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Efficient Approximation of Maximal Cliques in Ad-Hoc Networks

This research paper presents an algorithm to approximate maximal cliques in ad-hoc networks efficiently. By utilizing clique structures in conflict graphs, the algorithm generates all cliques around a link and heuristically shrinks them to maximal cliques. The running time depends on the chosen step size, impacting miss probability. Simulations show the algorithm's effectiveness over various node densities and network areas.

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Efficient Approximation of Maximal Cliques in Ad-Hoc Networks

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  1. Approximating Maximal Cliques in Ad-Hoc Networks Rajarshi Gupta and Jean Walrand{guptar,wlr}@eecs.berkeley.eduwww.eecs.berkeley.edu/~wlr Research funded in part by DARPA PIMRC 2004 - Barcelona, Spain, Sep 6 2004 Department of Electrical Engineering and Computer Sciences

  2. 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” PIMRC 2004

  3. Models interference in ad-hoc network Conflict Graph • Connectivity Graph G • Shows ad-hoc nodes • Link if nodes lie within transmission range • Conflict Graph CG • Link in connectivity graph = CG-node in CG • CG-Edge if links in G interfere with each other PIMRC 2004

  4. Approximate the interference of a link by a circle centered at mid-point Interference range of link L S D L Interference range of S Interference range of D Representing a Link by its Center • Since Ix > Tx, the extra area is small PIMRC 2004

  5. Cliques: What • Observe • Cliques in CG are local structures • Only one node in a clique may be active at once • Definitions • Clique = Complete Subgraph • Maximal Clique = Clique not a subset of any other Maximal Cliques: ABC, BCEF, CDF PIMRC 2004

  6. 1 2 Conflict Graph 5 3 Unit Disk Graphs: Scaling of 46% suffices Graph with radius in interval [x, 1]: scaling 4 Cliques: Examples 2 nodes can transmit at a time  40% Local constraints suggest 50% Gap between local (cliques) and global Here: scaling is 80% PIMRC 2004

  7. 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 PIMRC 2004

  8. C, D in same circle of diameter Ix => d(C, D) < Ix => C, D in same clique B in same clique as A => A, B interfere => d(A, B) < Ix Ix Ix D C A B Two Key Observations • All links sharing cliques with a link must lie within a circle of radius Ix (interference range) • All links that lie within a circle of diameter Ix must form a clique PIMRC 2004

  9. 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 0 Clique(L) contains all cliques of small disks PIMRC 2004

  10. 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 PIMRC 2004

  11. 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 Missing Cliques PIMRC 2004

  12. 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 PIMRC 2004

  13. 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 PIMRC 2004

  14. 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 PIMRC 2004

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