1 k-clustering in wireless ad hoc networks using local search rachel ben-eliyahu-zohary jce and bgu...
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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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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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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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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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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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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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k
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kkk
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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In general, number of nodes in a maximal cluster:• If K is even,
• If k is odd,
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)12(21)21(212/
1
2/
1
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kikikkk
i
k
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k
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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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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