Scalable Localization in Wireless Sensor Networks
Scalable Localization in Wireless Sensor Networks Muralidhar Medidi, Roger A. Slaaen, Yuanyuan Zhou, Christopher J. Mallery, and Sirisha Medidi? School of Electrical Engineering and Computer Science Washington State University Pullman, WA 99164-2752
Abstract. Localization, an important challenge in wireless sensor networks, is the process of sensor nodes self-determining their position. The difficulty encountered is in cost-effectively providing acceptable accuracy in localization. The potential for the deployment of high density networks in the near future makes scalability a critical issue in localization. In this paper we propose Cluster-based Localization (CBL), which provides effective localization suitable for large and highly-dense networks. CBL utilizes both a computationally-intensive localization technique (non-metric multidimensional scaling (MDS)) and a less intensive trilateration to achieve balance between performance and cost. Clustering is utilized to select a subset of nodes to perform MDS and then extend their localization to the remaining network. Besides providing scalability clustering overcomes local irregularities and provides good accuracy even in irregular networks with or without obstacles. Simulation results illustrate that CBL reduces both computation and communication, while still yielding acceptable accuracy.
1
Introduction
Wireless sensor networks (WSN) are typically densly populated ad-hoc networks composed of small, resource-constrained, immobile nodes. The ability for a sensor to self-determine its own position, enabling the node to correlate its data with a location, is critical in many domains [1]. Minimizing the cost of nodes is a critical consideration, which makes equipping all sensor nodes with GPS capabilities infeasible [2]. Furthermore, the number of sensor nodes deployed in a sensor network is typically high, e.g., on the order of thousands and possibly even millions, and the network density can reach a few hundred nodes per square meter [3],[4]. The need for position information in high-density WSN motivates the need for scalable localization. In this paper we propose a scalable hop-based localization technique called Cluster-based Localization (CBL). CBL’s use of clustering is motivated by the need for scalability and efficiency. Scalability is gained by, first using an expensive but accurate localization technique on a small subset of the nodes in the ?
This work was supported in part by NSF grant CNS-0454416.
network. The derived position estimates for the chosen subset will be used as references when localizing the remaining nodes. This approach leads to a significant reduction in the computation required to localize the entire network. Although scalability was the primary goal for performing the localization in different stages in CBL, employing clustering has also provided the benefit of smoothing over local variations within the network. This smoothing effect prevents the global localization results from being skewed by the affects of local aberrations. In particular, CBL is able to maintain accuracy even in the presence of RF-opaque obstacles and irregular network topologies. Although our CBL implementation uses non-metric MDS in localization of the representative nodes, this technique could be replaced so that CBL could benefit from other localization ones. The rest of this paper is organized as follows. Section 2 presents related work in the area of sensor network localization. CBL is described in Section 3. Section 4 contains performance evaluation and comparisons of CBL and Section 5 provides some concluding remarks.
2
Related Work
Previous attempts at localization in sensor networks can be categorized into two groups: range-aware and hop-based. In range-aware techniques a distance measure between neighboring nodes is used to estimate node positions. In hopbased ones ranging equipment is not necessary, and the estimated distances between nodes are typically approximated to the hops in the shortest path. Range-aware localization techniques typically derive inter-node distances based on received signal strength. APS [5], a distributed localization technique, extends both distance vector routing and GPS positioning. Similiar distance estimates are used in [6]. In [7], a similiar, but more coarse estimation process is described, in which nodes adjacent to at least one anchor are localized first. Received signal strength measurements of broadcasts from a single mobile beacon node are used in [8]. The beacon’s current position, will place constraints on the possible position of a node. Another technique using a mobile beacon node, utilizing time-of-arrival measurements and probablistic estimation, is proposed in [9]. In [10], four mobile beacons are used, creating a rectangle with the “un-localized” node in the middle, allowing for trilateration. PRI [11] provides improved performance by augmenting hop information with any available ranging information. In [12], known peer-to-peer communications are modeled as a set of geometric constraints on nodes’ positions. In [13], the Approximate Maximum-Likelihood method and Direction of Arrival estimation are reviewed. In [14], sensing constraints caused by mobile objects at several nodes are utilized to improve the accuracy of localization. RangeQ determines node positions by means of a distributed range quantization technique which is similar to quantization in image processing [11]. Ji and Zha present a distributed localization technique based on the estimation-comparison-correction paradigm. It applies multidimensional scaling (MDS) [15] to merge each individual node’s map of the network topology
into one global map of the network [16]. In [17], previous work from rigidity theory is extended to networks where not all nodes are localizable. Chan, Luk and Perrig [1] propose a scalable localization, where a complex clustering is used to produce a highly regular structure. This regularity can then be utilized, along with arbitrarily positioned anchor nodes, in the localization. The problem with calculating distances by means of signal strength measurements is that since all possible sources of signal interference cannot be accurately anticipated prior to sensor deployment, the estimated distances can become inaccurate due to multi-path interference, line-of-sight obstructions, etc. Hop-based localization techniques aim to remove the dependency on any form of ranging equipment. A theoretical analysis of network connectivity for node selflocalization is presented in [18]. An extension of the algorithm proposed in [5] is the differential Ad-Hoc Positioning System [19], which describes a differential error correction scheme designed to reduce the cumulative distances and positioning error over multiple hops. HOP-TERRAIN [20], similar to [5], utilizes an additional refinement phase to improve the localization accuracy. A low-powerdedicated hardware localization technique utilizing the HOP-TERRAIN is proposed in [21]. SHARP [22] technique adopts a hybrid hop-based and range-aware approach for localization. SHARP attempts to perform localization on a subset of the nodes in a network using inter-node distance estimates. This subset of nodes will be localized using MDS [2], and are then used to localize the remaining nodes with APS [5]. Shang et al. proposed MDS-MAP [23], an algorithm that utilizes MDS to perform global localization. Since MDS-MAP, there have been notable extensions of this technique: MDS-MAP(P)[2] and MDS-MAP(R)[24]. MDS-MAP(P) is a distributed algorithm that uses patches of relative maps, that can be computed in parallel, to estimate the absolute positions of nodes. On the otherhand, the distributed MDS-MAP(R) estimates the relative positions. A similar algorithm based on the estimation-comparison-correction paradigm and MDS [15] is described in [16].
