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20 On Boundary Recognition without Location Information in Wireless Sensor Networks OLGA SAUKH, ROBERT SAUTER, MATTHIAS GAUGER, and PEDRO JOS ´ E MARR ´ ON Universit ¨ at Bonn Boundary recognition is an important and challenging issue in wireless sensor networks when no coordinates or distances are available. The distinction between inner and boundary nodes of the network can provide valuable knowledge to a broad spectrum of algorithms. This article tackles the challenge of providing a scalabl
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  20 On Boundary Recognition without LocationInformation in Wireless Sensor Networks OLGA SAUKH, ROBERT SAUTER, MATTHIAS GAUGER,and PEDRO JOS´E MARR ´ONUniversit¨at Bonn Boundary recognition is an important and challenging issue in wireless sensor networks when nocoordinates or distances are available. The distinction between inner and boundary nodes of thenetwork can provide valuable knowledge to a broad spectrum of algorithms. This article tacklesthe challenge of providing a scalable and range-free solution for boundary recognition that doesnot require a high node density. We explain the challenges of accurately defining the boundaryof a wireless sensor network with and without node positions and provide a new definition of network boundary in the discrete domain. Our solution for boundary recognition approximatesthe boundary of the sensor network by determining the majority of inner nodes using geometricconstructions,whichguaranteethatforagiven d ,anodeliesinsideoftheconstructionfora d -quasiunit disk graph model of the wireless sensor network. Moreover, such geometric constructionsmake it possible to compute a guaranteed distance from a node to the boundary. We presenta fully distributed algorithm for boundary recognition based on these concepts and perform adetailed complexity analysis. We provide a thorough evaluation of our approach and show that itis applicable to dense as well as sparse deployments.Categories and Subject Descriptors: C.2.1 [ Computer-Communication Networks ]: Network Architecture and Design—  Network topology; Wireless communication ; G.2.2 [ Discrete Mathe-matics ]: Graph Theory— Graph algorithms ; F.2.2 [  Analysis of Algorithms and Problem Com-plexity ]: Nonnumerical Algorithms and ProblemsGeneral Terms: Algorithms, Theory Additional Key Words and Phrases: Boundary definition, boundary recognition, d-quasi unit diskgraphs, unit disk graphs, wireless sensor networks  ACM Reference Format: Saukh, O., Sauter, R., Gauger, M., and Marr´on, P. J. 2010. On boundary recognition withoutlocation information in wireless sensor networks. ACM Trans. Sensor Netw. 6, 3, Article 20 (June2010), 35 pages. DOI  =  10.1145/1754414.1754416 http://doi.acm.org/10.1145/1754414.1754416 A preliminary version of this article was presented at the 7th International Conference on Infor-mation Processing in Sensor Networks (IPSN). Authors’ address: Sensor Networks and Pervasive Computing Group, Institute for Com-puter Science IV, R¨omerstrasse 164, 53117 Bonn, Germany; email:  { saukh, sauter, gauger,pjmarron } @cs.uni-bonn.de.Permission to make digital or hard copies of part or all of this work for personal or classroom useis granted without fee provided that copies are not made or distributed for profit or commercialadvantage and that copies show this notice on the first page or initial screen of a display along with the full citation. Copyrights for components of this work owned by others than ACM must behonored. Abstracting with credit is permitted. To copy otherwise, to republish, to post on servers,to redistribute to lists, or to use any component of this work in other works requires prior specificpermission and/or a fee. Permissions may be requested from Publications Dept., ACM, Inc., 2 PennPlaza, Suite 701, New York, NY 10121-0701 USA, fax + 1 (212) 869-0481, or permissions@acm.org. C   2010 ACM 1550-4859/2010/06-ART20 $10.00DOI 10.1145/1754414.1754416 http://doi.acm.org/10.1145/1754414.1754416  ACM Transactions on Sensor Networks, Vol. 6, No. 3, Article 20, Publication date: June 2010.  