Upload
sheila-phelps
View
222
Download
1
Tags:
Embed Size (px)
Citation preview
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
3
4
106
2 5
7
89
10
11
2-30
-1 0
1
0 1 1 -2
-1 -1
7
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.
3
4
106
2 5 89
10
11
2-30
-1 0
1
0 1 1 -2
-1 -1
r-CDS
If v receives a BLACK message from u, v becomes grey and broadcasts a GREY message containing (v, u).
3
4
106
2 5
7
89
10
11
2-30
-1 0
1
0 1 1 -2
-1 -1
r-CDS
black node w receives a GREY message (v, u) w not connected to uColor v blue
3
4
106
2 5
7
89
10
11
2-30
-1 0
1
0 1 1 -2
-1 -1
(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
3
4
106
2 5
7
89
10
11
2-30
-1 0
1
0 1 1 -2
-1 -1