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Localization-Oriented Network Adjustment in Wireless Ad Hoc and Sensor Networks Tao Chen, Member, IEEE, Zheng Yang, Member, IEEE, Yunhao Liu, Senior Member, IEEE, Deke Guo, Member, IEEE, and Xueshan Luo Abstract—Localization is an enabling technique for many sensor network applications. Real-world deployments demonstrate that, in practice, a network is not always entirely localizable, leaving a certain number of theoretically nonlocalizable nodes. Previous studies mainly focus on how to tune network settings to make a network localizable. However, the existing methods are considered to be coarse-grained, since they equally deal with localizable and nonlocalizable nodes. Ignoring localizability induces unnecessary adjustments and accompanying costs. In this study, we propose a fine-grained approach, localizability-aided localization (LAL), which basically consists of three phases: node localizability testing, structure analysis, and network adjustment. LAL triggers a single round adjustment, after which some popular localization methods can be successfully carried out. Being aware of node localizability, all network adjustments made by LAL are purposefully selected. Experiment and simulation results show that LAL effectively guides the adjustment while makes it efficient in terms of the number of added edges and affected nodes. Index Terms—Localization, localizability, network deployment, wireless ad hoc networks, sensor networks
Ç 1
INTRODUCTION
A
the proliferation of wireless and mobile devices continues, a wide range of context-aware applications are deployed, including smart space, modern logistics and so on. In these applications, location information is the basis of other services, such as geographic routing, boundary detection, and network coverage control. In some other applications, such as military surveillance and environment monitoring, sensed data without location information are almost useless. Localization in wireless ad hoc and sensor networks is the problem in which every node determines its own location. In this work, we focus on 2D in-network localization [1], [2], [3], [4], in which some special nodes (called beacons or anchors) know their global locations and the rest determine their euclidean coordinates by measuring the euclidean distances to their neighbors. Due to hardware or deployment constraints, a network can be partially localizable given distance measurements and locations of beacons, that is to say, some nodes have unique locations while others do not. Theoretical analysis [5] indicate that, unless networks are highly dense and regular, in most cases, it is unlikely that all nodes in a network are localizable. In practice, a wireless ad hoc or sensor network cannot be excessively dense because the S
. T. Chen, D. Guo, and X. Luo are with the Key lab of Information System Engineering, College of Information Systems and Management, National University of Defense Technology, Room 603, Changsha, Hunan 410073, China. E-mail: {emilchenn, guodeke}@gmail.com,
[email protected]. . Z. Yang and Y. Liu are with the School of Software and TNList, Tsinghua University, Beijing 100084, China. E-mail: {yang, yunhao}@greenorbs.com. Manuscript received 14 Sept. 2012; revised 1 Jan. 2013; accepted 2 Jan. 2013; published online 11 Jan. 2013. Recommended for acceptance by J. Cao. For information on obtaining reprints of this article, please send e-mail to:
[email protected], and reference IEEECS Log Number TPDS-2012-09-0847. Digital Object Identifier no. 10.1109/TPDS.2013.17. 1045-9219/14/$31.00 ß 2014 IEEE
mechanism of topology control is usually used to alleviate collision and interference, ignoring the localizability of the network. We find supporting evidences from two real sensor network systems OceanSense [6] and GreenOrbs [7]. Networks are not entirely localizable in their initial deployments. Based on graph rigidity theory, network localizability [8] and node localizability [5], [9] are studied which solve the location-uniqueness of a network or a single node, respectively. So far one can distinguish localizable and nonlocalizable nodes in theory. To locate nonlocalizable nodes, the existing solutions mainly focus on how to tune network settings. The first attempt is to deploy additional nodes or beacons in application fields. Such incremental deployment increases node density and creates abundant internode distance constraints, thus, enhancing localizability. However, the attempt lacks feasibility, since the additional nodes should be placed in the vicinity of nonlocalizable nodes, whose locations are just unknown. Using mobile nodes (e.g., beacons) is another choice. The controlled motion of beacons provides thorough information for localization, but also incurs adjustment delay and controlling overheads [10], [11], [12]. Both incremental deployment and the use of mobile node require additional hardware cost which may not be available in many applications. A more practical and convenient way is to increase the distance ranging capability of sensor nodes. Enhanced nodes can measure the distances to a larger number of their nearby nodes. In popular ranging techniques, such as time of arrival (ToA) and received signal strength (RSS), the capability enhancement can be achieved by augmenting the transmitter power output. One approach is to augment the transmitting power of nodes stage-by-stage until all nodes become localizable, which causes multiple rounds of configuration dissemination and data collection in a network. A straight-forward Published by the IEEE Computer Society
CHEN ET AL.: LOCALIZATION-ORIENTED NETWORK ADJUSTMENT IN WIRELESS AD HOC AND SENSOR NETWORKS
single-round solution is maximizing the ranging capability. The principle drawback of power maximization is that it introduces many unnecessary distance measurements, which are obtained with costs. Anderson et al. [13] propose a single round approach to construct localizable networks. In their method, all nodes set their power doubly or triply of the original values. This method is coarse-grained, however, since it equally treats localizable and nonlocalizable nodes. Ignoring localizability induces a mass of meaningless adjustments and accompanying costs. In this study, we propose localizability-aided localization (LAL). We adopt sufficient condition of node localizability [9] to identify localizable and nonlocalizable nodes. A fine-grained configuration is computed systematically according to network topology and localizability information. Theoretical analysis guarantees that a tuned network is localizable and thus, well prepared for localization. The main contributions of this study are as follows: First, being aware of node localizability, adjustments made by LAL are purposefully selected, avoiding meaningless ranging and communication costs. Second, LAL is light weight, working properly with the existing localization methods (e.g., Sweeps [3], etc.) without incompatibility. Finally, we implement LAL on a real-sensor network consisting of 100 nodes. The deployment data are collected from in-situ measurement in a wild application field. The remainder of this paper is organized as follows: Related work is discussed in Section 2. We introduce graphrigidity theory briefly and discuss the state-of-the-art approach to construct localizable graph in Section 3. Theoretic foundations for LAL are provided in Section 4. Section 5 presents the design and implementation of LAL. Some discussions are provided in Section 6. Evaluation is discussed in Section 7. We conclude the work in Section 8. This is an extended work of a conference publication [14]. We have added the proofs of Theorem 4, 5 and Proposition 1. In Section 5, we have added Algorithm 1 and the explanations of Algorithm 2. Section 6 has been added to discuss the complexity and implementation issues. Realworld experiment and more simulation results have been added in Section 7.
