L. Li, J. Y. Halpern Cornell University

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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 Presented by Lu-chuan Kung For CS598hou Sp2006 University of Illinois at Urbana-Champaign

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

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

Example of No Topology Control with Smaller Transmission Radius Network may partition

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

Basic Cone-Based Algorithm 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 with “Ack”. Each node u increases power until each cone of degree αcontains a node, or u transmits with maximum power Who should be my neighbor in the graph? What is my transmission power?

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

Notation: 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- )

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

Properties of the Basic Algorithm Counter-example for  = 150 + 

Counter-example for  = 150 + 

Counter-example for  = 150 + 

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

Connectivity Lemma Sketch of Proof B A z z is in Nα(B) with minimal Angle(z,B,A) Case 1: Angle(z,B,A) < 60° Then d(A,z) < d(A,B), therefore the Lemma holds ( W=A, Y=z )

B A z y w x Case 2: Angle(z,B,A) > 60° Must exist y such that Angle(z,B,y) <= α Similarly there exists w and x st Angle(w,A,x) <= α Then either d(w,z) < d(A,B) or d(x,y) < d(A,B) Q.E.D

Connectivity Theorem Order the edges in ER by length and induction on the rank in the ordering For every edge in ER, there’s a corresponding path in G+ . If   150, then G+ preserves the connectivity of GR and the bound is tight.

Optimizations Shrink-back operation Asymmetric edge removal “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 e2 B e1  A e2 C

Properties of Asymmetric Edge Removal Counterexample for  = 120 + 

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).

Asymmetric Edge Theorem Two-step inductions on ER and then on E For every edge in ER , if it becomes an asymmetric edge in G , then there’s a corresponding path consisting of only symmetric edges. If   120, then G- preserves the connectivity of GR and the bound is tight.

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

Average Radius

Average Node Degree

Comparison with Other TC SMECN: small minimum-energy communication network (requires location information)

Reconfiguration In response to mobility, failures, and node additions Based on Neighbor Discovery Protocol (NDP) beacons Joinu(v) event: may allow shrink-back Leaveu(v) event: may resume “Hello” protocol AngleChangeu(v) event: may allow shrink-back or resume “Hello” protocol Careful selection of beacon power

Summary 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

Comments The requirement to measure angle-of-arrival is not practical CBTC doesn’t work better than distance-based topology control

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

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

The Why-150 Lemma 150 = 90 + 60