[email protected] Approximations for Min Connected Sensor Cover Ding-Zhu Du University of Texas...

lidongwuutdallasedu

Approximations for Min Connected Sensor Cover

Ding-Zhu DuUniversity of Texas at Dallas

Outline

I Introduction

II Two Approximations

III Final Remarks

Have you watched movie Twister

sensorBucket ofsensors

tornado

Where are all the sensors

Smartphone with a dozen of sensors


Wearable devices - Google Glass Applersquos iWatch

Buildings


Transportation systems etc


Sensor Web

Large of simple sensors Usually deployed randomly Multi-hop wireless link Distributed routing No infrastructure Collect data and send it to base station

Applications of Senor Web

observerAn example of sensor web

Whatrsquos Sensor

Small size Large number Tether- lessBUThellip

Whatrsquos limiting the task

Energy Sense Communication scale

CPU

Challenge

Target is Covered

Sensor system is Connected

Coverage amp Connectivity

Golden Rule then we say

System is alive


Communication Range

Sensing Range

d le Rs

sensor

target

communication radius

sensing radius

Rc

Rs


Communication Range

Sensing Range

d le Rs

d le Rc

sensor

target


sensing radius

Rc

Rs

Min-Connected Sensor Cover Problem

Figure Min-CSC Problem

A uniform set of sensors and a target area

Find a minimum of sensors

to meet two requirements

[Coverage] cover the target area and

[Connectivity] form a connected communication network

[Resource Saving]

communicationnetwork

sensing disks

Previous Work for PTAS

Itrsquos NP-hard

Ο(r ln n) ndash approximation given by Gupta Das and Gu [MobiHocrsquo03 2003] where n is the number of sensors and r is the link radius of the sensor network


Outline

I Introduction


III Final Remarks

Main Results

Random algorithm

Ο(log3n log log n)-approximation n is the

number of sensors

Partition algorithm

Ο(r)-approximation r is the link radius of the

network

Connected

Sensor Cover with Target

Area

Connected

Sensor Cover

with Target Points

With a random algorithm which with probability 1- ɛ produces an Ο(log3n log log n) - approximation

1

Algorithm 1

Group

Steiner Tree

2Min-CSC Min-CTC GST

1 2Min-CSC Min-CTC GST


How to map to GST








How to map to GST

Min-Connected Target Coverage Problem

A uniform set of sensors and a target POINTS



[Coverage] cover the target POINTS and



A graph G = (V E) with positive edge weight c for every edge e isin E

k subsets (or groups) of vertices G1Gk Gi sube V

Find a minimum total weight tree T contains at least one vertex in each Gi

Group Steiner Tree

Figure GST ProblemThis tree has minimum weight


Choose at least one sensor from each group

Coverage

b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4

Gi contains all sensors covering bi

S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7

Consider communication network

Connectivity

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7

Find a group Steiner tree in communication network

Min-Coverage amp Connectivity

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


Garg Konjevod and Ravi [SODA 2000] showed with probability 1- ε an approximation solution of GROUP STEINER TREE on tree metric T is within a factor of Ο(log2 n log log n log k) from optimal

What Is Link Radius

3 example In this onintersecti sensing

nonempty having sensors twoof distancefor boundupper least theis radiuslink

r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points

Connect output of Min-TC into Min-CTC It can be done in Ο(r) - approximation

1

Algorithm 2

2Min-CSC Min-CTC Min-TC

Refer to Lidong Wursquos paper [INFOCOM 2013rsquo]

There exists a polynomial-time (1 + ε)- approximation for MIN-TC

Green is an opt (CTC)

Red is an approx (TC)

lt (1+ε) opt (TC) lt (1+ε) opt (CTC)

Step 2 Target Coverage

Byrka et al [6] showed there exists a polynomial-time139-approximation of for Network Steiner Minimum Tree



Step 2 Network Steiner Tree

Let Sprime sube S be a (1 + ε)-approximation for MIN-TC Assign

weight one to every edge of G Interconnect sensors in Sprime to

compute a Steiner tree T as network Steiner minimum tree

All sensors on the tree form an approxfor min CTC

nodes approx for min CTC= edges +1 approx for Network STlt 139 opt (Network ST) +1lt 139 opt (CTC) + 1




Each orange line has distance lt r

opt (Network ST)

lt opt (CTC) -1 + r = opt (CTC) O(r)

Note lt (1+ε) opt (CTC)

Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)

n is the number of sensors

1 Unknown Relationship

2 Constant-appro for Min-CSC

Ο(r)

r is the link radius

THANK YOU

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What Is Link Radius

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Outline

I Introduction


III Final Remarks



tornado





Buildings




Sensor Web




Whatrsquos Sensor




CPU

Challenge

Target is Covered




System is alive


Communication Range

Sensing Range

d le Rs

sensor

target


sensing radius

Rc

Rs


Communication Range

Sensing Range

d le Rs

d le Rc

sensor

target


sensing radius

Rc

Rs








[Resource Saving]


sensing disks


Itrsquos NP-hard



Outline

I Introduction


III Final Remarks

Main Results

Random algorithm


number of sensors

Partition algorithm


network

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 1

Group

Steiner Tree




How to map to GST








How to map to GST











Group Steiner Tree




Coverage

b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7


Connectivity

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7



b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4



What Is Link Radius



r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

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What Is Link Radius

Slide 30

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Slide 32

Slide 33

Slide 34

Slide 35

Slide 36



tornado





Buildings




Sensor Web




Whatrsquos Sensor




CPU

Challenge

Target is Covered




System is alive


Communication Range

Sensing Range

d le Rs

sensor

target


sensing radius

Rc

Rs


Communication Range

Sensing Range

d le Rs

d le Rc

sensor

target


sensing radius

Rc

Rs








[Resource Saving]


sensing disks


Itrsquos NP-hard



Outline

I Introduction


III Final Remarks

Main Results

Random algorithm


number of sensors

Partition algorithm


network

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 1

Group

Steiner Tree




How to map to GST








How to map to GST











Group Steiner Tree




Coverage

b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7


Connectivity

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7



b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4



What Is Link Radius



r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

Slide 1

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Slide 28

What Is Link Radius

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36





Buildings




Sensor Web




Whatrsquos Sensor




CPU

Challenge

Target is Covered




System is alive


Communication Range

Sensing Range

d le Rs

sensor

target


sensing radius

Rc

Rs


Communication Range

Sensing Range

d le Rs

d le Rc

sensor

target


sensing radius

Rc

Rs








[Resource Saving]


sensing disks


Itrsquos NP-hard



Outline

I Introduction


III Final Remarks

Main Results

Random algorithm


number of sensors

Partition algorithm


network

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 1

Group

Steiner Tree




How to map to GST








How to map to GST











Group Steiner Tree




Coverage

b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7


Connectivity

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7



b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4



What Is Link Radius



r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

Slide 1

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Slide 28

What Is Link Radius

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36



Sensor Web




Whatrsquos Sensor




CPU

Challenge

Target is Covered




System is alive


Communication Range

Sensing Range

d le Rs

sensor

target


sensing radius

Rc

Rs


Communication Range

Sensing Range

d le Rs

d le Rc

sensor

target


sensing radius

Rc

Rs








[Resource Saving]


sensing disks


Itrsquos NP-hard



Outline

I Introduction


III Final Remarks

Main Results

Random algorithm


number of sensors

Partition algorithm


network

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 1

Group

Steiner Tree




How to map to GST








How to map to GST











Group Steiner Tree




Coverage

b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7


Connectivity

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7



b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4



What Is Link Radius



r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

Slide 1

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Slide 28

What Is Link Radius

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36



Whatrsquos Sensor




CPU

Challenge

Target is Covered




System is alive


Communication Range

Sensing Range

d le Rs

sensor

target


sensing radius

Rc

Rs


Communication Range

Sensing Range

d le Rs

d le Rc

sensor

target


sensing radius

Rc

Rs








[Resource Saving]


sensing disks


Itrsquos NP-hard



Outline

I Introduction


III Final Remarks

Main Results

Random algorithm


number of sensors

Partition algorithm


network

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 1

Group

Steiner Tree




How to map to GST








How to map to GST











Group Steiner Tree




Coverage

b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7


Connectivity

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7



b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4



What Is Link Radius



r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

Slide 1

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What Is Link Radius

Slide 30

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Slide 32

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Slide 35

Slide 36



CPU

Challenge

Target is Covered




System is alive


Communication Range

Sensing Range

d le Rs

sensor

target


sensing radius

Rc

Rs


Communication Range

Sensing Range

d le Rs

d le Rc

sensor

target


sensing radius

Rc

Rs








[Resource Saving]


sensing disks


Itrsquos NP-hard



Outline

I Introduction


III Final Remarks

Main Results

Random algorithm


number of sensors

Partition algorithm


network

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 1

Group

Steiner Tree




How to map to GST








How to map to GST











Group Steiner Tree




Coverage

b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7


Connectivity

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7



b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4



What Is Link Radius



r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

Slide 1

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What Is Link Radius

Slide 30

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Slide 32

Slide 33

Slide 34

Slide 35

Slide 36

Challenge

Target is Covered




System is alive


Communication Range

Sensing Range

d le Rs

sensor

target


sensing radius

Rc

Rs


Communication Range

Sensing Range

d le Rs

d le Rc

sensor

target


sensing radius

Rc

Rs








[Resource Saving]


sensing disks


Itrsquos NP-hard



Outline

I Introduction


III Final Remarks

Main Results

Random algorithm


number of sensors

Partition algorithm


network

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 1

Group

Steiner Tree




How to map to GST








How to map to GST











Group Steiner Tree




Coverage

b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7


Connectivity

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7



b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4



What Is Link Radius



r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

Slide 1

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What Is Link Radius

Slide 30

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Slide 36


Communication Range

Sensing Range

d le Rs

sensor

target


sensing radius

Rc

Rs


Communication Range

Sensing Range

d le Rs

d le Rc

sensor

target


sensing radius

Rc

Rs








[Resource Saving]


sensing disks


Itrsquos NP-hard



Outline

I Introduction


III Final Remarks

Main Results

Random algorithm


number of sensors

Partition algorithm


network

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 1

Group

Steiner Tree




How to map to GST








How to map to GST











Group Steiner Tree




Coverage

b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7


Connectivity

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7



b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4



What Is Link Radius



r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

Slide 1

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Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

