Connected Dominating Sets
Motivation for
Constructing CDS
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.
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.
Applications of CDS: Virtual backbone
Virtual Backbone
Flooding
Reduction of communication Redundancy
overhead
Contention
Collision
Reliability
Unreliability
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
 Only nodes in CDS maintain routing tables
 Routing information localized
 Save storage space
A B?
A: 
B:
C:
D: 
C
A B?
A:
B: 
C: 
D: 
B
A AB
D
Applications of CDS: Coverage
Area Coverage Problem
CDS provides connectivity
Applications of CDS: Coverage
Target Coverage Problem
CDS provides connectivity
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?
CDS Construction
Algorithms
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 nonadjacent 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 u
r(u) = the number of 2-hop-away neighbors – d(u)
where d(u) is the degree of node u
1
-1
2
0
3
5
-3
0
6
0
0
1
4
9
1
-2
11
2
1
8
-1
-1
10
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.
1
-1
2
0
3
5
-3
0
6
0
0
1
4
9
1
-2
11
2
1
8
-1
-1
10
7
r-CDS
If v receives a BLACK message from u, v
becomes grey and broadcasts a GREY
message containing (v, u).
1
-1
2
0
3
5
-3
0
6
0
0
1
4
9
1
-2
11
2
1
8
-1
-1
10
7
r-CDS
 black node w receives a GREY message (v, u)
 w not connected to u
Color v blue
(5, 0)
1
-1
2
0
3
5
-3
0
6
0
0
1
4
9
1
-2
11
2
1
8
-1
-1
10
7
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
1
-1
2
0
3
5
-3
0
6
0
0
1
4
9
1
-2
11
2
1
(8, 11)
8
-1
-1
10
7
Load-Balanced CDS (LBCDS)
1
2
3
4
5
6
7
8
Minimum-sized CDS
1

2
3
4
5
6
7
8
Load-Balanced CDS
Load-Balancedly Allocate
Dominatees (LBAD)
1
2
3
4
1

5
6
7
8
Unbalanced Allocation
2
3
4
5
6
7
8
Balanced Allocation
Load-Balanced Virtual Backbone
 LBCDS
+ LBAD
 First,
solve the LBCDS problem to find loadbalanced dominators
 And
then, solve the LBAD problem to find a
load-balanced job assignment scheme.
 Help
extend network lifetime
29
Measure load-balance (in General)
Feature vector
n

p-norm: | X | p  ( | xi | p )
1
p
p>1
i 1
52  52 
50
12  92 
82
p=2
• Smaller p-norm, more balanced of xi is
Measure load-balance of MCDS and LBCDS
degree of each dominator
mean degree of the graph
1
_
p=2
p p
M
 CDS p-norm:| D | p  ( | d i  d | )
i 1
1
2
3
5
4
6
1

d4 = 6
7
d7 = 3
MCDS
| D |MCDS
 (6  3) 2  (3  3) 2  9
p
8
d 3
2
3
d3 = 4
5
4
8
6
7
d6 = 4 d7 = 3
LBCDS
| D |LBCDS
 (4  3) 2  (4  3) 2  (3  3) 2  2
p
31
Greedy algorithm for LBCDS
degree of each dominator

mean degree of the graph
o Greedy criterion: min | d i  d |
d1 = 2
1
DO
{
Mark one node
si satisfying

min | d i  d | as black.
}
UNTIL {All black nodes form a CDS}
d3 =4
d2 = 2
2
3
4 d4 = 6
5
7
8
6
d5 = 2 d6 = 4 d7 = 3 d8 = 1

d 3
p-norm for the LBAD problem
 Valid Degree ( di’ )
1
M
_
'
 Allocation p-norm: | D | p  ( | d i  p | p ) p
i 1
d1 ’ = 1
1
d3’ = 3 3
d2 ’ = 1
2
4 d4’ = 1
d1’ = 1
1
83 5
p

3
3
_
5
7
8
6
d5’ = 1 d6’ = 1 d7’ = 1 d8’ = 1
Unbalanced Allocation
5
5
5
| D |MCDS
 (3  ) 2  (1  ) 2  (1  ) 2  2.67
p
3
3
3
d3’ = 2 3
d2’ = 1
2
4 d4’ = 1
5
7
8
6
d5’ = 1 d6’ = 2 d7’ = 1 d8’ = 1
Load-Balanced Allocation
5 2
5 2
5 2
| D | LBCDS

(
2

)

(
2

)

(
1

)  0.67
p
3
3
3
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.
2
1
3
1
2
1
1
1
1
4 d4’ = 5
1 1
2/8
d3’ = 3 3
4
2/7
3/8
5
6
5/7
1
7
d7’ = 2
8
5
1
6
d6’ = 2
3/8
1
7
d7’ = 2
p46 d 4'  p76 d 7'
p34d 3'  p64d 6'  p74d 7'
2/7 * 5 = 5/7 * 2
2/8 * 3 = 3/8 * 2 = 3/8 * 2
8