cs c = C a + c Cs b = C a +4 b 4 a C a s C a s Time t +4.λ after noe broacaste ADV () Fig. 7. Example of cost fiel establishment algorithm. C(s, ) =Cs a +, respectively (assuming the initial Cs, b Cs b an Cb s are as they o not have cost information to the initiator). Then each of them sets a timer for rebroacasting ADV. The expiring perio is proportional to the transmission cost between the sener an the receiver. For noe b, c an, the expiring perios are 4λ, λ an λ, respectively. At time t +λ, the timer of noe c expires. Noe c finalizes Cs c = C(s, c) an broacasts an ADV message incluing Cs c. When noe receives this ADV packet, as C(s, ) >Cs c+. =Ca s +4., noe upates C(s, ) to Cs a +4. an resets its timer to.λ (note that previous timer oes not expire by the time t+λ). For noe a an b, ascs a <Cs c + an C(s, b) <Cs c +, they simply iscar this ADV message. At time t +4λ, the timer of noe b expires. Noe b finalizes Cs b = C(s, b) an broacasts an ADV message containing Cs b. All other noes will iscar this ADV message because they have alreay ha a lower cost. At time t +4.λ, the timer of noe expires. Noe finalizes Cs = C(s, ) an broacasts an ADV message with Cs. For this cost fiel establishment algorithm, we can observe from Fig. 7 that each noe only broacasts an ADV message once with the optimal cost an eliminates the broacast of non-optimal ADV messages. The algorithm has the following two properties [7]: Each noe only broacasts the optimal cost to the initiator, an iscars all reunant or non-optimal ADV messages. Noe can get the minimum cost to the initiator by only one ADV message broacast at each noe. For the thir issue, we use the terminator to collect the minimum placement cost in the restricte area. Instea of forwaring the placement cost at each noe iniviually, we propose to aggregate the placement cost information en route an only forwar aggregate information to the terminator. Once a noe gets the minimum cost to the initiator, it can calculate its placement cost. So it is efficient to inclue the placement cost in the ADV packet rather than sen iniviual message to the terminator. However, as iscusse above in the cost fiel establishment algorithm, since the noe will abanon all the messages receive after the backoff timer expires, ADV packets are eliminate uring flooing. In orer to merge the establishment of the cost fiel for the initiator an the process of locating the operator placement together so as to reuce the message overhea collaboratively, we introuce the following process into the area-restricte flooing: The noe broacasts an ADV with the minimum C s sener an mthe inimum placement cost receive uring the backoff perio. After the backoff timer expire, the noe only rebroacasts its ADV packet, if the placement cost of which is less than the current obtaine one. During the flooing, the noe will cache the least placement cost an its sener. Or if the least placement cost is

9 LU et al.: EFFICIENT DISTRIBUTED ALGORITHMS FOR IN-NETWORK BINARY OPERATOR TREE PLACEMENT IN WIRELESS SENSOR NETWORKS 71 generate by itself, it woul cache itself as the generator of the least cost. This will be use to trace back the operator placement later. For completeness, the pseuo coe of the area-restricte flooing is shown in Algorithm 1. Since the floo area is set up base on the cost, at least two noes (initiator an terminator) are inclue in the floo area. Although the minimum cost to the initiator an the placement cost are both inclue in the ADV message that can be eliminate after the timer expires, we give the priority to the ADV message with less placement cost an make it forware to the terminator. After receiving the minimum placement cost, the terminator performs a trace-back process to fin the operator placement. Proposition 1. For a pair of initiator an terminator, arearestricte flooing can locate the heuristic operator placement within time λ C s. Proof. The proof can be ivie into three sequential steps. First, we will show that the noe that is operator placement is inclue in the floo area. Secon, the placement cost of a noe can be receive at the terminator. Finally, the elay is no more than λ Cs. For the first part, any noe p V with Cp s + C p Cs (the point in the Fermat area) will be inclue in the floo area as shown in Fig. 6. Therefore, the noe that is operator placement is certainly inclue in the flooing area as prove in Lemma 1. As iscusse above, the noe can get its minimum cost to the initiator with broacasting once an only once. Since before broacasting the ADV message the noe has receive the minimum cost to the initiator, the placement cost of the noe is accurately compute. Assuming noe q is the placement of operator o, cost q o is minimum. cost q o is inclue in ADV message an broacaste. When the neighbor noe p in the floo area, which is complie with Cq >C p (recall that is use to avoi evious path an loop), receives this ADV an forwar it towar the terminator no matter whether the time expires or not. Therefore, the minimum placement cost will be eventually message, it will replace the cost o with cost q o receive at the terminator. As Cq s + Cq Cs, the elay of the message incluing cost q o will arrive at the terminator no later than λ Cs after the