Geodesic
Coverage in sensor networks via persistent homology
de Silva, Vin1 ; Ghrist, Robert2
1Department of Mathematics and Computer Science, Pomona College, Claremont CA 91711, USA
2Department of Mathematics and Coordinated Sciences Laboratory, University of Illinois, Urbana, IL 61801, USA
Algebraic and Geometric Topology, Tome 7 (2007) no. 1, pp. 339-358
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We introduce a topological approach to a problem of covering a region in Euclidean space by balls of fixed radius at unknown locations (this problem being motivated by sensor networks with minimal sensing capabilities). In particular, we give a homological criterion to rigorously guarantee that a collection of balls covers a bounded domain based on the homology of a certain simplicial pair. This pair of (Vietoris–Rips) complexes is derived from graphs representing a coarse form of distance estimation between nodes and a proximity sensor for the boundary of the domain. The methods we introduce come from persistent homology theory and are applicable to nonlocalized sensor networks with ad hoc wireless communications.

DOI : 10.2140/agt.2007.7.339
Keywords: Rips complex, Cech complex, persistent homology, sensor network, coverage

de Silva, Vin  1   ; Ghrist, Robert  2

1 Department of Mathematics and Computer Science, Pomona College, Claremont CA 91711, USA
2 Department of Mathematics and Coordinated Sciences Laboratory, University of Illinois, Urbana, IL 61801, USA 
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de Silva, Vin; Ghrist, Robert. Coverage in sensor networks via persistent homology. Algebraic and Geometric Topology, Tome 7 (2007) no. 1, pp. 339-358. doi: 10.2140/agt.2007.7.339

[1] M Allili, K Mischaikow, A Tannenbaum, Cubical homology and the topological classification of 2D and 3D imagery, from: "IEEE Intl. Conf. Image Proc." (2001) 173

[2] A Ames, A homology theory for hybrid systems: hybrid homology, Lect. Notes in Computer Science 3414 (2005) 86

[3] R Bott, L W Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics 82, Springer (1982)

[4] V De Silva, Plex

[5] V D Silva, G Carlsson, Topological estimation using witness complexes, Symp. Point-Based Graphics (2004)

[6] V D Silva, R Ghrist, Coordinate-free coverage in sensor networks with controlled boundaries, Int. J. Robotics Research 25 (2006) 1205

[7] V D Silva, R Ghrist, A Muhammad, Blind swarms for coverage in 2–d, from: "Robotics: Systems and Science" (2005)

[8] J Eckhoff, Helly, Radon, and Carathéodory type theorems, from: "Handbook of convex geometry, Vol. A, B", North-Holland (1993) 389

[9] H Edelsbrunner, D Letscher, A Zomorodian, Topological persistence and simplification, from: "41st Annual Symposium on Foundations of Computer Science (Redondo Beach, CA, 2000)", IEEE Comput. Soc. Press (2000) 454

[10] S Fekete, A Kröller, D Pfisterer, S Fischer, Deterministic boundary recognition and topology extraction for large sensor networks (2006)

[11] M Gromov, Hyperbolic groups, from: "Essays in group theory", Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75

[12] A Hatcher, Algebraic topology, Cambridge University Press (2002)

[13] J C Hausmann, On the Vietoris–Rips complexes and a cohomology theory for metric spaces, from: "Prospects in topology (Princeton, NJ, 1994)", Ann. of Math. Stud. 138, Princeton Univ. Press (1995) 175

[14] T Kaczynski, K Mischaikow, M Mrozek, Computational homology, Applied Mathematical Sciences 157, Springer (2004)

[15] H Koskinen, On the coverage of a random sensor network in a bounded domain, from: "Proceedings of 16th ITC Specialist Seminar" (2004) 11

[16] X Y Li, P J Wan, O Frieder, Coverage in wireless ad-hoc sensor networks, IEEE Transaction on Computers 52 (2003) 753

[17] B Liu, D Towsley, A study of the coverage of large-scale sensor networks, from: "IEEE International Conference on Mobile Ad-hoc and Sensor Systems" (2004)

[18] S Meguerdichian, F Koushanfar, M Potkonjak, M Srivastava, Coverage problems in wireless ad-hoc sensor network, from: "IEEE INFOCOM" (2001) 1380

[19] K Mischaikow, M Mrozek, J Reiss, A Szymczak, Construction of symbolic dynamics from experimental time series, Phys. Rev. Lett. 82 (1999) 1144

[20] A Rao, C Papadimitriou, S Shenker, I Stoica, Geographic Routing without Location Information, from: "Proceedings of 9th Annual International Conference on Mobile Computing and Networking (Mobicom'03)" (2003)

[21] L Vietoris, Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann. 97 (1927) 454

[22] F Xue, P R Kumar, The number of neighbors needed for connectivity of wireless networks, Wireless Networks 10 (2004) 169

[23] H Zhang, J Hou, Maintaining Coverage and Connectivity in Large Sensor Networks, from: "International Workshop on Theoretical and Algorithmic Aspects of Sensor, Ad hoc Wireless and Peer-to-Peer Networks" (2004)

[24] A Zomorodian, G Carlsson, Computing persistent homology, Discrete Comput. Geom. 33 (2005) 249

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