What Is Link Radius

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36


Communication Range

Sensing Range

d le Rs

d le Rc

sensor

target


sensing radius

Rc

Rs








[Resource Saving]


sensing disks


Itrsquos NP-hard



Outline

I Introduction


III Final Remarks

Main Results

Random algorithm


number of sensors

Partition algorithm


network

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 1

Group

Steiner Tree




How to map to GST








How to map to GST











Group Steiner Tree




Coverage

b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7


Connectivity

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7



b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4



What Is Link Radius



r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

What Is Link Radius

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36








[Resource Saving]


sensing disks


Itrsquos NP-hard



Outline

I Introduction


III Final Remarks

Main Results

Random algorithm


number of sensors

Partition algorithm


network

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 1

Group

Steiner Tree




How to map to GST








How to map to GST











Group Steiner Tree




Coverage

b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7


Connectivity

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7



b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4



What Is Link Radius



r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

What Is Link Radius

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36


Itrsquos NP-hard



Outline

I Introduction


III Final Remarks

Main Results

Random algorithm


number of sensors

Partition algorithm


network

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 1

Group

Steiner Tree




How to map to GST








How to map to GST











Group Steiner Tree




Coverage

b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7


Connectivity

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7



b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4



What Is Link Radius



r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

What Is Link Radius

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36

Outline

I Introduction


III Final Remarks

Main Results

Random algorithm


number of sensors

Partition algorithm


network

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 1

Group

Steiner Tree




How to map to GST








How to map to GST











Group Steiner Tree




Coverage

b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7


Connectivity

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7



b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4



What Is Link Radius



r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

What Is Link Radius

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36

Main Results

Random algorithm


number of sensors

Partition algorithm


network

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 1

Group

Steiner Tree




How to map to GST








How to map to GST











Group Steiner Tree




Coverage

b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7


Connectivity

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7



b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4



What Is Link Radius



r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

What Is Link Radius

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 1

Group

Steiner Tree




How to map to GST








How to map to GST











Group Steiner Tree




Coverage

b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7


Connectivity

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7



b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4



What Is Link Radius



r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

What Is Link Radius

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36



How to map to GST








How to map to GST











Group Steiner Tree




Coverage

b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7


Connectivity

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7



b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4



What Is Link Radius



r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

What Is Link Radius

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36


How to map to GST











Group Steiner Tree




Coverage

b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7


Connectivity

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7



b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4



What Is Link Radius



r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

What Is Link Radius

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36





Group Steiner Tree




Coverage

b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7


Connectivity

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7



b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4



What Is Link Radius



r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

What Is Link Radius

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36



Coverage

b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7


Connectivity

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7



b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4



What Is Link Radius



r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

What Is Link Radius

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7


Connectivity

b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7



b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4



What Is Link Radius



r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

What Is Link Radius

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36


b3

b1

b2

b6

b5

b4

S1S2

S3S4

b7



b2 b6b3 b4b1 b5 b7

S1 S2

S1 S3

S1 S2 S3

S2 S3

S2 S4

S3 S4


S2 S3 S4



What Is Link Radius



r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

What Is Link Radius

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36



What Is Link Radius



r

r

Communication disk

Sensing disk

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

What Is Link Radius

Slide 30

Slide 31

Slide 32

Slide 33

Slide 34

Slide 35

Slide 36

Connected


Area

Connected

Sensor Cover

with Target Points


1

Algorithm 2





















opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

Slide 1

Slide 2

Slide 3

Slide 4

Slide 5

Slide 6

Slide 7

Slide 8

Slide 9

Slide 10

Slide 11

Slide 12

Slide 13

Slide 14

Slide 15

Slide 16

Slide 17

Slide 18

Slide 19

Slide 20

Slide 21

Slide 22

Slide 23

Slide 24

Slide 25

Slide 26

Slide 27

Slide 28

What Is Link Radius

Slide 30

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opt (Network ST)



Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

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What Is Link Radius

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Outline

I Introduction


III Final Remarks

Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

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What Is Link Radius

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Future Works

Ο(log3n log log n)




Ο(r)


THANK YOU

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What Is Link Radius

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THANK YOU

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What Is Link Radius

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[email protected] Approximations for Min Connected Sensor Cover Ding-Zhu Du University of Texas...

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