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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 5, MAY 2009
On the Trade-offs of Cooperative Data Compression in Wireless Sensor Networks with Spatial Correlations Tamer ElBatt, Senior Member, IEEE
Abstract—In this paper we study the problem of efficient data dissemination over one- and two-dimensional multi-hop wireless sensor grids with spatially correlated sample measurements. In particular, we investigate the trade-offs of exploiting correlations via cooperatively compressing the sensor data as it hops around the network. We focus on two performance metrics, namely transport traffic and scheduling latency. More specifically, we investigate using basic information theory the feasibility of sublinear scaling laws , with the number of nodes, under a variety of cooperation strategies ranging from naive non-cooperative forwarding to sophisticated hierarchical cooperation. First, we show that a simple cooperation scheme, namely forward/reverse cooperation, achieves a logarithmic growth rate for the transport traffic and a linear growth rate for the schedule length with the number of nodes. Thus, we √ shift our focus to multi-phase cooperation to show that: i) O( N ) schedule length is achievable using two-phase cooperation which is a combination of noncooperative and forward/reverse cooperation schemes and ii) Logarithmic schedule length and transport traffic are both achievable using hierarchical cooperation, yet at the expense of more complexity in coordinating nodes’ cooperation. This also opens room for optimizing these performance measures for a given network size. Finally, we analyze the impact of fixed bit rate and derive upper bounds on the scheduling latency. Index Terms—Wireless sensor networks, spatial correlations, data compression, entropy, scheduling latency, scaling laws.
I. I NTRODUCTION
F
UTURE wireless networks are envisioned to accommodate large numbers of embedded devices that operate cooperatively to achieve a pre-specified sensing/actuating task [3]. One of the main hurdles towards the realization of this objective is network scalability. It has been shown in [4] that the one-to-one transport capacity of wireless multi-hop √ networks scales as O( N ) (i.e. the per-node throughput scales as O( √1N )) where N is the number of nodes per unit area. Therefore, it has been concluded that focus should be confined to small size wireless multi-hop networks. In [5], the authors studied the problem of multi-hop broadcast in sensor networks where samples of a random field are recorded at each node in Manuscript received February 29, 2008; revised October 3, 2008; accepted December 3, 2008. The associate editor coordinating the review for this paper and approving it for publication was J. Zhang. T. ElBatt is with the Advanced Technology Center, Lockheed Martin Space Systems Company, Sunnyvale, CA, 94089 USA (e-mail:
[email protected]). This work was done while the author was with HRL Laboratories, LLC. Part of this paper has been presented at the IEEE/ACM International Conference on Information Processing in Sensor Networks (IPSN04) [1] and the IEEE International Conference on Communications (ICC04) [2]. Digital Object Identifier 10.1109/TWC.2009.080292
the network and disseminated to all other nodes. Unlike [4] where the network traffic is assumed to be spatially independent, they exploited the fact that spatially close sensors may experience correlations among their sample measurements in an attempt to reduce the traffic load and, hence, improve network scalability. Thus, they introduced the concept of joint routing and data compression which uses classical source codes at respective nodes and re-encode the data as it hops around the network. This involves a trade-off between traffic and latency which we analyze in a systematic fashion in this paper. In [6], a distributed algorithm for removing correlations among sensor data via computing wavelet transforms has been proposed. However, the scaling laws of traffic and latency were left as open problems. Distributed source coding (DSC) [7], [8] is an alternative approach that utilizes the theory of compression with sideinformation [10] to remove correlations without any communications among sensor nodes. However, implementing DSC in practical settings is still an open problem due to: i) The memory requirements and delays associated with long block codes and ii) The need for complete knowledge of the correlation structure at all source nodes a priori. The problem of efficiently gathering correlated data from dense sensor networks has been addressed in [11]–[13]. The impact of spatial correlations on the performance of routing and data compression is studied in [13]. The many-to-one capacity of wireless networks was characterized in [12], [14] under different sets of assumptions. The main difference between this work and [12], [14] is two-fold. First, they characterize the scaling behavior of the traffic generated under many-to-one applications whereas our focus here is on multi-hop broadcast applications.1 Second, we not only characterize the scaling laws for the traffic but for the scheduling latency as well. Our main contribution in this paper is two-fold: i) Show, with the aid of basic information theory, that there exists cooperation schemes, namely hierarchical cooperation, that achieve logarithmic scalings laws for the transport traffic and schedule length2 with the number of nodes and ii) Extend the results from one- to two-dimensional deployments as a representative of more realistic settings. We focus on onedimensional sensor deployments in the first part of the paper 1 It may also be referred to as flooding, many-to-many communications or all-to-all communications. 2 We use scheduling latency and schedule length (in # slots) interchangeably in this paper.
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ELBATT: ON THE TRADE-OFFS OF COOPERATIVE DATA COMPRESSION IN WIRELESS SENSOR NETWORKS WITH SPATIAL CORRELATIONS
due to its simplicity and ease of presentation [1] and then extend the study to two-dimensional settings in section V. We first motivate the work via illustrating the challenge of achieving sub-linear scaling laws for the transport traffic and schedule length, concurrently. We show that simple singlephase cooperation strategies, e.g. forward/reverse cooperation, can achieve sub-linear transport traffic scaling law, yet, at the expense of a linear schedule length. This has led to introducing the multi-phase framework. Two-phase √ cooperN , schedule ation is shown to exhibit a sub-linear, O √ √ length and O N log N transport traffic √ growth rates when the cooperation set size i is set to c N where c is a constant. Furthermore, we extend two-phase cooperation to the more sophisticated hierarchical cooperation which proceeds through two stages, namely cooperation and distribution, in an attempt to further reduce the schedule length to a logarithmic growth rate for any cooperation set size i. This strategy not only achieves logarithmic transport traffic and schedule length scaling laws but also creates room for trading one performance measure for the other via controlling the cooperation set size. We emphasize that cooperative data compression in sensor networks involves fundamental trade-offs among a number of performance metrics, namely transport traffic, latency and energy expenditure. This work goes beyond the focus of [5], [12] on transport traffic to systematically analyze the transport traffic-scheduling latency trade-off and characterize strategies to balance it. Extending this work to study the broader trafficlatency-energy trade-offs lies out of the scope of this paper and is a subject of future research. The paper is organized as follows: In section II, we describe the network model, correlation structure and performance metrics. Afterwards, the challenge of achieving sub-linear scaling laws is illustrated using naive cooperation strategies in section III. This is followed by a detailed analysis of the proposed multi-phase cooperation that achieves sub-linear scaling laws in section IV. In section V, the study is extended to two-dimensional grids. A discussion of the adaptive bit rate assumption and its relaxation is given in section VI. Finally, conclusions are drawn in section VII. II. P ROBLEM S ETUP A. Network Model In the first part of the paper, we limit our attention to onedimensional sensor deployments consisting of N stationary nodes which communicate only via the wireless medium and are uniformly spaced along a horizontal straight line of unit length. We assume that nodes are indexed in an ascending order from left to right. We assume that nodes are equipped with omni-directional antennas and can adapt the transmission power, for a given bit rate, to maintain a fixed transmission range (r), where N1−1 < r < N2−1 , i.e. nodes (m − 1) and (m + 1) are one-hop neighbors of an arbitrary node m. All nodes are assumed to share the same frequency channel and time is divided into equal size slots where each transmission fits exactly in a single slot via appropriately adjusting the bit rate. The implications and challenges of a fixed transmission rate, whereby joint sample transmissions need multiple slots, are discussed in greater detail in section VI. We also assume
