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K-clustering in Wireless Ad Hoc Networks

Fernandess and MalkhiHebrew University of Jerusalem

Presented by: Ashish Deopura

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Outline of the presentation

• Motivation for clustering in Mobile Ad hoc networks

• Problem Statement

• Algorithm Description

• Conclusions / Summary

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Wireless Ad-hoc networks

• Dynamic topology

• Power and bandwidth limitations

• Broadcast network

• Routing– How to determine path from source to

destination

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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.

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Cluster-based Routing Protocol (Cntd.)

• 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.

• Overcome mobility by adjusting cluster size (diameter) according to network stability.

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Cluster-heads

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IK

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B

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S E

F

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D

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CH Denote 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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Example K-clustering (for K = 3)

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Algorithm Description

• A two phase distributed algorithm for k-clustering where k > 1 that has a competitive worst case ratio of O(k)– First phase: construct a MCDS tree of the

network– Second phase: partition the spanning tree into

sub-trees with bounded diameter.

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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 (represents effective broadcast range).

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 = 1

The distance between non adjacent nodes is >= 2

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Preliminaries IGiven an undirected graph G = (V,E) consider

the following general definitions regarding k-clustering:

• Diameter:

• Dominating Set (DS):

• Connected Dominating Set (CDS): The induced sub graph G[D] is connected.

}.,:),(max{ Vvuvuddiam

./

.

DinnodesometoadjacentisDV

innodeeachtsVDSubset

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Preliminaries II

• Independent Set (IS):

• Maximal Independent Set (MIS): An independent set S where no proper superset of S is also an IS.

• A MIS is also DS.

.

.

Sinnodesofpairanybetween

edgenoistheretsVSSubset

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Maximal Independent Set (MIS)

B

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S E

F

H

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D

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G

IK

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CH Denote MIS & DS nodes

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First Phase: MCDS Tree Construction

Given an unit disk graph G = (V,E) the algorithm executes as follows:

• Step 1: Construct a spanning tree T.• partitions the nodes into disjoints sets Si

• Si is a set of nodes at level equal to i

• Every node knows its neighbors• A rank associated with every node

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Spanning tree (Cntd.)

• Maximal Independent Set Construction

D F

B

E G

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AA

B C

D E F G

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Spanning tree (Cntd.)

• Connected Dominating Set, parent child pointers

D F

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E G

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AA

B C

D E F G

INV INV

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Connected Dominating Set

5

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4root

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Denote MIS nodes

1

3

35

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DenoteNS nodes

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root

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35

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Denote spanning tree edge.

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Second Phase: K-sub tree

• Partition the spanning tree into sub-trees– Bounded Diameter

• Each node maintains – Height– Highest child

• Detach child if– H+ Height + 1 > k– Where H is the height received from a child

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K-sub-tree Converge-cast (K=4)

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root

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Denote MCDS spanning tree edge.

leaf

The tree rooted at this node exceeds k detach the highest child

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K-sub-tree Converge-cast (K=4)

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root

2 3

4

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6leaf

The tree rooted at this node exceeds k detach the highest child.

Denote MCDS spanning tree edge.

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K-sub-tree Converge-cast (K=4)

1

root

2

2

3

Denote MCDS spanning tree edge.

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Summary

• A distributed k-clustering algorithm

• Competitive worst case ratio of O(k)

• Building-block – essential for cluster-based routing protocols.

• Flexible - cluster diameter is a part of the algorithm parameter.

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Thanks