1. Connected Dominating Sets

2. Motivation for Constructing CDS

3. What Is 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.

4. 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.

5. Applications of CDS: Virtual backbone
Flooding
Reduction of communication
overhead
Redundancy
Contention
Collision
Reliability
Unreliability
CDS is used as a virtual backbone in wireless networks.

6. Applications of CDS: Broadcast
Only nodes in CDS relay messages
Reduce communication cost
Reduce redundant traffic

7. Applications of CDS: Unicast
Only nodes in CDS maintain routing tables
Routing information localized
Save storage space
A  B ? A:
B: 
C: 
D: 
A  B ?
A:  B: C:
D:  C D B
A  B A

8. Applications of CDS: Coverage
Area Coverage Problem
CDS provides connectivity

9. Applications of CDS: Coverage
Target Coverage Problem
CDS provides connectivity

10. Motivation for Constructing CDS
CDS plays an important role in wireless networks.
Challenges
How to construct a CDS?
How to make the size of a CDS small?

11. CDS Construction Algorithms

12. 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.

13. 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

14. 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))

15. 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)

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

17. Greedy Algorithm
MIS DS
Idea: connect MIS CDS

18. Centralized Algorithm
Alzoubi’s Algorithm
Construct a rooted spanning tree from the original network topology

19. 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).

20. 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.

21. Wu’s Algorithm

22. r-CDS For each node u r(u) = the number of 2-hop-away neighbors – d(u)
where d(u) is the degree of node u 2 -3 -1 1 -2 3 11 2 5 4 9 8 10 7 6 1

23. 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. 2 -3 -1 1 -2 3 11 2 5 4 9 8 10 7 6 1

24. r-CDS
If v receives a BLACK message from u, v becomes grey and broadcasts a GREY message containing (v, u). 2 -3 -1 1 -2 3 11 2 5 4 9 8 10 7 6 1

25. r-CDS black node w receives a GREY message (v, u) w not connected to u
Color v blue
(5, 0) 2 -3 -1 1 -2 3 11 2 5 4 9 8 10 7 6 1

26. 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
BLACK 2 -3 -1 1 -2 3 11 2 5 4
(8, 11) 9 8 10 7 6 1

27. Load-Balanced CDS (LBCDS)1 2 1 2 3 4 5 6 7 8 
Load-Balanced CDS 3 4
Minimized size + load balance 5 6 7 8
Minimum-sized CDS

28. Load-Balancedly Allocate Dominatees (LBAD)1 2 1 2 3 4 5 6 7 8  3 4 5 6 7 8
Unbalanced Allocation
Balanced Allocation

29. Load-Balanced Virtual Backbone
LBCDS + LBAD
First, solve the LBCDS problem to find load- balanced dominators
And then, solve the LBAD problem to find a load-balanced job assignment scheme.
Help extend network lifetime

30. Measure load-balance (in General)
Feature vector
p > 1
p-norm:
p = 2

31. Measure load-balance of MCDS and LBCDS
degree of each dominator
mean degree of the graph
CDS p-norm:
p = 2 1 2
d3 = 4 4 5
d6 = 4
d7 = 3 8
LBCDS 3
d4 = 6 6
MCDS 4 3 7 6 7

32. Greedy algorithm for LBCDS
degree of each dominator
mean degree of the graph
Greedy criterion:
d1 = 2
d2 = 2
d3 =4
d4 = 6
d5 = 2
d6 = 4
d7 = 3
d8 = 1 1 2
DO { Mark one node si satisfying as black. } UNTIL {All black nodes form a CDS} 3 4
Potential allocation more opportunity
Converge fast 5 6 7 8

33. p-norm for the LBAD problem
Valid Degree ( di’ )
Allocation p-norm:
d1’ = 1
d2’ = 1
d3’ = 2
d4’ = 1
d5’ = 1
d6’ = 2
d7’ = 1
d8’ = 1
Load-Balanced Allocation
d3’ = 3
d6’ = 1
Unbalanced Allocation 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

34. Probability-Based Allocation Scheme
Allocate dominatees to dominators deterministically cannot guarantee the load-balance.
Probability-based allocation: Expected allocated dominatees on each dominator are exactly the same. 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 1
3/8
2/8
d4’ = 5
d3’ = 3
2/7
5/7
d7’ = 2
d6’ = 2
d7’ = 2
2/7 * 5 = 5/7 * 2
2/8 * 3 = 3/8 * 2 = 3/8 * 2