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

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

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

4/42 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?

5/42 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

6/42 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 Construction Rules

7/42 Network Properties Connectivity Coverage Sparseness Degree Spanner Ratio

8/42 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 Network Properties - 1 Menger’s Thm

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

10/42 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)

11/42 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 Proximity Graphs

12/42 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 Definitions YG(θ): No node closer than Y in θ sector θ Proximity Graphs

13/42 Hierarchical Relationship Proximity Graphs Average degree, Connectivity

14/42 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 Construction Rules

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

16/42 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. Construction Rules X Y Y

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

18/42 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. Construction Rules

19/42 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. Construction Rules

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

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

22/42 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) NET Graphs

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

24/42 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 NET Graphs

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

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

27/42 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. NET Graphs

28/42 Significance Traditional approaches - 1.Sleep Scheduling - network is deployed with high density Nodes decide locally whether to stay awake 2.Power Control - Static & mobile ad-hoc networks Smallest transmission power Deployment 1.Incremental deployment Static nodes by a mobile agent 2.Distributed deployment Self-deployment of mobile nodes NET Graphs

29/42 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 Deployment Algo

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

31/42 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 Deployment Algo

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

33/42 Simulation Fast No negotiations Conservative Deployment Algo

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

35/42 Coverage Deployment Results

36/42 Connectivity Deployment Results

37/42 Degree Deployment Results

38/42 Constraint Satisfaction Deployment Results

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

40/42 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

41/42 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

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

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