area-restricte flooing is initialize, accoring to the scheme of the cost fiel establishment algorithm. B. Distribute Operator Tree Placement As iscusse in Section VI-A, before the initiator initializes the area-restricte flooing to fin the operator placement, the prerequisite is that all the noes within the search region know the minimum costs to the sink an the terminator. Unlike in [10], where the authors assumes the noes are knowlegable about the istance between noes, we o not assume the noes have any location information so as to make our istribute algorithm applicable. We also use the cost fiel establishment algorithm to obtain these minimum cost information. Algorithm 1: Area-restricte Flooing Algorithm Event: noe p receives ADV from noe q 1 begin if Cq >Cp && Cq s + Cp + C(p, q) Cs then if Cq s C s && Cs q + C(p, q) <Cp s then 4 reset timer to expire after λ C(p, q) Cp s = Cq s + C(p, q) 6 calculate cost p o 7 if cost o >cost p o then 8 cache p 9 else 10 cost p o = cost o 11 cache q 1 en 1 else if Cq s Cs then 14 if cost p o <cost o then 1 cost p o = cost o 16 cache q 17 en 18 en 19 en 0 en Event: noe p s timer expires 1 begin replace C s sener with Cs p replace cost o with cost p o 4 broacast ADV en Event: noe p receives ADV from q after p s timer expire 6 begin 7 if Cq s C s && C q >Cp && Cp s + Cp Cs then 8 if cost o <cost p o 9 replace cost p 0 cache q 1 broacast ADV en en 4 en o then with cost o Another issue about the area-restricte flooing is when the initiator shoul initiate the floo an when the terminator shoul finalize it. As the elay at each noe is proportional to the communication cost between the sener an the receiver in the cost fiel establishment algorithm, the longest elay of the cost fiel establishment is λc max, where C max is the largest minimum cost path in the network. In orer to avoi reunant message an to keep it from the effect of transmission, propagation an processing elay, λ is a constant value for certain network as shown in [7]; in turn λc max is also etermine. Therefore, for initializing arearestricte flooing, the initiator waits for a λc max elay after obtaining the minimum cost to the terminator. For finalizing area-restricte flooing, the terminator will receive the first

10 7 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 1, NO. 4, APRIL 01 ADV message λcs time later after the initiator starts the flooing an it will wait for a elay ( 1)λCsource before finalization. To prevent the escription from becoming cumbersome we have omitte the etails that can either be inferre or implemente using stanar techniques. These etails inclue the precise content of the messages, the manner of calculating the hosting cost, the elay before performing the initialization an finalization of area-restricte flooing an noe s behaviors uring cost fiel establishment flooing an area-restricte flooing. The istribute implementation of our heuristic algorithm is shown as follows accoring to ifferent noe types. It will fin the placement of an operator tree in a bottom-up manner. Note that noe type is not exclusive, that is, the noe can be more than one type at the same time. For example, when a noe is both the initiator an the terminator, it will select itself as the operator placement. SINK NODE: Event: When there is a in-network processing with operator tree (T ) nees to be performe. Action: Foreach operatorof T, assign its two chilren as initiator an terminator of area-restricte flooing, respectively. Broacast T into network using cost fiel establishment flooing. Thus, every noe will receive T an be aware of the minimum cost to sink. SENSOR NODES an OPERATOR PLACEMENT: Event: When sensor noe receives T or the noe is selecte as operator placement. Action: If the place operator is root operator, inform sink that the placement of operator tree T is complete, else perform the cost fiel establishment flooing if it is terminator. INITIATOR: Event: After being aware about the minimum cost to terminator. Action: Perform area-restricte flooing. TERMINATOR: Event: After obtaining the minimum placement cost for current operator. Action: Perform trace-back process to inform the noe with minimum placement cost an assign it as operator placement. As the cost fiel establishment flooing is use to obtain the minimum cost information in the istribute implementation, which establishes the minimum cost file with only one message broacast at each noe in one flooing roun, the message overhea at each noe for the minimum cost information is the number of floos. The message overhea of one area-restricte flooing roun is much smaller than one roun of the cost fiel establishment flooing, because the search region is much smaller than the entire network fiel an the message overhea of the area-restricte flooing is no more than / that of the cost fiel flooing for the same area as shown in [9]. As the number of the cost fiel establishment flooing roun an the number of the area-restricte floo rouns are (M +1)/ an (M 1)/, respectively. It follows that the message overhea per noe is O(M). As a comparison, for Sync an MCFA, the message overhea is of O( NM log M) an O( NM), respectively. VII. NUMERICAL EVALUATION In this section, we evaluate the performance of our