Maximal Cliques in UDG: Polynomial Approximation

Rajarshi Gupta, Jean WalrandDept of EECS, UC Berkeley

Olivier Goldschmidt, OPNET Technologies

International Network Optimization Conference (INOC 2005)

Lisbon, Portugal, March 2005

EECS, UC Berkeley INOC 2005

Unit Disk Graph

Geometric graph on a plane

Two vertices are connected iff their Euclidean distance is 1

Common application in wireless networks

EECS, UC Berkeley INOC 2005

UDG in Wireless Networks Wireless nodes are

connected if they are within a transmission radius

Assume all nodes have same transmission power

Then underlying graph model is UDG

2

31

45

A

CB

E

D

Connectivity Graph

EECS, UC Berkeley INOC 2005

Cliques

Capacity and cliques Clique = Set of nodes

that all interfere with each other

Observe: cliques in wireless graphs are local structures

A

B C

E F

D

Cliques in a graph Clique = Complete

Subgraph Maximal Clique is not a

subset of any other clique

Maximal Cliques:

ABC, BCEF, CDF

EECS, UC Berkeley INOC 2005

Problem Formulation

Given UDG on a plane Each vertex knows its position Also knows position of neighbors

Want to compute all maximal cliques in the network

EECS, UC Berkeley INOC 2005

General Clique Algorithms Well known problem in Graph Theory

Harary, Ross [1957] Bierstone [1960s] Bron, Kerbosch [1973]

Given any graph G=(V,E), generate all maximal cliques

Exponential number of maximal cliques in general graph

So these algorithms are exponential and centralized

Exponential number of maximal cliques even in UDG Hence want approximation algorithm that is

Localized, Polynomial and Distributed

EECS, UC Berkeley INOC 2005

Approximating Maximal Cliques

For each edge uv in UDG Length of edge uv =

duv

Output all cliques with edges duv

This will output all maximal cliques Football Fuv contains all

cliques

Fuv

Duv

u v

T1uv

duv

Fuv

Duv

u vduv

Fuv

u vduvu vduv

Disk Duv forms a clique

Curved Triangles T1uv & T2uv form cliques

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Bands

Consider Band of height duv within Fuv

For each vertex in Fuv position a band lying on the vertex Theorem: All cliques in Fuv included in set of bands {Buv}

Consider any clique q, and let x be its vertex farthest from uv Since x is farthest, all other vertices must lie on same side of x as uv But distance from x to all other vertices < duv

Hence all these vertices also lie in this band

Note: Band Buv may include extra vertices. Hence approx algo.

u v

Fuv

duv

Buv

duv u vht = duv

duv

x

yBandBuv

> duv

duv

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Basic Algorithm For small bands, single clique

includes all vertices Else we try the three cliques

we know Need to resort to bands only

as a last resort

Takes O() to generate clique Order of algorithm = O(m2)

m = number of edges = max degree of graph

Number of cliques = O(m)

if duv 1/3

output clique Fuv;else

output cliques Duv, T1uv, T2uv;

if all vertices in Duv, T1uv or T2uv we are done;

elseoutput {Buv} by positioning band at each vertex in Fuv;

Algorithm is localized and distributed

EECS, UC Berkeley INOC 2005

Modified Algorithm

Consider shapes D1uv, T11

uv, T21uv

of dimension 1 instead of duv These form cliques that are

supersets of Duv, T1uv, T2uv

If duv 3 – 1, every band is contained in either T11

uv or T21uv

Worst case running time same, but improves average case

1

duv

T1uv1

Duv1 T1uv

1

overlap

T2uv1

Duv

Modifications:if duv 3 - 1

cliques T11uv, T21

uv enough;else

if all vertices in D1uv, T11

uv, or T21

uv

we are done;else

use bands {Buv} as before;

EECS, UC Berkeley INOC 2005

Cliques per Edge Simulation details

10X10 field 100 to 2000 nodes Each point average

over 10 simulations Observations

increases linearly with node density

No. of cliques/edge also rises linearly

Actual # cliques only 1/8 or 1/10 of m

per

ed

ge

EECS, UC Berkeley INOC 2005

Clique Computation Methods

Four methods of computing

d < 1/3 d < 3-1

(modified) D, T1 and T2 Bands

Observations More bands at

denser networks Modified

algorithm reduces reliance on bands

Left bar = Basic algorithm Right bar = Modified

algorithm

EECS, UC Berkeley INOC 2005

Changes in Network Complexity analysis

Change affects neighborhood of one/two nodes May have O(2) edges Recomputing cliques at each edge takes O(2) time Total algorithm is O(4) Note that O(2) O(m), so no worse than O(m2)

Want all changes to be handled locally and efficiently New vertex: O(4) Delete Vertex: O(m) New Edge: O(4) Delete Edge: O(4) Move Vertex: O(4)

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Conclusions Motivation

Cliques in Unit Disk Graphs common in wireless networks

Background Number of maximal cliques is exponential Rely on approximation algorithm

Algorithm Summary Consider each edge, and find all cliques with this as

the longest edge Limit clique-forming vertices into characteristic shapes Runs in O(m2) time, generates O(m) cliques Distributed, localized and polynomial algorithm

Questions

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

guptar@eecs.berkeley.edu