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Analysis of a Cone-Based Distributed Topology Control Algorithm for Wireless Multi-hop Networks L. Li, J. Y. Halpern Cornell University P. Bahl, Y. M. Wang, and R. Wattenhofer Microsoft Research, Redmond

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The Aladdin Home Networking System Powerline Network Phoneline Ethernet LAN Home Gateway Alert Router IM Email Wireless Sensor Network

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OUTLINE Motivation Bigger Picture and Related Work Basic Cone-Based Algorithm –Summary of Two Main Results –Properties of the Basic Algorithm Optimizations –Properties of Asymmetric Edge Removal Performance Evaluation

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Example of No Topology Control with maximum transmission radius R (maximum connected node set) High energy consumption High interference Low throughput Motivation for Topology Control

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Network may partition Example of No Topology Control with smaller transmission radius

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Global connectivity Low energy consumption Low interference High throughput Example of Topology Control

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Bigger Picture and Related Work Routing MAC / Power-controlled MAC Selective Node Shutdown Topology Control Relative Neighborhood Graphs, Gabriel graphs, Sphere-of-Influence graphs, -graphs, etc. [GAF] [Span] [Hu 1993] [Ramanathan & Rosales- Hain 2000] [Rodoplu & Meng 1999] [Wattenhofer et al. 2001] Computational Geometry [MBH 01] [WTS 00]

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Basic Cone-Based Algorithm (INFOCOM 2001) Assumption: receiver can determine the direction of sender –Directional antenna community: Angle of Arrival problem Each node u broadcasts Hello with increasing power (radius) Each discovered neighbor v replies withAck.

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Cone-Based Algorithm with Angle Need a neighbor in every -cone. Can I stop? No! Theres an -gap!

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Notation E = { ( u, v ) V x V: v is a discovered neighbor by node u } –G = (V, E ) –E may not be symmetric (B,A) in E but (A,B) not in E

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Two symmetric sets E + = { ( u, v ): ( u, v ) E or ( v, u ) E } –Symmetric closure of E –G + = (V, E + ) E - = { ( u, v ): ( u, v ) E and ( v, u ) E } –Asymmetric edge removal –G - = (V, E - )

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Summary of Two Main Results Let G R = (V, E R ), E R = { ( u, v ): d( u, v ) R } Connectivity Theorem –If 150, then G + preserves the connectivity of G R and the bound is tight. Asymmetric Edge Theorem –If 120, then G - preserves the connectivity of G R and the bound is tight.

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The Why-150 Lemma 150 = 90 + 60

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Counterexample for = 150 + Properties of the Basic Algorithm

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Counterexample for = 150 +

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For 150 ( 5 /6 ) Connectivity Lemma –if d(A,B) = d R and (A,B) E +, there must be a pair of nodes, one red and one green, with distance less than d(A,B).

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Connectivity Theorem Order the edges in E R by length and induction on the rank in the ordering –For every edge in E R, theres a corresponding path in G +. If 150, then G + preserves the connectivity of G R and the bound is tight.

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Optimizations Shrink-back operation –Boundary nodes can shrink radius as long as not reducing cone coverage Asymmetric edge removal –If 120, remove all asymmetric edges Pairwise edge removal –If < 60, remove longer edge e 2 e1e1 e2e2 A B C

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Properties of Asymmetric Edge Removal Counterexample for = 120 +

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For 120 ( 2 /3 ) Asymmetric Edge Lemma –if d(A,B) R and (A,B) E, there must be a pair of nodes, W or X and node B, with distance less than d(A,B).

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Asymmetric Edge Theorem Two-step inductions on E R and then on E –For every edge in E R, if it becomes an asymmetric edge in G, then theres a corresponding path consisting of only symmetric edges. If 120, then G - preserves the connectivity of G R and the bound is tight.

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Performance Evaluation Simulation Setup –100 nodes randomly placed on a 1500m-by-1500m grid. Each node has a maximum transmission radius 500m. Performance Metrics –Average Radius –Average Node Degree

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Average Radius

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Average Node Degree

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In response to mobility, failures, and node additions Based on Neighbor Discovery Protocol (NDP) beacons –Join u (v) event: may allow shrink-back –Leave u (v) event: may resume Hello protocol –AngleChange u (v) event: may allow shrink-back or resume Hello protocol Careful selection of beacon power Reconfiguration

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Distributed cone-based topology control algorithm that achieves maximum connected node set –If we treat all edges as bi-directional 150-degree tight upper bound –If we remove all unidirectional edges 120-degree tight upper bound Simulation results show that average radius and node degree can be significantly reduced Summary

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Min Song 1, Yanxiao Zhao 1, Jun Wang 1, E. K. Park 2 1 Old Dominion University, USA 2 University of Missouri at Kansas City, USA IEEE ICC 2009 A High Throughput.

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