Querying Uncertain Minimum in Sensor Networks

Transcription

1 Querying Uncertain Minimum in Sensor Networks Mao Ye Ken C. K. Lee Xingjie Liu Wang-Chien Lee Meng-Chang Chen Department of Computer Science & Engineering, The Pennsylvania State University, University Park, USA Institute of Information Science, Academia Sinica, Taipei, Taiwan Abstract In-network aggregation is an important energysaving technique for wireless sensor networks. However, existing approaches designed for collecting precise sensor data do not work well when uncertain data is considered. In this paper, we study two types of probabilistic queries, namely, Probabilistic Minimum Value Queries (PMVQs and Probabilistic Minimum Node Queries (PMNQs. A PMVQ determines the possible minimum value(s among all sensed readings, while a PMNQ identifies sensor nodes which have the minimum readings. For PMVQ, in-network filtering is enabled by introducing the notion of MaxiMin, which identifies the largest possible minimum values. Besides, by exploring the decomposability of PMVQs, we propose a novel in-network minimum value aggregation algorithm, namely MVA, that facilitates efficient reading suppression and message compression within sensor networks. For PMNQs and its variants, namely, threshold-based PMNQ (p-pmnq and top-k PMNQ (k-pmnq, we develop two suites of algorithms based on two-phase and one-phase approaches. The two-phase algorithms first issue a PMVQ, followed by a simple probing query, while the one-phase algorithms make filtering decisions based on partial global knowledge in one run. We conducted a comprehensive performance evaluation via simulation. The experimental results validate our ideas and demonstrate the efficiency of our approaches. I. INTRODUCTION Wireless sensor networks (or simply sensor networks are powerful instruments to sense and to monitor remote physical environments such as coal mine, farmland, wildlife habitats etc. A typical sensor network is composed of a base station, a terminal host that issues queries to and collect query results from a collection of sensor nodes deployed over a spatial area. Sensor nodes are battery-powered computing devices equipped with sensor modules and wireless communication interfaces. With sensor modules, sensor nodes measure the environmental conditions, such as temperature, humidity, UV index, etc., of their surroundings. Through those collected readings, scientists can study the remote environments. Very often, the aggregated data, e.g., the average, minimum and maximum of sensed readings, rather than every single sensed reading are acquired. The reasons are two folded. First, aggregated data can provide a quick high-level information to the scientists. Second, to conserve scarce sensor node energy and to alleviate communication costs, aggregated readings are favorable since aggregated data are much smaller to transmit than the raw sensed readings [5], [], [], [], [5], [7]. Because of the dynamics of physical environments and possible hardware defeats, sensed readings collected and reported by sensor nodes are inherently inaccurate and imprecise. In other words, sensed readings can only reflect approximate measurements about environments that, to some extents, are considered to be uncertain [8], [7]. To deal with this kind of data uncertainty, research studies such as [], [], [9], [], [], [] focus on querying on centralized uncertain databases, assuming that all uncertain data are available in a single host. Indeed, it is an impractical assumption to have energy-scare sensor nodes convey all their sensed readings frequently. This raises a demand for in-network uncertain data aggregation approaches in sensor networks. To the best of our knowledge, this in-network uncertain data aggregation problem is new and has not yet been explored. In this paper, we study in-network uncertain data aggregation with a particular focus on applying the minimum aggregate function on sensed readings, because in many applications, knowledge about the minimum sensed reading and/or sensor nodes who providing the minimum values are very important. Take crop quality control as an example. To farmland management, keeping the soil moisture, i.e., one of growth factors to the crop quality is an important task. With sensor networks for soil moisture detection, the farmland management can examine (i whether the minimum soil moisture in the farm is above an alarming level, and (ii what portion of the farm needs watering. Assuming sensed readings are probabilistic such that possible sensed reading values are associated with likelihoods of being the actual measurements, we introduce Probabilistic Minimum Value Queries (PMVQs for (i to check on minimum values, and Probabilistic Minimum Node Queries (PMNQs for (ii to find the sensor nodes with minimum readings, in this paper. Both PMVQs and PMNQs determine the query results based on the minimum sensed reading. However, since their search objectives are not the same, the research issues and techniques to be developed for these queries are naturally different. To illustrate the challenges of PMVQs and PMNQs as well as to provide a quick view of our proposed solutions in this paper, let us consider an example sensor network (in Figure that consists of a base station and four sensor nodes, namely, n, n, n and n, and their sensed readings (see Table I. We consider that each uncertain sensed reading r i provided by a sensor node n i is a set of probabilistic reading values. For example, r (the sensed reading of n consists of two reading values, 5 and 6, with their associated probabilities.5 and.5, respectively. The probability associated with an value v indicates the likelihood that v is the actual measurement. Example : PMVQ. Suppose that a PMVQ is issued to determine the minimum value in this example sensor network. Here, there are reported sensor reading values, namely,,, 5 and 6 as listed in Table II. The maximum aggregate function can be treated similarly.

