approximating maximal cliques in ad-hoc networks

Approximating Maximal Cliques in Ad-Hoc Networks

Rajarshi Gupta and Jean Walrand{guptar, wlr}@eecs.berkeley.edu www.eecs.berkeley.edu/~wlr

Research funded in part by DARPA

PIMRC 2004 - Barcelona, Spain, Sep 6 2004

Department of Electrical Engineering and Computer Sciences

mailto:wlr%[email protected]

http://www.eecs.berkeley.edu/~wlr

EECS, U C Berkeley PIMRC 2004

Motivation Capacity in ad-hoc networks is a crucial

issue

Many approaches Information Theoretic Stochastic Graph Theoretic

Makes use of “clique” structures in “conflict graph”


Conflict Graph Models interference in

ad-hoc network

2

31

45

A

CB

E

Dinterference

Connectivity Graph

E

CD

B

A

Conflict Graph

Connectivity Graph GShows ad-hoc nodesLink if nodes lie within transmission range

Conflict Graph CGLink in connectivity graph = CG-node in CGCG-Edge if links in G interfere with each other


Representing a Link by its Center

Approximate the interference of a link by a circle centered at mid-point S

Since Ix > Tx, the extra area is small Interference

range of S

D

Interferencerange of D

L

Interferencerange of link L


Cliques: What

Observe Cliques in CG are local

structures Only one node in a

clique may be active at once

A

B C

E F

D

Maximal Cliques:

ABC, BCEF, CDF

Definitions Clique = Complete

Subgraph Maximal Clique =

Clique not a subset of any other


Cliques: Examples

2 nodes can transmit at a time 40%

Local constraints suggest 50%

Gap between local (cliques) and global

Here: scaling is 80%

2 nodes can transmit at a time 40%

Local constraints suggest 50%

Gap between local (cliques) and global

Here: scaling is 80%

1 2

3

4

5Conflict Graph

Unit Disk Graphs: Scaling of 46% suffices

Graph with radius in interval [x, 1]: scaling

Unit Disk Graphs: Scaling of 46% suffices

Graph with radius in interval [x, 1]: scaling


Cliques: Why and How Cliques in Ad-Hoc Networks

Puri (2002) – optimized traffic flows Jain et. al. (2003) – upper bound on ad-hoc

capacity Xue et. al. (2003) – clique-based pricing

General algorithms to compute cliques are centralized and exponential Harary, Ross (1957) Bierstone and Augustson et. al. (1960s) Bron, Kerbosch (1973)

We propose computationally simple heuristic approximation for unit-disk graphs


Two Key Observations All links sharing cliques with a link must lie

within a circle of radius Ix (interference range)

A

B

B in same clique as A => A, B interfere => d(A, B) < Ix

Ix

CD

C, D in same circle ofdiameter Ix => d(C, D) < Ix => C, D in same clique

Ix

All links that lie within a circle of diameter Ix must form a clique


Approximate Clique Algorithm

Use a disk of radius Ix/2 to scan a disk of radius Ix around link

Each position of scanning disk generates a clique

Move scanning disk in radial co-ordinate to avoid discontinuous jumps

Running time of algorithm depends on step size r

Clique(L) is subset of Circle 0Clique(L) contains all cliques of small disks

Clique(L) is subset of Circle 0Clique(L) contains all cliques of small disks


Shrink to Maximal Cliques Heuristically shrink set of cliques

Only remember one previous clique If newClique oldClique, discard newClique If oldClique newClique, overwrite oldClique Else save oldClique and remember newClique

Can further shrink to set of maximal cliques Brute force check against all remaining cliques Works on a much smaller set – hence quicker


Missing Cliques If step size r is too

large, might miss an intermediate clique Clique 1 = {1,2,3,4} Clique 2 = {3,4,5,6} Missed Clique = {2,3,4,5}

Worst probability of loss =

N = # of CG-nodes , where A =

area

r-xx

InitialPosition

NextPosition

IntermediatePosition

14

3

2

6

5


Expanded Scanning Disk Can ensure no cliques are lost

Use scanning disk of radius Covers area between two positions of scanning disk Generated clique may be super-maximal Used in simulations

Effect of approximation Number of cliques is exponential in general In such cases, our algorithm generates fewer

cliques, but they are super-maximal Ok for capacity purposes, since this is more

conservative


Computation Times Time taken to generate cliques that the link belongs to

~1 sec to get heuristically shrunk set of cliques <15 sec to shrink to set of maximal cliques


Conclusion Cliques in CG often used in ad-hoc networks Propose approximate algorithm

Generates all cliques around a link Heuristically shrinks set to maximal cliques

Analysis Running time depends only on chosen step size Effect of step size in miss probability

Simulation Over various node densities and network area Can generate all maximal cliques quickly

approximating maximal cliques in ad-hoc networks

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