20:2  ã  O. Saukh et al. 1. INTRODUCTION In wireless sensor networks (WSN) it is often necessary to know the topol-ogy of the deployed network as well as specifics on the coverage of the targetarea. Such information can be easily extracted if a significant subset of nodesis equipped with GPS receivers or has other means of determining their re-spective locations. However, such hardware is expensive and requires a lot of energy. In this article, we present a scalable solution that, without the use of position information, recognizes certain topological properties such as innerand boundary nodes in a two-dimensional space. Our approach sets no con-straints on the node distribution and the node density. Additionally, each innernode is assigned a minimum guaranteed distance to the boundary.Many WSN applications demonstrate the need for the extraction of suchtopological information [Rao et al. 2003; Fekete et al. 2004; Funke and Klein2006]. Network lifetime is perhaps the most important issue in WSN. As thefailure of boundary nodes results in reduced coverage, the load of these nodesshould be reduced. For example, routing algorithms can exhaust the energy of the nodes that lie on the boundary of a hole [Kr¨oller et al. 2006]. The result-ing coverage reduction might lead to a partitioning of a WSN communicationgraph or missed events in the monitored region. However, it is possible to de-velop algorithms that adapt to the actual deployment using the distinctionbetween boundary nodes and inner nodes. The knowledge about holes can alsobeusedafteraninitialdeployment ornodefailurestotarget specificareas withadditional nodes.In a large number of scenarios, for example in the environmental monitor-ing of vineyards, forests, or large warehouses, sensor readings differ heavilybetween the boundary and the center of the network. Sensor readings from theboundary nodes might influence the aggregation results considerably becausethey also capture events occurring outside of the monitored region of inter-est. Therefore, single sensor readings should be interpreted differently andaggregation should not be performed across inner and boundary nodes. A finerdifferentiation among nodes based on their distance to the boundary makes itpossible to detect such an influence.Whether a node is an inner or a boundary node might be crucial in objecttracking scenarios. For example, when tracking events entering and leaving a region, boundary nodes might be involved in more complex sensing taskswhereas inner nodes might spend more energy on performing routing tasks.This lets boundary nodes play a special role that cannot be assigned prior todeployment. In addition, grouping nodes by nesting levels allows the definitionof further security perimeters with different alert degrees. Accurately defining the boundary of a wireless network is a challenge thatshould not be underestimated. Funke and Klein [2006] provide an implicitdefinition of a network boundary in terms of nodes being close to the bound-ary of the continuous domain. Kr¨oller et al. [2006] and Wang et al. [2006]define the boundary with the help of cycles. In this article we discuss the prob-lem of defining the boundary in the discrete domain and its relation to twointuitive properties: uniqueness and continuity. We illustrate that this is a  ACM Transactions on Sensor Networks, Vol. 6, No. 3, Article 20, Publication date: June 2010.  On Boundary Recognition without Location Information in WSN  ã  20:3 hard problem even with known node positions, and show that the definitionsfound in prior work are incomplete. We then generalize the definition of bound-ary for the case where no location information is available, and discuss itsproperties.Our approach to the problem of boundary recognition presented in this arti-cleprovidesacloseapproximationofthegeneralizedboundarywithoutproduc-ingfalsenegatives.Additionally,thisboundaryrecognitionapproachisthefirstthatisabletoconsiderablyrelaxtheassumptionsonnodedensityandprovidesa good solution for both sparse and dense networks. Moreover, our result is ageneralization of the approach described in Kr¨oller et al. [2006]. Similarly toKr¨oller et al. [2006], our approach is based on the recognition of inner nodes of the network and considers all other nodes to be part of the outer boundary orthe boundary of a hole. We introduce geometric constructions called patterns,which guarantee that a particular node lies inside of the pattern for all de-ployments of the sensor network with a given connectivity graph. Our patternsare generic, simple, parameterized, and no global knowledge of the networkconnectivity graph is required to recognize them. These properties make ourapproach scalable and applicable to sparse networks. Moreover, many of thepatterns allow additionally assigning to a node a guaranteed minimum dis-tance to the boundary. Nodes with the same guaranteed distance from theboundary form a nesting level. Although our patterns do not cover all nodes that lie inside of the networkin all cases, simulation results show that the patterns are powerful enoughto detect almost all inner nodes of the network, and therefore provide a goodapproximation of the network boundaries. We provide pattern rules that con-siderably simplify the recognition of patterns and allow the algorithm to runin polynomial time.The remainder of this article is structured as follows. First, we discuss re-lated work in Section 2. In Section 3, we describe the problems related to thedefinition of the network boundary. We then present in Section 4, geometricpatterns that are the foundation of our approach. The algorithmic aspects andthe time, space, and message complexities are discussed in Section 5, followedby simulation results in Section 6. Our conclusions and an outlook to futureresearch directions complete this article. 