Distributed computation of coverage in sensor networks by homological methods

Recent work on algebraic-topological methods for verifying coverage in planar sensor networks relied exclusively on centralized computation: a limiting constraint for large networks. This paper presents a distributed algorithm for homology computation over a sensor network, for purposes of verifying coverage. The techniques involve reduction and coreduction of simplicial complexes, and are of independent interest. Verification of the ensuing algorithms is proved, and simulations detail the improved network efficiency and performance.

[1]  R. Ho Algebraic Topology , 2022 .

[2]  Sonia Martínez,et al.  Coverage control for mobile sensing networks , 2002, IEEE Transactions on Robotics and Automation.

[3]  Vin de Silva,et al.  Coordinate-free Coverage in Sensor Networks with Controlled Boundaries via Homology , 2006, Int. J. Robotics Res..

[4]  Michael Robinson,et al.  Inverse problems in geometric graphs using internal measurements , 2010, 1008.2933.

[5]  Kazunori Hayashi,et al.  Mayer–Vietoris sequences and coverage problems in sensor networks , 2011 .

[6]  B. Eckmann Harmonische Funktionen und Randwertaufgaben in einem Komplex , 1944 .

[7]  Robert Ghrist,et al.  Applications of sheaf cohomology and exact sequences to network coding (諸分野との協働による数理科学のフロンティア--RIMS研究集会報告集) -- (情報・通信における符号理論と数学) , 2011 .

[8]  Sándor P. Fekete,et al.  Deterministic boundary recognition and topology extraction for large sensor networks , 2005, SODA '06.

[9]  Sriram V. Pemmaraju,et al.  Local approximation schemes for topology control , 2006, PODC '06.

[10]  Gunnar E. Carlsson,et al.  Zigzag Persistence , 2008, Found. Comput. Math..

[11]  S. I. Gelʹfand,et al.  Methods of Homological Algebra , 1996 .

[12]  Alireza Tahbaz-Salehi,et al.  Distributed Coverage Verification in Sensor Networks Without Location Information , 2008, IEEE Transactions on Automatic Control.

[13]  Gaurav S. Sukhatme,et al.  Connecting the Physical World with Pervasive Networks , 2002, IEEE Pervasive Comput..

[14]  Erin W. Chambers,et al.  Vietoris–Rips Complexes of Planar Point Sets , 2009, Discret. Comput. Geom..

[15]  Panganamala Ramana Kumar,et al.  The Number of Neighbors Needed for Connectivity of Wireless Networks , 2004, Wirel. Networks.

[16]  Michael Robinson,et al.  Asynchronous Logic Circuits and Sheaf Obstructions , 2010, GETCO.

[17]  Vin de Silva,et al.  HOMOLOGICAL SENSOR NETWORKS , 2005 .

[18]  Jennifer C. Hou,et al.  Maintaining Sensing Coverage and Connectivity in Large Sensor Networks , 2005, Ad Hoc Sens. Wirel. Networks.

[19]  Xiang-Yang Li,et al.  Coverage in wireless ad-hoc sensor networks , 2002, 2002 IEEE International Conference on Communications. Conference Proceedings. ICC 2002 (Cat. No.02CH37333).

[20]  Johannes Gehrke,et al.  Gossip-based computation of aggregate information , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

[21]  Fabian Kuhn,et al.  Ad-hoc networks beyond unit disk graphs , 2003, DIALM-POMC '03.

[22]  Dmitriy Morozov,et al.  Zigzag persistent homology and real-valued functions , 2009, SCG '09.

[23]  Marian Mrozek,et al.  Coreduction Homology Algorithm , 2009, Discret. Comput. Geom..

[24]  Marian Mrozek,et al.  Coreduction homology algorithm for inclusions and persistent homology , 2010, Comput. Math. Appl..

[25]  Miodrag Potkonjak,et al.  Exposure in wireless Ad-Hoc sensor networks , 2001, MobiCom '01.

[26]  Yasuaki Hiraoka,et al.  Applications of Sheaf Cohomology and Exact Sequences to Network Coding (Frontiers in mathematical science through collaborations with other disciplines) , 2011 .

[27]  Lali Barrière,et al.  Robust position-based routing in wireless Ad Hoc networks with unstable transmission ranges , 2001, DIALM '01.

[28]  Henri Koskinen,et al.  ON THE COVERAGE OF A RANDOM SENSOR NETWORK IN A BOUNDED DOMAIN , 2004 .

[29]  Miodrag Potkonjak,et al.  Coverage problems in wireless ad-hoc sensor networks , 2001, Proceedings IEEE INFOCOM 2001. Conference on Computer Communications. Twentieth Annual Joint Conference of the IEEE Computer and Communications Society (Cat. No.01CH37213).

[30]  Erin W. Chambers,et al.  Rips Complexes of Planar Point Sets , 2007, 0712.0395.

[31]  Vin de Silva,et al.  Coverage in sensor networks via persistent homology , 2007 .

[32]  A. Jadbabaie,et al.  Decentralized Computation of Homology Groups in Networks by Gossip , 2007, 2007 American Control Conference.

[33]  Robert G. Gallager,et al.  A new distributed algorithm to find breadth first search trees , 1987, IEEE Trans. Inf. Theory.

[34]  Magnus Egerstedt,et al.  Control Using Higher Order Laplacians in Network Topologies , 2006 .

[35]  Yasuaki Hiraoka,et al.  Applications of Sheaf Cohomology and Exact Sequences on Network Codings , 2011 .