maximal cliques in udg: polynomial approximation rajarshi gupta, jean walrand dept of eecs, uc...
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Maximal Cliques in UDG: Polynomial Approximation
Rajarshi Gupta, Jean WalrandDept of EECS, UC Berkeley
Olivier Goldschmidt, OPNET Technologies
International Network Optimization Conference (INOC 2005)
Lisbon, Portugal, March 2005
EECS, UC Berkeley INOC 2005
Unit Disk Graph
Geometric graph on a plane
Two vertices are connected iff their Euclidean distance is 1
Common application in wireless networks
EECS, UC Berkeley INOC 2005
UDG in Wireless Networks 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
31
45
A
CB
E
D
Connectivity Graph
EECS, UC Berkeley INOC 2005
Cliques
Capacity and cliques Clique = Set of nodes
that all interfere with each other
Observe: cliques in wireless graphs are local structures
A
B C
E F
D
Cliques in a graph Clique = Complete
Subgraph Maximal Clique is not a
subset of any other clique
Maximal Cliques:
ABC, BCEF, CDF
EECS, UC Berkeley INOC 2005
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
EECS, UC Berkeley INOC 2005
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 approximation algorithm that is
Localized, Polynomial and Distributed
EECS, UC Berkeley INOC 2005
Approximating Maximal Cliques
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
Fuv
Duv
u v
T1uv
duv
Fuv
Duv
u vduv
Fuv
u vduvu vduv
Disk Duv forms a clique
Curved Triangles T1uv & T2uv form cliques
EECS, UC Berkeley INOC 2005
Bands
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.
u v
Fuv
duv
Buv
duv u vht = duv
duv
x
yBandBuv
> duv
duv
EECS, UC Berkeley INOC 2005
Basic Algorithm 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(m2)
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;
elseoutput {Buv} by positioning band at each vertex in Fuv;
Algorithm is localized and distributed
EECS, UC Berkeley INOC 2005
Modified Algorithm
Consider shapes D1uv, T11
uv, 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 T11
uv or T21uv
Worst case running time same, but improves average case
1
duv
T1uv1
Duv1 T1uv
1
overlap
T2uv1
Duv
Modifications:if duv 3 - 1
cliques T11uv, T21
uv enough;else
if all vertices in D1uv, T11
uv, or T21
uv
we are done;else
use bands {Buv} as before;
EECS, UC Berkeley INOC 2005
Cliques per Edge 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
per
ed
ge
EECS, UC Berkeley INOC 2005
Clique Computation Methods
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
Left bar = Basic algorithm Right bar = Modified
algorithm
EECS, UC Berkeley INOC 2005
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(m2)
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)
EECS, UC Berkeley INOC 2005
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(m2) time, generates O(m) cliques Distributed, localized and polynomial algorithm