3
Cluster-based Localization
To localize all sensor nodes accurately, the localization process usually involves computationally-intensive procedures, e.g., MDS that has O(n3 ) computational complexity [2],[25]. Since sensor nodes are expected to be resource-constrained and networks to be large and dense, light-weight localization algorithms that achieve decent accuracy and high scalability are desirable. Therefore, we attempt to address the trade-offs between accuracy and scalability, so as to make our localization algorithm, CBL, a practical solution for sensor network applications. CBL adopts a hierarchical idea to perform the whole localization procedure, i.e., instead of regarding the whole network as a flat topology in which all nodes share the same localization procedure as in most existing localization algorithms, we deliberately separate the nodes into two types, and apply different localization approaches on them to improve scalability as well as ensure accuracy. The first type of nodes are representatives that account for a small portion of the nodes in the whole network. Representatives will be selected to reflect a good
abstraction of the network and localized by using complex but accurate localization algorithms, which essentially improves the overall localization accuracy. The remaining nodes, which account for the majority of the network, can employ a lighter-weight localization process that uses the representatives as reference nodes. By applying this mechanism, CBL achieves decent accuracy with relatively low computation and communication overhead. 3.1
Representative Selection
Since representatives will be localized first and used as reference nodes for estimating the remaining nodes’ positions, they should provide a good abstraction of the whole network. Further, the representative selection process itself should be light-weight to ensure the overall scalability. Therefore, in CBL we apply a simple single-hop and size-bounded clustering algorithm to select representatives, or Cluster Heads (CHs), from the network. The single-hop feature of the clustering algorithm ensures the algorithm’s efficiency, and the size-bounded feature enables each cluster to be formed fairly uniformly and makes cluster heads represent a good abstraction of the whole network. We assume each node has a unique ID. To prepare for clustering, initially each node performs neighbor discovery to become aware of its single-hop neighborhood information. After that, a node either assumes to be a CH if it has the largest ID among its neighbors, or waits to be contacted if any of its neighbors have larger IDs. A CH will start to contact its neighbors to build its own cluster whose size is bounded by a threshold θ. A node becomes a Cluster Member (CM) of the first neighboring CH that contacts it, or becomes a CH if all its larger-ID neighbors have already spoken by either becoming a clusterhead with full size-bounded cluster or becoming a CM in some other cluster. To enable the network formed by CHs to provide a good abstraction of the whole network, a CH c will attempt avoiding CMs that cover overlapping transmission areas with c’s existing CMs. After deciding its own role (CH/CM), a node informs its neighbors about its decision. Eventually, within a few rounds each node will finish determining its role (CH/CM) and each cluster forms a star topology with a CH at the center. In Fig. 1(a), we illustrate a random network and the connectivity between nodes. In Fig. 1(b), we show the resulting clusters in which a CM is represented as a line from its CH to itself, and in Fig. 1(c) we show the connectivity for each cluster. It can be observed that the CMs of each cluster are reasonably well distributed around the CH; and these CHs, or representatives, provide a good abstraction of the original topology. 3.2
Cluster Head Localization
CHs are taken as representatives and their localization accuracies have great impact on the overall localization accuracy. Since CHs usually account for a small portion of the whole network, it is affordable to apply complex localization algorithms to estimate CHs’ positions. Further, we want to apply a hop-based algorithm to localize CHs so as to improve CBL’s applicability. Based on these
(a) Network topology
(b) Resulting clusters
(c) Cluster network
Fig. 1. Example clustering
three considerations, we chose MDS as the basic technique to localize CHs, since hop-based MDS localization algorithm MDS-MAP(P) [2] was shown to achieve a good accuracy. CBL uses similar approaches to localize CHs, however, in CBL only CHs are involved in the MDS computation, therefore the computation overhead is contained. CBL does not involve any pre-installed anchor nodes as opposed to MDS-MAP(P), which reduces the cost of sensor hardware. Further, in CBL we employ the non-metric MDS technique that has a weaker requirement on the input data than the classical MDS used in MDS-MAP(P). To apply MDS for local map construction, each CH c needs to collect the distance information from its “two-step” clusters’ CHs (we call neighboring clusters “one-step” clusters). After that, c computes shortest distances (in hops) between each pair of the involved CHs, which will be taken as the input for the MDS algorithm to estimate positions of the involved CHs and to create a local map. MDS is a set of well-known data analysis techniques for geometrical position estimation and information visualization; see [15] for details. In our localization, we utilize the non-metric MDS, which assumes a less-stringent monotonicity constraint than the classical metric MDS deployed in [2]. After obtaining a local map through MDS calculation, each CH will attempt to merge its local map with its neighboring CHs’. Similar to MDS-MAP(P), the merging is a completely distributed process, in which the CHs with larger IDs will have higher priorities to choose one of their neighboring CHs to merge. Further, neighboring maps will be merged based on their common nodes, i.e. the maps with the highest number of common nodes should be merged first. When merging, we apply the best linear transform technique to transform one map onto another, and the new coordinates are computed based on the average of the common nodes’ coordinates. 3.3
Cluster Member Localization
After determining each CH’s coordinates, we will take CHs as reference nodes to estimate the remaining nodes’, or CMs’, positions. We choose the least-square triangulation technique for CMs’ localization to obtain decent accuracy with low computation overhead. In particular, each CH first calculates the euclidean distances to the CHs in its neighboring clusters, then estimates the average hop distance by using these euclidean distances and corresponding hop length. After
that, each CH broadcasts a LOCATION message, which contains its coordinates and the estimated hop distance, to the CMs within α hops (α should be chosen to balance the message overhead and the localization accuracy). Each CM waits for at least three LOCATION messages from different CHs, and then perform a triangulation to determine its own coordinates. If a CM receives more than three LOCATION messages, a least square error technique is applied.
