Chapter 6 Relaxation (1) CDS in unit disk graph

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Presentation transcript:

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

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

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

Military applications Example 1 Military applications

Environmental Monitoring Example 2 Environmental Monitoring

Example 3 Biological Systems

Example 4 Traffic Control

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.

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:  A  B ? A:  B: C: D:  C D B A  B A

Unit Disk Graph

Unit Ball Graph

Connected Dominating Set

CDS in unit disk graphs

CDS in unit ball graphs

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

Stage 1

MCDS (opt) MIS

Disk Packing

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

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

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

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

Sphere Packing

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

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

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

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

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

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

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

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

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

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)

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

Connect all nodes in an MIS with greedy algorithm

Theorem

Proof

Operations Research Dominating Packing Wireless Networking mathematics Computer Science

Thanks, End