Chapter 6 Relaxation(1) CDS in unit disk graph

Ding-Zhu Du

Sensor NetworksSensor NetworksA sensor network is an ad hoc A sensor network is an ad hoc wireless networkwireless network which consists of a which consists of a huge amount of static or mobile huge amount of static or mobile sensorssensors. The sensors collaborate to . The sensors collaborate to sense, collect, and process the raw information of the sense, collect, and process the raw information of the phenomenonphenomenon in in the sensing area (in-network), and transmit the processed information the sensing area (in-network), and transmit the processed information to the to the observersobservers..

Sensing AreaSensing Area

phenomenonphenomenon

User1User1

SinkSink

Internet / Internet / SatelliteSatellite

Sensor networkSensor network

User2User2

Sensor Networks (Cont.)

Sensor Node

• Sensing + Computation + Communication

• Small size• Limited power

Military applications

Applications

Example 1

Environmental Monitoring

Example 2

Biological Systems

Example 3

Example 4

Traffic Control

Virtual Backbone Flooding

Reduction of communication

overhead

Redundancy

Contention

Collision

Reliability Unreliability

Applications of CDS: Virtual backbone

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

B

A

C D

A B ?A: B:C:D:

A B ?A:B: C: D:

A B

Only nodes in CDS maintain routing tables Routing information localized Save storage space

Unit Disk Graph

Unit Ball Graph

Connected Dominating Set

connected. is by inducedsubgraph if

set dominating connected a called is e,Furthermor

. oadjacent tor in either is

nodeevery ifset dominating a is subset nodeA

).,(graph aConsider

C

C

CC

C

EVG

Dominating set

Connected dominating set

CDS in unit disk graphs

y.cardinalit minimumset with dominating

connected a find graph,disk unit aGiven

CDS in unit ball graphs

y.cardinalit minimumset with dominating

connected a find graph, ballunit aGiven

Two Stage Algorithm

Dominating set

Connected dominating set

Stage 1. Compute a dominating set D.

Stage 2. Connect D into a connected dominating set.

Stage 1

set. dominating a isset t independen maximalEvery

MCDS (opt)

MIS

23|mis| opt

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?

8!(Wu et al, 2006)

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

set. dominating

connected minimum theof size theis and

sett independen maximal a is mis where

14|mis|

opt

opt

2.18.3|mis| opt

9748.3|mis| opt

3

11

3

23|mis| opt

(Wu et al. 2006)

(Funke et al. 2006)

(Yao et al. 2008)

(Wan et al, 2002)

Sphere Packing

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

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?

1.5

.5

1

Relationship between problems 1, 2 and 3?

For balls not touched each other, 12!! (Hoppe, 1874)

Allow balls to touch, 12!!

icosahedron

11!

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

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

1

>1

22!

How many unit balls can kiss two intersecting unit balls?

20?!

In unit ball graph

111|mis| opt

12

11

11

(Butenko, et al, 2007)

??|mis| opt (Zhang, et al, 2008)

Connect all nodes in an MIS with a spanning tree

Stage 2

122 optfor unit ball graphs

(Butenko, 2007)

3

22

3

120 opt for unit disk graphs

(Wan-Yao)

Stage 2: Connect all nodes in an MIS D.

Consider a greedy method.

)(}){()(

by inducedsubgraph

of components connected of #)(

.return

};{

and )( maximize to node a choose

do 2)( while

;

CfxCfCf

C

Cf

C

xCC

Cfx

Cf

DC

x

x

Connect all nodes in an MIS with greedy algorithm

Theorem

.10ln13ln2 and 11 space,In

.3ln6)1ln(2 and 4 plane,In

.1))1ln(2(

mostat size with CDS

a produces algorithmgreedy then the

,1 |mis| If

opt

opt

Proof

.1)()*(

Then

}.,...,{* and }{, Denote

subgraph. connected a induces },...,{ ,any for and

set dominating connected minimum a be }{Let

algorithm.Greedy by in turn selected are ,..., Suppose

1

1110

1

1

1

iyjiy

jjiii

i

opt

g

CfCCf

yyCxCCDC

yyi

, ...,yy

xx

jj

1y1jy jy

)1

1(

Then ).(1 Denote

.)(1

)()(

)()(

/))((

/))(*)(1(

/))*(1(

/))(()( Thus,

1 allfor

)()( So,

1

1

11

1

1

1

optaa

Cfopta

opt

CfoptCf

opt

CfoptCfCf

optCfopt

optCfCCfopt

optCCfyopt

optCfyCfx

optj

CfCfx

ii

ii

i

i

iii

i

ii

optjji

optjii

iyi

j

ji

ji

.

2 Then

.)12 hence and 2||

then, i.e., exist,t doesn' such If(

.

satisfying onelargest thebe to Choose

)/11(

/0

0

/00

opti

i

optiii

eaopt

optig

optgoptD

optai

aopt

i

eaoptaa

.1))1ln(2(|mis|

Therefore,

.)1 |mis| :(

.1 /|)mis|1(/

)/(ln 2

2

Thus,

)/(ln

0

0

0

/0

optg

optnote

optoptopta

optaoptopt

ioptg

optaopti

eaopt opti

mathematics

Operations Research

Computer Science

Packing

Dominating

Wireless Networking

Thanks, End