A Clustering Scheme for Hierarchical Control in Wireless Networks - - PowerPoint PPT Presentation

A Clustering Scheme for Hierarchical Control in Wireless Networks

Suman Banerjee, Samir Khuller Department of Computer Science, University of Maryland at College Park Slides by James Flemer Department of Computer Science, Rensselaer Polytechnic Institute

October 21, 2002 – p.1

Introduction

Miniaturization and proliferation of wireless devices Creation of rapidly deployable wireless sensor networks Smart Dust project at UC Berkeley Wireless Integrated Network Services (WINS) at UCLA Sensors placed arbitrarily Mobility of network nodes create changing topology/connectivity Dynamic addition/removal of sensor nodes

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Problems

How can we ... Address nodes Control nodes Route messages to/between nodes Maintain scalability Existing large scale networks use a hierarchical design Need hierarchical clustering of wireless nodes

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Layered Hierarchy

Layer 0 Layer 1 Layer 2

B K A B C D F E J H K G G G

Figure 1: Example layered hierarchical clustering.

October 21, 2002 – p.4

Clustering Goals

Complete coverage of the graph. No missed nodes. Each cluster must be connected. Clusters should be moderately sized. Small clusters are inefficient. Large clusters overburden the controlling node. e.g. k ≤ n < 2k, for some k. Clusters should have minimal overlap. Each node should be in a minimal number of clusters.

October 21, 2002 – p.5

Clustering in a Star Graph

Each of such k-1 vertices and the center vertex make a cluster

• f size k.

Figure 2: The center vertex is a member of all clusters, violating requirement (5).

October 21, 2002 – p.6

Graphs for Wireless Networks

Connectivity based on transmission range. Nodes may have different transmission ranges. Only consider nodes that are mutually in range. This type of graph is called a Disk Graph, because the nodes are disks with a certain radius, and edges are determined by overlapping disks. For a unit disk graph (all radii the same), the maximum independent set in the neighborhood of u, N(u), has at most 5 vertices. For a disk graph with radii bounded by Rmax and Rmin, the maximum independent set has at most O

• log Rmax

Rmin

• vertices.

October 21, 2002 – p.7

Clustering in a Disk Graph

A B C L2 L3 L1 L4 Each of L1, L2 and L3 k = 0.45n segments has 0.03n vertices 0.3n vertices 0.3n vertices 0.3n vertices Total n vertices Segment L4 has 0.01n vertices

Figure 3: All the vertices in L2 are shared, violating clus- ter overlap requirement (4).

October 21, 2002 – p.8

Formal Clustering Goals

Given a graph G = (V,E), and a positive integer, k (1 ≤ k ≤ |V|), find a collection of subsets V1,...,Vl

• f V, such that:
• 1. The clusters cover the graph. ∪l

i=1Vi = V

• 2. G[Vi], the subgraph of G induced by Vi, is

connected.

• 3. The subsets are bounded in size. ∀i,|Vi| < 2k

∀i (except one),k ≤ |Vi|

• 4. There is minimal cluster overlap. |Vi∩Vj| ∼ O(1)
• 5. Each vertex is in a minimal number of clusters.

|S(v)| ∼ O(1), where S(v) = {Vi|v ∈ Vi}

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Theoretical Solution

• 1. Find a (rooted) spanning tree, T, of the graph G.

Let T(v) be the subtree of T rooted at v, and C(v) be the children of v in T.

• 2. Find a vertex, u, such that |T(u)| ≥ k and

∀v ∈ C(u),|T(v)| < k.

• 3. Group the subtrees rooted at the children of u into

clusters of size between k and 2k, including u if necessary for connectivity.

• 4. Remove T(u) from T.
• 5. Terminate if T is empty, else goto 2.

October 21, 2002 – p.10

Cluster Forming

Cluster formed at v has been deleted from the tree during processing at v

• ✁

0.8k 0.6k 0.8k 1.3k 0.7k u Cluster with 1.5k members Cluster with 1.4k+1 members 0.7k

D B A

z y v Partial cluster with 0.7k+1 members retained in the tree

F

0.6k 0.5k

C

Cluster with 1.1k members |T(u)| = 4.7k + 1 after v is processed x q r p s

Figure 4: Example cluster forming operation.

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Distributed Solution

Tree Discovery Protocol (TDP) Constructs a spanning tree. Similar to Spanning Tree Protocol (STP) used by Ethernet switches. All nodes have unique identifiers (ID). All advertisements send the senders node ID, their distance from the root, and the ID of the root. All nodes start by assuming they are the root, and advertise themselves as the root. Nodes change their parent if they hear a distance less than their current distance or hear of another root with a lower ID.

October 21, 2002 – p.12

Distributed Solution (cont.)

Cluster Formation Protocol (CFP) Leaves begin by forming trivial clusters containing only themselves. Nodes then report their partial clusters (less than k in size) to their parents. Each node, receives reports from its children and merges the partial trees into clusters if possible. After a node has processed reports from all of its children, it then reports itself to its parent including a partial cluster of its children if one exists.

October 21, 2002 – p.13

Distributed Protocol Layers

Tree Discovery Protocol Wireless MAC Protocol Cluster Formation Protocol

October 21, 2002 – p.14

Cluster Maintenance

Relax cluster size upper bound to 3k. Causes the one remaining partial cluster to merge immediately. New node, v, joins: If any of its neighbors, u ∈ N(v), belong to a cluster with less than 3k −1 nodes, it joins that cluster. Else, one neighboring cluster of size 3k −1 splits, and the CFP is rerun on those nodes and the new node.

October 21, 2002 – p.15

Cluster Maintenance (cont.)

Existing node, v, leaves: May cause a cluster to become disconnected. Any of the created connected components w/ size ≥ k is made into a cluster. All other nodes in that cluster disband, and then join as if they were new nodes. Link outages: Only matters if it partitions cluster, in which case use the node-leaves maintenance method.

October 21, 2002 – p.16

Questions?

Original Paper Available at:

http://www.cs.umd.edu/users/samir/grant/bk01.ps http://www.cs.umd.edu/Library/TRs/CS-TR-4103/CS-TR-4103.ps.Z

Slides produced with Prosper and L

AT

EX.

http://prosper.sourceforge.net/

October 21, 2002 – p.17