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Chapter 6 Relaxation (1) CDS in unit disk graph

Chapter 6 Relaxation (1) CDS in unit disk graph. Ding-Zhu Du. Sensor network. Internet / Satellite. User1. User2. Sink. phenomenon. Sensing Area. Sensor Networks.

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Chapter 6 Relaxation (1) CDS in unit disk graph

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

  2. Sensor network Internet / Satellite User1 User2 Sink phenomenon Sensing Area 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.

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

  4. Applications Example 1 Military applications

  5. Example 2 Environmental Monitoring

  6. Biological Systems Example 3

  7. Traffic Control Example 4

  8. Applications of CDS: Virtual backbone 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. 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

  11. Unit Disk Graph

  12. Unit Ball Graph

  13. Connected Dominating Set Dominating set 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 >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. For balls not touched each other, 12!! (Hoppe, 1874) icosahedron Allow balls to touch, 12!!

  30. 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. 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

  39. Operations Research Dominating Packing Wireless Networking mathematics Computer Science

  40. Thanks, End

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