2. Approximating Maximal Cliques in Ad-Hoc Networks Rajarshi Gupta and Jean Walrand {guptar, wlr}@eecs.berkeley.edu www.eecs.berkeley.edu/~wlrwlr}@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

3. EECS, U C BerkeleyPIMRC 2004 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”

4. EECS, U C BerkeleyPIMRC 2004 Conflict Graph Models interference in ad- hoc network 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

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

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

7. EECS, U C BerkeleyPIMRC 2004 Cliques: Examples 2 nodes can transmit at a time  40% Local constraints suggest 50% Gap between local (cliques) and global Here: scaling is 80% 2 nodes can transmit at a time  40% Local constraints suggest 50% Gap between local (cliques) and global Here: scaling is 80% 12 3 4 5 Conflict Graph Unit Disk Graphs: Scaling of 46% suffices Graph with radius in interval [ x, 1]: scaling Unit Disk Graphs: Scaling of 46% suffices Graph with radius in interval [ x, 1]: scaling

8. EECS, U C BerkeleyPIMRC 2004 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

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

10. EECS, U C BerkeleyPIMRC 2004 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 Clique(L) is subset of Circle 0 Clique(L) contains all cliques of small disks

11. EECS, U C BerkeleyPIMRC 2004 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

12. EECS, U C BerkeleyPIMRC 2004 Missing Cliques 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

13. EECS, U C BerkeleyPIMRC 2004 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

14. EECS, U C BerkeleyPIMRC 2004 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

15. EECS, U C BerkeleyPIMRC 2004 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