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that the network spatial coverage is always fixed to 1 unit length, i.e. as N grows the network gets more dense due to smaller inter-node spacing. Each sensor is assumed to record periodic samples of the sensed field, scalar quantize, encode and transmit them such that the random field can be reconstructed at all nodes in the network up to a certain level of distortion. We assume that successive samples taken by the same sensor are temporally uncorrelated and, hence, we focus on the set of samples recorded by all sensors at a given time instant and drop the time index. On the other hand, sensor measurements are assumed to be spatially correlated according to a onedimensional stationary random process S(y), where S(y) is a real-valued random variable representing the field value at location 0 ≤ y ≤ 1. This is motivated by the fact that the vast majority of physical phenomena are analog where sensors are better modeled as continuous rather than discrete-valued sources. Moreover, we assume that the random process S(y) has the property that the correlation between samples increases as the sensors get dense. We assume that the reading of sensor m, denoted Sm , is quantized by a fixed quantizer Q(.) subject to a constraint on the average distortion per sample (i.e. 1 N m=1 E(d(Sm , Xm )) ≤ D ) where Q(Sm ) = Xm , d(.,.) N is a distortion measure and D is a prescribed requirement on distortion. Thus, the minimum number of bits required to represent the output of the quantizer of node m (i.e. the expected codeword length per sample) is given by the entropy H(Xm ) of the associated discrete random variable Xm . It should be noted that the average of H(Xm ) is attained for each packet transmitted by node m with high probability if it is the outcome of encoding large numbers of samples. This is based on the fact that the expected codeword length per sample becomes arbitrarily close to the entropy of the source if large block lengths are encoded using classical source codes, e.g. Huffman and Arithmetic coding[9]. In this paper, we focus on many-to-many communication scenarios due to their theoretical and practical merits. From a theoretical perspective, the traffic and latency associated with this scenario constitute upper bounds on their one-to-one and many-to-one counterparts. It also arises under a wide variety of applications ranging from collaborative video surveillance to data gathering where the collector node is an unmanned aerial vehicle (UAV) that can only be reached via distributed sensor beam forming (sensor reachback problem). B. Performance Metrics In this paper, we rank cooperative data compression schemes based on two performance measures, namely the transport traffic and scheduling latency. We measure the traffic generated by a cooperation scheme using the notion of bitdistance product (or transport traffic in bit.meters) first introduced in [4], yet, with a slight modification to handle the case of multiple destinations encountered in many-to-many communications. More precisely, for each transmission, we multiply the number of bits carried by the distance traveled by that transmission, irrespective of the number of receivers within that distance, and then sum over all transmissions. Formally, it is defined as follows,
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Definition 1: The transport traffic (TT) of a cooperation scheme that involves NT transmissions is given by: TT =
NT
b j ∗ aj
bit.meters
j=1
where bj is the number of bits carried by transmission j and aj is the hop distance, in meters, traveled by transmission j. The significance of this metric stems from the fact that it not only quantifies the traffic volume communicated by the sensor nodes but also accounts for the spatial progress of this traffic towards the destinations. This is of paramount importance in multi-hop networks to distinguish scenarios that transport the same bit volume, yet, over vastly different spatial scales. The second metric, scheduling latency, is measured in the number of slots required by a given cooperation scheme to complete the multi-hop broadcast task. Hence, it depends on the total number of transmissions required by the cooperation scheme and the maximum number of non-conflicting transmissions that can be packed per slot. The latter depends on the interference model and number of nodes N and may vary from slot to slot. Since we are interested in characterizing the minimum schedule latency for the cooperation strategies, we define SL as the length of the shortest schedule a cooperation strategy would need to complete the multi-hop broadcast task. Definition 2: The shortest schedule length (SL) of a cooperation scheme that involves NT transmissions is given by: SL =
NT NT P S
where NT is the total number of transmissions needed for flooding and NT P S is the maximum number of nonconflicting transmissions per slot which, in turn, depends on the interference model as discussed in the next section. III. M OTIVATION : S UB -L INEAR S CALING L AWS In this section, we characterize, with the aid of basic information theory, the scaling laws of the transport traffic and scheduling latency associated with the sensor broadcast problem under naive cooperation strategies, namely noncooperative forwarding and single-phase cooperation. We show that sub-linear transport traffic scaling is achievable, however, the schedule length is shown to scale, at best, linearly with the number of nodes. This motivates the paper and multiphase cooperation analyzed in section IV. A. Non-Cooperative Forwarding (NCF) Under this strategy, each sensor transmits a quantized version of its sample to its neighbors. A node who receives a sample of a neighbor is supposed to rebroadcast it blindly without re-encoding it, which explains the non-cooperative nature of this strategy. Next, we derive the many-to-many transport traffic scaling law with the number of nodes. Lemma 1: For the model given in section II.A, the many-tomany TT under non-cooperative forwarding scales as O(N ). Proof: In order to achieve many-to-many communication, the minimum number of bits generated by node 1, that is H(X1 ), is forwarded over (N − 1) hops. By the same token, node j, where 2 ≤ j ≤ N − 1, generates H(Xj ) bits which
go over (N − 2) hops in order to be received by all other nodes in the network, namely nodes 1, 2, ..., j −1, j +1, ..., N . This process is repeated for all nodes until node N goes through the same procedure. In order to quantify TT scaling law, we assume that each node sends only one sample (its own sample) per transmission, i.e. it does not forward samples from other sensors along with its own sample.3 Accordingly, the bit.distance product contributed by node 1 is given by H(X1 ) (N − 1) N −1 , where the factor N1−1 represents the distance traveled over 1-hop and the factor (N − 1) represents forwarding over (N − 1) hops as discussed earlier. Hence, the bit.distance product contributed by all nodes is given by, 1 (N − 1)H(X1 ) T TN CF = N −1
N −1 + (N − 2) H(Xj ) + (N − 1)H(XN ) j=2
= O(N ) bit.meters
(1)
Next, we quantify the schedule length scaling behavior. Lemma 2: For the model given in section II.A, the many-tomany SL under non-cooperative forwarding scales as O(N ). Proof: It is straightforward to show that NT is given by N (N − 1). To quantify NT P S , we need to specify an interference model. In this paper, we adopt a simple interference model that is widely employed in the multi-hop packet radio networks literature (e.g.[16]), whereby a collision arises whenever multiple transmissions are heard by a receiver in the same slot. Otherwise, a transmission is deemed successful. Accordingly, broadcast transmissions of nodes who are more than two-hops away are considered non-conflicting and may share the same slot. For one-dimensional networks, NT P S is given by 2N 3 when each node broadcasts its own sample to its left and right neighbors. On the other hand, when a node forwards a sample of another node received through one of its neighbors, its broadcast yields one useful sample transfer to the other neighbor. Hence, NT P S is given by N3 under this scenario. Notice that the former scenario occurs only once for each node whereas the latter one arises several times throughout the packet forwarding process. This, in turn, yields N3 ≤ NT P S ≤ 2N 3 and, hence, proves the claim. We conclude this section with the observation that the transport traffic and schedule length grow linearly with the number of nodes when the spatial correlations are simply ignored. Next, we investigate the feasibility of sub-linear scaling using simple cooperation schemes. B. Single-phase Cooperation: Benefits and Limitations In this section, we prove the existence of simple cooperation schemes that achieve sub-linear TT scaling and, at best, linear SL scaling. These cooperation schemes are not optimal in a particular information- or queuing-theoretic sense, yet, they 3 Even if each sensor forwards others’ samples along with its own sample and fits all of them in a single slot, via increasing the link bit rate, it can be shown that the scaling laws derived in this section would still hold.
