analysis of a cone-based distributed topology control algorithm for wireless multi-hop networks l....

Analysis of a Cone-Based Distributed Topology Control

Algorithm for Wireless Multi-hop Networks

L. Li, J. Y. HalpernCornell University

P. Bahl, Y. M. Wang, and R. WattenhoferMicrosoft Research, Redmond

The Aladdin Home Networking System

PowerlineNetwork

PhonelineEthernet

LAN

HomeGateway

AlertRouter

IM

Email

WirelessSensorNetwork

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

SelectiveNode

Shutdown

TopologyControl

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]

ComputationalGeometry

[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 with “Ack”.

Cone-Based Algorithm with Angle

Need a neighbor in every -cone.

Can I stop?

No! There’s 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

R A B 70

60

50

= 145

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.

The Why-150 Lemma

150 = 90 + 60

Both circles have max radius R

A

N

B

• Counterexample for = 150 +

Properties of the Basic Algorithm

Both circles have max radius R

A

W

N

K

J

B

Y

WAN = 150 WAK = 150

• Counterexample for = 150 +

Both circles have max radius R

A

N

B W

K

J

Y

WAN = 150 WAK = 150 Z

X 150 < WAX < α

d(A,X) < R < d(X,B)

• 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).

A B W

Y

Z

X

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– “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

e1

e2

A B

C

Properties of Asymmetric Edge Removal

• Counterexample for = 120 +

R A B

60+/3

60

60-/3

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).

A B

W

X

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

0

100

200

300

400

500

600

Basic With opt1 Withopt1&2

With allopts

Ave

rag

e ra

diu

s

Max power

150-deg

120-deg

Average Node Degree

0

5

10

15

20

25

30

Basic With opt1 Withopt1&2

With allopts

Ave

rag

e n

od

e d

egre

e

Max power

150-deg

120-deg

• 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

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