4
Performance Evaluation
CBL was implemented in ns-2 [26], and our main goals when evaluating its performance were to determine the accuracy, scalability, and how irregularities in a network topology will affect performance. The simulation environment parameters are: ns2 version 2.29; area = 100 × 100m2 ; 100 nodes; transmission range = 15m and cluster size varied from 1 to 10. We utilize a commonly used metric, accuracy, to measure how much estimated positions deviate from actual positions. Two maps, estimated and actual, are compared by finding the best linear transformation (rotating, shifting and/or scaling) of the estimated map onto the actual and calculating the average Euclidean difference between each corresponding point. In the simulations the node-density is controlled by varying the enclosing network area while maintaining the same number of nodes. We also varied the upper-bound for cluster-size θ to determine how changes in clustersizes affect the performance of CBL. For comparison, we have included both MDS-MAP(P) and DV-hop results at similar densities for reference. The data for MDS-MAP(P) is obtained from [2] and are based on MATLAB simulations; a CAML implementation of DV-hop was used to obtain results for comparison. During cluster-head (CH) localization, all hop values used are perturbed by adding noise σ (0 ≤ σ ≤ 10−5 ) before the non-metric MDS is employed to eliminate ties and help the non-metric MDS converge, because the non-metric MDS technique is dependent on the differences in the distance estimations. 4.1
Random network
The topologies in our random networks are square-shaped n × n ( n controls density ) networks, with 100 randomly placed nodes. As a key step, the clustering we deployed directly affects the quality of localization as well as the computation overhead. We observed that the average cluster-size stays close to the given upper-bound, which reflects the clusters’ high quality in spite of our lightweight clustering. Further, as expected, cluster-density drops significantly given a larger upper bound. In Figure 2 we show the resulting localization accuracy for random networks. Because we extend cluster-head (CH) localizations to obtain member (CM) localizations, we present CHs’ accuracy results in Figure 2(a) and all nodes’ accuracy results in Figure 2(b). Our results show that increasing the cluster size does not lead to significant performance degradation. Although MDS-MAP(P) and DV-hop achieve slightly better accuracy than CBL, CBL does not utilize nor depend on any anchor nodes or extra refinement of the estimates as DV-hop and
MDS-MAP(P) require. Furthermore, it should also be observed that the difference between the accuracy of all nodes and that of only cluster-heads is very small. This reflects the effectiveness of the clustering technique we applied, and shows that if the CH localization is accurate enough the low cost trilateration technique provides good results. 25
15
15
10
10
5
5
0
5
10
15
20
25
30
=4 =10 =1 =∞ MDS-MAP(P) DV-hop
20 Accuracy
20
Accuracy
25
θ=4 θ=10 θ=1 θ=∞ MDS-MAP(P) DV-hop
0
35
5
10
15
20
25
30
35
Node Density
Node Density
(a) Cluster-heads
(b) Entire network
Fig. 2. Accuracy for localization random topologies
30 25 20 15
20 15
10
10
5
5
0
5
10
15
20
25
Node Density
(a) Cluster-heads
30
=4 =10 =1 = MDS-MAP(P) DV-hop
25
Accuracy
Accuracy
30
=4 =10 =1 = MDS-MAP(P) DV-Hop
35
0
5
10
15
20
25
30
35
Node Density
(b) Entire network
Fig. 3. Accuracy for localization in C-shaped topologies
4.2
C-shaped network
C-shaped topologies are commonly employed to stress any localization technique [2],[5]. In this irregular topology the estimated distances between nodes can de-
viate greatly from the actual Euclidean distances. The resulting cluster-densities and average cluster-size are very similar to those for random networks. In Figure 3, we show the resulting accuracy for both the cluster-heads Fig. 3(a) and all nodes Fig. 3(b). As in the random topologies, changes in cluster-sizes do not affect the resulting accuracy. Furthermore, compared to MDS-MAP(P), CBL’s accuracy is only slightly worse, while providing the same lowered computation as in the random networks. Compared to DV-hop, CBL provides better accuracy for irregular topologies. This is mostly because of DV-hop’s dependence on the uniformity and regularity of the network. 4.3
Irregular node densities
To test how well CBL will perform in topologies with changing node densities, we created an irregular topology. The irregular topologies, also referred to as biased in the literature, are another fairly common test. An illustration of an irregular topology can be found in Figure 5(a), where the area containing the higher node-density is indicated by the square in the lower left corner. We can see from Figure 4 that, as expected, CBL is not greatly affected by the irregularity in node-densities; verifying that in CBL innacurate estimates do not propagate and affect a significant part of a network. 25
25
=4 =10 =∞ Uniform desity, =10
20 Accuracy
Accuracy
20
15
10
5
=4 =10 =∞ Uniform desity, =10
15
10
5
10
15
20
25
Node Density
(a) Cluster-heads
30
35
5
5
10
15
20
25
30
35
Node Density
(b) Entire network
Fig. 4. Accuracy for localization in irregular node-density topologies
4.4
Obstacles
Obstacles usually create great difficulty for any localization technique because any network with obstacles will have a much more irregular structure and the inter-node distance estimates are likely to be more inaccurate than those without obstacles. We performed simulations on random topologies with two different types of RF-opaque obstacles; four line-shaped (Figure 5(b)) and one H-shaped (Figure 5(c)) obstacle similar to the ones used in literature. As shown in Figure
5(c) the nodes close to the horizontal line of the H-shaped obstacle will be ones most affected and these nodes will derive severely overestimated inter-node distances between themselves and nodes on the other side of the horizontal line.
(a) Irregular densities
node-
(b) With obstacles
(c) With H-shaped obstacle
Fig. 5. Irregular network topologies
Figure 6 shows the resulting accuracy for the entire network with both four obstacles Fig. 6(a) and one H-shaped obstacle Fig. 6(b). We omit the graphs for the cluster-heads due to space constraints, but the same trend seen in other topologies is also evident for networks with obstacles. The localization accuracy achieved is very similar to that of networks without obstacles. This is because the use of clusters produces a technique that is less sensitive to any type of irregularities, including obstacles, and by covering a larger geographical area the clusters can in effect reach around many of the obstacles. Although the inter-cluster distances at times might be overestimated the overall accuracy is not significantly affected. The same trend as in the random networks is observed, where increasing node-densities improves the accuracy and there is a small degradation in accuracy when comparing that of all nodes and that of CHs. 25
25
=4 =10 =∞ No obstacles, =10
20 Accuracy
Accuracy
20
15
10
5
=4 =10 =∞ No obstacles, =10
15
10
5
10
15
20
25
Node Density
(a) Four obstacles
30
35
5
5
10
15
20
25
Node Density
(b) H-shaped obstacle
Fig. 6. Accuracy for localization in random topologies with obstacles
30
35
4.5
Scalability and overhead
CBL is explicitly designed to improve scalability by reducing computation overhead. We show its time and message overhead, which are two important metrics that reflect its scalability, in Figure 7. In Figure 7(a), we show the running-time in seconds for ns-2 simulations with different-sized clusters. As expected, CBL significantly reduces the running time if clustering is applied. The running-time is reduced to 6% when using clusters of size five. The reason for this reduction is mostly that the computation overhead of MDS techniques greatly depends on the network density, and this density is reduced by the clustering. As shown in Figure 7(b), we see a significant decrease in the messages needed to perform the localization as the cluster size is increased from 1. This is mainly because the messages exchanged when merging local maps are greatly reduced, and the complexity of each merge is less as each local map is smaller. Because of the very high running-time of ns-2, when using single-node clusters, we were not able to complete simulations at the higher densities. 3
10
35000
=5 =10 =1
=5 =10 =1
30000
10
25000 Messages
Running Time
2
1
10
20000 15000 10000
0
10
5000 -1
10
8
10
12
14
16
18
Node Density
(a)
0
5
10
15
20
25
30
Node Density
(b)
Fig. 7. Computation and communication overhead
In Figure 8, we plotted the accuracy of CBL as the network size increases: we maintain the same density and increase network area and nodes correspondingly. We have included DV-hop as a reference. For all DV-hop simulations we used 4 anchor nodes, and we can see that CBL (which uses no anchors) degrades more gracefully in accuracy as the number of nodes and network size increases, than DV-hop.