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ELBATT: ON THE TRADE-OFFS OF COOPERATIVE DATA COMPRESSION IN WIRELESS SENSOR NETWORKS WITH SPATIAL CORRELATIONS
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Fig. 2.
Forward/Reverse cooperation (F/R) over a 4 node sensor network.
Sequential cooperation over a 4 node sensor network.
provide valuable insights about the problem. Hence, they motivate cooperation strategies that achieve sub-linear scaling for, both, the TT and SL concurrently. We first analyze sequential cooperation which exhibits sub-linear TT scaling law, yet, at the expense of O(N 2 ) SL. Next, we show how the latency performance of a variant strategy, called forward/reverse cooperation, can be improved to O(N ). 1) Sequential Cooperation: As shown in Fig.1, each node takes turn in a round-robin fashion to encode its sample given the samples of left nodes received through its left neighbor. This not only slows down TT growth rate as suggested by intuition but also degrades SL growth rate as shown by the following two lemmas. Lemma 3: For the network model given in section II.A, the many-to-many TT under sequential cooperation grows as H(X1 , X2 , ..., XN ). Proof: Node 1 generates H(X1 ) as shown in Fig. 1. Next, node 2 sends back H(X2 |X1 ) (denoted “2|1” in Fig. 1) to node 1 and then sends a joint version of the two samples, i.e. H(X1 , X2 ) (denoted “1, 2” in Fig. 1), to node 3. The scheme proceeds in the same manner with node m sending H(Xm |X1 , ..., Xm−1 ) to node (m − 1) followed by H(X1 , X2 , ..., Xm ) to node (m+1) until m = N as shown in Fig. 1 for N = 4. Notice that the encoded sample of node m sent to node (m−1) should be propagated in the left direction such that each node gets the samples of all other nodes. Thus, the TT generated by the sequential strategy is given by, 1 T Tseq = H(X1 ) + H(X2 |X1 ) + H(X1 , X2 ) N −1
+2H(X3 |X1 , X2 ) + ... + (N − 1)H(XN |X1 , ..., XN −1 )
1 = (N − 1)H(X1 , X2 , ..., XN ) N −1
(2)
= H(X1 , X2 , ..., XN ) ≤ O(N )
(3)
Again, the factor N 1−1 in (2) represents the contribution of the inter-node distance in the transport traffic metric. The equality in (2) follows directly from the chain rule for entropies whereas equality in (3) holds when the sensor nodes are dispersed to render their sample measurements independent. Next, we need to characterize the asymptotic behavior of the joint entropy of N quantized random variables. This open problem has been recently addressed in [12] for a stationary Gaussian random process and a scalar quantizer with uniform
step size and infinite number of levels. It was assumed that the distortion measure is mean square error (MSE) and the autocorrelation function of the spatial process S(y) is expo2 nential of the form Rs (y) = e−y . For this setup, [12] showed that H(X1 , X2 , ..., XN ) is upper bounded by O(log N ) as N → ∞ which immediately yields the following result. Corollary 1: For the network model given in section II.A 2 and spatial correlation function Rs (y) = e−y , T Tseq is upper bounded by O(logN ). Lemma 4: For the model given in section II.A, the manyto-many SL under sequential cooperation grows as O(N 2 ). Proof: From Fig. 1, it is evident that node 1 participates in 1 transmission to node 2 and (N − 1) receptions from node 2, node 2 participates in 1 transmission to node 3 and (N − 2) receptions from node 3, and node (N − 1) participates in 1 transmission to node N and 1 reception from +2) node N . Hence, NT is given by (N −1)(N , i.e. O(N 2 ). 2 By construction, sequential cooperation enforces certain order on nodes’ communication since node m has to wait for the joint samples of nodes 1,2,...,m − 1 in order to generate its own conditional and joint samples. This significantly limits the spatial slot reuse which causes NT P S to be one and SLseq to scale quadratically with N , that is, much faster than NCF. The considerable degradation in schedule length scaling may suggest that there is a trade-off between TT and SL, i.e. traffic reduction has to come at the expense of longer schedules. However, we show next that logarithmic TT and linear SL scaling laws are both attainable, if we transmit conditional samples more efficiently and reuse slots. 2) Improving Sequential Cooperation Latency: Given the sequential cooperation strategy in Fig. 1, we observe that there is no need to have dedicated transmissions for individual conditional samples to send them back to left nodes. Instead, we gather the joint samples of all nodes at node N through cooperation in the forward direction. Afterwards, send conditional-joint samples in the reverse direction at higher bit rates, compared to the transmission rate of individual conditional samples, in order for each transmission to fit in a single slot.4 We refer to this new cooperation scheme as forward/reverse cooperation (F/R) as shown in Fig. 2. The following two lemmas show that F/R cooperation achieves a logarithmic TT scaling while preserving a linear SL scaling. Lemma 5: For the network model given in section II.A, forward/reverse cooperation exhibits O H(X1 , X2 , ..., XN ) transport traffic and O(N ) schedule length growth rate. Proof: From Fig. 2, it is evident that F/R cooperation entertains a logarithmic TT growth rate similar to sequen4 Relaxing
this assumption is addressed in section VI.
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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 5, MAY 2009
tial cooperation. Moreover, if the broadcast nature of omnidirectional antennas is exploited, the transport traffic of sequential cooperation and F/R can be further reduced, yet, only by a multiplicative factor. This is attributed to the fact that a joint sample transmission meant for the right neighbor is overheard by the left neighbor (e.g. H(X1 , X2 ) sent from node 2 to node 3 is heard by node 1). In addition, it should be noted that F/R cooperation consumes (N − 1) transmissions in the forward direction and (N − 1) transmissions in the reverse direction, i.e. NT scales as O(N ). Thus, even if slot reuse is not exploited, then SLF/R scales as O(N ), well below the quadratic growth rate of plain sequential cooperation. Lemma 6: For the network model given in section II.A, forward/reverse cooperation exhibits linear schedule length scaling law, even if slot reuse is exploited. Proof: The transmissions of nodes m + 3j, where j takes integer values, can share the same slot and, hence, cooperation N3 3j slots in this case. in the forward direction consumes j=1 This is due to the fact that cooperation is limited to reuse clusters (of size 3 nodes)5 and is repeated over those clusters in order for all samples to reach the rightmost node N . On the other hand, cooperation in the reverse direction remains unchanged (i.e. O(N )) since it involves propagating conditional joint samples throughout the linear network. Hence, we conclude that, with slot reuse, NT scales as O(N 2 ) for F/R cooperation. Moreover, NT P S scales as O(N ) based on arguments similar to section III.A. Therefore, SLF/R still scales as O(N ), even if slot reuse is exploited. The reason is that slot reuse does not only increase NT P S from one to O(N ), but it also increases NT from O(N ) to O(N 2 ) to achieve forward cooperation. The best scaling laws obtained thus far (i.e. logarithmic TT and linear SL) give rise to the following fundamental question: Is it possible to achieve sub-linear scaling laws for the TT and SL simultaneously? We show next that this is indeed attainable using multi-phase cooperation. IV. M ULTI -P HASE C OOPERATION : ACHIEVING S UB -L INEAR S CALING In this section, we establish two results that constitute major contributions of this paper. First, two-phase√cooperation achieves sub-linear SL scaling law, namely O( N ), while preserving a sub-linear TT scaling law. Second, hierarchical cooperation further slows the growth rate of the SL and TT, with N , to logarithmic and poly-logarithmic respectively. A. Two-Phase Cooperation The essence of two-phase cooperation is to localize cooperation within regions of the network, where nodes cooperate to compress each other’s data, and beyond those regions no cooperation is performed. This constitutes a simple approach for trading traffic for latency via introducing the cooperation set size i degree of freedom. Accordingly, a one-dimensional network of N nodes is partitioned into Ni cooperation sets, each accommodating i nodes as shown in Fig. 3 for i = 2. As the name suggests, this strategy proceeds through two phases. 5A
reuse cluster is defined as a group of nodes where no slots are reused within this group.
