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K-clustering in Wireless Ad Hoc Networks

using local search

Rachel Ben-Eliyahu-ZoharyJCE and BGU

Joint work with Ran Giladi (BGU) and Stuart Sheiber and Philip Hendrix (Harvard)

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Cluster-based Routing Protocol

The network is divided to non overlapping sub-networks (clusters) with bounded diameter.

• Intra-cluster routing: pro-actively maintain state information for links within the cluster.

• Inter-cluster routing: use a route discovery protocol for determining routes. Route requests are propagated via peripheral nodes.

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Cluster-based Routing Protocol

+ Limit the amount of routing information stored and maintained at individual hosts.

+ Clusters are manageable. Node mobility

events are handled locally within the clusters. Hence, far-reaching effects of topological changes are minimized.

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Cluster-heads

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CH Denote Cluster-heads

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K-Clustering

• Topology: the objective is to partition the network into minimum number of sub networks (clusters) with bounded diameter, k.

• A more symmetric topology than cluster heads.

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Problem Statement

• Minimum k-clustering: given a graph G = (V,E) and a positive integer k, find the smallest value of ƒ such that there is a partition of V into ƒ disjoint subsets V1,…,Vƒ and diam(G[Vi]) <= k for i = 1…ƒ.

• The algorithmic complexity of k-clustering is known to be NP-complete for simple undirected graphs.

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K-clustering K = 3

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System Model

Two general assumptions regarding the state of the network’s communication links and topology:

1. The network may be modeled as an unit disk graph.

2. The network topology remains unchanged throughout the execution of the algorithm.

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Unit Disk Graph

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The distance between adjacent nodes <= 2

The distance between non adjacent nodes is > 2

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Contribution of Fernandess and

Malkhi

A two phase distributed asynchronous polynomial approximation for k-clustering where k > 1 that has a competitive worst case ratio of O(k):

First phase – constructs a spanning tree of the network.

Second phase – partitions the spanning tree into sub-trees with bounded diameter.

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Second Phase: K-sub-treeGiven a tree T=(V,E) the algorithm finds a sub-tree

whose diameter exceeds k, it then detaches the highest child of the sub-tree and repeats over on the reduced tree.

k

k- r

detach highest sub-tree

root of the sub-tree

sub-treesub-tree

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Random DecentRANDOM_DESCENT(problem,terminate)returns solution stateinputs: problem, a problem

termination condition, a condition for stoppinglocal vars: current, a solution state

next, a solution state

current ← Initial State (problem(

while (not terminate) next ← a selected neighbor of current

∆ E← Value(next) - Value(current) if ∆ E <0 then current ←next

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Initial State

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Initial State (k=2)

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K is even (e.g. 2)

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K = 2 (cont.)

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K = 2 (cont.)

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K = 2 (cont.)

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K = 2 (cont.)

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K = 2 Total: 8 clusters

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A better State

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Random DecentRANDOM_DESCENT(problem,terminate)returns solution stateinputs: problem, a problem

termination condition, a condition for stoppinglocal vars: current, a solution state

next, a solution state

current ← Initial State (problem(

while (not terminate) next ← a selected neighbor of current

∆ E← Value(next) - Value(current) if ∆ E <0 then current ←next

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Building the neighbor

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Building the neighbor

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Building the neighbor

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Building the neighbor

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Building the neighbor

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Building the neighbor

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For Odd k:

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K is odd (e.g. 3)

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K is odd (e.g. 3)

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Random DecentRANDOM_DESCENT(problem,terminate)returns solution stateinputs: problem, a problem

termination condition, a condition for stoppinglocal vars: current, a solution state

next, a solution state

current ← Initial State (problem(

while (not terminate) next ← a selected neighbor of current

∆ E← Value(next) - Value(current) if ∆ E <0 then current ←next

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Experimental Evaluation

• Randomly Generated Graphs

• Grid Graphs

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Randomly Generated Graphs

• Parameters: – n – number of nodes– l – length of a unit

• Graph Generation: - n points are placed randomly on a 1X1

square- two vertices are connected iff the distance between them is less than l.

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400 nodes, k=5

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Results

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Experiments on Grids

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Theorem: The number of nodes in a maximal cluster in a greed:

• If K is even,

• If k is odd,

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)12(21)21(212/

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2/

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kikikkk

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kikikk kk

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)1)(1(21221)21(21

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e.g. 13 if k=4

e.g. 8 if k=3

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A maximal cluster on grid

x+y=r

x+y=r+k

x-y=s

x-y=s-k

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A maximal cluster on grid

x+y=r

x+y=r+k

x-y=s

x-y=s-k

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A maximal cluster – k is even

x+y=r

x+y=r+4

x-y=s

x-y=s-4

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A maximal cluster – k is odd

x+y=r

x+y=r+3

x-y=s

x-y=s-3

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In general, number of nodes in a maximal cluster:• If K is even,

• If k is odd,

12

)12(21)21(212/

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2/

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kikikkk

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k

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kkk

kikikk kk

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k

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)1)(1(21221)21(21

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e.g. 13 if k=4

e.g. 8 if k=3

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Optimal Clustering for k=4

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Optimal Clustering for k=3

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Related Work

• Local search techniques were used for network partitioning

• Simulated annealing and genetic algorithms

• Was tested on a very limited network size : 20-60 nodes.

• We present solid criteria for evaluating the local search

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Conclusions

• A new local search algorithm for k-clustering was introduced

• It outperforms existing distributed algorithm for large k and dense networks.

• Grids can be built using optimal clustering

• Clustering on grids needs improvement.

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Future Work

• Check a distributed version of local search

• Change the algorithm for local search• Find an efficient way to fix a solution –

e.g. by merging small clusters• Use local search for other optimization

problems in networking