5
Conclusions & Future Work
In CBL, we utilized clustering to obtain an abstraction of the network, differentiating between the localization of cluster-heads and that of cluster-members. This
180
=11 =∞ DV-hop
160
Accuracy
140 120 100 80 60 40 20 0
60
80
100
120
140
160
180
200
220
240
n
Fig. 8. Accuracy in n × n networks
reduces the computation and communication cost while providing decent localization accuracy. The use of clustering also makes the localization more resilient to irregularities like obstacles: CBL performed well in regular networks, irregular networks and in networks with irregular node-densities. In all our simulations we used a simplified transmission model with no irregularities. This model may not be reflective of all realistic scenarios and the irregular transmission model may create more complex topologies, which might reduce the accuracy of any localization technique. We have shown that localizing only a small subset of nodes and using this to quickly and efficiently localize the remaining nodes is feasible and produces good results with lower computation complexity. Our goal was to create a scalable and more feasible localization algorithm; CBL achieves this scalability, even when using a high-cost technique such as non-metric MDS for localizing cluster-heads. CBL can provide a parameterized (by controlling the upper bound of cluster size) abstraction of networks. Although CBL utilizes non-metric MDS, a computationally-expensive but accurate localization, other localization techniques can also be used in CBL instead; the effect of changing the technique needs further investigation. Feasible localization is achieved by not introducing any new constraints on the network or sensor nodes: CBL does not rely on any kind of specialized, extra capable nodes such as anchors or beacons, or predetermined information about distributions, size, etc.
References 1. Chan, H., Luk, M., Perrig, A.: Using clustering information for sensor networks localization. In: DCOSS. (2005) 2. Shang, Y., Ruml, W.: Improved MDS-based localization. In: INFOCOM. Volume 4. (2004) 2640–2651 3. Akyildiz, I., Su, W., Sankarasubramaniam, Y., Cayirci, E.: A survey on sensor networks. IEEE Communications Magazine (2002) 102–114 4. Shih, E., S.Cho, Ickes, N., Min, R., Sinha, A., Wang, A., Chandrakasan, A.: Physical layer driven protocol and algorithm design for energy-efficient wireless sensor networks. In: MobiCom. (2001)
5. Niculescu, D., Nath, B.: Ad hoc positioning system (APS). In: GLOBECOM. Volume 5. (2001) 2926–2931 6. Savarese, C., Rabaey, J., Beutel, J.: Locationing in distributed ad-hoc wireless sensor networks. In: ICASSP. Volume 4. (2001) 2037–2040 7. Yao, Q., Tan, S., Ge, Y., Yeo, B., Yin, Q.: An area localization scheme for large wireless sensor networks. In: VTC. Volume 2. (2005) 2835–2839 8. Sichitiu, M., Ramadurai, V.: Localization of wireless sensor networks with a mobile beacon. In: MSN. (2004) 174–183 9. Sun, G., Guo, W.: Comparison of distributed localization algorithms for sensor networks with a mobile beacon. In: ICNSC. Volume 1. (2004) 536–540 10. Patro, R.: Localization in wireless sensor network with mobile beacons. In: IEEE Convention of Electrical and Electronics Engineers in Israel. (2004) 22–24 11. Li, X., Shi, H., Shang, Y.: A partial-range-aware localization algorithm for ad hoc wireless sensor networks. In: LCN. (2004) 77–83 12. Doherty, L., Pister, K., El Ghaoui, L.: Convex position estimation in wireless sensor networks. In: INFOCOM. Volume 3. (2001) 1655–1663 13. Bergamo, P., Asgari, S., Wang, H., Maniezzo, D., Yip, L., Hudson, R., Yao, K., Estrin, D.: Collaborative sensor networking towards real-time acoustical beamforming in free-space and limited reverberance. TMC 3(3) (2004) 211–224 14. Galstyan, A., Krishnamachari, B., Lerman, K., Pattem, S.: Distributed online localization in wireless sensor networks using a moving target. In: IPSN. (2004) 61–70 15. Coxon, A.: The Users Guide to Multi Dimensional Scaling. Heinemann Educational Books (1982) 16. Ji, X., Zha, H.: Robust sensor localization algorithm in wireless ad hoc sensor networks. In: INFOCOM. (2003) 527–532 17. Eren, T., Whiteley, W., Belhumeur, P.: Further results on sensor network localization using rigidity. In: EWSN. (2005) 405–409 18. Liu, K., Wang, S., Ji, Y., Yang, X., Hu, F.: On connectivity for wireless sensor networks localization. In: IWCCC. Volume 2. (2005) 879–882 19. Perkins, D., Tumati, R.: Reducing localization errors in sensor ad hoc networks. In: IPCCC. (2004) 723–729 20. Savarese, C., Rabaey, J., Langendoen, K.: Robust positioning algorithms for distributed ad-hoc wireless sensor networks. In: USENIX Annual Technical Conference. (2002) 317–327 21. Karalar, T., Yamashita, S., Sheets, M., Rabaey, J.: A low power localization architecture and system for wireless sensor networks. In: SIPS. (2004) 89–94 22. Ahmed, A., Hongchi, S., Shang, Y.: Sharp: A new approach to relative localization in wireless sensor network. In: ICDCS. (2005) 892–898 23. Shang, Y., Ruml, W., Zang, Y., Fromhertz, M.: Localization from mere connectivity. In: MobiHoc, Annapolis, MD (2003) 202–212 24. Shang, Y., Meng, J., Shi, H.: A new algortihm for relative localization in wireless sensor networks. In: IPDPS. (2004) 25. Morrison, A., Ross, G., Chalmers, M.: Fast multidimensional scaling through sampling, springs and interpolation. Information Visualization 2 (2003) 68–77 26. URL: ns-2, discrete event simulator. http://www.isi.edu/nsnam/ns/ (2006)
Abstract. Localization, an important challenge in wireless sensor networks, is the process of sensor nodes self-determining their position. The difficulty encountered is in cost-effectively providing acceptable accuracy in localization. The potential for the deployment of high density networks in the near future makes scalability a critical issue in localization. In this paper we propose Cluster-based Localization (CBL), which provides effective localization suitable for large and highly-dense networks. CBL utilizes both a computationally-intensive localization technique (non-metric multidimensional scaling (MDS)) and a less intensive trilateration to achieve balance between performance and cost. Clustering is utilized to select a subset of nodes to perform MDS and then extend their localization to the remaining network. Besides providing scalability clustering overcomes local irregularities and provides good accuracy even in irregular networks with or without obstacles. Simulation results illustrate that CBL reduces both computation and communication, while still yielding acceptable accuracy.