heuristic algorithm through simulations, an compare their performance among the heuristic algorithm, the greey algorithm, Sync an MCFA. Two basic network topologies are use in simulation, which are the Manhattan Graph (MG) an Controlle Ranom Graph (CRG). These graphs represent both abstract an realistic sensor network topologies. In orer to make the network topologies more realistic, we choose to egrae the connectivity. This can be one for MG by ranomly removing a percentage of the noes an for CRG by increasing the area in which the network is eploye so as to reuce the number of reachable neighbors. The CRG is generate by ranomly istributing network noes in a square area. When placing a noe, we aopt the approach in [19] an we require the placement of the noe is at least half of raio range to any other noe so as to avoi unrealistically close noes. The length of a sie of the square area can be calculate by Nrc f, where N is the number of network noes, r c is raio range (0m in simulation) an f is a factor to control the connectivity. In the simulation, three particular network topologies each with 00 noes are use in simulation as shown in Fig. 8. In simulation, we use pq as the communication cost between two ajoining noes p an q, where pq is the istance between noe p an q an pq 0m, i.e., signals attenuate inversely proportional to the square of istance. For real applications, the sener inclues its broacasting power in its ADV message. When a receiver hears the message, it can calculate the minimum energy neee an the minimum transmission power neee for the sener to just reach the receiver by measuring the signal strength an employ one chosen signal attenuation moel, then acknowlege the sener with these information. As mentione before, we o not have any constraint for the communication cost, which can take any common form, such as hop count or consume energy. We firstly evaluate the impact of parameters incluing α an network topology on the operator placement for both heuristic an greey algorithms, then we compare message overheas of istribute implementation among heuristic, Sync an MCFA. A. In-network Processing Cost As analyze mathematically in V-B, α impacts the approximate ratio of the heuristic algorithm. To verify our mathematic analysis, we simulate all three algorithms in three network topologies an varie α from 1 to.. Notice that we use Sync as the optimal solution to obtain the minimum in-network processing cost.

11 LU et al.: EFFICIENT DISTRIBUTED ALGORITHMS FOR IN-NETWORK BINARY OPERATOR TREE PLACEMENT IN WIRELESS SENSOR NETWORKS 7 (a) Controlle ranom graph (f=0.6) (b) Controlle ranom graph (f=0.8) (c) Manhattan graph with % holes Fig. 8. Network Topologies As shown in Fig. 9, we can observe that, the ratio between the processing cost resulte from both heuristic an greey algorithms an the resulte from the optimal algorithm, ecreases overall as α increases from 1 to. It is expecte from our mathematical analysis in Section V-A. Moreover, the heuristic algorithm performs better than the greey algorithm(at least no worse than the greey algorithm) in all these three topologies. In CRG (f=0.6) an CRG (f=0.8) shown in Fig. 9(a) an 9(b), the heuristic algorithm an the greey algorithm achieve the same optimal cost when α reaches a threshol, α =.4 an α =.6, respectively. In Fig. 9(c), the heuristic algorithm obtains the optimum cost much earlier (at α =.) than the greey algorithm (at α =.0). As analyze in Section V, when a an the ata reuction ratio of operator is no more than 1/, the optimal operator placement is the noe with the largest ata output, as verifie by our simulation results. The intersection point of these three algorithms is that when a, all of them choose the noe with the largest ata output as the operator placement, so both heuristic an greey algorithms can obtain the optimal cost at most when α is. From Fig. 9, we can also notice that the worst-case innetwork process cost for the greey algorithm varies accoring to ifferent network topologies, while the worst case of the heuristic algorithm in each topology is when α =1,which is not compliant with the mathematical analysis in Section V. That is because we always choose the worst case of the operator placement as the placement of our heuristic algorithm in mathematical analysis, which woul never occur in real network scenarios. B. Message Overhea The message overhea is the metric to measure the cost of performing istribute implementation of operator placement algorithms. In this subsection, we compare the message overheas for our heuristic algorithm, Sync an MCFA. Fig. 10(a) illustrates the message overheas per noe for these three algorithms, as a function of operator tree size for a given network size (i.e., N = 00). From Fig. 10(a), we can see the message overhea increase as the operator tree size increases. We can also see that the heuristic algorithm has much less message overhea than that of Sync an at least 0% less than that of MCFA. Finally, the gap increases as the operator tree increases. In Fig. 10(b), we compare the message overhea per noe for the heuristic algorithm, MCFA an Sync, as a function of the network size with a fixe-size operator tree (i.e., M =7). As analyze