2 Base Station p.6. n n p probability distribution.5.5 of sensed reading values r n r 5 6 v p p v 5 6 v 5 6 v r r Sensor nodes Fig.. n An example sensor network Sensor node Sensed reading Sensed reading values (v i,j, p i,j n r {(5,.5, (6,.5} n r {(,., (,.6} n r {(,., (5,., (6,.} n r {(,., (,., (5,.6, (6,.} TABLE I EXAMPLE SENSED READINGS Since there is no precise sensed reading values, the exact minimum value, v min, can be any value among {,, 5, 6}. We approach this issue by determining the probability of every value v to be the minimum, denoted by P r[v min = v], and then return the values which have non-zero probabilities. The possible minimum values v along with their corresponding probabilities P r[v min = v] in this example are listed in Table II. As shown, the result of the PMVQ is {(,.6, (,.5}. In Example, the difference of PMVQs from conventional minimum aggregation queries is clearly shown. Conventional ones return a single minimum value while data are certain. Differently, PMVQs need to transmit a lot of sensed reading values if the probabilities for those values to be the minimum are not zero. In addition to comparison between values, probability computation is invoked. Although PMVQs are more complex than conventional aggregation queries, we can see that in-network processing of PMVQs is feasible. For example, reading r, that does not have any possible value likely to be the minimum due to the presence of r, can be safely discarded within the network. To quickly identify redundant readings, we introduce a notion of MaxiMin, which represents the maximum value among all the possible minimum values. Therefore, any sensed reading value, which is greater than MaxiMin, should have a zero probability to be the minimum and can be dropped. As will be explained later, the evaluation of a PMVQ is decomposable so that partial results can be directly used to derive a final query result. Thus, these partial results can be transmitted in place of raw sensed readings during the in-network processing. In light of these, we devise an innetwork minimum value aggregation algorithm, which enables in-network reading suppression and message compression. Example : PMNQ. In the example sensor network, a PMNQ is issued to search for those sensor nodes which provide the minimum values. The formula for computing the probability that a node n i has the minimum value, denoted by P r[v min r i ], will be detailed later in Section II. Here, the sensor nodes that provide the minimum sensed reading values are listed, along with their probabilities, in Table III. Sensed reading values (v 5 6 Probability P r[v min = v] TABLE II THE VALUE SET AND THEIR PROBABILITIES BEING THE MINIMUM As shown, those sensor nodes with probabilities greater than zero form the PMNQ result set, i.e., {(n,.9, (n,.6, (n,.6}. Sensor node (n i n n n n Probability P r[v min r i ] TABLE III THE PROBABILITIES OF SENSOR NODES PROVIDING THE MINIMUM VALUE Further, in a large-scale sensor network, multiple sensor nodes may have the same sensed reading value simultaneously. As such, a number of sensor nodes can be the result of a PMNQ. In case that all sensor nodes may not have equal likelihood to provide the minimum sensed reading value, we consider two extensions of PMNQs that can reduce the result set size and thus the energy expenditure by filtering out those with small probabilities to provide the minimum value. The first extended PMNQs are called threshold-based PMNQs that returns result sensor nodes only when their probabilities exceed a specified threshold. The second extended PMNQs are called top-k PMNQs. A top-k PMNQ searches for k sensor nodes whose probabilities to provide the minimum value are the greatest. To deal with these PMNQs and two types of extended PMNQs, we consider two suites of algorithms. By extract the decomposable part from the PMNQ aggregate, we propose the first algorithm in a two-phase fashion. In the first phase, the minimum value of the whole sensor network is determined as a global knowledge. Then, in this second phase, this global knowledge is fed back to all sensor nodes which decide their probabilities locally to provide the minimum and reports themselves to the base station based on the constraint conditions. This approach involves two rounds of messages. The second algorithm evaluates a PMNQ (or its variants by combining the determination of the minimum value of a sensor network and sensor nodes that provide minimum sensed readings within a single round of messaging. Finally, we conduct an extensive set of experiments through simulations to evaluate the performance of our proposed innetwork algorithms for PMVQs and PMNQs. The experimental results demonstrate the effectiveness of our algorithms. The remainder of the paper is organized as follows. Section II formulates the problem and formally defines PMVQs and PMNQs. Section III presents our algorithms for PMVQs. Section IV describes two algorithms for PMNQs, including the variants, i.e., threshold-based PMNQs and top-k PMNQs. Section V evaluates the performance of our algorithms. Section VI reviews existing research works in uncertain data management and in-network query processing in sensor networks related to this research. Finally, the paper is concluded in Section VII. II. PRELIMINARIES In this section, we state notations and assumptions for this work, and then we formally define Probabilistic Minimum Value Queries (PMVQs and Probabilistic Minimum Node

3 Queries (PMNQs. Besides, we describe the tree-based network topology on which our in-network algorithms for both PMVQs and PMNQs operate. A. Assumptions and Notations Without lose of generality, a sensor network is considered to consist of (i one base station where queries are initiated and query results are collected and (ii a set of sensor nodes N = {n, n, n N } deployed over a spatial area. Each sensor node n i N maintains a reading r i, which can be uncertain and captured as a set of possible values. We denote reading r i as {v i,, v i,, v i, ri }. Collectively, there exists a set of sensed reading values V = {v v r i, n i N}. We assume v i, < v i, < < v i, ri, and thus the value range of r i is denoted by [v i,, v i, ri ]. In this work, we assume only one of the sensed reading values represents the actual measurement at any time and denote the probability of v i,j to be the actual measurement for r i by p i,j. As a whole, the sum of all probabilities is one, i.e., v i,j r i p i,j =. Besides, given a value v, we use P r[r i = v] and P r[r i > v] to represent the probability that r i s value is equal to v and r i s value is greater than v. In