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Approximating Maximal Cliques in Ad-Hoc Networks

Approximating Maximal Cliques in Ad-Hoc Networks. Rajarshi Gupta and Jean Walrand {g uptar, 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. Motivation.

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Approximating 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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