2013 IEEE Wireless Communications and Networking Conference (WCNC): NETWORKS
Compressed Topology Tomography in Sensor Networks Yao Liang and Rui Liu Department of Computer and Information Science Indiana University-Purdue University Indianapolis Indiana, USA
[email protected],
[email protected] Abstract—Wireless sensor network (WSN) topology tomography is essential for routing improvement, topology control, anomaly detection and load balance. Previous studies on WSN topology tomography are restricted to static routing tree estimation, which is unrealistic in real-world WSNs due to wireless channel dynamics. We study general WSN routing topology tomography from indirect measurements observed at the sink, where routing structure is dynamic. We formulate the problem as a novel compressed sensing problem, and present our decoding algorithm. We provide rigorous complexity analyses of our algorithm. Thorough simulations validate the effectiveness of our approach and algorithm. Keywords—network tomography; wireless sensor networks; compressed sensing; algorithm complexity
I.
INTRODUCTION
Wireless sensor networks (WSNs) are growing rapidly in both size and complexity in wide deployments. Consequently, it becomes increasingly important to monitor the WSN structure and dynamics using indirect measurements obtained at the WSN sink(s). Referred to as network tomography (e.g., [1-4]), this capability plays a significant role in routing improvement, topology control, anomaly detection and load balance, which enables effective management and optimized operations for deployed WSNs consisting of unattended wireless sensor nodes. While most existing research on WSN tomography has focused on link loss and delay monitoring [59], studies on WSN topology tomography were restricted to static routing tree estimation [10, 11] due to its simplicity. However, wireless channel dynamics (e.g., fading and interference) inevitably leads to time-varying routing topology in real-world WSN deployments, making the current static routing tree estimation problematic and not useful. This motivates our work.
the first of its kind to addresses WSN topology tomography regarding time-varying routing topology. The remainder of the paper is organized as follows. Section II briefly gives the network topology model adopted in this work. Section III presents the problem formulation. Section IV presents our fast WSN topology recovery algorithm. In Section V, we provide the complexity analyses of our devised algorithm. Section VI reports our simulations. Finally, Section VII gives the conclusions and outlines our future work. II.
We assume WSN routing is dynamic in a cycle of data or measurements collection due to wireless link dynamics, and we consider routing topology without looped routes. A routing topology for WSN is modeled by a directed acyclic graph G = (V, E), where V is a set of n nodes (the sink s and n-1 sensor nodes), i.e., its cardinality |V| = n, and E is a set of edges. The sensor nodes are battery-operated while the sink is assumed to be not power-limited. A directed edge , , represents the wireless communication link from node to node . Each edge is associated with a unique label , , given by a labeling function L: E→N, where N denotes the set of positive integers. A routing path originated from sensor node , , , where t1, to the sink is denoted as , ,, , , t2, …, tj are relay sensor nodes. Let yi denote an indirect path , , is measurement of path pi at the sink. Then, Y a measurement vector ( 1), representing a complete set of path measurements for all sensor nodes in the G of the WSN. For a static WSN routing, the routing topology is represented as a directed spanning tree , , with the 1. is the edge set and | | sink being the root, where
We study the general WSN topology tomography aimed to recover the dynamic routing structure. We take a compressed sensing (CS) [12-14] based approach, and formulate the problem as a novel CS problem [15]. We devise a fast topology recovery algorithm based on compressed indirect measurement at the sink. In this paper, we extend our work in [15] and present the rigorous complexity analyses of our topology recovery algorithm. In addition, we provide thorough simulations to validate and evaluate our dynamic topology recovery algorithm. To our knowledge, our work represents
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ROUTING MODEL
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Figure 1. An illustration of an A-‘Tree’.
Note that in a static routing tree, each sensor has the unique path toward the sink, which allows for the very simple recovery of the tree topology. Now consider a general acyclic dynamic routing topology G ( ), which can be decomposed into a (directed) spanning tree and some additionally attached edge(s). These additionally attached directed edges are referred to as ‘shortcuts’ in G. In this sense, a G is referred to as a (directed) Augmented ‘Tree’ (A-‘Tree’), as illustrated in Fig. 1. Let denote the set of shortcuts, then we have , 1 | | . Thus, each sensor with | | | | | | node may have different paths to the sink at different time instances due to the dynamic routing under wireless channel dynamics.
| | ), which is a reasonable assumption in WSN practice for one cycle of data/measurements collection. Thus, we innovatively formulate the dynamic routing topology tomography problem as follows. Given a measurement vector Y at the WSN sink, recover the X and measurement matrix , so that || || subject to , (2) where l0-norm ||X|| is the number of nonzero elements in the | |. vector X, that is || ||
We consider to recovery the dynamic routing topology structure evolving along time. Given a set of M (M=n-1) path measurements originated from individual n-1 sensor nodes received at the sink, the entire dynamic routing topology G(V, E) will be exactly reconstructed with .