mathematically, the message overhea of Sync an MCFA increases with the network size, while for the heuristic algorithm, the message overhea stays flat for increasing network size. Notice that, although the heuristic algorithm causes slightly more in-network processing cost for the operator tree than that of the optimal solution, the istribute implementation of our heuristic algorithm has much less message overhea than that of istribute optimal solutions (i.e., Sync an MCFA). The complexity reuction in managing the message overhea woulbetranslateintoenergysavinginwsns,whichin turn, woul justify the rationale of eploying our heuristic algorithm. VIII. CONCLUSION The problem of operator tree placement is crucial for innetwork processing in wireless sensor networks. In the paper, we esigne a heuristic algorithm to place the operator tree on network noes. First we presente the mathematical analysis of our propose heuristic algorithm compare with the optimal solution. We then followe with a low-cost istribute implementation of our istribute heuristic algorithm an analyze its message overhea, which is of O(M) an is much less than that of other istribute algorithms. Finally, the simulation results verifie that our heuristic algorithm has better performance than the greey algorithm an incurs slightly more cost than the optimum solution. The results also inicate that the heuristic algorithm has much less message overhea than that of other istribute placement algorithms (i.e., Sync an MCFA). Our research, with its practical algorithm an funamental analysis, she new lights onto the in-network processing paraigm, which are pervasive in moern communication an network infrastructure. As future works, three alternative venues are uner consierations. First, we woul like to exten our binary tree structure into a M-ary tree structure, which provies an elevate capability for ata fusion at operators. Secon, we plan to explore the possibility of joint optimization of energy efficiency an loa balancing in large wireless sensor networks. Finally, we are also exploiting this funamental research into

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Ye, A. Chen, S. Lu, an L. Zhang, A scalable solution to minimum cost forwaring in large sensor networks, in Computer Communications an Networks, 001. Proceeings. Tenth International Conference on. IEEE, 001, pp Yonggang Wen (S99-M08) was born in Nanchang, Jiangxi, China in He receive his PhD egree in Electrical Engineering an Computer Science (with minor in Western Literature) from Massachusetts Institute of Technology (MIT), Cambrige, MA, USA, in 008; his MPhil egree (with honor) in Information Engineering from Chinese University of Hong Kong (CUHK), Hong Kong, China, in 001; an his BEng egree (with honor) in Electronic Engineering an Information Science from University of Science an Technology of China (USTC), Hefei, Anhui, China, in His major fiel of stuy focuses on information an communication technologies (ICT). He is currently an Assistant Professor with School of Computer Engineering at Nanyang Technological University, Singapore. Previously, he has worke in Cisco as a Senior Software Engineer for content networking proucts. He has also worke as a Research Intern at Bell Laboratories, Sycamore Networks an Mitsubishi Electric Research Laboratory (MERL). He has publishe more than 0 papers in top journals an prestigious conferences. His system research on Clou Social TV has been feature by international meia (e.g., The Straits Times, The Business Times, Lianhe Zaobao, Channel News Asia, ZDNet, CNet, Unite Press International, ACM Tech News, Times of Inia, Yahoo News, etc). His research interests inclue clou computing, mobile computing, multimeia network, cyber security an green ICT. Dr. Wen is a member of Sigma Xi (the Scientific Research Society), IEEE an SIAM. Rui Fan is currently an assistant professor with School of Computer Engineering at Nanyang Technological University. He receive BSc from California Institute of Technology, MSc an Ph.D. both from Massachusetts Institute of Technology. Zongqing Lu is currently a Ph.D. caniate with the School of Computer Engineering in Nanyang Technological University, Singapore. He receive bachelor an master egree both from Southeast University, China. His research interests inclue wireless sensor networks, mobile a hoc networks, social networks, elay tolerant networks, mobile computing, network privacy an security. He is a stuent member of IEEE. Su-Lim Tan is currently an assitant professor with Singapore Institute of Technology. He receive bachelor egree from computer engineering in Nanyang Technological University an Ph.D. egree from University of Warwick, UK. Jit Biswas is a Senior Scientist in the Neural an Biomeical Technology (NBT) epartment at the Institute of Infocomm Research, A*STAR (Agency for Science, Technology an Research), Singapore. He is currently working on vital signs monitoring using ambient sensing approaches an wireless sensor networks. Dr. Biswas has an unergrauate egree in Electrical an Electronics Engineering from Birla Institute of Technology an Science, Pilani (Inia) an a Ph.D. egree in Computer Science from the University of Texas at Austin (USA).