other words, P r[r i = v] = p i,j if v i,j = v; and P r[r i > v] = v i,j r i v i,j>v P r[r i = v i,j ]. Further, we use P r[r i v] to represent the probability that r i s value is greater than or equal to v, i.e., P r[r i v] = P r[r i = v] + P r[r i > v]. To facilitate our discussion, Table IV summarizes the notations used in this paper. Notation Description N a set of sensor nodes i.e., {n,, n N } r i the reading of n i including values {v i,,, v i, ri } v i,j a possible reading value of r i. p i,j the probability that v i,j is the actual measurement R a set of readings i.e., {r,, r N } V a set of possible values {v v r i, r i D} P r[r i = v] the probability that v i,j ( r i equals v P r[r i > v] the probability that v i,j ( r i greater than v P r[r i v] P r[r i = v] + P r[r i > v] P r[v min = v] the probability that v is the minimum P r[v min r i ] the probability that n i has the minimum reading value par(n a parent node of n ( N in the tree topology chi(n a set of child nodes of n ( N in the tree topology T (n a set of nodes within the subtree ed at n ( N TABLE IV NOTATIONS AND DESCRIPTIONS B. Definition of Probabilistic Minimum Value Queries Due to probabilistic nature of sensed reading values, the minimum value is uncertain. Let v min be the minimum value. Then, the probability that a value v ( V is the minimum, denoted by P r[v min = v], is defined in Definition. Definition : Minimum Value Probability. Given a set of sensor nodes, N, and correspondingly, a set of sensed reading values, V, the probability that a value v ( V is the minimum, denoted by P r[v min = v], is determined by Equation (. P r[v min = v] = ( P r[r i = v] P r[r j > v] N (N φ n j (N N ( where N is the power set of N, i.e., N N, N N. We use x to represent the cardinality of a set x. Referring to Example, we can determine the probability that a sensed reading value being minimum as follows. P r[v min = ] = (P r[r > ] P r[r = ] P r[r > ] P r[r > ]+ (P r[r > ] P r[r > ] P r[r > ] P r[r = ]+ (P r[r > ] P r[r = ] P r[r > ] P r[r = ] = (..9+ (.6.+ (.. = =.6. While all the probabilities have been shown in Table II, we omit all detailed calculations due to space limitation. Based on Definition, Probabilistic Minimum Value Queries (PMVQs is defined in Definition. Definition : Probabilistic Minimum Value Query (PMVQ. In a sensor network with a set of sensed reading values, V, a PMVQ searches possible minimum values v V where P r[v min = v] (see Equation ( is greater than zero. Formally, PMVQ(V = { v v V, P r[v min = v] > }. As already discussed in Example, the result set for the PMVQ is {, }, and the corresponding confidence probabilities are listed in Table II. In the next section, we shall analyze and tackle the challenge arising in in-network PMVQ processing and present our algorithms. C. Definition of Probabilistic Minimum Node Queries Let minimum node (or min-node in short denote a node that provides a minimum reading value. Here, we first define the probability for a node to have the minimum reading value in Definition. Based on the probabilities for all sensor nodes, Probabilistic Minimum Node Queries is defined in Definition. Definition : Minimum Node Probability. Given a set of sensor nodes, N, the probability that a sensor node n i ( N provides the minimum sensed reading value v min, denoted by P r[v min r i ], is computed as the probability for r i to have some reading values not greater than those of all the other sensor nodes. Formally, it is stated in Equation (. P r[v min r i] = ( P r[r i = v i,j] P r[r x v i,j] v i,j r i n x (N {n i } ( As shown in Example, the probability for node n to be a min-node is P r[v min r ] = ( (.5..= + =. Next, the probability for node n to be a min-node is P r[v min r ] = (. +(.6.9 =. +.5 =.9. Table III lists all the probabilities. Definition : Probabilistic Minimum Node Query (PMNQ. Given a set of sensor nodes N, a PMNQ returns a set of sensor nodes n i which have their minimum node probabilities greater than zero. Formally, PMNQ(N = { } n i ni N, P r[v min r i ] >. As mentioned in Example, sensor nodes n, n and n are returned by the PMNQ. It is likely for some users or Actually, each value in the result set is associated with a confidence probability as the query answer.

4 applications to be interested only in min-nodes that have minimum node probabilities greater than certain defined threshold or those that have top-ranked minimum node probabilities. Thus, we extend PMNQs to threshold-based PMNQs and topk PMNQs in Definition 5 and Definition 6, respectively. Definition 5: Threshold-based Probabilistic Minimum Node Query (δ-pmnq. Given a set of sensor nodes N and a user specified threshold, δ ( < δ, a δ-pmnq returns a set of min-nodes n i whose P r[v min r i ] is greater than δ. Formally, { } δ-pmnq(n = n i ni N, P r[v min r i ] > δ. Definition 6: Top-k Probabilistic Minimum Node Query (k-pmnq. Given a set of sensor nodes N and a specified number k (where k, a k-pmnq returns k sensor nodes n i whose P r[v min r i ] are the highest. Formally, k-pmnq(n = {n i ni N, nx N N P r[vmin ri] > P r[vmin rx] } where N N and N = k. Continue with Example. A δ-pmnq with δ =.5 returns {n } and k-pmnq with k = returns {n, n }. As we can observe from Definition, computing the minimum node probability of a node requires a knowledge of the probabilities of all other sensor nodes. Thus, it is very challenging to determine the minimum node probabilities within the network based on partial knowledge from only a few sensor nodes. In Section IV, we shall further address the challenges and present our in-network PMNQ processing algorithms. D. Tree-Based Network Topology Due to limited radio communication ranges, sensor nodes can only communicate with spatially close sensor nodes. As a result, sensor nodes relay messages to disseminate queries and collect sensed readings. A query is disseminated from the node to all its neighbor nodes, which in turn forward the query to their neighbor nodes. Through this process, a treebased network topology is also formed. A node n that passes a query to other nodes N ( N is called a parent node and those nodes in N are called child nodes. We use par(n to denote the parent node of a node n and chi(n to denote a set of child nodes whose parent is n. When the query reaches the leaf of the tree, sensed readings/aggregated data are propagated up to the. When a node n receives data from its chi(n, it will process its data and propagate the intermediate result to its parent par(n. When a node is reached, the result is sent to the base station. Through the tree-like topology, queries and query results can be disseminated