Connected Dominating Sets

Motivation for Constructing CDS

A dominating set (DS) is a subset of all the nodes such that each node is either in the DS or adjacent to some node in the DS.

What Is CDS?

A connected dominating set (CDS) is a subset of the nodes such that it forms a DS and all the nodes in the DS are connected.

What Is CDS?

Virtual Backbone Flooding

Reduction of communication overhead

RedundancyContentionCollision

Reliability Unreliability

Applications of CDS: Virtual backbone

CDS is used as a virtual backbone in wireless networks.

Applications of CDS: Broadcast

Only nodes in CDS relay messages Reduce communication cost Reduce redundant traffic

Applications of CDS: Unicast

B

A

C D

A B ?A: B:C:D:

A B ?A:B: C: D:

A B

Only nodes in CDS maintain routing tables Routing information localized Save storage space

Applications of CDS: Coverage

Area Coverage Problem

CDS provides connectivity

Target Coverage Problem

Applications of CDS: Coverage

CDS provides connectivity

Motivation for Constructing CDS

How to construct a CDS?

How to make the size of a CDS small?

CDS plays an important role in wireless networks.

Challenges

CDS Construction CDS Construction AlgorithmsAlgorithms

Definition & Preliminaries

Minimum connected dominating set Given: a graph G=(V,E).

Goal: find the smallest CDS. NP-hard Approximation algorithms Performance ratio (PR) = |C|/|C*| Smaller PR, better algorithm.

Definition and Preliminaries (Cont.)

Notations Given a graph G and a DS C, all nodes in G can be

divided into three classes.

Black nodes: Nodes belong to C.

Grey nodes: Nodes are not in C but adjacent to C. White nodes: Nodes are neither in C nor adjacent to C.

C

Greedy Algorithm in General Graph

Guha’s algorithm 1

Select the node with the max

number of neighbors as a dominating node.

Iteratively scans the grey nodes and their white neighbors. Select the grey node or the pair of nodes with the max number of white neighbors.

PR = 2(1 + H(Δ))

Greedy Algorithm in General Graph

Guha’s algorithm 2 Iteratively select the node with the

max number of white neighbors as a dominating node.

The first phase terminates when there are no white nodes.

Color some grey nodes black to connect all the black nodes.

PR = 3 + ln(Δ)

Greedy Algorithm

Maximal Independent Set (MIS) is a maximal set of pair-wise non-adjacent nodes.

MIS DS

Greedy Algorithm

MIS DS Idea: connect MIS CDS

Centralized Algorithm

Alzoubi’s Algorithm

Construct a rooted spanning

tree from the original network topology

Centralized Algorithm

Alzoubi’s Algorithm

Color each node to be black

or grey based on its rank (level. ID). The node with the lowest rank marks itself black. All the black nodes form an Maximal Independent Set (MIS).

Wu’s Algorithm Each node exchanges its neighborhood

information with all of its one-hop neighbors. Any node with two unconnected neighbors

becomes black. The set of all the black nodes form a CDS.

Wu’s Algorithm

r-CDS

For each node ur(u) = the number of 2-hop-away neighbors – d(u)where d(u) is the degree of node u

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r-CDS

Node u with the smallest <r, deg, id> within its neighborhood becomes black and broadcast a BLACK message where deg is the effective degree.

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r-CDS

If v receives a BLACK message from u, v becomes grey and broadcasts a GREY message containing (v, u).

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r-CDS

black node w receives a GREY message (v, u) w not connected to uColor v blue

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(5, 0)

BLACK

(8, 11)

r-CDS v has received a GREY message (x, y) v receives a BLACK message from u y & u not connected

Color v and x blue

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