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RELATED WORK
There are two major categories of localization methods: range-free and range-based methods. Range-free localization methods [15], [16] merely use neighborhood information (such as node connectivity and hop count) to determine node locations. Range-based approaches [1], [2], [3], [17], [18], [19], [20] assume nodes are able to measure internode distances, and can provide more accurate localization results. Many range-based algorithms adopt distanceranging techniques, such as radio signal strength (RSS) [1], [19] and time difference of arrival (TDoA) [17], [20], [21]. The RSS maps received signal strength to distance according to a signal attenuation model, while TDoA measures the signal propagation time for distance calculation. The majority of localization algorithms [2], [21] assume a dense network such that iterative trilateration (or multilateration) can be conducted. Other methods [3] record all possible locations in each positioning step and prune incompatible ones whenever possible. Those localization
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algorithms only try to localize as many nodes as possible, however, do not concern the rest nonlocalizable part of the network. The existence of nonlocalizable networks leads to the research on location-uniqueness problem (a.k.a, localizability problem). Goldenberg et al. [5] first propose nontrivial, necessary, and sufficient conditions of locationuniqueness of specific node. Yang and Liu [9] further derive currently the best necessary and sufficient conditions of node localizability. In recent years, a few works are published on localization in nonlocalizable networks. Pathirana et al. use a mobile robot to localize nodes [22]. They obtain distance information between robot and fixed nodes using RSS data, so as to reduce the number of beacons required to uniquely localize a network. However, they require the availability of precise velocity and acceleration of mobile robot. Sichitiu and Ramadurai use a GPS-equipped mobile node to localize fixed nodes by measuring the distance between the mobile node and fixed ones [23]. Different from the above approaches, which use a mobile node with known location information, Priyantha et al. propose an approach, which only requires the distance information between mobile node and fixed ones [10]. Wu et al. propose a similar approach with the assumption that each node can move around and measure the distances to its neighbors and the relative distances between successive positions along its trajectory [12]. However, the assumption of availability of mobile nodes is costly and not scalable, and restricts the application of the approach. In contrast to these mobilitybased approaches, Anderson et al. propose a graph manipulation method to assure the network localizability [13]. They identified graphical properties which can ensure unique localizability, and propose a G2 approach to make the two-connected graph localizable and a G3 approach to obtain a trilateration graph based on a connected graph. Both approaches treat all vertices in the graph as a whole, which is more coarse-grained than LAL.
3
PRELIMINARIES
3.1 Localizability The ground truth of a network can be modeled by a distance graph G ¼ ðV ; EÞ, where V denotes the set of vertices (wireless devices, e.g., sensor node, RFID) and E denotes the set of edges. For i; j 2 V ; ði; jÞ 2 E if the distance between i and j can be measured or both of them are in known locations (e.g., beacon nodes). We assume G is connected and has at least four vertices and focus on 2D space in the following analysis. Localization is to find a map p (a.k.a., realization) from vertices in G to points in a euclidean space. If p respects the distance constraints between any pair of vertices as depicted in E, p is a feasible realization of graph G. Two realizations are equivalent if they are identical under translations, rotations, and reflections. A distance graph G has at least one feasible realization which represents the ground truth of the corresponding network. Formally, G is embeddable in 2D space and all pairwise distances are compatible. A realization is generic if the vertex coordinates are algebraically independent. Like many previous work on graph rigidity theory [24], [25] and network localization [3], [5], [8], [12], [13], [26], we focus on generic cases in our
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study, since the set of generic realizations is dense in the realization space and almost all realizations are generic [27]. We omit the word generically, hereafter. A graph is called rigid if one cannot continuously deform its realizations while preserving distance constraints, and redundantly rigid if it remains rigid upon removal of any single edge [28]. A graph is globally rigid if it is uniquely realizable [27]. Jackson and Jorda´n [24] summarize the reasons of realization nonuniqueness, and present the following theorem. Theorem 1 ([24]). A graph with n 4 vertices is globally rigid in two dimensions if and only if it is three-connected and redundantly rigid. A graph G ¼ ðV ; EÞ is called k-connected (for k 2 IN) if jV j > k and G X is connected for every set X V with jXj < k. In other words, no two vertices in G are separated by removing fewer than k other vertices. Eren et al. [8] further prove that a network is uniquely localizable if and only if its distance graph is globally rigid and it contains at least three beacons. Yang and Liu [9] present necessary and sufficient conditions of node localizability, which can be used to identify localizable and nonlocalizable nodes in a network. Theorem 2 ([9]). In a distance graph G ¼ ðV ; EÞ with a set B V of k 3 vertices at known locations, a vertex is localizable if it is included in the redundantly rigid component inside which there are three vertex-disjoint paths to three distinct vertices in B. A redundantly rigid component is a maximal redundantly rigid subgraph in G. Theorem 2 is a sufficient condition of node localizability, denoted by RR3P for short, where RR and 3P stand for redundant rigidity and three vertex-disjoint paths, respectively.