2. RELATED WORK There is a strong need to extract spatial information of deployed sensor and adhoc networks without node coordinates. Even if distance information betweenthe nodes is available, the problem of accurate node localization is NP-hard[Aspnes et al. 2004]. However, a number of approaches provide reasonableapproximations of different topological characteristics of the network such asthe outer boundary of the network and boundaries of holes [Funke and Klein2006; Funke 2005; Wang et al. 2006; Kr¨oller et al. 2006], isolines or contours[Funke and Klein 2006], and medial axis lines [Wang et al. 2006; Bruck et al.2005], or streets [Kr¨oller et al. 2006], which express topological levels based onthe number of hops to the boundary.  ACM Transactions on Sensor Networks, Vol. 6, No. 3, Article 20, Publication date: June 2010.  20:4  ã  O. Saukh et al. Related work from the area of boundary recognition in sensor networkswithout location information can be classified into three groups based on therespective assumptions on node distribution, node density, and the underlying communication model.The approaches in the first group rely on a certain node distribution of the sensor nodes in the non-hole regions. For example, the approach de-scribed in Fekete et al. [2004] requires a uniform distribution of sensornodes.In the second group, a number of approaches [Funke and Klein 2006; Funke2005; Wang et al. 2006] present solutions for boundary recognition based onthe assumption that the length of the shortest path between two nodes pro-vides a reasonable approximation of the geodesic distance between the nodes.However, this assumption requires a rather high average node degree in thenetwork (in the range of 25) for the approaches to perform reasonably well[Funke and Klein 2006; Funke 2005]. The required node degree can be reducedto 10 if the topology conforms to a more regular node distribution like a grid ora perturbed grid [Funke and Klein 2006]. Under the unit disk graph assump-tion, sufficient node density and further assumptions on hole size and holeplacement, the algorithm marks the nodes close to the boundary with certainguarantees [Funke and Klein 2006]. In their previous work [Funke 2005], theauthors additionally indicate that the success rate of their method decreaseswith the decrease of the parameter  d  when the network topology follows the  d -quasi unit disk graph model. Another approach in this group [Wang et al. 2006]also assumes that the distances among nodes can be approximated reasonablywell, based on the shortest path length, and requires the lowest density of allapproaches of this group with a node degree of 10 to 16. Such node densitiesare realistic for dense deployments [Blough et al. 2003].The last group of approaches does not constrain the node distribution ormakeassumptionsregardingnodedensitybutonlysetsconstraintsontheradiomodel of the sensor nodes. The unit disk graph model is a weak approximationof the properties of the wireless radio. Therefore the more general  d -quasi unitdisk graph model is preferable [Schmid and Wattenhofer 2006]. The approachpresentedinKr¨olleretal.[2006]istheonlyworksofarthattacklestheproblemof boundary recognition based on the single assumption that the input networkfollows a  d -quasi unit disk graph for a given  d ≥ √  2   2  . The algorithm searches forseveral types of patterns, so called “flowers,” which are further extended andmerged in the augmenting phase of the algorithm to form a boundary of thenetwork. However, the presented flowers are extremely comple and, in randomtopologies, they only exist with a high probability if the average node degreeis very high (20–30) [Kr¨oller et al. 2006; Fekete and Kr¨oller 2006]. Moreover,a sensor node requires knowledge of its 8-hop neighborhood to be able to startsearching for a flower. Evaluation results showed that this algorithm did notfind a flower for network topologies with an average node degree smaller orequal to 10 [Wang et al. 2006].The approach presented in this article belongs to the last group and intro-duces the concept of patterns—generalization of “flowers”—that are generic,  ACM Transactions on Sensor Networks, Vol. 6, No. 3, Article 20, Publication date: June 2010.
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