Set 1 11 00 00 11
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Set N/2 .
.
.
.
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N
1 Fig. 3.
Two-phase cooperation with set size i = 2.
In the first phase, members of each set compress their data using F/R cooperation. Once this is done, a node in an arbitrary set would have only the samples of all nodes in its set but not the samples of nodes in other sets. The role of the second phase is to exchange the sample measurements among various cooperation sets. This is achieved via inter-set exchange among representative nodes in respective cooperation sets (e.g. nodes 1, i + 1, 2i + 1,..., N − i + 1) using NCF as described in section III.A.6 This is followed by data distribution within each set in order to disseminate the samples gathered at the representative nodes to other members in their respective sets. Next, we present the TT and SL scaling results for two-phase cooperation. Theorem 1: For the network model given in section II.A and cooperation set size i, the many-to-many TT under twophase cooperation is given by, T Ttpc = O H(X1 , X2 , ..., XN ) +
N H(X1 , X2 , ..., Xi ) i
(4)
Proof: F/R cooperation takes place in phase 1 within Ni sets simultaneously. This could be achieved assuming that neighboring nodes at the edges of two neighboring sets (e.g. nodes i and i + 1) do not interfere with each other (possibly using directional antennas). Accordingly,
N i−1 H(X1 , X2 , ..., Xi ) T Tphase1 = i N −1 = O H(X1 , X2 , ..., Xi ) (5) On the other hand, the TT generated in phase 2 is given by, N i−1 H(X1 , X2 , ..., Xi ) + T Tphase2 = O i N −1 N ∗H(Xi+1 , Xi+2 , ..., XN |X1 , X2 , ..., Xi ) (6) i The first term in (6) corresponds to the NCF among the representative nodes of various cooperation sets, whereas the second term represents the transport traffic associated with distributing the sample measurements to other members within each set. The summation of (5) and (6) proves the claim. The scaling law in (4) can be easily verified through the extreme cases of i = 1 and i = N which simply reduce to the NCF and F/R cooperation policies respectively. Moreover, it is 6 We assume that the transmission power of representative nodes could be raised to reach each other over a single-hop. This assumption is discussed further in section VI.
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ELBATT: ON THE TRADE-OFFS OF COOPERATIVE DATA COMPRESSION IN WIRELESS SENSOR NETWORKS WITH SPATIAL CORRELATIONS
evident that the transport traffic monotonically increases as the cooperation set size i decreases. This is attributed to the fact that the first term in (4) does not depend on i and constitutes the minimum amount of transport traffic when i = N . For 1 < i < N , the two terms in (4) persist and hence the transport traffic becomes greater than the first term alone. Finally, it is straightforward to show that as i decreases the second term increases. Next, we analyze SLtpc . Theorem 2: For the network model given in section II.A and cooperation set size i, the many-to-many SL under twophase cooperation grows as O(i + Ni ). Proof: It is evident that phase 1 consumes O(i) slots to be completed. In addition, phase 2 consumes O( Ni ) throughout the non-cooperative sample exchange task and O(i) for distributing the gathered sample measurements throughout each set. Hence, the O(i + Ni ) SL follows. Notice that the parameter i provides a degree of freedom for optimizing the schedule length for a given network size. This result reveals the following two observations. First, for a fixed cooperation set size i, the linear growth rate of SL with N still persists. Second, for a given N , the non-linear dependence of SL on i suggests that this function√has an extremum which turns out to be a minimum at i∗ = c N where c is a constant. Corollary 2: If the√cooperation set size is set to i∗ for all N , then SLtpc = O( N ). This comes at the expense of increasing TT compared to the extreme case of i = N . Hence, we conclude that twophase cooperation constitutes a√ simple approach for trading TT for SL. It achieves O( N ) schedule length at the expense the transport traffic from O(logN ) √ of increasing √ to O( N log N ). Hierarchical (logi N -phase) cooperation proposed in the next section overcomes this problem and achieves logarithmic SL and poly-logarithmic TT scaling laws, for any set size i. B. Hierarchical Cooperation In this section, we explore hierarchical cooperation as a potential avenue for further reducing the SL and TT growth rates for any i. 1) Concept of Operation: Hierarchical cooperation proceeds through two stages, namely cooperation and distribution. The cooperation stage goes through k = logi N phases as shown in Fig. 4. On the other hand, the distribution stage goes through (k − 1) phases as shown in Fig. 5. Next, we illustrate the operation of this strategy with the aid of an example. Consider a one-dimensional sensor network consisting of N = 8 nodes. In phase p = 1 of the cooperation stage, the network is partitioned into Ni cooperation sets, each accommodating i nodes as shown in Fig. 4 for i = 2. Members of each set cooperatively compress their sample data similar to phase 1 of the two-phase cooperation scheme. At the beginning of phase 1 < p < k, any node in an arbitrary set would have access only to the samples of all other nodes in its phase (p − 1) set as shown in Fig. 4. Thus, a single sensor node is elected from each cooperation set in phase (p − 1) as a representative node of that set. Accordingly, at the beginning N representative nodes who need of phase p, there will be ip−1 to disseminate the samples gathered during phase (p − 1). At this point, the N node sensor network at the beginning of
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Fig. 4. Example of the cooperation stage for N = 8 and i = 2. It shows the samples at various nodes at the beginning of each cooperation phase. Phase 2
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Distribution Stage
1,2,...8 5,6,7,8
Fig. 5. Example of the distribution stage for N = 8 and i = 2. It shows the samples at various nodes at the beginning of each distribution phase.
N phase 1 reduces to a network of ip−1 nodes at the beginning of phase p. Once more, we assume that only representative nodes can raise their transmission powers appropriately in each phase in order to construct the shown cooperation sets. At the beginning of phase p = 2, there are four representative nodes (1,3,5,7) where node 1 has access to the samples of its phase p = 1 set members, namely node 2 and node 3 has access to node’s 4 sample and so on. Likewise, at the beginning of phase p = 3, there would be two representative nodes, namely 1 and 5, who have access to samples from their phase p = 2 set members. Hence, the cooperation stage proceeds recursively through k = logi N phases until i nodes gain access to an estimate of the entire field (nodes 1 and 5 in the example). At this point, the role of the distribution stage commences in order to distribute the entire field estimate to the nodes whose access is limited to a partial view of the entire field (nodes 2,3,4,6,7,8 in the example). As shown in Fig. 5, the distribution stage proceeds recursively in a hierarchical manner from phase 1 up to phase (k − 1) where k = 3. In the first distribution phase, node 1 distributes its field estimate to node 3 and node 5 distributes its own estimate to node 7. In a similar fashion, node 1 distributes its estimate to node 2, node 3 to node 4, node 5 to node 6, and node 7 to node 8 throughout the second distribution phase. Thus, by the completion of the distribution stage, all N nodes would have access to the entire field estimate as required by the many-to-many communications objective. 2) Scaling Results: In this section, we characterize the scaling laws of the transport traffic and schedule length under hierarchical cooperation, which is one of the main contributions of this paper. To simplify notation, we denote joint sample transmissions H(Xu , Xu+1 , ..., Xv ) as C(u)(v) (N ) and the conditional joint sample transmissions H(X1 , ..., Xu−1 , Xv+1 , ..., XN |Xu , Xu+1 , ..., Xv ) as D(u)(v) (N ). We will use the notation C(u)(v) and D(u)(v) throughout the paper where the dependence on N is implied. Theorem 3: For the network model given in section II.A and cooperation set size i, the many-to-many TT under hier archical cooperation grows as O logi N H(X1 , X2 , ..., XN ) .