We note that in the traditional CS formulation [12-14], measurement matrix is known a priori whether randomly or deterministically generated. In contrast, the in our problem formulation of (2) above is completely unknown and determined by the underlying routing algorithm operated in an undeterministic real-world communication environment. However, we know each potential link’s value a priori by the use of labeling function (see Section II). That is, our formulation of (2) is to recover and therefore to reconstruct the sparseness pattern of the X, given a Y.
We formulate the problem of WSN routing topology tomography from a compressed sensing perspective. The standard CS framework can be represented as , where is an 1 sparse discrete signal vector, is an measurement matrix and is the 1 measurement vector. The CS theory enables, under certain conditions, to recover X from Y where , as long as signal is sparse. This can be achieved by solving the following optimization:
A path measurement is calculated as the packet routed through the network towards the sink. Unlike the traditional CS approaches, we employ modular summation (with mod m) (SUMm) rather than regular summation, for efficient WSN innetwork computing and communications for scalability. In addition, we adopt exclusive or (i.e., XOR) as the second metric to reduce the probability of inaccurate topology reconstruction.
III.
FORMULATION
|| || subject to
,
IV.
(1)
where ||X||p (p = 0, 1) denotes lp-norm of X. In this work, we introduce the new concept of so-called , of WSN upstream routing, where Base Topology | | , and is a set of all possible edges in the WSN upstream routing. Since outgoing links from the sink are excluded in WSN upstream routing, the total number of all possible directed wireless links (considering asymmetry wireless channel property) for G* should be | | 1 1 1 . Our insight is that the number of wireless links actually used in a WSN routing topology Gi for a measurement/data collection cycle would be much fewer than the total potential choices in the upstream base topology G*, because reliable wireless links are likely to be reused whenever possible. Let | | 1 . Let be a link (labeling value) vector of dimension, which shall be sparse. When the sink receives a set of M different path measurements, denoted as an 1 vector , , where 1. Thus the (1 , 1 ) can measurement matrix , represent a routing matrix in the WSN where the th row represents the th path while the th column represents the th link, whose elements , are defined as 1, ; , 0, . Note that | | 1 | | , where | | is the number of shortcuts in G (i.e., A-‘tree’), as discussed in Section II. Since is sparse, | | should be a relatively small number (e.g.,
TOPOLOGY RECOVERY ALGORITHEM
We present our algorithm, referred to as fast routing topology recovery (FRTR) algorithm, to solve the topology tomography problem of (2). As data/measurement packets are received at the sink in sequence, we can assume that due to the sparsity of link vector X in WSN, any routing path originated from a new sensor node will not introduce more than two new wireless links not used before in a collection cycle. This assumption is indeed the most sparseness assumption regarding the dynamic routing, for in static routing tree, any routing path from a new sensor node will not introduce more than one new wireless link. In our FRTR algorithm, each measurement packet originated from a sensor node i contains the node’s unique ID i, and two path measurement yi and yi’ according to SUMm and XOR respectively. The sink receives these packages in sequence and will form a sequence vector S={i1,i2, ,iM}, and the corresponding measurement vector Y={{yi1,yi1’},{yi2,yi2’}, , {yiM ,yiM’}}. FRTR algorithm is based on the observation that for a candidate path originated from node i in the search space, if its path calculation results based on SUMm and XOR metrics match the given two compressed path measurements{yi , yi’}, then this candidate path is the real routing path with a very high probability. The basic idea of FRTR algorithm is to check each node child in the sequence vector in the receiving order. The sink and all the previously recovered nodes will be considered as its potential parent node candidates Candidates. For each candidate node parent, finding its all possible paths without
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Notation getSize(s): return the size of the set s; s1 s2: join the two sets s1 and s2; group(s): group the same topologies in the set s. findPaths(n, t): find all possible paths with at most one shortcut from node n to the root r in topology t; prepend(n, p): add node n to path p and return new path; getPathSum(p): compute the module sum of all edge labels along path p; getPathXor(p): compute the exclusive or of all edge labels along path p; update(t, p): add new edge(s) along path p into topology t and return the new topology.
