Connected Dominating Sets

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 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. A  B ? A:  B: C: D:  A  B ? A: B:  C:  D:  Applications of CDS: Unicast • Only nodes in CDS maintain routing tables • Routing information localized • Save storage space 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 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. 1 1 1 -2 -1 0 0 -1 -1 2 -3 0 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 3 11 2 5 4 9 8 10 7 6 0 1

23. 1 1 1 -2 -1 0 0 -1 -1 2 -3 0 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 11 2 5 4 9 8 10 7 6 0 1

24. 1 1 1 -2 -1 0 0 -1 -1 2 -3 0 r-CDS If v receives a BLACK message from u, v becomes grey and broadcasts a GREY message containing (v, u). 3 11 2 5 4 9 8 10 7 6 0 1

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

26. BLACK 1 1 1 -2 -1 0 0 -1 -1 2 -3 0 (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 11 2 5 4 9 8 10 7 6 0 1