1
Introduction
Wireless sensor networks (WSN) are typically densly populated ad-hoc networks composed of small, resource-constrained, immobile nodes. The ability for a sensor to self-determine its own position, enabling the node to correlate its data with a location, is critical in many domains [1]. Minimizing the cost of nodes is a critical consideration, which makes equipping all sensor nodes with GPS capabilities infeasible [2]. Furthermore, the number of sensor nodes deployed in a sensor network is typically high, e.g., on the order of thousands and possibly even millions, and the network density can reach a few hundred nodes per square meter [3],[4]. The need for position information in high-density WSN motivates the need for scalable localization. In this paper we propose a scalable hop-based localization technique called Cluster-based Localization (CBL). CBL’s use of clustering is motivated by the need for scalability and efficiency. Scalability is gained by, first using an expensive but accurate localization technique on a small subset of the nodes in the ?
This work was supported in part by NSF grant CNS-0454416.
network. The derived position estimates for the chosen subset will be used as references when localizing the remaining nodes. This approach leads to a significant reduction in the computation required to localize the entire network. Although scalability was the primary goal for performing the localization in different stages in CBL, employing clustering has also provided the benefit of smoothing over local variations within the network. This smoothing effect prevents the global localization results from being skewed by the affects of local aberrations. In particular, CBL is able to maintain accuracy even in the presence of RF-opaque obstacles and irregular network topologies. Although our CBL implementation uses non-metric MDS in localization of the representative nodes, this technique could be replaced so that CBL could benefit from other localization ones. The rest of this paper is organized as follows. Section 2 presents related work in the area of sensor network localization. CBL is described in Section 3. Section 4 contains performance evaluation and comparisons of CBL and Section 5 provides some concluding remarks.
2
Related Work
Previous attempts at localization in sensor networks can be categorized into two groups: range-aware and hop-based. In range-aware techniques a distance measure between neighboring nodes is used to estimate node positions. In hopbased ones ranging equipment is not necessary, and the estimated distances between nodes are typically approximated to the hops in the shortest path. Range-aware localization techniques typically derive inter-node distances based on received signal strength. APS [5], a distributed localization technique, extends both distance vector routing and GPS positioning. Similiar distance estimates are used in [6]. In [7], a similiar, but more coarse estimation process is described, in which nodes adjacent to at least one anchor are localized first. Received signal strength measurements of broadcasts from a single mobile beacon node are used in [8]. The beacon’s current position, will place constraints on the possible position of a node. Another technique using a mobile beacon node, utilizing time-of-arrival measurements and probablistic estimation, is proposed in [9]. In [10], four mobile beacons are used, creating a rectangle with the “un-localized” node in the middle, allowing for trilateration. PRI [11] provides improved performance by augmenting hop information with any available ranging information. In [12], known peer-to-peer communications are modeled as a set of geometric constraints on nodes’ positions. In [13], the Approximate Maximum-Likelihood method and Direction of Arrival estimation are reviewed. In [14], sensing constraints caused by mobile objects at several nodes are utilized to improve the accuracy of localization. RangeQ determines node positions by means of a distributed range quantization technique which is similar to quantization in image processing [11]. Ji and Zha present a distributed localization technique based on the estimation-comparison-correction paradigm. It applies multidimensional scaling (MDS) [15] to merge each individual node’s map of the network topology
into one global map of the network [16]. In [17], previous work from rigidity theory is extended to networks where not all nodes are localizable. Chan, Luk and Perrig [1] propose a scalable localization, where a complex clustering is used to produce a highly regular structure. This regularity can then be utilized, along with arbitrarily positioned anchor nodes, in the localization. The problem with calculating distances by means of signal strength measurements is that since all possible sources of signal interference cannot be accurately anticipated prior to sensor deployment, the estimated distances can become inaccurate due to multi-path interference, line-of-sight obstructions, etc. Hop-based localization techniques aim to remove the dependency on any form of ranging equipment. A theoretical analysis of network connectivity for node selflocalization is presented in [18]. An extension of the algorithm proposed in [5] is the differential Ad-Hoc Positioning System [19], which describes a differential error correction scheme designed to reduce the cumulative distances and positioning error over multiple hops. HOP-TERRAIN [20], similar to [5], utilizes an additional refinement phase to improve the localization accuracy. A low-powerdedicated hardware localization technique utilizing the HOP-TERRAIN is proposed in [21]. SHARP [22] technique adopts a hybrid hop-based and range-aware approach for localization. SHARP attempts to perform localization on a subset of the nodes in a network using inter-node distance estimates. This subset of nodes will be localized using MDS [2], and are then used to localize the remaining nodes with APS [5]. Shang et al. proposed MDS-MAP [23], an algorithm that utilizes MDS to perform global localization. Since MDS-MAP, there have been notable extensions of this technique: MDS-MAP(P)[2] and MDS-MAP(R)[24]. MDS-MAP(P) is a distributed algorithm that uses patches of relative maps, that can be computed in parallel, to estimate the absolute positions of nodes. On the otherhand, the distributed MDS-MAP(R) estimates the relative positions. A similar algorithm based on the estimation-comparison-correction paradigm and MDS [15] is described in [16].