3.2 Construction of Localizable Graph Given a nonlocalizable graph, it is important to make it localizable through incremental construction. Define G2 as ðV ; E [ E 2 Þ, where ði; jÞ 2 E 2 if i 6¼ j and 9k 2 V such that ði; kÞ and ðj; kÞ 2 E. Similarly, define G3 as ðV ; E [ E 2 [ E 3 Þ, where ði; jÞ 2 E 3 if i 6¼ j and 9k 2 V such that ði; kÞ 2 E and ðj; kÞ 2 E 2 . Based on Theorem 1, Anderson et al. present the following result. Theorem 3 ([13]). Let G ¼ ðV ; EÞ be a two-edge-connected graph in 2D space. Then G2 is globally rigid. G is called l-edge-connected, if jV j > 1 and G F is connected for every set F E of fewer than l edges. A kconnected graph is also k-edge-connected. G2 is obtained from G by adding edges between twohop vertex pairs. Note that in some ranging techniques (e.g., RSS and ToA), RF signal is used for distance measurement. Thus, the communication range approximately represents the maximum ranging distance. In a wireless network, doubling the communication range will produce a new graph which has G2 as a subgraph. Similarly, G3 can be obtained by tripling the communication range. Anderson et al. also prove that, if G ¼ ðV ; EÞ is
Fig. 1. Proof of Theorem 5. Red dots are beacons.
connected, G3 is globally rigid (specifically, G3 can be localized by iterative trilateration).
4
THEORETIC FOUNDATION
4.1 Component Tree To support fine-grained network adjustment, we decompose a graph into components according to connectivity. A two-connected component in a graph G is a maximal subgraph of G without articulation vertex whose removal disconnects G. A graph G is minimally rigid if G is rigid, and G e is not rigid for all e 2 E. Jackson and Jorda´n [24] summarize the link between rigidity and two-connectivity. Lemma 1 ([24]). Let G ¼ ðV ; EÞ be minimally rigid with jV j 3. Then G is two-connected. Based on Lemma 1, we obtain the following theorem. Theorem 4. Given a connected distance graph G with more than three vertices, all beacons and localizable vertices recognized by RR3P in G are in the same two-connected component. Proof. According to Theorem 2, beacons and localizable nodes are included in a redundantly rigid component which is two-connected suggested by Lemma 1. u t We call the two-connected component in Theorem 4 anchor component (denoted by GA ) for convenience. Theorem 5. Suppose G ¼ ðV ; EÞ is a two-connected graph with a set B of k 3 beacons and B V . Let VN denote the set of nonlocalizable vertices, and EN denote the set of edges ði; jÞ, i 2 VN and ði; jÞ 2 E 2 . Then, G ¼ ðV ; E [ EN Þ is localizable. Proof. The theorem holds for the trivial case that VN ¼ ;. For VN 6¼ ;, if there is no localizable vertex in G other than beacons, adding EN to G results in G2 , which is localizable according to Theorem 3. Otherwise, there exists a localizable vertex vl 2 V B. For any vn 2 VN , two vertex-disjoint paths p1 and p2 can be found connecting vn and vl since G is two-connected. As illustrated in Fig. 1a, p1 and p2 form a cycle C. Suppose v1 and v2 are two direct neighbors of vl in C. Consider the worst case that C contains only one localizable vertex, thus, v1 and v2 are nonlocalizable. As vl is
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Fig. 2. Two categories of two-connected components.
localizable, it has three vertex-disjoint paths to three beacons, and all three paths are inside the redundantly rigid component (denoted by RRC) of B. Adding EN to G will connect v1 to all neighbors of vl . Thus, v1 is included by the RRC since it connects four vertices that are originally in RRC. And v1 has three vertex-disjoint paths to three beacons in RRC no matter whether or not it stands on one of the three paths of vl (an example is shown in Fig. 1b). Theorem 2 ensures that v1 is localizable in G ¼ ðV ; E [ EN Þ. For the same reason, v2 is localizable. Hence, C 2 is localizable due to Theorem 3 as C is two-connected and contains three localizable vertices. As a result, vn is localizable in G ¼ ðV ; E [ EN Þ. Since, vn is arbitrarily chosen, G ¼ ðV ; E [ EN Þ is localizable entirely. u t Note that beacons are not required to be deployed together in one hop distance in Theorem 5. A direct result of the theorem is that, if we add EN to the anchor component, GA becomes localizable. GA can be seen as localizable in the following analysis. We decompose a graph into two-connected components. Except for the anchor component, all of them are nonlocalizable. We classify them into two categories, as illustrated in Fig. 2. Category I has two interconnected vertices, and is trivially two-connected. Category II has at least three vertices. The anchor component is a category II component. Two components sharing a common vertex are adjacent. They cannot have more than one common vertex, or they should be merged into one. We construct component tree T ¼ ðVT ; ET Þ of G ¼ ðV ; EÞ as follows: Any vertex i 2 VT corresponds to a two-connected component of G and ði; jÞ 2 ET if the correspondents of i and j are adjacent in G. For convenience, let GA be the root of T . Fig. 3 illustrates the construction of a component tree T from G. We decompose G into several two-connected components and connect neighboring ones in T . Following such construction, T contains no cycle. Location information, as well as localizability, diffuses from the root of T to leaf vertices along tree edges. Another issue needs to be addressed here is the path from the root to any tree vertex in T . The category permutation of the components along a path determines how to make adjustments in our solutions. For simplicity, we use a string such as “II þ þ I” to describe the categories of components on a path, from the root to a destination component. Taking Fig. 3 as an example, the path to G4 is denoted by “II þ I þ II þ I”. We call such a string “path string” henceforth.
4.2 Component-Based Adjustment Before discussing component-based adjustment, we define the operation of vertex augmentation.
Fig. 3. Construction of component tree.
Definition 1 (Vertex Augmentation). In a graph G ¼ ðV ; EÞ, the vertex augmentation of v 2 V (denoted by vk , k 2 IN) is to connect v and its i-hop neighbors in G (i k). In a distance graph G ¼ ðV ; EÞ of a network, for any v 2 V , vn can be achieved by increasing the ranging capability of v (more accurately, v’s correspondents) by n times. Since there are two categories of two-connected component according to the classification in this work, a path string of a component can be decomposed into a series of overlapping segments of two consecutive categories. There are four cases: “I þ I,” “I þ II,” “II þ I,” and “II þ II.” If we provide adjustment solutions for all four cases to make the corresponding components localizable, the entire network becomes localizable by applying these solutions from the root to leaves in a component tree. When discussing the solutions of the four cases, we assume the former component has already been manipulated and thus localizable. The assumption is reasonable because GA is the root and made localizable according to Theorem 5. .
