Distributed Topology Control in Mobile Ad-hoc Networks using Minimal Connected Dominating Sets (mCDS)

1. The University of Texas at Dallas
Distributed Topology Control
in Mobile Ad-hoc Networks
By
Siddanagouda Khot
Advisor: Dr. S. Venkatesan

2. The University of Texas at Dallas
Outline
 MANETs
 Problem
 Related work
 Problem definition
 Algorithm
 Properties and proofs
 Simulation results
 Conclusion and future work

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MANETs
 Mobile independent nodes
 Communication with neighbors
A B
C

4. Challenges to address:
To form a backbone or skeletal network
 Routing messages
 Efficient use of channel by reducing the broadcast
messages
 Better communication without interference
Approach:
Connected Dominating Set (CDS)
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5. A B
C
D
G
F
E
A B
C
D
G
F
E
Example of CDS
Ad-hoc Network Nodes in Green
Ways of forming a CDS
 Self-selected dominating set.
 Neighbor designated dominating set.
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Problem definition
Objective: Construct a sub-graph (SG) for a given connected
graph (G) s.t.
 Nodes in SG are connected.
 Nodes in SG form a dominating set.
 Nodes in SG satisfy shortest path property.
 SG so formed is minimal.
Nodes of SG are called skeletal nodes.
SG also called minimal Connected Dominating Set (mCDS)

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Our approach
Many mCDS for a connected graph G (Nodes in Green)
Our algorithm, mCDS, finds one of them.
Finding a minimum CDS is NP-hard.
A
B
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D
E
F
G
H
A
B
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D
E
F
G
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A
B
C
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H

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Assumptions
 While running the algorithm, topology does not change.
 Local information, communicate only with neighbors.
 Graph is not completely connected, else no CDS required.
 Same transmission range, links are bidirectional.
 Every node knows its one hop neighborhood information
(provided as input to the algorithm).

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Flow of the algorithm with respect to stages and states
First stage:
 Undecided nodes
 Stable skeletal nodes
 Stable non-skeletal nodes
Second stage:
 Bi-stable undecided nodes
 Bi-stable non-skeletal
 Bi-stable skeletal
Third stage (iterative stage):
 Dynamic undecided nodes
 Dynamic non-skeletal
 Dynamic skeletal
Undecided node
Bi-stable undecided node
Dynamic undecided node
Stable skeletal node Stable non-skeletal node
Bi-stable skeletal node Bi-stable non-skeletal node
Dynamic skeletal node Dynamic non-skeletal node

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First stage:
 Undecided nodes
 Exchange neighbor information with
two hop neighbors
 Stable skeletal node rule
 Inform two hop neighbors about its
status
 If undecided node, apply Stable non-
skeletal node rule
 Inform two hop neighbors about its
status
01
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01
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Non-skeletal nodes in red
01
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Bi-stable undecided nodes in blue ring

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Second stage:
 Bi-stable undecided nodes
 Bi-stable non-skeletal node rule
 Inform two hop neighbors about its
status
 If bi-stable undecided, apply Bi-stable
skeletal node rule
 Inform two hop neighbors about its
status
01
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Nodes 2 and 4 become bi-stable non-skeletal Node 3 becomes bi-stable skeletal

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Shortest Path Set (SPS)
The set of two hop shortest path provided by x that are not
provided by any stable or bi-stable skeletal nodes.
Exchange SPS and then apply the dynamic non-
skeletal and dynamic skeletal node rule.
SPS(0) = [{1,6},{2,7},{2,6}]
SPS(1) = [{0,3},{2,7},{3,7}]
SPS(2) = [{0,3},{0,4},{1,4}]
SPS(3) = [{1,4},{1,5},{2,5}]
SPS(4) = [{2,5},{2,6},{3,6}]
SPS(5) = [{3,6},{3,7},{4,7}]
SPS(6) = [{0,5},{0,4},{4,7}]
SPS(7) = [{0,5},{1,5},{4,7}]
0
1
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4
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7

