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Maximal Cliques in UDG: Polynomial Approximation

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. Unit Disk Graph. Geometric graph on a plane

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Maximal Cliques in UDG: Polynomial Approximation

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  1. 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

  2. Unit Disk Graph • Geometric graph on a plane • Two vertices are connected iff their Euclidean distance is  1 • Common application in wireless networks INOC 2005

  3. 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 C B A 1 3 D 5 4 E Connectivity Graph UDG in Wireless Networks INOC 2005

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

  5. 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 INOC 2005

  6. 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 approximationalgorithm that is • Localized,Polynomial and Distributed INOC 2005

  7. 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 Approximating Maximal Cliques • Disk Duv forms a clique • Curved Triangles T1uv & T2uv form cliques INOC 2005

  8. 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. Bands INOC 2005

  9. 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 duv 1/3 output clique Fuv; else output cliques Duv, T1uv, T2uv; if all vertices in Duv, T1uv or T2uv we are done; else output {Buv} by positioning band at each vertex in Fuv; Basic Algorithm Algorithm is localized and distributed INOC 2005

  10. Consider shapes D1uv, T11uv, 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 T11uv or T21uv Worst case running time same, but improves average case Modified Algorithm Modifications: if duv3 - 1 cliques T11uv, T21uv enough; else if all vertices in D1uv, T11uv, or T21uv we are done; else use bands {Buv} as before; INOC 2005

  11. 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 Cliques per Edge per edge INOC 2005

  12. 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 Clique Computation Methods • Left bar = Basic algorithm • Right bar = Modified algorithm INOC 2005

  13. 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) INOC 2005

  14. 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 INOC 2005

  15. Questions  http://www.eecs.berkeley.edu/~guptar guptar@eecs.berkeley.edu

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