and collected efficiently. III. IN-NETWORK PMVQ PROCESSING Communication is the most energy consuming operation for sensor nodes. The design and implementation of energyefficient in-network query processing algorithm should minimize the communication costs. In general, communication costs can be accounted as the number of messages relayed If N < k, then N = N. and the sizes of messages sent between sensor nodes. Thus, redundant data should be suppressed or compressed whenever appropriate. As defined in Definition, the core part of PMVQs is to determine the probabilities for values to be the minimum (as in Equation (. For any value v to be examined, the P r[r i = v] and P r[r i > v] for all individual n i s sensed readings needs to be collected. Obviously, transmitting all the possible sensed reading values and their associated probabilities from sensor nodes to the base station for processing is very expensive and should be avoided. We may avoid transmitting those that are for sure not in the result set and not involved in the probability computation. Based on this idea, we develop an in-network minimum value screening (MVS algorithm that effectively identifies and discards redundant values. We discuss it in Subsection III-A. Further, in-network minimum value screening algorithm only performs screening on the data propagating to the base station. It does not perform any minimum value probability computation. To effectively reduce communication overhead, some partial results (which are smaller to transmit than raw sensed readings should be computed in sensor nodes. However, it is a challenge due to indivisible computation of the minimum value probability (as stated in Equation (, i.e., partial results cannot be determined by sensor nodes, thereby forcing all sensed readings to be delivered to the base station. To address this, we reformulate Equation ( into a decomposable minimum value probability computation. By doing so, the minimum value probability can be determined in a divide-and-conquer fashion. For individual sensed reading values, intermediate minimum value probabilities can be derived, processed and transmitted in place of raw sensed reading values. Based on the above ideas, we devise an innetwork minimum value aggregation (MVA algorithm in Subsection III-B. Besides, based on the decomposable minimum value probability computation, we observe two optimization techniques that suppress and compress data transmitted in a sensor network, and present them in Subsection III-C. A. In-Network Minimum Value Screening Algorithm Intuitively, given two sensed readings r i and r j and their respective value ranges, [v i,, v i, ri ] and [v j,, v j, rj ], if v i, ri < v j,, then r i s possible reading values are all smaller than r j s. Thus, r j s sensed reading values should not be the minimum and r j can be completely dropped without propagating it to the base station. As shown in Example, r s values are for sure not the minimum values as r s values are all smaller than r s. Thus n drops its own sensed reading from forwarding to the base station. Thus, the basic idea of in-network minimum value screening (MVS algorithm is to screen out some sensed readings at nonleaf sensor nodes, if their reading values are all greater than the largest possible minimum value. Accordingly, we define a notion of MaxiMin as stated in Definition 7. MaxiMin is the largest possible minimum values among possible minimum values from individual sensed readings. Then, whenever a sensed reading r i has its smallest possible value v i, greater than MaxiMin, it can be dropped.

5 Definition 7: MaxiMin. Given a set of sensor nodes N, i.e., n, n, n N, the largest possible minimum values among the reading values of sensor nodes in N, is denoted by MaxiMin. Here, MaxiMin can be determined as min ni N{v i, ri }. Figure gives the pseudo code of the in-network minimum value screening algorithm performed at sensor nodes to filter the candidate minimum values for PMVQs. In Figure, if a running sensor node is a leaf node (who does not have child node in a tree-based network topology, it reports its reading values to its parent (line -. Otherwise, the running node should be a non-leaf node. Then, it collects all children sensed readings. Based on those and its sensed readings as R, it derives a MaxiMin (line -5. Then, all sensed reading values greater than MaxiMin are discarded (line 6-7. Finally, the screened values are propagated to the node s parent (line 8 and this algorithm completes. If a node is the, we refer the parent here to as the base station. Notice that in this algorithm, no computation on minimum value probability is invoked. Due to space constraint, we informally describe the algorithm of PMVQ evaluation at the base station. Once the base station collects all sensed readings as R. Then, the algorithm identifies all possible sensed reading values as V based on R. Thereafter, it examines the P r[v min = v] for each value v ( V, according to Equation (. Notice that some readings may be screened out from the network. Thus, V covers only the candidate values to be the minimum. Meanwhile, probabilities P r[r i = v] and P r[r i > v] for those discarded readings are assumed be and, respectively. At last, the algorithm outputs each value v whose P r[v min = v] greater than zero. The algorithm (and thus the evaluation of a PMVQ completes when all values in V are examined. Fig.. n Algorithm In-Network Minimum Value Screening Local. a sensor node identity n i; a local minimum value range [l, u]; Begin. if ( chi(n i = then. send r i to par(n i;. else. R {r i} c chi(n i Rc; 5. MaxiMin = min ri Rv i, ri ; 6. foreach ( r i R do 7. if ( MaxiMin < v i,j then R R {ri}; 8. send R to par(n i; End. n In-network minimum value screening algorithm for sensor nodes n (a Network snapshot n Sensor node MaxiMin n n n 6 n (b MaxiMin value table Fig.. Running example for MVS Here we illustrate the algorithm based on our running example shown in Figure (a. Since n and n are leaf nodes, thus they both transmit their own readings to their parents. Once n gets the reading r, it calculates the MaxiMin and find both r and r are qualified to be sent to the parent, i.e., v, = < MaxiMin = 6 and v, = < MaxiMin = 6. But at n, the MaxiMin is found equal to, thus r is dropped at node n. B. In-network PMVQ Processing In this subsection, we present an idea of decomposable minimum value probability computation, by which P r[v min = v] can be incrementally computed in the network. As a result, only partial results are delivered to the parent sensor nodes. This effectively saves communication costs. We detail the idea as follows. Decomposable Minimum Value Probability Computation. As explained in Lemma, we rewrite Equation ( into an equivalent form as stated in Equation (. P r[v min = v] = P r[r i v] P r[r i > v] ( This equivalent form provides exactly the same result as Equation (. Recall Example. With Equation (, P r[v min = ] can be computed as ( (.6.9 =.6. Similarly, P r[v min = ] can be computed as (.6.9 (.6.8 =.5. Lemma : Given a set of sensor nodes, N, for computation of P r[v min = v], the formula N N ( P r[r i = v] n i (N N P r[r i > v] (i.e., Equation ( is equivalent to another formula P r[r i v] ( P r[r i > v] (5 (i.e., Equation (. Proof. First, by removing the empty set from N, we reexpress Equation ( as P r[v min = v] = ( P r[r i = v] N N n i (N N P r[r i > v] P r[r i > v]. (6 Let N be {n, n,, n N }, N be N {n }, N be N {n }, N i be N i {n i }. Then, we can elaborate the