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Case “II þ I.” Suppose two corresponding components are G1 and G2 , as shown in Fig. 4. The vertex v1 is localizable and has two localizable neighbors v2 and v3 in G1 . Suppose the neighbor of v1 in G2 is v4 . The operation v24 adds two edges ðv4 ; v2 Þ and ðv4 ; v3 Þ in E 2 , which makes v4 localizable by connecting it to three localizable vertices. Case “II þ II.” Fig. 5 shows two category II components G1 and G2 . The vertex v1 is localizable, and has two localizable neighbors v2 and v3 in G1 , and two nonlocalizable neighbors v4 and v5 in G2 . G22 is globally rigid according to Theorem 3. Similar to the case “II þ I”, v4 and v5 are made localizable by v24 and v25 . Then G22 is localizable since it is globally rigid and has three vertices whose location can be uniquely determined by calculations.
Fig. 4. Case “II þ I”: Localize the category I component by adding two edges. Solid lines are edges in E and dashed lines are edges in E 2 .
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Fig. 5. Case “II þ II”: G22 is localizable through v24 and v25 . Solid lines are edges in E, and dashed lines are edges in E 2 .
v33
Fig. 6. Case “I þ I”: is needed to make G3 localizable. Solid lines are edges in E. The dashed line is an edge in E 2 , and the dotted line is an edge in E 3 .
Fig. 7. Case “I þ II”: v33 (or v34 ) is needed to make G23 localizable.
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Case “I þ I.” Suppose the two category I components are G2 and G3 . v2 is the articulation vertex of G2 and G3 , and two vertices v1 and v3 reside in two components, respectively, as shown in Fig. 6. Because all paths start with a category II component GA , G2 must have a parent component G1 . No matter the category of G1 , v1 has at least one localizable neighbor v4 . v23 only adds a single edge ðv1 ; v3 Þ, and v3 still does not fulfill the sufficient condition in Theorem 2. To make v3 localizable, at least one edge such as (v3 , v4 ) should be added. That is to say, we should do v33 in the original topology to make G3 localizable. Case “I þ II.” Suppose the two components are G2 and G3 as shown in Fig. 7. Similar to the case “I þ I”, v33 is needed to make v3 localizable. Then, v24 makes v4 localizable. Since G23 is globally rigid according to Theorem 3 and the locations of three vertices v2 , v3 , and v4 can be uniquely determined, all vertices in G3 are localizable.
LAL DESIGN AND IMPLEMENTATION
The previous section established the theoretical foundation of LAL. In this section, we discuss LAL design and implementation. Being aware of node localizability, LAL can effectively guide the adjustment of network, while traditional approach could only make indistinctive augmentations. Basically, LAL consists of three major steps, as depicted in Fig. 8. Step 1: Localizability testing. When a network is deployed in an application field, due to some systematic or environment factors unpredictable in the design phase, it may be not ready for localization. Hence, node localizability testing is conducted foremost in LAL, which identifies localizable and nonlocalizable nodes in a network for further adjustment.
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Fig. 8. Workflows of traditional approaches and LAL.
Fig. 9. Proof of Proposition 1, when v 2 GA and more than one nodes are localizable. Solid lines are edges in E, and dashed lines are edges in E 2 .
Step 2: Structure analysis. To support fine-grained manipulation, we decompose a distance graph into two-connected components. These components are organized in a tree structure and the one containing beacons is the root. Adjustments are conducted along tree edges from the root to leaves. Step 3: Distinctive adjustment. LAL treats nodes differently according to their localizability and places in the component tree. Through vertex augmentation, LAL converts all nonlocalizable in one round. The networks tuned by LAL are localizable and can be localized by the existing localization approaches. Step 1 can be done by applying Theorem 2. Given a specific node, its localizability relies on the property of disjoint paths and redundant rigidity, which can be tested in polynomial time by network flow algorithms and the pebble game algorithm [25], respectively. In step 2, a graph is decomposed into two-connected components using depth-first search [29]. In step 3, node adjustments described in Section 4 are conducted along the paths of the component tree starting at the root. We present Algorithm 1 to detail step 3. Algorithm 1. LAL_Basic. Require: A two-connected component Gi , the component tree CT ree, and localizability vector. 1: if Gi is GA then 2: for each nonlocalizable vertex vi 2 GA do 3: Vertex augmentation v2i 4: end for 5: else parent½Gi 6: Gj 7: Find out which category do Gi and Gj belong to. 8: Apply corresponding solution in Section 4 to add edges. 9: end if 10: for each child component Gk of Gi in CT ree do Gi 11: parent½Gk 12: LAL BasicðGk ; CT reeÞ 13: end for In Algorithm 1, edges are added by vertex augmentation of all nonlocalizable vertices in GA , and GA is localizable according to Theorem 5. For a component other than GA , the categories of it and its parent component are used to
CHEN ET AL.: LOCALIZATION-ORIENTED NETWORK ADJUSTMENT IN WIRELESS AD HOC AND SENSOR NETWORKS