3
Cluster-based Localization
To localize all sensor nodes accurately, the localization process usually involves computationally-intensive procedures, e.g., MDS that has O(n3 ) computational complexity [2],[25]. Since sensor nodes are expected to be resource-constrained and networks to be large and dense, light-weight localization algorithms that achieve decent accuracy and high scalability are desirable. Therefore, we attempt to address the trade-offs between accuracy and scalability, so as to make our localization algorithm, CBL, a practical solution for sensor network applications. CBL adopts a hierarchical idea to perform the whole localization procedure, i.e., instead of regarding the whole network as a flat topology in which all nodes share the same localization procedure as in most existing localization algorithms, we deliberately separate the nodes into two types, and apply different localization approaches on them to improve scalability as well as ensure accuracy. The first type of nodes are representatives that account for a small portion of the nodes in the whole network. Representatives will be selected to reflect a good
abstraction of the network and localized by using complex but accurate localization algorithms, which essentially improves the overall localization accuracy. The remaining nodes, which account for the majority of the network, can employ a lighter-weight localization process that uses the representatives as reference nodes. By applying this mechanism, CBL achieves decent accuracy with relatively low computation and communication overhead. 3.1
Representative Selection
Since representatives will be localized first and used as reference nodes for estimating the remaining nodes’ positions, they should provide a good abstraction of the whole network. Further, the representative selection process itself should be light-weight to ensure the overall scalability. Therefore, in CBL we apply a simple single-hop and size-bounded clustering algorithm to select representatives, or Cluster Heads (CHs), from the network. The single-hop feature of the clustering algorithm ensures the algorithm’s efficiency, and the size-bounded feature enables each cluster to be formed fairly uniformly and makes cluster heads represent a good abstraction of the whole network. We assume each node has a unique ID. To prepare for clustering, initially each node performs neighbor discovery to become aware of its single-hop neighborhood information. After that, a node either assumes to be a CH if it has the largest ID among its neighbors, or waits to be contacted if any of its neighbors have larger IDs. A CH will start to contact its neighbors to build its own cluster whose size is bounded by a threshold θ. A node becomes a Cluster Member (CM) of the first neighboring CH that contacts it, or becomes a CH if all its larger-ID neighbors have already spoken by either becoming a clusterhead with full size-bounded cluster or becoming a CM in some other cluster. To enable the network formed by CHs to provide a good abstraction of the whole network, a CH c will attempt avoiding CMs that cover overlapping transmission areas with c’s existing CMs. After deciding its own role (CH/CM), a node informs its neighbors about its decision. Eventually, within a few rounds each node will finish determining its role (CH/CM) and each cluster forms a star topology with a CH at the center. In Fig. 1(a), we illustrate a random network and the connectivity between nodes. In Fig. 1(b), we show the resulting clusters in which a CM is represented as a line from its CH to itself, and in Fig. 1(c) we show the connectivity for each cluster. It can be observed that the CMs of each cluster are reasonably well distributed around the CH; and these CHs, or representatives, provide a good abstraction of the original topology. 3.2
Cluster Head Localization
CHs are taken as representatives and their localization accuracies have great impact on the overall localization accuracy. Since CHs usually account for a small portion of the whole network, it is affordable to apply complex localization algorithms to estimate CHs’ positions. Further, we want to apply a hop-based algorithm to localize CHs so as to improve CBL’s applicability. Based on these
(a) Network topology
(b) Resulting clusters
(c) Cluster network
Fig. 1. Example clustering
three considerations, we chose MDS as the basic technique to localize CHs, since hop-based MDS localization algorithm MDS-MAP(P) [2] was shown to achieve a good accuracy. CBL uses similar approaches to localize CHs, however, in CBL only CHs are involved in the MDS computation, therefore the computation overhead is contained. CBL does not involve any pre-installed anchor nodes as opposed to MDS-MAP(P), which reduces the cost of sensor hardware. Further, in CBL we employ the non-metric MDS technique that has a weaker requirement on the input data than the classical MDS used in MDS-MAP(P). To apply MDS for local map construction, each CH c needs to collect the distance information from its “two-step” clusters’ CHs (we call neighboring clusters “one-step” clusters). After that, c computes shortest distances (in hops) between each pair of the involved CHs, which will be taken as the input for the MDS algorithm to estimate positions of the involved CHs and to create a local map. MDS is a set of well-known data analysis techniques for geometrical position estimation and information visualization; see [15] for details. In our localization, we utilize the non-metric MDS, which assumes a less-stringent monotonicity constraint than the classical metric MDS deployed in [2]. After obtaining a local map through MDS calculation, each CH will attempt to merge its local map with its neighboring CHs’. Similar to MDS-MAP(P), the merging is a completely distributed process, in which the CHs with larger IDs will have higher priorities to choose one of their neighboring CHs to merge. Further, neighboring maps will be merged based on their common nodes, i.e. the maps with the highest number of common nodes should be merged first. When merging, we apply the best linear transform technique to transform one map onto another, and the new coordinates are computed based on the average of the common nodes’ coordinates. 3.3
Cluster Member Localization
After determining each CH’s coordinates, we will take CHs as reference nodes to estimate the remaining nodes’, or CMs’, positions. We choose the least-square triangulation technique for CMs’ localization to obtain decent accuracy with low computation overhead. In particular, each CH first calculates the euclidean distances to the CHs in its neighboring clusters, then estimates the average hop distance by using these euclidean distances and corresponding hop length. After
that, each CH broadcasts a LOCATION message, which contains its coordinates and the estimated hop distance, to the CMs within α hops (α should be chosen to balance the message overhead and the localization accuracy). Each CM waits for at least three LOCATION messages from different CHs, and then perform a triangulation to determine its own coordinates. If a CM receives more than three LOCATION messages, a least square error technique is applied.
