Maximal Cliques in UDG: Polynomial Approximation Rajarshi Gupta, Jean Walrand Dept of EECS, UC Berkeley Olivier Goldschmidt, OPNET Technologies International Network Optimization Conference (INOC 2005) Lisbon, Portugal, March 2005

EECS, UC BerkeleyINOC 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 BerkeleyINOC 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 A C B E D Connectivity Graph

EECS, UC BerkeleyINOC 2005 Cliques Capacity and cliques Clique = Set of nodes that all interfere with each other Observe: cliques in wireless graphs are local structures Cliques in a graph Clique = Complete Subgraph Maximal Clique is not a subset of any other clique Maximal Cliques: ABC, BCEF, CDF

EECS, UC BerkeleyINOC 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 BerkeleyINOC 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 c entralized Exponential number of maximal cliques even in UDG  Hence want approximation algorithm that is Localized, Polynomial and Distributed

EECS, UC BerkeleyINOC 2005 Approximating Maximal Cliques For each edge uv in UDG Length of edge uv = d uv Output all cliques with edges  d uv This will output all maximal cliques Football F uv contains all cliques Disk D uv forms a clique Curved Triangles T1 uv & T2 uv form cliques

EECS, UC BerkeleyINOC 2005 Bands Consider Band of height d uv within F uv For each vertex in F uv position a band lying on the vertex Theorem: All cliques in F uv included in set of bands {B uv } 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 < d uv Hence all these vertices also lie in this band Note: Band B uv may include extra vertices. Hence approx algo.

EECS, UC BerkeleyINOC 2005 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(m  2 ) m = number of edges  = max degree of graph Number of cliques = O(m  ) if d uv  1/  3 output clique F uv ; else output cliques D uv, T1 uv, T2 uv ; if all vertices in D uv, T1 uv or T2 uv we are done; else output {B uv } by positioning band at each vertex in F uv ; Algorithm is localized and distributed

EECS, UC BerkeleyINOC 2005 Modified Algorithm Consider shapes D 1 uv, T1 1 uv, T2 1 uv of dimension 1 instead of d uv These form cliques that are supersets of D uv, T1 uv, T2 uv If d uv   3 – 1, every band is contained in either T1 1 uv or T2 1 uv Worst case running time same, but improves average case Modifications: if d uv   cliques T1 1 uv, T2 1 uv enough; else if all vertices in D 1 uv, T1 1 uv, or T2 1 uv we are done; else use bands {B uv } as before;

EECS, UC BerkeleyINOC 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 edge

EECS, UC BerkeleyINOC 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 BerkeleyINOC 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(m  2 ) 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 )

EECS, UC BerkeleyINOC 2005 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(m  2 ) time, generates O(m  ) cliques Distributed, localized and polynomial algorithm

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