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Proof: Please refer to Appendix A. Based on the logarithmic growth rate of H(X1 , X2 , ..., XN ) discussed earlier for a Gaussian random process with Rs (y) = 2 e−y , we conclude that the TT under hierarchical cooperation scales faster than F/R cooperation, yet, much slower than the linear scaling laws of NCF. Performance degradation compared to F/R cooperation should be weighed against the significant reduction in the SL growth rate achieved by hierarchical cooperation as we show next. Theorem 4: For the network model given in section II.A and cooperation set size i, the many-to-many SL under hierarchical cooperation grows as O(i logi N ). Proof: Each phase in the cooperation stage consumes O(i) slots to be completed. Accordingly, the cooperation stage consumes O(i logi N ) slots. Similarly, the distribution stage consumes O(i logi N ) slots. Thus, the schedule length for this strategy, SLhier , scales as O(i logi N ). The above result reveals two key observations: i) Hierarchical cooperation yields a logarithmic growth rate for the schedule length with N , that is well below F/R and twophase cooperation and ii) The cooperation set size constitutes a degree of freedom for trading TT for SL for a given network size. Clearly, SL increases with the parameter i whereas Theorem 3 shows that TT decreases with i. Hence, for a given network size, there exists an optimum set size that strikes a balance between these two conflicting objectives. In the next section, we extend our results to the more realistic two-dimensional grids which highlight the key role of inter-node spacing in the transport traffic metric. V. T WO -D IMENSIONAL S ENSOR G RIDS In this section, we shift our attention to two-dimensional grids in order to: i) compare one- and two-dimensional settings and ii) compare our scaling results to the network flow constructed in [5]. We assume N nodes are uniformly spaced over a regular two-dimensional grid of unit area and nodes are indexed in an ascending order from left to right and from top to bottom such that node 1 is at the top-left corner and node N is at the bottom-right corner. Nodes have fixed transmission range where √N1−1 < r < √N2−1 , i.e. each node has up to four neighbors. The spatial correlation function is a stationary two-dimensional random process S(y, z), where S(y, z) is a real-valued random variable representing the field value at location 0 ≤ y ≤ 1, 0 ≤ z ≤ 1 whose auto-correlation function increases as the sensors get spatially close. Next, we show how NCF is different over two-dimensional grids. Afterwards, we extend the analysis to single- and multi-phase cooperation studied earlier. A. Two-dimensional Non-cooperative Forwarding Our objective in this section is to examine the scaling behavior of non-cooperative sensor broadcast over two-dimensional grids (e.g. N = 16 as shown in Fig. 6). Quantifying the transport traffic in this case differs from the one-dimensional case due to: i) The distance between nodes is √N1−1 as opposed to 1 N −1 which has direct impact on the spatial progress of singlehop transmissions and ii) The node arrangement allows up to four nodes to re-broadcast overheard messages as opposed to
11 00 00 11
1
11 00 00 11
3
1 0 0 1
11 00 00 11
5 00 DN1 ={1}11
6 11 00 00 11 00 11
7
1 0 0 1 0 1
8 11 00 00 11 00 11
10
11
11 00 00 11
11 00 00 11
DN2 ={2,5}
DN3 ={3,6,9}
11 00 00 11
9
11 00 00 11
13
11 00 00 11
11 00 00 11
DN4 ={4,7,10,13}
2
14
DN5 ={8,11,14}
1 0 0 1
15
1 0 0 1
4
12
16
11 00 00 11
DN6 ={12,15}
Fig. 6. Two-dimensional regular grid with N = 16 nodes illustrating the NCF of node 1’s sample throughout the grid using the concept of Diagonal Nodes at level l denoted, DNl .
at most two node re-broadcasts in the one-dimensional case. Therefore, studying the problem over two-dimensional grids turns out to be more involved even in the simple case of no cooperation. Hence, we classify nodes to three types, namely corner nodes having only 2 neighbors (e.g. node 1) and interior nodes with 4 neighbors (e.g. node 10). Edge nodes, which have 3 neighbors (e.g. node 5), can be handled in a similar fashion. The rationale behind this distinction is to account for the different number of quadrants over which a broadcast sample is forwarded which has direct impact on the number of transmissions. For instance, node 1’s sample spreads over the South-East (SE) quadrant only whereas node 10’s sample is forwarded over all four quadrants. Lemma 7: For regular two-dimensional sensor grids, the TT associated with non-cooperatively disseminating the sample√of a corner node throughout the entire network grows as O( N ). Proof: Please refer to Appendix B. Next, we quantify the TT associated with disseminating the sample of an interior node m at coordinates (hm , vm ), where hm and vm are the horizontal and vertical coordinates relative to node 1 (e.g. (h10 , v10 )=(1,2)). The sample of an interior node m should be forwarded throughout all four quadrants. The transport traffic associated with the dissemination of H(Xm ) is upper bounded by the sum of covering each quadrant separately due to the role of nodes on quadrant edges whose broadcast transmissions are heard by nodes in multiple quadrants, i.e. T Tsample
m
< T TSE + T TSW + T TN E + T TN W √ √ H(Xm ) = √ O ( N − hm )( N − vm ) N −1 √ +O (hm + 1)( N − vm ) √ +O ( N − hm )(vm + 1)
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+O (hm + 1)(vm + 1) = O
√ N H(Xm )
(7)
Theorem 5: For regular two-dimensional sensor grids, √ the many-to-many TT under NCF is upper bounded by O(N N ). Proof: Combining the results of Lemma 7 and (7) and 2−dim applying them for the appropriate √ nodes, T TN CF turns out to be upper bounded by O(N N ). Notice that the two-dimensional TT scales √ faster than its one-dimensional counterpart by a factor of N . This is primarily due to the role of inter-node spacing in the bit.distance product. Nevertheless, the traffic volume (in bits) exhibits the same scaling behavior irrespective of the node arrangement. Finally, the following theorem quantifies the SL associated with NCF over two-dimensional grids. Intuition might suggest that SL should be smaller over two-dimensional grids due to the fact that the number of nodes which overhear and forward a transmission is up to four nodes as opposed to two in the onedimensional case. In fact, NT P S turns out to be higher in the two-dimensional case, yet, only by a multiplicative constant which has no impact on the scaling behavior. Theorem 6: For regular two-dimensional sensor grids, the many-to-many SL under NCF grows as O(N ). Proof: NT P S grows linearly for non-cooperative forwarding, similar to the one-dimensional setting. From (7) and (24) and ignoring the inter-node spacing √N1−1 factor which does not play any role in quantifying SL, it can be seen that the total number of transmissions associated with disseminating the sample of a single node to all other nodes is O(N ) and, hence, NT scales as O(N 2 ). Hence, SLN CF exhibits linear scaling with N irrespective of the node arrangement. B. Cooperation over Two-Dimensional Grids 1) Forward/Reverse Cooperation: The two-dimensional F/R strategy cooperatively compresses the data in two rounds; first, it disseminates joint samples in the forward direction (i.e. from top left corner (node 1) to bottom right corner (node N )). Afterwards, it disseminates conditional-joint samples in the reverse direction from node N towards node 1. Unlike the non-cooperative strategy, all diagonal nodes have to participate in the forward and reverse transmissions since each node has access to different subsets of samples (depending on which broadcast transmissions it hears). For instance, the diagonal nodes at level 2, DN2 = {2, 5} in Fig. 6, transmit different samples, namely H(X1 , X2 ) and H(X1 , X5 ) respectively. The transport traffic of F/R cooperation scales √ faster over two-dimensional grids, again by a factor of N , due to the larger distance traveled over each hop. Next, we present the result which can be established along the lines of section III.B.2. Theorem 7: For regular two-dimensional sensor grids, many-to-many TT under F/R cooperation grows as √ O( N H(X1 , X2 , ..., XN )). 