Approximating Maximal Cliques in Ad-Hoc Networks
Rajarshi Gupta and Jean Walrand{guptar, 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
EECS, U C Berkeley PIMRC 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”
EECS, U C Berkeley PIMRC 2004
Conflict Graph Models interference in
ad-hoc network
2
31
45
A
CB
E
Dinterference
Connectivity Graph
E
CD
B
A
Conflict Graph
Connectivity Graph GShows ad-hoc nodesLink if nodes lie within transmission range
Conflict Graph CGLink in connectivity graph = CG-node in CGCG-Edge if links in G interfere with each other
EECS, U C Berkeley PIMRC 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
Interferencerange of D
L
Interferencerange of link L
EECS, U C Berkeley PIMRC 2004
Cliques: What
Observe Cliques in CG are local
structures Only one node in a
clique may be active at once
A
B C
E F
D
Maximal Cliques:
ABC, BCEF, CDF
Definitions Clique = Complete
Subgraph Maximal Clique =
Clique not a subset of any other
EECS, U C Berkeley PIMRC 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%
1 2
3
4
5Conflict 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
EECS, U C Berkeley PIMRC 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
EECS, U C Berkeley PIMRC 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
CD
C, D in same circle ofdiameter 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
EECS, U C Berkeley PIMRC 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 0Clique(L) contains all cliques of small disks
Clique(L) is subset of Circle 0Clique(L) contains all cliques of small disks
EECS, U C Berkeley PIMRC 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
EECS, U C Berkeley PIMRC 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
r-xx
InitialPosition
NextPosition
IntermediatePosition
14
3
2
6
5
EECS, U C Berkeley PIMRC 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
EECS, U C Berkeley PIMRC 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
EECS, U C Berkeley PIMRC 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