Figure 3. An illustration example for FRTR. The thick arrows show the recovered path for the new incoming node. The blue dashed edge is the new shortcut introduced by the new incoming node.
in the ascending order of their levels and the path candidates could be sorted by the number of hops. A. Example Fig. 3 illustrates how FRTR algorithm works for a network with 4 nodes. In this network, the sink is node 0; the sequence vector is S={1,2,3}; and the indirect path measurement vector Y, based on both module Sum and Xor measurement metrics, is Y={{7,7},{3,3},{9,7}}. The labels assigned on edges are given in Fig. 3. For convenience, “recovering node i” is used to refer “recovering the path originated from node i”. These two terms are exchangeable. Fig. 3 (a) is the initial topology which only contains the sink node 0. For recovering the first child node 1, the sink node is its only parent node candidate and the path {e1,0} is its only path candidate. Since {e1,0} match the path measurement {7,7}, node 1 is the child of the sink node 0 as shown in (b). For recovering the next child node 2, either node 0 or node 1 could be its parent. If node 0 is examined first, the possible path is {e2,0} which fits the path measurement {3,3}, and thus the parent of node 2 would be node 0 as shown in (c). In this step for node 2, FRTR will no longer examine the candidate node 1 as node 2’s potential parent node. Similarly for node 3, its parent node candidates are {0, 1, 2}. If the label of the edge e3,0 is not 9, node 0 cannot be its parent. The next parent node candidate is node 1 where the path {e3,1, e1,2, e2,0} fits, then the recovered topology for this step is shown as (d) in which there is a shortcut e1,2 marked by the blue dotted line.
Function FRTR(S, Y, r) 1: TP←{{r}}; /*initial topology TP*/ 2: for (i = 1; i ≤ getSize(S); i++) 3: child←S[i]; y1←Y[i,1]; y2←Y[i,2]; 4: Candidates←{r} S[1, …, i-1]; 5: for all candidate parent Candidates do 6: newTP←findTP(child, parent, y1, y2, TP); 7: /*if a valid newTP found, break the inner for loop*/ 8: if (newTP ≠Null) then break; 9: end for 10: TP←newTP; 11: end for Function findTP(child, parent, y1, y2, TP) 1: newTP ←Null; /* initial newTP as Null */ 2: PS←findPaths(parent, TP); 3: for all path p PS do 4: p←prepend(child, p); 5: if (getPathSum(p) == y1 && getPathXor(p)==y2) 6: then 7: newTP←update(TP, p); 8: return newTP; 9: end for
V.
Figure 2. FRTR algorithm.
new shortcut or with one new shortcut based on the recovered topology TP and check whether the SUMm aggregation and the XOR aggregation of a path candidate p matches both of the received indirect path measurement y1 and y2. If they match, update the topology as newTP and return it. Once a valid parent node is found for the child node, other unchecked parent candidate nodes will not be examined. The returned topology newTP will be considered as the recovered topology for the next child node. Fig. 2 gives the FRTR algorithm and its main helper function findTP. Since FRTR algorithm will stop further searching after it finds the first valid candidate, its performance could be improved by sorting the parent candidates Candidates and the path candidates PS according to the properties of a given WSN routing mechanism. For instance, if nodes in a given network prefer to choose the shortest available paths, the parent candidates could be sorted
COMPLEXITY ANALYSIS
We analyze the complexity of our devised FRTR algorithm in a rigorous manner. The complexity of FRTR will also be compared with some traditional CS algorithms. A. Complexity of FRTR We first show that the complexity of Function findTP given in Fig. 2 is O(n2) based on the following three theorems, where n is the size of WSN (i.e., the total number of WSN nodes). Theorem 1 Given a directed spanning tree SPj-1 consisting of j1 nodes, adding the jth node into SPj-1 to create a new spanning tree SPj, if the topology of the formed SPj is linear, the number of possible additional paths for the jth node to the root by introducing a shortcut edge will be maximized. Proof Let PNj denote the number of possible additional paths towards the root with one shortcut for the jth node in the spanning tree SPj. When j