2) Multi-Phase Cooperation: The transport traffic of twophase cooperation over two-dimensional grids can be derived from the NCF and F/R cooperation results in Theorems 5 and 7. Hence, we present the results and skip the derivation. Our
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main result confirms that the TT of two-phase cooperation scales faster over two-dimensional grids, √ 2−dim =O N H(X1 , X2 , ..., XN ) T Ttpc √ N N √ + H(X1 , X2 , ..., Xi ) i i
(8)
√ Notice that (8) is exactly similar to (4) except for the . factors that reveal the contribution of the larger node spacing in two-dimensional regular grids. In addition, it can be shown 2−dim is lower bounded by the T Ttpc generated that the T Ttpc over one-dimensional deployments given in (4). Finally, along the same lines of section IV.B.2, it can be shown that the transport traffic of hierarchical cooperation over two-dimensional grids is also lower bounded by its one√ dimensional counterpart, due to the . factors, i−1 2−dim H(X1 , X2 , ..., XN ) T Thier = √ N −1
k−1 N N √ N + i 2 + i 3 + ... + i 2 ∗ i i i
(9)
Unlike the one-dimensional case, the transport traffic of twodimensional hierarchical cooperation √does not reduce to a closed form. This is attributed to the i factor that accounts for the larger distance between representative nodes. Therefore, we establish upper and lower bounds on (9). First, an upper bound can be obtained via substituting all terms by Ni ,
i−1 N 2−dim H(X1 , X2 , ..., XN ) T Thier < √ k i N −1 √ =O N logi N H(X1 , X2 , ..., XN ) (10) It is worth noting that the TT of two-dimensional hierarchical cooperation is upper bounded by √ the one-dimensional growth rate in Theorem 3 scaled by N . Second, we show that the TT of one-dimensional hierarchical cooperation lower bounds its two-dimensional √ counterpart. This is achieved via substituting (i − 1) by ( i − 1) and all terms by the smallest N term, k+1 , in (9), i 2 √
i−1 N 2−dim √ H(X1 , X2 , ..., XN ) k k+1 T Thier > N −1 i 2 = O logi N H(X1 , X2 , ..., XN ) Similar to one-dimensional settings, the previous results, summarized in Table I, show that hierarchical cooperation exhibits the best sub-linear scaling laws for TT and SL over two-dimensional grids. Moreover, they confirm that TT scales faster over two-dimensional grids, yet, SL does not depend on the node arrangement. Finally, it has been argued in [5] that the transport traffic associated with a network flow constructed over a two-dimensional grid scales as O(log N ). However, this result does not account for: i) Traffic generated to gather samples of each subnet at nodes on the boundary of a cut and ii) Traffic needed for distributing the data within each subnet. Our hierarchical cooperation results confirm logarithmic scaling for the TT over one-dimensional networks
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TABLE I T RANSPORT T RAFFIC AND S CHEDULE L ENGTH S CALING L AWS Cooperation Strategy NCF Sequential F/R Two-phase √ i∗ = c N Hierarchical
Transport Traffic one-dimensional O(N ) O(logN ) O(logN ) √ √ O( N log N ) O(logi N.logN )
Transport Traffic two-dimensional √ < O(N √ N) < O( √ N logN ) O( N logN ) √ 3 O(N 4 log N ) √ < O( N logi N.logN )
Schedule Length O(N ) O(N 2 ) O(N ) √ O( N ) O(ilogi N )
For F/R cooperation, it can be noticed from Fig. 2 that C(1)(m) slots are consumed by node m’s transmission to node (m + 1) in the forward direction and D(1)(m) slots are consumed by the transmission from node (m+1) to node m in the reverse direction. Summing the number of slots consumed by the N nodes yields, SLF/R =
N −1
C(1)(m) + D(1)(m)
For two-phase cooperation with set size i and integer √ and faster scaling by a factor of N over two-dimensional grids due to the contribution of the inter-node distance in the bit.distance product. Moreover, hierarchical cooperation achieves a logarithmic schedule length irrespective of the node arrangement. Thus, hierarchical cooperation is more efficient than the flow √ studied in [5] which generates schedules of length O( N ) slots. VI. D ISCUSSION : F IXED R ADIO R ESOURCES In this section, we relax the adaptive bit rate assumption adopted throughout the paper and discuss the high transmit power assumption adopted for representative nodes in multiphase cooperation. First, we discuss the rationale behind the adaptive bit rate assumption. Second, we quantify the SL of the studied cooperation schemes under fixed link bit rates. Central to this issue is the open problem of characterizing the joint entropy of highly correlated samples studied in [17], namely H(X1 , X2 , ...., Xm ), for finite m. We show that the SL scaling laws will not reduce to closed form expressions because of this fundamental hurdle. Hence, we establish an upper bound on SL, assess its utility and limitations and, finally, identify the requirements for establishing tighter bounds. It is evident that the SL depends on the cooperation strategy and link bit rate. Motivated by the fundamental role cooperation plays in reducing TT and SL, we chose to focus on its inherent trade-offs. Therefore, we assumed so far the availability of radio resources, namely bit rate and transmit power, that could be appropriately adjusted in order to always fit joint and conditional-joint samples in a single time slot. This assumption facilitates the scaling laws derived and summarized in Table I which, in turn, reveal interesting trafficlatency trade-offs attributed to cooperative data compression and ways to balance it. Therefore, this study constitutes an important step towards fully understanding the TT-SL tradeoffs in resource-limited scenarios. Finally, this assumption has been adopted consistently for all cooperation schemes to make sure the results are solely attributed to the construction and operation of each cooperation scheme. A. The Case of Fixed Bit Rate In this section, we assume that the transmission rate is fixed to 1 bit/slot. Accordingly, joint sample transmissions (e.g. C(1)(m) ) would need multiple slots. This has direct impact on the SL scaling results derived in sections III and IV. Hence, we revisit one-dimensional F/R, two-phase and hierarchical cooperation in case of fixed link bit rate.