6 above equation into P r[v min = v] ( = P r[r i = v] N (N {,{n }} P r[r i > v] = (P r[r = v] + P r[r > v] ( P r[r i = v] N N P r[r i > v] n i (N N = P r[r v] ( P r[r i = v] N (N {,{n }} P r[r i > v] = P r[r v] P r[r v] ( P r[r i = v] N (N {,{n }} P r[r i > v] = = P r[r i v] P r[r i > v] n i (N N P r[r i > v] n i (N N n i (N N P r[r i > v] P r[r i > v] P r[r i > v] This final step equals Equation (. The proof completes. In Equation (, both n P r[r i N i v] and n P r[r i N i > v] can be factorized; thus for N which is divided into x subsets, N, N,, N x, they become 5, P r[r i v] = P r[r i v] (7 j and P r[r i > v] = N j {N,N,,N x} N j {N,N,,N x} j P r[r i > v]. (8 Thus, in a tree-based network, whenever a sensor node n j receives n P r[r i N i v] and n P r[r i N i > v] from sensor nodes in its subtree (i.e., N = T (n j {n j }, it can directly determine P r[r j v] n P r[r i N i v] and P r[r j > v] n P r[r i N i > v], and propagate them to n j s parent sensor node. In other words, by transmitting values v and their aggregated probabilities, P r[v min v] and P r[v min > v], decomposable minimum value probability computation can be performed within the network. Finally, this decomposable form not only reduces communication cost but also incurs a smaller computation cost, compared with Equation (. Based on Equation (, we devise the in-network minimum value aggregation (MVA algorithm for sensor nodes as outlined in Figure. If a sensor node n i running the algorithm is a leaf node, it reports its every possible value v and P r[r i v] and P r[r i > v] to its parent node (line -. On the other hand, if a sensor node is a non-leaf node, it collects all reading values and probabilities from its child nodes. Then it multiplies all those probabilities (line -. Finally, all values v and their aggregated probabilities, P r[v min v] and P r[v min > v] to its parent can be another sensor node or the base station (line. When the base station receives the values and probabilities, by Equation (, it deduces the 5 Here, N = i x N i and i, j x, i j N i N j = minimum values and associated probabilities as the result of the PMVQ. Due to limited space, the pseudo-code for this base station is omitted. Algorithm In-Network Minimum Value Aggregation Local. a local node n i, a sensed reading r i = {v i,, v i,, v i, ri } Begin. if ( chi(n i then /* n i is a leaf node */. send (v j, P r[r i v j ], P r[r i > v j ], v j r j to par(n i ;. else /* n i is a non-leaf node */. R c chi(n i Rc; 5. V { v v R } {v v r i }; P r[v min v], v V ; P r[v min > v], v V ; forall ( v r i do 8. P r[v min v] P r[v min v] P r[r i v]; P r[v min > v] P r[v min > v] P r[r i > v]; 9. forall ( n j chi(n i do. forall ( v r j do. P r[v min v] P r[v min v] P r[r j v]; P r[v min > v] P r[v min > v] P r[r j > v];. send (v, P r[v min v], P r[v min > v], v V to par(n i ; End Fig.. In-network minimum value aggregation algorithm for sensor nodes C. Further Optimization In this section, we discuss two optimizations, namely, transmission suppression and message compression, that can further reduce the transmission cost within the network. The former enables some sender nodes to suppress transmission of some data if the values of those suppressed data can be deduced by the receivers. The latter compress messages to be delivered between sensor nodes. Transmission Suppression. Since only the possible minimum values which have non-zero probabilities are taken in the PMVQ result, those values with probabilities being zero can be suppressed from transmission in the network. According to Equation (, we can observe that P r[v min = v] equals when one of the following two conditions is satisfied: ( ni NP r[r i v] = and nj NP r[r j > v] =, and ( P r[r i v] = P r[r i > v]. n i N n i N The first condition happens when there is a sensor node n i whose P r[r i v] equals to. Notice that whenever P r[r i v] equals, P r[r i > v] should be ; thus the second part of the condition should be satisfied if the first part holds. Therefore, when a sensor node finds n P r[r i N i v] for a subset of sensor nodes N equal to, (that also implies n P r[r i N i > v] =, it can suppress the transmission of this aggregated probability to its parent. As a result, in our design, if a sensor node does not receive n P r[r i N i v] and n P r[r i N i v] from its child nodes, it can deduce that those missed probabilities are zero. For the second condition, the sensor node that has both n P r[r i N i v] and n P r[r i N i > v] can identify this condition and suppress the reporting of P r[v min = v] to the base station if they are equal. In this case, the base station that does not receive P r[v min = v] for a given value v can assume that P r[v min = v] is zero. Message Compression. As discussed above, to compute P r[v min = v] for a single value v, both terms n P r[r i N i v] and n P r[r i N i v] for a subset of sensor nodes N are needed. Intuitively, if there are x possible