14. Third stage (iterative stage):
 Dynamic undecided nodes
 Dynamic non-skeletal node rule
 Inform two hop neighbors about its status
 If dynamic undecided, apply Dynamic
skeletal node rule
 Inform two hop neighbors about its status
 Update |SPS|
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0
1
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3
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Iteration 1: |SPS| = 3;

16. 0
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Node 7 => dynamic non-skeletal node
Nodes 0, 3 and 6 => dynamic skeletal
nodes
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Iteration 2: |SPS(1)| = 2, |SPS(2)| = 1, |
SPS(4)| = 1 and |SPS(5)| = 2 0
1
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2
1
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1

18. Node 4 => dynamic non-skeletal node
Node 5 => dynamic skeletal node
0
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2
1
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19. 0
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Iteration 3: |SPS(2)| = 1 and |SPS(1)| = 1
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20. 0
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Node 2 => dynamic non-skeletal node
Node 1 => dynamic skeletal node
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Properties and proofs
SG of graph G satisfies the following properties:
 Nodes in SG are connected.
 Nodes in SG form a dominating set.
 Nodes in SG satisfy shortest path property.
 SG so formed is minimal.

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Lemma 1
Consider a shortest path Pxy between nodes x and y in G. Let
(x,…,wi-1,wi,wi+1,…,y) be the sequence of nodes in Pxy. If node
wi does not belong to SG, then the predecessor and
successor nodes, wi-1 and wi+1, of wi in Pxy must be connected
by a 2-hop path going through a (skeletal) node in SG.
x yWi-1 wi
Wi+1
∉

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Connected set proof
Theorem 1. Given a connected graph G, the sub-graph SG
derived from our algorithm, is also connected.
Dominating set proof
Theorem 2. A node v in graph G is either part of SG or
adjacent to a node in SG.
Shortest path proof
Theorem 3. Consider a pair of non-neighboring nodes a and
b in G. From among the shortest paths between a and b,
there exists a shortest path p such that all intermediate
nodes of p are skeletal nodes.

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Minimality proof
Theorem 4. There is no proper subset of SG that satisfies
the connected, dominating and shortest paths properties.
Termination proof
Theorem 5. In every iteration of the dynamic skeletal and
dynamic non-skeletal rule, at least one node among the
dynamic undecided nodes changes its state to either (i)
dynamic skeletal node or (ii) dynamic non-skeletal node.

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a. 20 nodes b. 30 nodes
0
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1 2 3 4 5 6 7
Density
Nodes
mCDS nodes
MPR
MPR Prune
0
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25
30
35
1 2 3 4 5 6 7 8
Density
Nodes
mCDS nodes
MPR
MPR Prune
Simulation results
Comparison with MPR and MPR prune algorithms:

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c. 40 nodes d. 50 nodes
0
5
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40
1 2 3 4 5 6 7 8 9 10 11
Density
Nodes
mCDS nodes
MPR
MPR Prune
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10 11
Density
Nodes
mCDS nodes
MPR
MPR Prune

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e. 60 nodes f. 70 nodes
0
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70
1 2 3 4 5 6 7 8 9 10 11 12 13
Density
Nodes
mCDS nodes
MPR
MPR Prune
0
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40
50
60
70
80
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
Density
Nodes
mCDS nodes
MPR
MPR Prune

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Complexity for number of iterations
For clusters n1’, n2’,…,nm’ having n1, n2,…,nm dynamic undecided
nodes:
= max {n1, n2,…,nm} + c where c is the constant number of
rounds after stable skeletal, stable non-skeletal, bi-stable non-
skeletal and bi-stable skeletal node rules.
Computation complexity per node
Overall computation complexity per node is O( 3
+ . 4
)
Ψ
∆ Ψ ∆

29. Conclusion
 Shortest path property
 mCDS has minimal nodes
Future work
Make comparison with the optimum CDS algorithms
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Thanks