Maximal Cliques in UDG: Polynomial Approximation

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Maximal Cliques in UDG Polynomial Approximation

• Rajarshi Gupta, Jean Walrand
• Dept of EECS, UC Berkeley
• Olivier Goldschmidt, OPNET Technologies
• International Network Optimization Conference
  (INOC 2005)
• Lisbon, Portugal, March 2005

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Unit Disk Graph

• Geometric graph on a plane
• Two vertices are connected iff their Euclidean
  distance is ? 1
• Common application in wireless networks

3
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

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Cliques

• Cliques in a graph
• Clique Complete Subgraph
• Maximal Clique is not a subset of any other
  clique

• Capacity and cliques
• Clique Set of nodes that all interfere with
  each other
• Observe cliques in wireless graphs are local
  structures

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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

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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

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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

• Disk Duv forms a clique

• Curved Triangles T1uv T2uv form cliques

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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 lt duv
• Hence all these vertices also lie in this band
• Note Band Buv may include extra vertices. Hence
  approx algo.

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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(m?2)
• 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
• else
• output Buv by positioning band at each
  vertex in Fuv

Algorithm is localized and distributed
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Modified Algorithm

• Consider shapes D1uv, T11uv, 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 T11uv or T21uv
• Worst case running time same, but improves
  average case

Modifications if duv ? ?3 - 1 cliques T11uv,
T21uv enough else if all vertices in D1uv,
T11uv, or T21uv we are done else use bands
Buv as before
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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 edge
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Clique Computation Methods

• Four methods of computing
• d lt 1/?3
• d lt ?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

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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(m?2)
• 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)

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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(m?2) time, generates O(m?) cliques
• Distributed, localized and polynomial algorithm

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Questions ?

• http//www.eecs.berkeley.edu/guptar
• guptar_at_eecs.berkeley.edu