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

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

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

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

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

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

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]

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.

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

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

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 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.

The Why-150 Lemma 150 =

Counterexample for = Properties of the Basic Algorithm

Counterexample for = 150 +

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).

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.

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

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 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.

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

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

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