(11)
m=1
SLtpc =
N i ,
i−1 C(1)(m) + D(1)(m) m=1 N
i C((m−1).i+1)(m.i) + (i − 1) D(1)(i) (12) +
m=1
where the first term gives the number of slots for F/R cooperation within each set (phase 1), the second term represents non-cooperative exchange of joint samples in phase 2 and the last term gives the number of slots for distributing other sets’ data within each set. For hierarchical cooperation with set size i and N is power of i, it can be shown that, SLhier =
log iN p=1
i−1 C(1)(m.ip−1 ) + D(1)(m.ip−1 ) m=1
+(i − 1)
log i N −1 p=1
D(1)( Np ) i
(13)
where the first sum term represents the number of slots consumed by k phases of the cooperation stage where each phase involves F/R cooperation over gradually expanding regions. The second sum term accounts for the distribution stage which is a generalization of the last term in (12) over gradually shrinking regions. The common challenge shared by (11), (12) and (13) is quantifying C(u)(v) and D(u)(v) . According to [17], these quantities are not known for finite v. Therefore, we have to either approximate or bound them to be able to bound SL under the fixed bit rate assumption. In the rest of this section, we focus on F/R cooperation for the purpose of developing bounds and assessing their utility. Similar bounds could be derived for the other schemes as shown later in Theorems 9 and 10. For the exponential spatial 2 correlation Rs (y) = e−y , it has been shown in [12] that, log N lim H(X2 |X1 ) = O (14) N→ ∞ N Notice that the above result is asymptotic in N and, hence, cannot be applied to finite number of samples, namely C(1)(m) and D(1)(m) in (11). However, it can be used to establish bounds on C(1)(m) , for all m, via using the following asymptotic bound on the joint entropy of N samples, C(1)(N ) , H(X1 , X2 , ..., XN ) = H(X1 ) + H(X2 |X1 ) + H(X3 |X2 , X1 ) +... + H(XN |XN −1 , ..., X1 )
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(15)
ELBATT: ON THE TRADE-OFFS OF COOPERATIVE DATA COMPRESSION IN WIRELESS SENSOR NETWORKS WITH SPATIAL CORRELATIONS
≤ H(X1 ) + (N − 1)H(X2 |X1 ) = O(log N )
(16)
where the inequality is obtained via substituting higherdimensional conditional entropies with H(X2 |X1 ) and the upper bound in (16) is obtained using the asymptote of H(X2 |X1 ) in (14). Accordingly, the following upper bound on SLF/R can be established. Theorem 8: If the transmission bit rate is fixed to 1 bit/slot, then the SL for F/R cooperation would scale as, SLF/R = O(N logN )
(17)
Proof: If we ignore the role of conditioning in D(1)(m) , then the claim is proved by substituting C(1)(m) and D(1)(m) in (11) with the asymptotic upper bound in (16). The above result reveals a number of valuable insights. First, fixing the bit rate, such that joint sample transmissions require multiple slots, gives rise to faster SL scaling law for F/R cooperation by a factor upper bounded by logN . Second, we need to assess the tightness of this bound. Notice that the logN factor in (17) is well below the SL scaling laws derived under the adaptive bit rate assumption. Recall from sections III and IV that, except for √ hierarchical cooperation, the SL scaling laws range from O( N ) to linear and super-linear.7 This, in turn, justifies the utility of this bound to F/R and two-phase cooperation, yet, renders it of limited utility for hierarchical since the logN scaling factor in (16) is of the same order as the SL in Theorem 4 which is also logarithmic. Therefore, tighter bounds for C(1)(m) should be sought. We revisit (15) and substitute H(X4 |X3 , X2 , X1 ) and higher-dimensional conditioning with H(X3 |X2 , X1 ) which yields the following bound, H(X1 , X2 , ..., XN ) ≤ H(X1 ) +H(X2 |X1 ) + (N − 2)H(X3 |X2 , X1 ) (18) = O N H(X3 |X2 , X1 ) This, in turn, yields the following upper bound on SL, SLF/R = O N 2 H(X3 |X2 , X1 ) Clearly, this bound is tighter than the one in (17) due to the known information-theoretic fact that conditioning reduces entropy. However, it can only be given in terms of conditional entropy due to the fundamental hurdle of characterizing higher dimensional conditional entropies which is an open problem [17]. Thus, we conclude that the upper bound in (17) is the best achievable given the known trends of conditional entropies. Once higher dimensional conditioning is characterized, tighter bounds could be established. Theorem 9: If the transmission bit rate is fixed to 1 bit/slot, then the SL for two-phase cooperation would scale as, N log i (19) SLtpc = O i (log i + log N ) + i 7 For R (y) = e|−y| , the upper bound on SL F /R in (17) would be √ s O(N N logN ).
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Proof: Similar to SLF/R scaling law under fixed bit rate in Theorem 8, we utilize the upper bound in (16) to prove the claim. Thus, the three terms comprising SLtpc in (12) would scale as follows, N log i) + O(i log N ) i Re-arranging the terms yields (19). Theorem 10: If the transmission bit rate is fixed to 1 bit/slot, then the SL for hierarchical cooperation scales as, SLhier = O i logi N (log i + log N ) (20) SLtpc = O(i log i) + O(
Proof: Along the lines of Theorems 8 and 9, we utilize the upper bound in (16). Therefore, the two terms comprising SLhier in (13) would scale as follows, SLhier = O (i log i) logi N + O (i log N ) logi N Re-arranging the terms yields (20). B. Fixed Transmission Power in Multi-phase Cooperation In section IV, we assumed that the transmission power of representative nodes can be appropriately raised in order to construct the cooperation sets. The rationale behind this assumption is two-fold. First, it conforms with the hierarchical structure of multiphase cooperation whereby specially-designated resource-rich representative nodes have higher transmission power capabilities than others and, hence, have the potential to serve as gateways between various cooperation sets. This yields the sub-linear SL scaling laws witnessed under two-phase and hierarchical cooperation. On the other hand, limiting the transmit power of representative nodes to phase 1 levels not only deviates from the hierarchical mode of operation but also defeats its purpose. This is due to the fact that multi-phase cooperation was originally proposed to attain sub-linear SL scaling law while preserving the sub-linear TT scaling law as shown in section IV. It can be shown that if we limit the transmit power of representative nodes to reach their direct neighbors only then an O(N ) term would always persist in the SL scaling law of multi-phase cooperation. Consider two-phase cooperation as an example. Phase 1 will not be affected since it involves intra-set cooperation only and still consumes O(i) slots. On the other hand, Phase 2 is affected as follows: i) NCF between any neighboring sets, e.g. sets 1 and 2, is now carried out through neighboring members of those sets, namely nodes i and i + 1 (requires 1 slot) and ii) Neighboring nodes on set boundaries distribute exchanged samples within their respective sets, e.g. node i disseminates set 2 data throughout set 1, node i + 1 disseminates set 1 data throughout set 2, etc. (requires O(i) slots). These two activities are repeated Ni − 1 times to achieve many-to-many communications. Therefore, SLtpc under fixed transmission power for representative nodes turns out to scale as O i + N + Ni . Unlike Theorem 2, no SL scaling gains are witnessed due to the persistence of an O(N ) term contributed by flat operation which defeats the purpose of multi-phase cooperation.
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Second, it passes on hierarchy and recursion to the underlying mathematical model due to: i) The fixed number of nodes i per set in all cooperation and distribution phases and ii) The fixed number of transmissions per set, irrespective of the phase index. A byproduct of this hierarchy is an elegant analysis which yields formal results confirming intuition. VII. C ONCLUSION In this paper, we quantified the traffic and latency scaling laws associated with many-to-many communication over oneand two-dimensional wireless sensor networks with spatially correlated sample measurements. First, we showed the existence of simple cooperation schemes that achieve a logarithmic growth rate for the transport traffic with the number of nodes, yet, the schedule length scales linearly. √ Second, we √ showed that√ two-phase cooperation achieves O( N log( N )) and O( N ) for the TT and SL respectively. Moreover, we extended two-phase cooperation to hierarchical cooperation, which limits cooperation to gradually increasing regions of the network, in order to achieve the best, namely logarithmic, TT and SL scaling laws. In addition, we observed that the transport traffic scales faster over two-dimensional grids whereas the scheduling latency was unaffected. Finally, we investigated the impact of fixed transmission bit rate, established bounds on the SL and identified the need to characterize higher dimensional conditional entropies, in future research, as a prerequisite for establishing tighter bounds. Characterizing the many-to-many capacity of wireless multi-hop networks is an essential complement to this work in order to assess the scalability of various cooperation schemes. This work can be extended to random grids and more realistic interference models where the signal-to- interference-and-noise-ratio serves as the criterion for successful reception. A PPENDIX A P ROOF OF T HEOREM 3 We first consider the cooperation stage where cooperation in phase p takes place within each of the N ip cooperation sets simultaneously. Hence, the transport traffic generated throughout the cooperation stage (cs) is given by, (N −i+1) i−1 C(j)(j+i−1) T Tcs = N −1 j=1,(i+1),...