7 minimum values (e.g., v, v, v x, the message needs to cover x pairs of n P r[r i N i v j ] and n P r[r i N i v j ] ( j x values. In fact, here exists an optimization opportunity to reduce some values as they can be derived from others in the same message. Consider a set of values {v, v, v x }, where v j < v j+. We can see n P r[r i N i > v j ] = n P r[r i N i v j+ ], for j x. As such, we can eliminate either n P r[r i N i > v j ] or n P r[r i N i v j+ ] to reduce the message payload size. In this work, our design propagates (i {( n P r[r i N i v ], n P r[r i N i v ], n P r[r i N i v x ]} and (ii n P r[r i N i > v x ] from a sensor node to its parent node. Meanwhile, the parent node can recover n P r[r i N i > v i+ ] from n P r[r i N i v i ], correspondingly. r i P r[r i v] r {(,., (,.5} r {(,., (,.6} n r {(,., (,.9, (5,.8, (6,.8} r {(,., (,.9, n n n (5,.8, (6,.} (b ri value table (a Network snapshot Fig. 5. Running example for MVA Finally, to illustrate of the idea of in-network aggregation, and the possible optimizations. We show the running example in Figure 5 as follows. Notice that, in order to conserve energy, only n P r[r i N i v] information is transmitted, denoted by ri in Figure 5(a. Since both n and n are leaf nodes, r and r can be easily derived by transforming r and r as shown in Figure 5(b. After n gets r from n, it first recovers n i T (n P r[r i > v] from n i T (n P r[r i v], such as n i T (n P r[r i > ] = n i T (n P r[r i ] =.9, and so on. Then n delivers r, which merges the information from both T (n and itself. Finally, n transmits r to the. Notice that only non-zero probability minimum values are included in ri, e.g., only and are included in r from n. The compactness of r i make this MVA approach more efficient in terms of energy conservation against MVS algorithm. In this running example, the transmission cost of MVS is 8 messages (see Figure (a; while MVA costs messages in transmission (see Figure 5(a. IV. IN-NETWORK PMNQ PROCESSING Different from PMVQs that look for possible minimum values, PMNQs take a further step to search for sensor nodes that provide the possible minimum values, namely, the minnodes. An intuitive approach for processing a PMNQ is to first find the possible minimum values within the network (e.g., by issuing a PMVQ and then use the PMVQ result to probe for the min-nodes. We classify algorithms based on this idea as the two-phase, or global knowledge-based approach because the PMVQ result here represents a global knowledge useful for PMNQ processing. However, this twophase approach incurs two rounds of messaging between the base station and sensor nodes, resulting in significant energy consumption within the sensor network. Thus, we propose another approach that requires only one round of messaging. Algorithms developed based on this one-phase approach not only avoid excessive communication overhead but also have a shorter access latency than two-phase algorithms. In the following, we discuss these two types of algorithms for handling PMNQs and their variants, threshold-based PMNQs and top-k PMNQs. A. Two-Phase In-Network PMNQ Processing While the two-phase approach sounds intuitive, a key challenge arising here is the computation of the minimum node probability. According to Equation (, it basically requires all sensor nodes to deliver aggregated probabilities on all possible minimum values if the computation is to be performed locally at sensor nodes. To allow sensor nodes decide whether they are the min-nodes and to compute their minimum node probabilities locally, we transform Equation ( as follows to obtain Equation (9. P r[v min r i ] = v i,j r i = v i,j r i ( P r[r i = v i,j ] ( P r[ri = v i,j ] P r[r i v i,j ] }{{} local factor P r[r x v i,j ] n x (N {n i } (9 P r[r x v i,j ] n x N }{{} global factor From this equation, we can see that the term P r[ri=vi,j] P r[r i v i,j] can be determined by a sensor node n i locally. Thus, we call it a local factor. On the other hand, the term n x N P r[r x v i,j ] needs to obtain probabilities from all sensor nodes, and thus is called a global factor. Notice that this global term is part of the minimum value probability formula in Equation (, i.e., this term is handily available at the base station if innetwork minimum value aggregation algorithm is used in the first phase. Therefore, in two-phase in-network PMNQ processing, we use the in-network minimum value aggregation algorithm in the first phase. In the following, we assume that after the first phase, a set of possible minimum values V min = {v, v, v Vmin }, together with their probabilities are determined at the base station. In the following, we discuss the determination of result sensor nodes and computation of their minimum node probabilities for PMNQ, threshold-based PMNQ and top-k PMNQ. Evaluation of PMNQ. In the second phase, the base station simply delivers the PMVQ result to all sensor nodes. Then, upon reception of the PMVQ result, the sensor node n i checks whether any of its sensed reading value is covered by V min. If so, n i computes its minimum node probability based on Equation (9 and reports its ID with corresponding probability to the base station. Evaluation of Threshold-based PMNQ. For threshold-based PMNQs, only those sensor nodes, whose minimum node probabilities are greater than a specified threshold δ, reports its ID and the corresponding probability to the base station as the result of the PMNQ. Evaluation of Top-k PMNQ. Based on the discussed approach, P r[v min r i ] for each sensor node n i can be

8 determined. Following the tree-based network topology, the IDs of min-nodes and their probabilities are propagated up to the base station. Along the propagation path, every non-leaf node only sends to its parent node the IDs and probabilities of those that are ranked in top-k in their probabilities. Eventually, the base station receives k min-node IDs and their probabilities from the node. n n n (a PMNQ n n n n n (b δ-pmnq (δ =.8 n n n n (c k-pmnq (k = Fig. 6. Running example for phase- of the two-phase algorithms for PMNQ variants, where r i = (n i, P r[v min r i ] (i.e., r = (n,., r = (n,.9, r = (n,.6 and r = (n,.6. To demonstrate the above ideas, we use the running example in Figure 6 for illustration. For simplicity, only the query answer propagation in the second phase is shown. With the PMVQ result obtained in the first phase, each node is able to compute the minimum node probability for itself, denoted by r i. For PMNQ, all non-zero probability nodes are returned to the base station, i.e., r, r and r are delivered. Next, for δ-pmnq with δ =.8, r and r are filtered locally, because their probabilities are lower than.8. Finally, for k-pmnq with k =, each sensor node only transmit r i with the largest probability. Thus n transmits r instead of r, and n delivers r to the base station. B. One-Phase In-Network PMNQ Processing The main challenge faced in one-phase in-network PMNQ processing is attributed to the absence of prior global knowledge about the possible minimum values among all sensor nodes. Thus, in the one-phase approach, a sensor node can only determine candidate min-nodes based on its best knowledge. However, given a subtree that consists of a set of sensor nodes, N ( N, the sensor node of the subtree may identify candidate min-nodes within the subtree and discard others based on the estimated minimum possible values/probabilties of nodes in N. Based on this