4
Performance Evaluation
CBL was implemented in ns-2 [26], and our main goals when evaluating its performance were to determine the accuracy, scalability, and how irregularities in a network topology will affect performance. The simulation environment parameters are: ns2 version 2.29; area = 100 × 100m2 ; 100 nodes; transmission range = 15m and cluster size varied from 1 to 10. We utilize a commonly used metric, accuracy, to measure how much estimated positions deviate from actual positions. Two maps, estimated and actual, are compared by finding the best linear transformation (rotating, shifting and/or scaling) of the estimated map onto the actual and calculating the average Euclidean difference between each corresponding point. In the simulations the node-density is controlled by varying the enclosing network area while maintaining the same number of nodes. We also varied the upper-bound for cluster-size θ to determine how changes in clustersizes affect the performance of CBL. For comparison, we have included both MDS-MAP(P) and DV-hop results at similar densities for reference. The data for MDS-MAP(P) is obtained from [2] and are based on MATLAB simulations; a CAML implementation of DV-hop was used to obtain results for comparison. During cluster-head (CH) localization, all hop values used are perturbed by adding noise σ (0 ≤ σ ≤ 10−5 ) before the non-metric MDS is employed to eliminate ties and help the non-metric MDS converge, because the non-metric MDS technique is dependent on the differences in the distance estimations. 4.1
Random network
The topologies in our random networks are square-shaped n × n ( n controls density ) networks, with 100 randomly placed nodes. As a key step, the clustering we deployed directly affects the quality of localization as well as the computation overhead. We observed that the average cluster-size stays close to the given upper-bound, which reflects the clusters’ high quality in spite of our lightweight clustering. Further, as expected, cluster-density drops significantly given a larger upper bound. In Figure 2 we show the resulting localization accuracy for random networks. Because we extend cluster-head (CH) localizations to obtain member (CM) localizations, we present CHs’ accuracy results in Figure 2(a) and all nodes’ accuracy results in Figure 2(b). Our results show that increasing the cluster size does not lead to significant performance degradation. Although MDS-MAP(P) and DV-hop achieve slightly better accuracy than CBL, CBL does not utilize nor depend on any anchor nodes or extra refinement of the estimates as DV-hop and
MDS-MAP(P) require. Furthermore, it should also be observed that the difference between the accuracy of all nodes and that of only cluster-heads is very small. This reflects the effectiveness of the clustering technique we applied, and shows that if the CH localization is accurate enough the low cost trilateration technique provides good results. 25
15
15
10
10
5
5
0
5
10
15
20
25
30
=4 =10 =1 =∞ MDS-MAP(P) DV-hop
20 Accuracy
20
Accuracy
25
θ=4 θ=10 θ=1 θ=∞ MDS-MAP(P) DV-hop
0
35
5
10
15
20
25
30
35
Node Density
Node Density
(a) Cluster-heads
(b) Entire network
Fig. 2. Accuracy for localization random topologies
30 25 20 15
20 15
10
10
5
5
0
5
10
15
20
25
Node Density
(a) Cluster-heads
30
=4 =10 =1 = MDS-MAP(P) DV-hop
25
Accuracy
Accuracy
30
=4 =10 =1 = MDS-MAP(P) DV-Hop
35
0
5
10
15
20
25
30
35
Node Density
(b) Entire network
Fig. 3. Accuracy for localization in C-shaped topologies
4.2
C-shaped network
C-shaped topologies are commonly employed to stress any localization technique [2],[5]. In this irregular topology the estimated distances between nodes can de-
viate greatly from the actual Euclidean distances. The resulting cluster-densities and average cluster-size are very similar to those for random networks. In Figure 3, we show the resulting accuracy for both the cluster-heads Fig. 3(a) and all nodes Fig. 3(b). As in the random topologies, changes in cluster-sizes do not affect the resulting accuracy. Furthermore, compared to MDS-MAP(P), CBL’s accuracy is only slightly worse, while providing the same lowered computation as in the random networks. Compared to DV-hop, CBL provides better accuracy for irregular topologies. This is mostly because of DV-hop’s dependence on the uniformity and regularity of the network. 4.3
Irregular node densities
To test how well CBL will perform in topologies with changing node densities, we created an irregular topology. The irregular topologies, also referred to as biased in the literature, are another fairly common test. An illustration of an irregular topology can be found in Figure 5(a), where the area containing the higher node-density is indicated by the square in the lower left corner. We can see from Figure 4 that, as expected, CBL is not greatly affected by the irregularity in node-densities; verifying that in CBL innacurate estimates do not propagate and affect a significant part of a network. 25
25
=4 =10 =∞ Uniform desity, =10
20 Accuracy
Accuracy
20
15
10
5
=4 =10 =∞ Uniform desity, =10
15
10
5
10
15
20
25
Node Density
(a) Cluster-heads
30
35
5
5
10
15
20
25
30
35
Node Density
(b) Entire network
Fig. 4. Accuracy for localization in irregular node-density topologies
4.4
Obstacles
Obstacles usually create great difficulty for any localization technique because any network with obstacles will have a much more irregular structure and the inter-node distance estimates are likely to be more inaccurate than those without obstacles. We performed simulations on random topologies with two different types of RF-opaque obstacles; four line-shaped (Figure 5(b)) and one H-shaped (Figure 5(c)) obstacle similar to the ones used in literature. As shown in Figure
5(c) the nodes close to the horizontal line of the H-shaped obstacle will be ones most affected and these nodes will derive severely overestimated inter-node distances between themselves and nodes on the other side of the horizontal line.
(a) Irregular densities
node-
(b) With obstacles
(c) With H-shaped obstacle
Fig. 5. Irregular network topologies
Figure 6 shows the resulting accuracy for the entire network with both four obstacles Fig. 6(a) and one H-shaped obstacle Fig. 6(b). We omit the graphs for the cluster-heads due to space constraints, but the same trend seen in other topologies is also evident for networks with obstacles. The localization accuracy achieved is very similar to that of networks without obstacles. This is because the use of clusters produces a technique that is less sensitive to any type of irregularities, including obstacles, and by covering a larger geographical area the clusters can in effect reach around many of the obstacles. Although the inter-cluster distances at times might be overestimated the overall accuracy is not significantly affected. The same trend as in the random networks is observed, where increasing node-densities improves the accuracy and there is a small degradation in accuracy when comparing that of all nodes and that of CHs. 25
25
=4 =10 =∞ No obstacles, =10
20 Accuracy
Accuracy
20
15
10
5
=4 =10 =∞ No obstacles, =10
15
10
5
10
15
20
25
Node Density
(a) Four obstacles
30
35
5
5
10
15
20
25
Node Density
(b) H-shaped obstacle
Fig. 6. Accuracy for localization in random topologies with obstacles
30
35
4.5
Scalability and overhead
CBL is explicitly designed to improve scalability by reducing computation overhead. We show its time and message overhead, which are two important metrics that reflect its scalability, in Figure 7. In Figure 7(a), we show the running-time in seconds for ns-2 simulations with different-sized clusters. As expected, CBL significantly reduces the running time if clustering is applied. The running-time is reduced to 6% when using clusters of size five. The reason for this reduction is mostly that the computation overhead of MDS techniques greatly depends on the network density, and this density is reduced by the clustering. As shown in Figure 7(b), we see a significant decrease in the messages needed to perform the localization as the cluster size is increased from 1. This is mainly because the messages exchanged when merging local maps are greatly reduced, and the complexity of each merge is less as each local map is smaller. Because of the very high running-time of ns-2, when using single-node clusters, we were not able to complete simulations at the higher densities. 3
10
35000
=5 =10 =1
=5 =10 =1
30000
10
25000 Messages
Running Time
2
1
10
20000 15000 10000
0
10
5000 -1
10
8
10
12
14
16
18
Node Density
(a)
0
5
10
15
20
25
30
Node Density
(b)
Fig. 7. Computation and communication overhead
In Figure 8, we plotted the accuracy of CBL as the network size increases: we maintain the same density and increase network area and nodes correspondingly. We have included DV-hop as a reference. For all DV-hop simulations we used 4 anchor nodes, and we can see that CBL (which uses no anchors) degrades more gracefully in accuracy as the number of nodes and network size increases, than DV-hop.