(N −i +1)
+i
C(j)(j+i2 −1) + ...
(N −i(k−1) +1)
+i
C(j)(j+ik−1 −1)
j=1,(i(k−1) +1),...
(N −ik +1)
+i
k−1
(N −ik−1 +1)
D(j)(j+ik−1 −1) + ....
j=1,ik−1 +1,..
(N −i2 +1)
+i
D(j)(j+i2 −1)
j=1,(i2 +1),...
(N −i+1)
+
D(j)(j+i−1)
(22)
j=1,(i+1),...
In (22), the q-th term represents phase q in the distribution stage. From Fig. 4 and 5, it is evident that phase 1 in the cooperation stage matches phase (k − 1) in the distribution stage in the sense that they carry out cooperation and distribution respectively among the same set of nodes and they operate over the same number of sets. The same argument applies to phase 2 in the cooperation stage and phase (k − 2) in the distribution stage, and more generally, to phases p and (k − p) in respective stages. Thus, summing up the matching terms in (21) and (22) (except for the last term in (21) which has no matching term) yields the TT of hierarchical cooperation, N i−1 C(1)(N ) T Thier = N −1 i N i(i − 1) + C(1)(N ) 2 + ... N −1 i k−2 i (i − 1) N + C(1)(N ) k−1 N −1 i k−1 i (i − 1) + (23) C(1)(N ) N −1 The above equation can be reduced to,
i−1 N H(X1 , X2 , ..., XN ) T Thier = k N −1 i
A PPENDIX B P ROOF OF L EMMA 7
j=1,(i2 +1),...
k−2
i−1 ik−2 T Tds = N −1
which proves the claim.
2
the k-th cooperation phase. In addition, an arbitrary term p consists of the summation of N ip terms which corresponds to the number of cooperation sets in that phase. Accordingly, the first summation consists of Ni terms whereas the last summation collapses to a single term since N = ik . Next, we quantify the transport traffic generated throughout the distribution stage (ds) which consists of (k − 1) phases,
C(j)(j+ik −1)
(21)
j=1,(ik +1),...
Notice that the first term in (21) represents the TT generated throughout phase 1 in the cooperation stage, the p-th term represents the TT generated throughout phase p, and the last term (i.e. k-th term) represents the TT associated with
We focus, without loss of generality, on a particular corner node (node 1) and determine the minimum number of nodes required to rebroadcast a message subject to the constraint that all nodes eventually receive the message. To this end, we introduce the notion√of diagonal nodes at level l, denoted DNl , where 1 ≤ l ≤ 2 N − 2 as shown in Fig. 6. The next step is to determine the minimum subset of diagonal nodes per level which guarantee reception of the message at all nodes in the next level. We refer to these nodes (hashed in Fig. 6) as “forwarding nodes” (FN) where F Nl ⊆ DNl ∀l. It is evident that the cardinality of FN, |F N |, depends on the diagonal level
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ELBATT: ON THE TRADE-OFFS OF COOPERATIVE DATA COMPRESSION IN WIRELESS SENSOR NETWORKS WITH SPATIAL CORRELATIONS
relative to the principal diagonal shown below, ⎧ l+1 : ⎪ ⎪ 2 ⎪ l+2 ⎪ ⎪ : ⎨ 2 l : |F Nl | = 2 √ ⎪ 2 N −l−1 ⎪ ⎪ : ⎪ 2 √ ⎪ ⎩ 2 N −l : 2
characterized by l = √ l < √N l < √N l= N √ l> N √ l> N
√ N as
and odd and even and odd and even
Accordingly, the TT associated with the dissemination of node 1’s sample over the grid, √N −1 l + 1
H(X1 ) T Tnode1 = √ 2 N −1 l=1,3,..
√ N −2
l=2,4,.. √ 2 N−3
+
+
+
√
l+2 N + 2 2 √
2 N −l−1 2
√ l= N +1,.. 2√ N −2
√
2 N −l 2
√ l= N+2,..
=O
√ N H(X1 )
(24)
where the outcome of the arithmetic series summations yields the result in (24) and proves the claim. ACKNOWLEDGMENT The author would like to thank the reviewers and the editor for their comments and suggestions which improved the presentation of this paper. We also thank Hesham El Gamal for insightful discussions at early stages of this work. R EFERENCES [1] T. ElBatt, “On the scalability of hierarchical cooperation for dense sensor networks," in Proc. IEEE/ACM International Conf. Inform. Processing Sensor Networks (IPSN), Apr. 2004. [2] T. ElBatt, “On the cooperation strategies for dense sensor networks," in Proc. IEEE International Conf. Commun. (ICC), June 2004. [3] D. Estrin, D. Culler, K. Pister, and G. Sukhatme “Connecting the physical world with pervasive networks," IEEE Pervasive Computing, vol. 1, no. 1, pp. 59-69, Jan.-Mar. 2002. [4] P. Gupta and P. R. Kumar “The capacity of wireless networks," IEEE Trans. Inform. Theory, vol. 46, no. 2, pp. 388-404, Mar. 2000. [5] A. Scaglione and S. Servetto “On the interdependence of routing and data compression in multi-hop sensor networks," in Proc. ACM International Conf. Mobile Computing Networking (MOBICOM), Sept. 2002. [6] S. Servetto, “Distributed signal processing algorithms for the sensor broadcast problem," in Proc. Conf. Inform. Sciences Systems (CISS), Mar. 2003.
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Tamer ElBatt (M’00-SM’06) received the B.S. and M.S. degrees in Electrical Engineering from Cairo University, Cairo, Egypt, in 1993 and 1996, respectively, and the Ph.D. degree in Electrical and Computer Engineering from the University of Maryland at College Park, MD, in 2000. From 1993 to 1996, he was a Teaching Assistant with the Department of Electronics and Communications Engineering, Cairo University, Cairo, Egypt. From 1996 to 2000, he was a Research Assistant in the Department of Electrical and Computer Engineering, University of Maryland at College Park, MD. From 2000 to 2006, he was with HRL Laboratories, LLC, Malibu, CA as a Research Scientist in the Information and System Sciences Laboratory. From 2006 to 2008, he was with San Diego Research Center as a Senior Research Staff Member. In January 2008 he joined the Advanced Technology Center (ATC) of the Lockheed Martin Space Systems Company, Sunnyvale, CA where he is currently a Senior Research Scientist and leads Communications and Networking R&D. He is a recipient of HRL Achievement Award in 2002, 2004. He has over 35 publications in refereed journals and international conferences, holds 4 U.S. patents and four more applications pending. He is a Senior Member of the IEEE, currently serves on the editorial board of the W ILEY I NTERNATIONAL J OURNAL OF S ATELLITE C OMMUNICATIONS AND N ETWORKING, and has served on the program committees of major IEEE and ACM conferences, e.g. INFOCOM, MOBIHOC, MOBICOM, ICC, IPSN, SECON, and VANET. His research interests lie in the broad areas of performance analysis and design of wireless networks with emphasis on clean-slate networking architectures, cross-layer design, MAC, MIMO networking, sensor and vehicular networks.
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