guide adjustments as described in Section 4. Algorithm 1 recursively repeats until all components get manipulated. After applying Algorithm 1, the entire network is localizable. Popular localization algorithms can then be used seamlessly to localize all nodes in the network without incompatibility. Proposition 1. Given the locations of localizable nodes, for any nonlocalizable node v in a connected graph G ¼ ðV ; EÞ, v can be localized by Sweeps [3] after the execution of LAL_Basic. Proof. Case 1: v 2 GA . If there is no localizable node in GA , LAL_Basic builds G2A . As proved in [3], G2A can be localized by Sweeps since GA is two-connected. If there is only one localizable node vl in GA , for any nonlocalizable node vn , there exists a cycle C which contains both vn and vl as shown in Fig. 1. So all nodes including vn can be localized by Sweeps according to [3], because C 2 is a bilateration graph. If there are more than one localizable nodes, for any nonlocalizable node vn , we can find a path between two localizable nodes which passes vn . Suppose there are i > 2 nodes on the path, as shown in Fig. 9. v1 and v2 are the two localizable nodes, and v3 , v4 ; . . . ; vi are nonlocalizable. As discussed in the proof of Theorem 5, v3 is localizable after v23 . Since all nonlocalizable nodes are augmented, the subgraph with added edges has a bilateration ordering and can be localized by Sweeps. Thus, we have proved the proposition for the cases that v 2 GA . Case 2: v 62 GA . v is either in a category II component with three localizable nodes or in a category I component, but connected to three localizable nodes after the execution of LAL_Basic according to the analysis in Section 4. In both cases, a bilateration ordering exists, so v can be localized by Sweeps. u t To reduce the number of redundant edges added in category II component, we analyze the graph properties of these components and find the following observations in the original communication graph. Observation 1. In GA , some nonlocalizable vertices have one localizable neighbor. As shown in Fig. 1, the localizable vertex vl has at least three vertex-disjoint paths to three beacons. Adding two edges which connect two neighbor vertices on different vertex-disjoint paths to the nonlocalizable vertex (e.g., v1 ) is enough to make the vertex localizable according to Theorem 2. Moreover, only one edge is needed if a nonlocalizable vertex has two localizable neighbors in GA . Observation 2. If a nonlocalizable vertex has three or more localizable vertices within two-hop distance, connecting it to three localizable vertices makes it localizable. Observation 3. Some globally rigid components are not localizable in the original network topology due to the lack of beacons. If three nodes are adjusted to be localizable in a globally rigid component, the component is immediately localizable without extra manipulation. We revise LAL_Basic and propose a heuristic algorithm LAL_Heuristic. The difference between the two algorithms lies in the way they add edges in category II components. LAL_Heuristic uses a function named Add_Heuristic to add
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edges instead of vertex augmentation. The pseudo-code of the function is shown in Algorithm 2. We omit the implementation of applying three observations in the pseudo-code, and provide descriptions in the following explanation. If GA is used as an input of Add_Heuristic, the algorithm first finds all nonlocalizable neighbors of localizable vertices, and adds edges for each nonlocalizable one according to Observation 1. Because disjoint paths from a localizable vertex to beacons are found and memorized in Step 1, only a search for nonlocalizable neighbor with linear complexity is required to implement line 8. Those nonlocalizable neighbors are then marked as localizable. Each time a new vertex is marked localizable, Observation 3 is applied by re-examining the localizability of the corresponding component. From line 14 to line 25, Observation 2 and 3 are repeated to add edges to make more vertices localizable. Note that not all vertices in the component are localizable after the iterative process. From line 26 to 28, the same square operation as used in LAL_Basic is adopted to make those nonlocalizable vertices localizable. The correctness of Algorithm 2 is guaranteed by Theorems 1 and 5. Algorithm 2. Add_Heuristic. Require A two-connected component Gi , and localizability vector of vertices. 1: if Gi is GA then 2: for each nonlocalizable vertex vj 2 Gi do 3: if vj has localizable neighbor then 4: Add vj into a set VD . 5: end if 6: end for 7: for each vertex vj 2 VD do 8: Add edges based on Observation 1. 9: Mark vj localizable. 10: Apply Observation 3. 11: end for 12: end if 13: flag 1 14: while not all vertices in Gi are localizable and flag ¼¼ 1 do 15: flag 0 16: for nonlocalizable vertex vj 2 Gi do 17: num amount of localizable two-hop neighbors 18: if num 3 then 19: Add edges based on Observation 2. 20: Mark vj as localizable. 21: flag 1 22: Apply Observation 3. 23: end if 24: end for 25: end while 26: for each vertex vk marked nonlocalizable in Gi do 27: Vertex augmentation v2k 28: end for
6
DISCUSSION
We find that the network topology is a byproduct of some basic services in ad hoc and wireless sensor networks, such
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TABLE 1 Experiment Results on GreenOrbs
Fig. 10. GreenOrbs system in a campus. The communication range is about 12 meters. After localizability test, a great portion of nodes are recognized as localizable (black), while a few are nonlocalizable (red). Green ones are beacons.
with large errors) are sifted by these approaches, and only used accurate ranging results in LAL design.