idea, we develop the one-phase algorithms for PMNQs, threshold-based PMNQs and top-k PMNQs. Evaluation of PMNQ. Inspired by in-network minimum value screening, we devise a one-phase in-network screening algorithm for PMNQ evaluation. In this algorithm, all sensor nodes report their IDs and sensed reading values to their parent nodes, which in turn collect and process data from child nodes for reporting to their parent nodes. Thus, at a non-leaf node, the MaxiMin of reading values collected from all sensor nodes within a subtree is derived. The sensor nodes whose smallest reading values are greater than the MaxiMin should not be min-nodes and thus can be dropped from further propagation. Evaluation of Threshold-based PMNQ. For threshold-based PMNQs, only min-nodes whose minimum node probabilities are greater than δ are returned as query result 6. According to 6 When δ is set to, this approach naturally supports PMNQs. Fig. 7. n n n n (a δ-pmnq with δ =.8 n n n n (b k-pmnq with k = Running examples of one-phase algorithms for PMNQ variants Equation (9, the minimum node probability for a sensor node involves both local and global factors. Let a set of sensor nodes N be divided into two non-empty subsets N and (N N. We can re-express the global factor in the equation as follows: P r[r x v] = P r[r x v] P r[r x v] n x N n x N n x (N N P r[r x v] n x N Let P r[v min r i ] N denote the minimum node probability of node n i N. We have P r[v min r i ] N = ( P r[ri=v i,j] P r[r i v i,j] P r[r x v i,j ]. v i,j r i n x N Thus, the relationship between P r[v min r i ] N and P r[v min r i ] N can be obtained as follows. P r[v min r i] N = ( P r[r i=v i,j ] P r[r i v i,j ] n P x N r[rx vi,j] v i,j r i n P r[r i v i,j ] x N P r[r x v i,j] v i,j r i ( P r[r i=v i,j ] P r[v min r i] N Hence, as long as P r[v min r i ] N < δ (i.e., P r[v min r i ] N < δ, n i can be dropped from further propagation. Based upon this idea, in our in-network threshold-based PMNQ processing, each sensor node propagates its reading values to its parent node. While the sensor readings are collected at the node of a subtree that consists of a set of sensor nodes N, P r[v min r i ] N for every sensor node n i is determined. If P r[v min r i ] N is smaller than the threshold, the sensor node ID of n i and its sensor readings are dropped. Finally, the IDs and sensed readings of the remaining nodes are sent to the parent, along with the global factor n P r[r x N x v i,j ] corresponding to nodes in N (denoted by r in Figure 7. Note that the global factor plays an important role in estimating the minimum node probabilities. It incorporates more information from nodes as the propagation moves towards the base station, resulting more accurate estimation of the minimum node probabilities and better filtering capability. Continue with the running example shown in Figure 7(a, where δ equals to.8. Since n and n are both leaf nodes, they transmit r and r, together with r and r to their parents, respectively. Once n receives data from n, it computes the min-node probability for each candidate node, i.e., n and n 7. Let T (n be the set of nodes within the subtree ed at n. Both n and n are discarded at node n because P r[v min r ] T (n and P r[v min r ] T (n are both smaller than.8. At node n, n is the only qualified candidate node. Thus, r and r are transmitted to the base station. 7 Due to page limitation, we omit the detailed computation and list all computed probabilities in Table V.

9 n y r x P r[v min r x] T (ny n y r x P r[v min r x] T (ny n r. n r. n r. n r.9 n r.6 n r.6 n r.6 n r.6 TABLE V MINIMUM NODE PROBABILITIES Evaluation of Top-k PMNQ. In-network evaluation of top-k PMNQs is more challenging than other types of PMNQs because the actual minimum node probabilities for sensor nodes cannot be determined when the global factor in Equation (9 is not available. Meanwhile, when P r[v min r i ] N of node n i is found to be greater than P r[v min r j ] N of node n j at the node of N, we cannot be sure that n i s P r[v min r i ] N will still be greater than n j s P r[v min r i ] N at the base station. This is because the global factor changes while more sensor readings are collected. According to Table V, we find that at n, P r[v min r ] T (n < P r[v min r ] T (n. However, when propagating to n, P r[v min r ] T (n > P r[v min r ] T (n. In other words, sensor nodes cannot be discarded simply based on their estimated minimum node probabilities at an intermediate node. In this subsection, we address this challenge by identifying a condition (as stated in Lemma that allows a sensor node n x be dropped if it is not one of min-node with topk probabilities. Lemma : Given an intermediate node n (its subtree is denoted by N where data from two sensor nodes n x and n y are collected. Let v t be the MaxiMin determined at n and V consists of possible values smaller than or equal to v t, i.e., V = {v i v i V, i t}. We can assess that the minimum node probability of n x is greater than that of n y if the following condition is satisfied. P r[r x = v ] P r[r x v ] P r[ry = v ] P r[r y v ] v V ( Proof. Based on the definition of MaxiMin, we know that any value beyond MaxiMin would have zero probability to be the minimum value. Let V = V V, then we have the global factor of n x N P r[r x v ] =, for v V. Therefore P r[v min r x ] P r[v min r y ] = ( ( P r[r x = v] P r[r x v] P r[r x = v] P r[r x v] v V n i N = ( ( P r[r x = v ] P r[r x v ] P r[r x = v ] P r[r x v ] v V n i N P r[r i v] P r[r i v ] ( Combining the condition in Equation, we have P r[v min r x ] > P r[v min r y ].. Based on the above lemma, we develop an in-network onephase k-pmnq algorithm, as shown in Figure 8. Starting from the leaf nodes (line -, every sensor node propagates its reading values and associated probabilities to its parent node, which in turn sends processed data to its parent. At a nonleaf node, which is the node of a subtree that consists of a set of sensor nodes N, after calculation of global factor (line -, the MaxiMin is determined from its collected data (line. Then, for each pair of sensor nodes, say n x and n y, we check the condition presented in Lemma (line 6-. If the condition is satisfied, we can determine that P r[v min r x ] N is greater than P r[v min r y ] N. Through the relationship between pairs of sensor nodes, we can deduce whether a min-node is qualified as a top-k candidate (line - 5. If it is not, its ID and sensed readings are pruned from further propagation. Finally, the IDs and sensed readings of the rest of nodes are sent along with n x N P r[r x v i,j ] to the parent. This process will eventually terminate at the of the whole network. Algorithm One-phase k-pmnq Local. a local node n i, a reading r i including {v i,, v i,, v i, ri }, k, counter j, n j N Begin. if ( chi(n i then /* n i is a leaf node */. send R = {r i } and (v j, P r[r i v j ], v j r i to par(n i ;. else /* n i is a non-leaf node */. R n c chi(n i Rc; 5. V { v v R } {v v r i }; P r[v min v], v V ; forall ( v r i do 8. P r[v min v] P r[v min v] P r[r i v]; 9. forall ( n j chi(n i do. forall ( v r j do. P r[v min v] P r[v min v] P r[r j v];. R R {r i }. MaxiMin min rj R{v j, rj }. V {v v V v MaxiMin } counter j, n j N; forall ( r x, r y R, x y do 7. inc ; 8. forall ( v V do 9. if ( P r[rx=v ] P r[r x v ] P r[ry=v ] P r[r y v ] < then. inc ;. break;. counter y counter y + inc;. forall ( r x R do. if ( counter x > k then 5. R R {r x} 6. send R and (v, P r[v min v], v V to par(n i ; End Fig. 8. One-phase k-pmnq algorithm for sensor nodes Sensor node Local factor n {(,., (,., (5,.5, (6,.