5
Conclusions & Future Work
In CBL, we utilized clustering to obtain an abstraction of the network, differentiating between the localization of cluster-heads and that of cluster-members. This
180
=11 =∞ DV-hop
160
Accuracy
140 120 100 80 60 40 20 0
60
80
100
120
140
160
180
200
220
240
n
Fig. 8. Accuracy in n × n networks
reduces the computation and communication cost while providing decent localization accuracy. The use of clustering also makes the localization more resilient to irregularities like obstacles: CBL performed well in regular networks, irregular networks and in networks with irregular node-densities. In all our simulations we used a simplified transmission model with no irregularities. This model may not be reflective of all realistic scenarios and the irregular transmission model may create more complex topologies, which might reduce the accuracy of any localization technique. We have shown that localizing only a small subset of nodes and using this to quickly and efficiently localize the remaining nodes is feasible and produces good results with lower computation complexity. Our goal was to create a scalable and more feasible localization algorithm; CBL achieves this scalability, even when using a high-cost technique such as non-metric MDS for localizing cluster-heads. CBL can provide a parameterized (by controlling the upper bound of cluster size) abstraction of networks. Although CBL utilizes non-metric MDS, a computationally-expensive but accurate localization, other localization techniques can also be used in CBL instead; the effect of changing the technique needs further investigation. Feasible localization is achieved by not introducing any new constraints on the network or sensor nodes: CBL does not rely on any kind of specialized, extra capable nodes such as anchors or beacons, or predetermined information about distributions, size, etc.
References 1. Chan, H., Luk, M., Perrig, A.: Using clustering information for sensor networks localization. In: DCOSS. (2005) 2. Shang, Y., Ruml, W.: Improved MDS-based localization. In: INFOCOM. Volume 4. (2004) 2640–2651 3. Akyildiz, I., Su, W., Sankarasubramaniam, Y., Cayirci, E.: A survey on sensor networks. IEEE Communications Magazine (2002) 102–114 4. Shih, E., S.Cho, Ickes, N., Min, R., Sinha, A., Wang, A., Chandrakasan, A.: Physical layer driven protocol and algorithm design for energy-efficient wireless sensor networks. In: MobiCom. (2001)
5. Niculescu, D., Nath, B.: Ad hoc positioning system (APS). In: GLOBECOM. Volume 5. (2001) 2926–2931 6. Savarese, C., Rabaey, J., Beutel, J.: Locationing in distributed ad-hoc wireless sensor networks. In: ICASSP. Volume 4. (2001) 2037–2040 7. Yao, Q., Tan, S., Ge, Y., Yeo, B., Yin, Q.: An area localization scheme for large wireless sensor networks. In: VTC. Volume 2. (2005) 2835–2839 8. Sichitiu, M., Ramadurai, V.: Localization of wireless sensor networks with a mobile beacon. In: MSN. (2004) 174–183 9. Sun, G., Guo, W.: Comparison of distributed localization algorithms for sensor networks with a mobile beacon. In: ICNSC. Volume 1. (2004) 536–540 10. Patro, R.: Localization in wireless sensor network with mobile beacons. In: IEEE Convention of Electrical and Electronics Engineers in Israel. (2004) 22–24 11. Li, X., Shi, H., Shang, Y.: A partial-range-aware localization algorithm for ad hoc wireless sensor networks. In: LCN. (2004) 77–83 12. Doherty, L., Pister, K., El Ghaoui, L.: Convex position estimation in wireless sensor networks. In: INFOCOM. Volume 3. (2001) 1655–1663 13. Bergamo, P., Asgari, S., Wang, H., Maniezzo, D., Yip, L., Hudson, R., Yao, K., Estrin, D.: Collaborative sensor networking towards real-time acoustical beamforming in free-space and limited reverberance. TMC 3(3) (2004) 211–224 14. Galstyan, A., Krishnamachari, B., Lerman, K., Pattem, S.: Distributed online localization in wireless sensor networks using a moving target. In: IPSN. (2004) 61–70 15. Coxon, A.: The Users Guide to Multi Dimensional Scaling. Heinemann Educational Books (1982) 16. Ji, X., Zha, H.: Robust sensor localization algorithm in wireless ad hoc sensor networks. In: INFOCOM. (2003) 527–532 17. Eren, T., Whiteley, W., Belhumeur, P.: Further results on sensor network localization using rigidity. In: EWSN. (2005) 405–409 18. Liu, K., Wang, S., Ji, Y., Yang, X., Hu, F.: On connectivity for wireless sensor networks localization. In: IWCCC. Volume 2. (2005) 879–882 19. Perkins, D., Tumati, R.: Reducing localization errors in sensor ad hoc networks. In: IPCCC. (2004) 723–729 20. Savarese, C., Rabaey, J., Langendoen, K.: Robust positioning algorithms for distributed ad-hoc wireless sensor networks. In: USENIX Annual Technical Conference. (2002) 317–327 21. Karalar, T., Yamashita, S., Sheets, M., Rabaey, J.: A low power localization architecture and system for wireless sensor networks. In: SIPS. (2004) 89–94 22. Ahmed, A., Hongchi, S., Shang, Y.: Sharp: A new approach to relative localization in wireless sensor network. In: ICDCS. (2005) 892–898 23. Shang, Y., Ruml, W., Zang, Y., Fromhertz, M.: Localization from mere connectivity. In: MobiHoc, Annapolis, MD (2003) 202–212 24. Shang, Y., Meng, J., Shi, H.: A new algortihm for relative localization in wireless sensor networks. In: IPDPS. (2004) 25. Morrison, A., Ross, G., Chalmers, M.: Fast multidimensional scaling through sampling, springs and interpolation. Information Visualization 2 (2003) 68–77 26. URL: ns-2, discrete event simulator. http://www.isi.edu/nsnam/ns/ (2006)