7 as data collection [30], thus, its collection induces none or little additional overhead. Inspired by this fact, this paper naturally adopts the centralized scheme. Actually, a distributed implementation may cause even more communication overhead, which mainly comes from the localizability testing part. In distributed implementation of Pebble Game algorithm, each node must send a message to neighbors once the pebble number changes. So the message complexity is OðjV kEjÞ. The most recent distributed algorithm for maximum flow problem has OðjV j2 jEjÞ message complexity [31]. So we can learn that the distributed implementation of LAL has at least OðjV j2 jEjÞ complexity. In addition, the main communication overhead of centralized LAL comes from the collection of topology data and the dissemination of adjustment solution. Even if we use flooding for both tasks, the message complexity is only OðjV j2 Þ. We discuss the computational complexity of LAL according to the design and implementation described in Section 5. In Step 1, we use the pebble game algorithm [25], which has complexity of OðjV j2 Þ, to find redundantly rigid components, then use the classical Ford-Fulkerson algorithm, which has complexity of OðjEkfjÞ, and jfj is the maximum flow, to find paths in redundantly rigid components from each vertex to beacons. In Step 2, the network is decomposed to two-connected components by depth-first search with complexity of OðjV j þ jEjÞ and organized as component tree with complexity of OðjV jÞ. In Step 3, Algorithm 1 has complexity of OðjV jÞ, since the solutions for all four cases discussed in Section 4 require OðnÞ running time, where n is the number of nonlocalizable nodes in current component. In Algorithm 2, applying Observation 3 is the most computationally intensive operation, which requires the examination of the conditions in Theorem 1. The examination is also implemented by using pebble game and Ford-Fulkerson algorithm. Observation 3 is applied whenever a nonlocalizable node becomes localizable, so in the worst case, the complexity of Algorithm 2 is OðjV j3 þ jV kEkfjÞ. Because jEj < jV j2 and jfj is restrained to a small constant value, (e.g., the number of beacons), the complexity of Algorithm 2 can be considered as OðjV j3 Þ. Noisy measurement results may influence the results of localization approaches based on the graph-rigidity theory. To alleviate such influence, some researchers are trying to sift the outliers of ranging results, and only use the reliable ones [32], [33]. We assume that noisy results (the outliers
EVALUATION
7.1 Experiment To examine the effectiveness of our solution, we implement LAL on the data trace collected from the ongoing wireless sensor network system GreenOrbs [7]. One mission of GreenOrbs is to realize all-year ecological surveillance in the forest, so it is important to reduce energy consumption. Combining with duty cycling schemes, the transmission power is also well controlled under the highest level to provide just enough connectivity for data collection or other services. GreenOrbs consists of two working systems. In this paper, we use the trace collected from the deployed system consists of 100 TelosB motes in the campus of Zhejiang Agricultural and Forestry University, Hangzhou, China. Fig. 10 shows the topology of the system when communication range is about 12 meters. RSSI is used to measure the distance between nodes, so the topology can be used as distance graph if edges between beacons are added. As shown in Fig. 10, a great portion of them are localizable while a few nodes near the border of the deployed area are nonlocalizable. In the experiment, we compare LAL with the indistinctive approach proposed by Anderson et al. [13], denoted by IND. We use LAL_B and LAL_H denote LAL_Basic and LAL_Heuristic, respectively. There are 285 edges in the original topology in Fig. 10. Comparisons are conducted from the aspect of the number of edges and nodes with different transmitting power output in the result topology. We denote nodes with original communication range as Power 1 nodes, the ones with doubled and tripled range are denoted as Power 2 and Power 3 nodes, respectively. As shown in Table 1, LAL can significantly reduce the number of edges added to the graph. The three observations in Section 5 can also help to reduce 31 (about 50 percent) edges added by LAL_B. LAL also helps to reduce energy consumption and interference with less increased communication range. IND has to double the communication range of all nodes because the network in Fig. 10 is edge twoconnected, but not localizable, while only 39 and 27 nodes need to double their communication range in LAL_B and LAL_H, respectively. None of the three algorithms need to triple the range of any node in Fig. 10. 7.2 Simulation To further examine the scalability and efficiency of LAL, simulations are conducted under different network instances and varied network parameters. We randomly
CHEN ET AL.: LOCALIZATION-ORIENTED NETWORK ADJUSTMENT IN WIRELESS AD HOC AND SENSOR NETWORKS
Fig. 11. Number of edges in the graph after adding edges.
Fig. 12. Number of added edges.
generate networks of 400 nodes which are uniformly distributed in a unit square ½0; 12 . The unit disk model with a radius is adopted for communication and distance measurement. For each setting, we integrate results from 100 network instances, with three beacons in each instance. The original communication range is denoted by R. For each instance, we change the value of R from 0.06 to 0.98 stage-by-stage with a step length of 0.002. As shown in Figs. 11 and 12, our approach needs to add much less edges to the original distance graph than IND in networks with various densities. We can also find that LAL_H needs fewer extra edges than LAL_B does in most cases, which is in accordance with our experiment result. In Figs. 11 and 12, the curves of IND first increase to a peak value, then decrease as the value of R increases and keeps stable for the last a few settings. Such a phenomenon can be explained with Fig. 13. When the value of R is small, such as R ¼ 0:06, the average size of anchor component is about 340, which means the network is not two-connected in most instances. IND needs to triple the communication range of all nodes in these instances. As R increases, most network instances are still not two-connected, but the average degree
Fig. 14. Number of nodes at different power levels.
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Fig. 13. Number of localizable nodes and nodes in anchor component.
is increased, so the number of edges keeps increasing. When R increases to a larger value such as 0.074, the average size of GA are over 395, which implies most instances are two-connected. Nodes in two-connected instances only need to double the range, and at the same time more instances become localizable, so the number of edges of IND decreases as the value of R increases. When R > 0:085, most instances has no nonlocalizable nodes and no edges need to be added. The curves of LAL_B and LAL_H follow the same trend. They reach the peak value when R ¼ 0:064, because R and the number of edges are keep increasing while the number of localizable nodes in GA keeps almost unchanged with a very small value as shown in Fig. 13. We also study the transmitting power requirements of LAL_B, LAL_H, and IND. Fig. 14 plots the number of nodes at different power levels. As shown in Figs. 14a and 14b, results of LAL_H and LAL_B have much more Power 1 nodes than IND except when R < 0:066, and much less Power 3 nodes than IND except when R > 0:09. Curves in Fig. 14b have a similar trend as curves in Fig. 12. The numbers of Power 2 nodes of LAL_B and LAL_H keep high in only the first three settings, and then it decreases sharply to a very small value. The results in Fig. 14 also explain why LAL can save much extra edges than IND because higher power level tends to induce more edges than a lower one. The curves of LAL_H are below those of LAL_B in most settings. It suggests that LAL_H need to adjust the transmitting power of fewer nodes than that of LAL_B, which is in accordance with the results in Figs. 11 and 12. To better understand how much LAL_B and LAL_H outperform IND as far as transmitting power is concerned, we further provide an example to show the distribution of nodes at different power levels. Because IND treat the entire
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Fig. 15. Testing LAL and IND on network instance consists of 400 nodes with a “T” hole (R ¼ 0:08). Gray dots denote Power 1 nodes, and blue dots denote Power 2 nodes while red dots denote Power 3 nodes.
network indistinctly and obviously needs more adjustments than LAL, an improved approach (Improved IND) is implemented instead. Improved IND first uses iterative trilateration to localize as many nodes as possible in a graph G, and then the rest nodes take vertex augmentation to increase their ranging capability. If the graph is 2-edgeconnected, these nodes augment their ranging capability by two times. Otherwise, the ranging capability should be augmented by three times. These augmented nodes and their two-hop neighbors form connected subgraphs in G3 , so they can also be localized through trilateration as discussed in Section 3. As shown in Fig. 15, a network instance with a “T” hole is generated in which 400 nodes are randomly distributed and R ¼ 0:08. Both LAL_B and LAL_H need to adjust the power of fewer nodes with lower power levels than improved IND. All augmented nodes in Fig. 15c are Power 3 nodes, while there are only three Power 3 nodes in Figs. 15a and 15b. It suggests that LAL_B and LAL_H are finer grained than improved IND. We also notice that there are less Power 2 nodes in Fig. 15b than that in Fig. 15a, which is in accordance with the experiment result and simulation results in Fig. 14.