} n {(,., (,., (5,., (6,.} n {(,., (,., (5,., (6,.} n {(,., (,., (5,.75, (6,.} TABLE VI LOCAL FACTORS Let s see the running example in Figure 7(b. Due to page limit, we omit the detailed computation here, which can be obtained with the global factors listed in Figure 5(b. As shown, at n, the MaxiMin equals to 6. Therefore, V = {,, 5, 6}. We examine the checking condition (see Lemma on every pair of candidate nodes. Take nodes n and n as an example. We find the condition is not satisfied and thus we cannot determine the relationship between n and n at

10 the intermediate node n. Later, when r, r and r all arrive to node n, MaxiMin is reduced to. Within the value set V = {, }, we find that n has larger local factors than n on all values in V. Thus, we can conclude that n has a higher min-node probability than n. Since we are interested in the -PMNQ in this example, r can be discarded. Similarly, n and n are pruned from further propagation. Transmission Cost (byte Synthetic Data 7 (in log scale 6 5 Naive MVS MVA P Transmission Cost (byte Real Data x Naive MVS MVA P V. PERFORMANCE EVALUATION In this section, we evaluate the performance of the proposed approaches for in-network processing of PMVQ, PMNQ and their variants. In the following, we first discuss the simulation model, followed by the experimental results. A. Simulation Model Wireless transmission is a dominant factor to energy consumption in wireless sensor networks. In our experiments, we consider the total amount of data transmission, including propagation of the query, sensed reading values, probabilities, and other required information, as the performance metric. Since there is no existing techniques for in-network processing of PMVQs and PMNQs, we use the centralized processing approach, which collects all sensed readings to the base station for processing, as the baseline. We implement a discrete-event simulator for a sensor network that consists of N sensor nodes. According to TAG [], a tree-based topology is adopted for message routing and data collection. A base station is directly connected to the of the TAG tree. We use both synthetic data and real traces of sensor readings in the experiments. For synthetic data, the reading values of r i at node n i is generated randomly within the range of [µ min, µ max ]. Let λ be the average size of the possible value set for a sensed reading. Without loss of generality, we set the granularity g of those possible values to be. Thus, the whole value set of reading r i can be generated based on λ and g. The uncertainty of synthetic dataset is introduced by assigning confidence values to sensed reading values randomly. On the other hand, the real sensor traces are extracted from the Intel Lab Data []. In order to introduce uncertainty into the dataset, the reading trace corresponding to a sensor node is used as statistics to generate the possible reading values and confidence probabilities for the node. The parameter settings used in our experiments are summarized in Table VII. Parameters Default Settings k: # of top ranked tuples 5 δ: probability threshold.5..9 λ: mean cardinality of sensed reading 5 N: # of sensor nodes 5 [µ min, µ max]: value range of a sensed reading µ min = µ max = g: granularity of discrete pdf representation s q: PVMQ/PNMQ message size (bytes s n: node segment size (bytes s v: reading value segment size (bytes 8 s p: probability value segment size (bytes 8 TABLE VII PARAMETER SETTINGS Transmission Cost (byte 8 x 6 (a Synthetic Data Fig. 9. (b Real Data Performance improvement on general PMVQ/PMNQ Synthetic Data P PMNQ δ PMNQ k PMNQ (a Synthetic Data Fig.. B. Experimental Results Transmission Cost (byte 6 Real Data P PMNQ δ PMNQ k PMNQ (b Real Data Performance improvement on PNMQ We conduct a series of experiments to evaluate the proposed algorithms in terms of performance improvement over the baseline approach and sensitivity tests on a number of system and query parameters. Unless otherwise specified, experimental results shown below are obtained from the average of independent simulation runs where each run uses a new network topology together with the new sensed readings. Performance Improvement: We first validate our algorithms by comparing our algorithms against the baseline approach (labeled as Naive. Figure 9 shows the performance saving of our algorithms, including the minimum value screening algorithm (labeled as MVS and the minimum value aggregation algorithm (labeled as MVA for PMVQs as well as the one-phase algorithm (labeled as and two-phase algorithm (labeled as P for PMNQs. To provide a general comparison of all situations, here we randomize all parameter settings within the ranges specified in Table VII. The experiments have used both the synthetic data and real trace. The results are plotted in Figure 9(a and Figure 9(b, respectively. As shown, all the algorithms proposed in this paper reduce a significant amount of data transmissions against the naive approach. We find that MVA outperforms MVS significantly for PMVQs, while P performs slightly better than for PMNQs 8. Between PMVQs and PMNQs, the processing cost of PMNQs is higher than that of PMVQs because the former is basically more complex than the latter. Moreover, for a largescale sensor network, PMNQs may include several sensor nodes in the results and thus incurring extra transmission overhead. Next, we take a step further to compare the performance of the one-phase and two-phase approaches on the various variants of PMNQs, including the original PMNQs, δ-pmnqs and k-pmnqs. Results in Figure indicate that has a 8 Note that the approach has a much shorter query latency than the P approach.