1. Chapter 6 Relaxation (1) CDS in unit disk graph
Ding-Zhu Du

2. Sensor Networks
A sensor network is an ad hoc wireless network which consists of a huge amount of static or mobile sensors. The sensors collaborate to sense, collect, and process the raw information of the phenomenon in the sensing area (in-network), and transmit the processed information to the observers.
Sensing Area
phenomenon
User1
Sink
Internet / Satellite
Sensor network
User2

3. Sensor Networks (Cont.)
Sensor Node
Sensing + Computation + Communication
Small size
Limited power

4. Military applications
Example 1
Military applications

5. Environmental Monitoring
Example 2
Environmental Monitoring

6. Example 3
Biological
Systems

7. Example 4
Traffic Control

8. 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.

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

10. 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

11. Unit Disk Graph

12. Unit Ball Graph

13. Connected Dominating Set

14. CDS in unit disk graphs

15. CDS in unit ball graphs

16. Two Stage Algorithm Stage 1. Compute a dominating set D.
Stage 2. Connect D into a connected
dominating set.
Dominating set
Connected dominating set

17. Stage 1

18. MCDS (opt)
MIS

19. Disk Packing

20. How many independent points can be contained
by a disk with radius 1? 5!

21. How many independent points can be contained
by two disks with radius 1 and
center distance < 1?
(Wu et al, 2006) 8!

22. How many independent points can be packed Into
four disks that one contains centers of other three?
< 15!
(Yao et al, 2008)

23. In unit disk graph (Wan et al, 2002) (Wu et al. 2006)
(Funke et al. 2006)
(Yao et al. 2008)

24. Sphere Packing

25. 1. How many independent points can be packed by a ball with radius 1?
>1

26. 2. How many (untouched) unit balls can be packed into
a ball with radius 1.5?
0.5
1.5

27. 3. Gregory-Newton Problem (1694)
How many unit balls (not touch each other)
can kiss a unit ball?
                              

28. Relationship between problems 1, 2 and 3?
1.5 1 .5

29. 12!! (Hoppe, 1874) icosahedron 12!! For balls not touched each other,
Allow balls to touch,
12!!

30. 11! How many independent points can be contained
In a ball subtracting another ball?
11!

31. How many independent points can be contained
by two balls with radius 1 and center distance < 1?
22! 1
>1

32. How many unit balls can kiss two intersecting unit balls?
20?!

33. In unit ball graph (Butenko, et al, 2007) 11 12 11
(Zhang, et al, 2008)

34. Connect all nodes in an MIS with a spanning tree
Stage 2
Connect all nodes in an MIS with a spanning tree
for unit disk graphs
(Wan-Yao)
for unit ball graphs
(Butenko, 2007)

35. Stage 2: Connect all nodes in an MIS D.
Consider a greedy method.

36. Connect all nodes in an MIS with greedy algorithm

37. Theorem

38. Proof

42. Operations Research
Dominating
Packing
Wireless
Networking
mathematics
Computer Science

43. Thanks, End