8
Program 61125202, China 973 Program under grant nos. 2012CB316200 and 2011CB302705, China 863 Program under grant no. 2011AA010100.
REFERENCES [1] [2] [3]
[4]
[5]
[6] [7]
CONCLUSION
We analyze the limitations of the existing studies on localization in nonlocalizable networks, and propose a localizability-aided approach named LAL. LAL make adjustments according to node localizability results, other than indistinctively consider the network as a whole. By deriving LAL, we can answer the questions on localization in nonlocalizable networks: which nodes need to augment their ranging capability and which new edges need to be measured. Our designs not only excel previous works theoretically, but also have some good characteristics for practical application. We implement LAL and demonstrate its effectiveness through working system experiment and extensive simulations.
ACKNOWLEDGMENTS This work was supported in part by the NSFC Youth Program no. 61202487, NSFC General Program nos. 61170284, 61171067, 61133016, 61272482, and 61272466, NSFC Major Program no. 61190110, NSFC Distinguished Young Scholars
[8]
[9] [10] [11] [12] [13]
[14] [15]
P. Bahl and V.N. Padmanabhan, “RADAR: An In-Building RFBased User Location and Tracking System,” Proc. IEEE INFOCOM, 2000. N.B. Priyantha, A. Chakraborty, and H. Balakrishnan, “The Cricket Location-Support System,” Proc. Sixth Ann. Int’l Conf. Mobile Computing and Networking, pp. 32-43, 2000. D.K. Goldenberg, P. Bihler, M. Cao, J. Fang, B.D.O. Anderson, A.S. Morse, and Y.R. Yang, “Localization in Sparse Networks Using Sweeps,” Proc. 12th Ann. Int’l Conf. Mobile Computing and Networking, pp. 110-121, 2006. D. Moore, J. Leonard, D. Rus, and S. Teller, “Robust Distributed Network Localization with Noisy Range Measurements,” Proc. Second Int’l Conf. Embedded Networked Sensor Systems, pp. 50-61, 2004. D.K. Goldenberg, A. Krishnamurthy, W.C. Maness, Y.R. Yang, A. Young, A.S. Morse, and A. Savvides, “Network Localization in Partially Localizable Networks,” Proc. IEEE INFOCOM, pp. 313326, 2005. “Oceansense Project,” http://www.cse.ust.hk/liu/Ocean/index. html, 2013. L. Mo, Y. He, Y. Liu, J. Zhao, S.-J. Tang, X.-Y. Li, and G. Dai, “Canopy Closure Estimates with GreenOrbs: Sustainable Sensing in the Forest,” Proc. Seventh ACM Conf. Embedded Networked Sensor Systems, pp. 99-112, 2009. T. Eren, D.K. Goldenberg, W. Whiteley, Y.R. Yang, A.S. Morse, B.D.O. Anderson, and P.N. Belhumeur, “Rigidity, Computation and Randomization in Network Localization,” Proc. IEEE INFOCOM, Apr. 2004. Z. Yang and Y. Liu, “Understanding Node Localizability of Wireless Ad-Hoc Networks,” IEEE Trans. Mobile Computing, vol. 11, no. 8, pp. 1249-1260, Aug. 2012. N.B. Priyantha, H. Balakrishnan, E.D. Demaine, and S. Teller, “Mobile-Assisted Localization in Wireless Sensor Networks,” Proc. IEEE INFOCOM, pp. 172-183, 2005. K.F. Ssu, C.H. Ou, and H.C. Jiau, “Localization with Mobile Anchor Points in Wireless Sensor Networks,” IEEE Trans. Vehicular Technology, vol. 54, no. 3, pp. 1187-1197, May 2005. C. Wu, Y. Zhang, W. Sheng, and S. Kanchi, “Rigidity Guided Localization for Mobile Robotic Sensor Networks,” Int’l J. Ad Hoc and Ubiquitous Computing, vol. 6, no. 2, pp. 114-128, 2010. B.D. Anderson, P.N. Belhumeur, T. Eren, D.K. Goldenberg, A.S. Morse, W. Whiteley, and Y.R. Yang, “Graphical Properties of Easily Localizable Sensor Networks,” Wireless Network, vol. 15, no. 2, pp. 177-191, 2009. T. Chen, Z. Yang, Y. Liu, D. Guo, and X. Luo, “Localization in Non-Localizable Sensor and Ad-Hoc Networks: A LocalizabilityAided Approach,” Proc. IEEE INFOCOM, pp. 276-280, 2011. T. He, C. Huang, B.M. Blum, J.A. Stankovic, and T. Abdelzaher, “Range-Free Localization Schemes for Large Scale Sensor Networks,” Proc. ACM MobiCom, pp. 81-95, 2003.
