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Using local geometry for Topology Construction in Wireless Sensor Networks

Using local geometry for Topology Construction in Wireless Sensor Networks . Sameera Poduri Robotic Embedded Systems Lab(RESL) http://robotics.usc.edu/resl University of Southern California Joint work with Prof. Gaurav Sukhatme (RESL, USC),

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Using local geometry for Topology Construction in Wireless Sensor Networks

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  1. Using local geometry for Topology Construction in Wireless Sensor Networks Sameera Poduri Robotic Embedded Systems Lab(RESL) http://robotics.usc.edu/resl University of Southern California Joint work with Prof. Gaurav Sukhatme (RESL, USC), Sundeep Pattem & Prof. Bhaskar Krishnamachari (ANRG, USC)

  2. Motivation Different Coverage & Connectivity requirements local control, global requirements 2/42

  3. Problem • Given a set of nodes, construct an efficient topology • Control instruments • Power control • Sleep scheduling • Position control Local conditions that influence global network properties 3/42

  4. Approach • What are the desirable properties? (global/local?) • What topologies have these properties? • Can they be constructed with local rules? • How can we design deployment algorithms to implement these rules? 4/42

  5. Talk Outline • Network properties • Proximity graphs • Local rules for construction • Neighbor-Every-Theta graphs • Connectivity Properties • Coverage optimization • Deployment Algorithms • Results • Related Work • Summary & Future directions 5/42

  6. Construction Rules Model • Communication • binary disk • Different communication ranges • Coverage • binary disk • Nodes can sense the angle and distance of neighbors • Very large network • No localization/GPS 6/42

  7. Network Properties • Connectivity • Coverage • Sparseness • Degree • Spanner Ratio 7/42

  8. Network Properties - 1 Connectivity • 0/1 : Path between any two given nodes • “degree” of connectivity (k-connectivity) • Path Connectivity = minimum (vertex disjoint) paths between any two given nodes • Vertex Connectivity = minimum vertices to disconnect the network • Edge Connectivity = minimum edges to disconnect the network Menger’s Thm 8/42

  9. Network Properties - 2 Coverage • Net area “sensed” Degree • # neighbors Sparseness • #edges = O(#nodes) • Spanner • efficiency of paths • , c = spanner ratio 9/42

  10. Proximity Graphs • Encode spatial arrangement of nodes. • Can model network communication graph • Popular graphs • Minimum Spanning Tree (MST) • Relative Neighborhood Graph (RNG) • Gabriel Graph (GG) • Delaunay Graph (DG) • Yao Graph (YG) 10/42

  11. Proximity Graphs Properties • All are connected and sparse • RNG: low power consumption, low degree and good connectivity • GG & DG: optimal power spanners • GPSR derives it’s scalability from the RNG and GG (routing decisions based on local state only) • YG: low spanner 11/42

  12. Proximity Graphs Definitions RNG: No node closer to both X and Y GG: No node in the circle of minimum radius passing through X and Y DG: No node in the circumcircle of X, Y, Z θ YG(θ): No node closer than Y in θ sector 12/42

  13. Proximity Graphs Hierarchical Relationship Average degree, Connectivity 13/42

  14. Construction Rules Model • Communication • binary disk • Different communication ranges • Coverage • binary disk • Nodes can sense the angle and distance of neighbors • Very large network • No localization/GPS 14/42

  15. Construction Rules GOAL: Communication graph = Proximity graph • Problem: Comm Graph is Disk graph • (Only edges < Rc) RNG Comm. Graph 15/42

  16. Construction Rules Relative Neighborhood Graph Theorem1: If each node has at least one neighbor in every 2/3 sector around it, the communication graph is a super-graph of RNG. Y Y X 16/42

  17. Construction Rules RNG… 2/3 result - • Sufficient but not necessary • Best you can do with no global knowledge • “tight” bound 17/42

  18. Construction Rules Gabriel Graph Theorem 2: If each node has at least one neighbor in every θ = arccos(r/R)sector around it, the communication graph is a super-graph of GG. 18/42

  19. Construction Rules Delaunay Graph • Corollary : If each node has at least one neighbor in every θ = arccos(r/R)sector around it, the communication graph is a super-graph of DG. 19/42