CHEN ET AL.: LOCALIZATION-ORIENTED NETWORK ADJUSTMENT IN WIRELESS AD HOC AND SENSOR NETWORKS
[16] D. Niculescu and B. Nath, “DV Based Positioning in Ad Hoc Networks,” Telecomm. Systems, vol. 22, no. 1, pp. 267-280, 2003. [17] A. Savvides, C.-C. Han, and M.B. Strivastava, “Dynamic FineGrained Localization in Ad-Hoc Networks of Sensors,” Proc. ACM MobiCom, pp. 166-179, 2001. [18] X. Wang, J. Luo, S. Li, D. Dong, and W. Cheng, “Component Based Localization in Sparse Wireless Ad Hoc and Sensor Networks,” Proc. IEEE 16th Int’l Conf. Network Protocols, 2008. [19] S.Y. Seidel and T.S. Rappaport, “914 MHZ Path Loss Prediction Models for Indoor Wireless Communications in Multifloored Buildings,” IEEE Trans. Antennas and Propagation, vol. 40, no. 2, pp. 207-217, Feb. 1992. [20] K. Whitehouse and D. Culler, “Calibration as Parameter Estimation in Sensor Networks,” Proc. First ACM Int’l Workshop Wireless Sensor Networks and Applications, pp. 59-67, 2002. [21] C. Peng, G. Shen, Y. Zhang, Y. Li, and K. Tan, “Beepbeep: A High Accuracy Acoustic Ranging System Using Cots Mobile Devices,” Proc. Fifth Int’l Conf. Embedded Networked Sensor Systems, pp. 1-14, 2007. [22] P.N. Pathirana, N. Bulusu, A.V. Savkin, and S. Jha, “Node Localization Using Mobile Robots in Delay-Tolerant Sensor Networks,” IEEE Trans. Mobile Computing, vol. 4, no. 3, pp. 285296, May/June 2005. [23] M.L. Sichitiu and V. Ramadurai, “Localization of Wireless Sensor Networks with a Mobile Beacon,” Proc. IEEE Int’l Conf. Mobile Ad-Hoc and Sensor Systems, pp. 174-183, 2004. [24] B. Jackson and T. Jorda´n, “Connected Rigidity Matroids and Unique Realizations of Graphs,” J. Combinatorial Theory Series B, vol. 94, no. 1, pp. 1-29, 2005. [25] D.J. Jacobs and B. Hendrickson, “An Algorithm for TwoDimensional Rigidity Percolation: The Pebble Game,” J. Computational Physics, vol. 137, pp. 346-365, 1997. [26] J. Aspnes, T. Eren, D.K. Goldenberg, A.S. Morse, W. Whiteley, Y.R. Yang, B.D.O. Anderson, and P.N. Belhumeur, “A Theory of Network Localization,” IEEE Trans. Mobile Computing, vol. 5, no. 12, pp. 1663-1678, Dec. 2006. [27] B. Hendrickson, “Conditions for Unique Graph Realizations,” SIAM J. Computing, vol. 21, no. 1, pp. 65-84, 1992. [28] G. Laman, “On Graphs and Rigidity of Plane Skeletal Structures,” J. Eng. Math., vol. 4, pp. 331-340, 1970. [29] J. Hopcroft and R. Tarjan, “Algorithm 447: Efficient Algorithms for Graph Manipulation,” Comm. ACM, vol. 16, no. 6, pp. 372-378, 1973. [30] O. Gnawali, R. Fonseca, K. Jamieson, D. Moss, and P. Levis, “Collection Tree Protocol,” Proc. Seventh ACM Conf. Embedded Networked Sensor Systems, pp. 1-14, 2009. [31] T. Pham, I. Lavallee, M. Bui, and S. Do, “A Distributed Algorithm for the Maximum Flow Problem,” Proc. Fourth Int’l Symp. Parallel and Distributed Computing, 2005. [32] Z. Yang, L. Jian, C. Wu, and Y. Liu, “Beyond Triangle Inequality: Sifting Noisy and Outlier Distance Measurements for Localization,” ACM Trans. Sensor Networks, vol. 9, no. 2, p. 26, Mar. 2013. [33] Z. Yang, C. Wu, T. Chen, Y. Zhao, W. Gong, and Y. Liu, “Detecting Outlier Measurements Based on Graph Rigidity for Wireless Sensor Network Localization,” IEEE Trans. Vehicular Technology, vol. 62, no. 1, pp. 374-383, Jan. 2013. Tao Chen received the MS and PhD degrees in military operational research from the National University of Defense Technology, Changsha, P.R. China, in 2006 and 2011, respectively. He is an assistant professor at the College of Information System and Management, National University of Defense Technology. His research interests include wireless sensor networks and data center networking. He is a member of the IEEE and the ACM.
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Zheng Yang received the BE degree in computer science from Tsinghua University, Beijing, in 2006 and the PhD degree from Hong Kong University of Science and Technology (HKUST), in 2010. He is currently an assistant professor at Tsinghua University. His main research interests include wireless ad hoc/sensor networks and pervasive computing. He has published a number of research papers in highly recognized journals and conference, including IEEE/ACM Transactions on Networking, IEEE Transactions on Parallel and Distributed Systems, IEEE Transactions on Mobile Computing, IEEE INFOCOM, IEEE ICDCS, IEEE RTSS, ACM SenSys and so on. He is a member of the IEEE and the ACM. Yunhao Liu received the BS degree in automation from Tsinghua University, Beijing, China, in 1995, and the MS and PhD degrees in computer science and engineering from Michigan State University, in 2003 and 2004, respectively. He is now a professor at Tsinghua University. His research interests include wireless sensor network, peer-to-peer computing, and pervasive computing. He is a senior member of the IEEE.
Deke Guo received the BS degree in industry engineering from the Beijing University of Aeronautics and Astronautics, China, in 2001, and the PhD degree in management science and engineering from National University of Defense Technology, Changsha, China, in 2008. He is an associate professor with the College of Information System and Management, National University of Defense Technology. His research interests include distributed systems, wireless and mobile systems, P2P networks, and interconnection networks. He is a member of the IEEE and the ACM. Xueshan Luo received the BE degree in information engineering from Huazhong Institute of Technology, Wuhan, China, in 1985, and the MS and PhD degrees in system engineering from the National University of Defense Technology, Changsha, China, in 1988 and 1992, respectively. Currently, he is a professor at College of Information System and Management, National University of Defense Technology. His research interests include the general areas of information system and operational research. His current research focuses on architecture of information system.
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