  20. Neighbor-Every-Theta Condition NET Graph: A graph in which every node satisfies NET condition 20/42

  21. NET Graphs Connectivity of NET graph Theorem3: An infinite NET graph is at least 2/ connected for  <  #nodes > 2 Every polygon has at least 3 exterior angles >  #Edges cut  3 /  2/ #nodes = 2 #Edges cut  2 2/ - 1  k #nodes = 1 #Edges cut  2/ 21/42

  22. NET Graphs Connectivity of NET graph.. For  = , NET graph is guaranteed to be 1-connected Result by D’Souza et al. *, If each node has at least one neighbor in every  sector around it, then the graph is guaranteed to be connected. * R. M. D'Souza, D. Galvin, C. Moore, D. Randall. A local topology control algorithm guaranteeing global connectivity and greedy routing. (Working paper) 22/42

  23. NET Graphs NET graphs • Each node has at least one neighbor in every  sector • Single parameter family of graphs • Connectivity ≥ 2/ •  = 2/3 RNG 23/42

  24. NET Graphs Coverage Optimization • Suppose that a node needs k neighbors to satisfy the sector conditions for the proximity graphs • To maximize coverage from the node’s local perspective: - All neighbors must lie on the perimeter of the communication range - They should be placed symmetrically around the node 24/42

  25. NET Graphs Theorem 3 For , the area coverage is maximized when the nodes are placed at the edges of disjoint sectors of . 25/42

  26. NET Graphs Tiling Graphs • When k = 3, 4, 6, the locally optimal symmetric placement can be replicated globally • This results in Tiling graphs 26/42

  27. NET Graphs Tiling Graph properties • Globally optimal in terms of coverage • A number of other global properties: • While the RNG and GG have spanning ratios of and in general, the spatial arrangement of nodes in the tilings result in constant spanning ratios. 27/42

  28. NET Graphs Significance Traditional approaches - • Sleep Scheduling - • network is deployed with high density • Nodes decide locally whether to stay awake • Power Control - • Static & mobile ad-hoc networks • Smallest transmission power Deployment • Incremental deployment • Static nodes by a mobile agent • Distributed deployment • Self-deployment of mobile nodes 28/42

  29. Deployment Algo Incremental Deployment • Deploy nodes one at a time • Pick new position based on geometry of existing nodes, cost of travel, etc • Can be implemented for mobile nodes too • Works best when the topology is known a priori 29/42

  30. Deployment Algo Incremental Deployment - topologies Gaussian error 3o and 15% range No Error Tiling angle (/3) Non- tiling angle(2/5) 30/42

  31. Deployment Algo Distributed Deployment • Nodes make decisions independently • Potential Field Approach Algorithm • Start state • all constraints satisfied • all edges are preserved • Spread out and trim unnecessary edges 31/42

  32. Deployment Algo otherwise If edge is not required 0 (m=1) Distributed Deployment 32/42

  33. Deployment Algo Simulation • Fast • No negotiations • Conservative 33/42

  34. Deployment Algo Distributed Deployment - topologies Distributed Incremental No Error Tiling angle (/3) Non- tiling angle(2/5) 34/42

  35. Deployment Results Coverage 35/42

  36. Deployment Results Connectivity 36/42

  37. Deployment Results Degree 14 12 10 8 6 4 37/42

  38. Deployment Results Constraint Satisfaction 38/42

  39. Deployment Results Comparison with RNG Comm. graph Difference RNG 39/42

  40. Related Work • Topology Control: • X. Li’05, Santi’03 (surveys) • Power Control: • Wattenhofer’05, Brendin’05, Jennings’02, Borbash’02 • Sleep scheduling: • Zhang’05, Wang’03 • Deployment of static network by mobile agent: • Batalin’04, Corke’04 • Deployment of mobile network: • Howard’02, Cortes’04, Poduri’03 40/42

  41. Summary • NET graphs • based on purely local geometric conditions • single parameter • range of coverage-connectivity trade-offs • Applications • Power control, Sleep scheduling (dense networks) • Controlled deployment • Assumptions: • Disk model for communication (but ranges could be different) • Directional information about neighbors 41/42

  42. Extensions Relax assumptions: • Irregular communication range • Vary Rs/Rc • Formalize notion of boundary Deployment Algorithm: • Improve Sparseness • Negotiations? - Coloring • Rendezvous problem 42/42

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