# Approximations for Min Connected Sensor Cover Ding-Zhu Du University of Texas at Dallas.

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lidong.wu@utdallas.edu Approximations for Min Connected Sensor Cover Ding-Zhu Du University of Texas at Dallas

Outline I. Introduction II. Two Approximations III. Final Remarks

Have you watched movie Twister? sensor Bucket of sensors tornado

Where are all the sensors? Smartphone with a dozen of sensors

Where are all the sensors? Wearable devices - Google Glass, Apple’s iWatch

Buildings Where are all the sensors?

Transportation systems, etc Where are all the sensors?

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

observer An example of sensor web

What’s Sensor? Small size Large number Tether- less BUT…

What’s limiting the task? Energy, Sense, Communication scale, CPU...

Challenge Target is Covered? Sensor system is Connected? Coverage & Connectivity Golden Rule, then we say System is alive!!

Coverage & Connectivity Communication Range Sensing Range d ≤ Rs sensor target communication radius sensing radius Rc Rs

Coverage & Connectivity Communication Range Sensing Range d ≤ Rs d ≤ Rc sensor target communication radius 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] communication network sensing disks

Previous Work for PTAS It’s NP-hard! Ο(r ln n) – approximation given by Gupta, Das and Gu [MobiHoc’03, 2003], where n is the number of sensors and r is the link radius of the sensor network. Min-Connected Sensor Cover Problem

Outline I. Introduction II. Two Approximations III. Final Remarks

Main Results Random algorithm: Ο(log 3 n log log n)-approximation, n is the number of sensors. Partition algorithm : Ο(r)-approximation, r is the link radius of the network.

C onnected S ensor C over with Target Area C onnected Sensor C over with T arget Points With a random algorithm which with probability 1- ɛ, produces an Ο(log 3 n log log n) - approximation. 1 Algorithm 1 G roup S teiner T ree 2 Min-CSCMin-CTCGST

1 2 Min-CSCMin-CTCGST

1 2 Min-CSCMin-CTCGST How to map to GST? Min-Connected Sensor Cover 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.

1 2 Min-CSCMin-CTCGST How to map to GST? Min-Connected Target Coverage Problem A uniform set of sensors, and a target POINTS Find a minimum # of sensors to meet two requirements: [Coverage] cover the target POINTS, and [Connectivity] form a connected communication network.

1 2 Min-CSCMin-CTCGST A graph G = (V, E) with positive edge weight c for every edge e ∈ E. k subsets (or groups ) of vertices G 1,..., G k, G i ⊆ V Find a minimum total weight tree T contains at least one vertex in each G i. Group Steiner Tree: Figure: GST Problem. This tree has minimum weight.

1 2 Min-CSCMin-CTCGST Choose at least one sensor from each group. Coverage b3b3 b1b1 b2b2 b6b6 b5b5 b4b4 S1S1 S2S2 S3S3 S4S4 b7b7 b2b2 b6b6 b3b3 b4b4 b1b1 b5b5 b7b7 S 1 S2 S 1 S3 S 1 S2 S 3 S 2 S3 S 2 S4 S 3 S4 * G i contains all sensors covering b i. S 2 S3 S4

1 2 Min-CSCMin-CTCGST b3b3 b1b1 b2b2 b6b6 b5b5 b4b4 S1S1 S2S2 S3S3 S4S4 b7b7 Consider communication network. Connectivity b2b2 b6b6 b3b3 b4b4 b1b1 b5b5 b7b7 S 1 S2 S 1 S3 S 1 S2 S 3 S 2 S3 S 2 S4 S 3 S4 * G i contains all sensors covering b i. S 2 S3 S4

1 2 Min-CSCMin-CTCGST b3b3 b1b1 b2b2 b6b6 b5b5 b4b4 S1S1 S2S2 S3S3 S4S4 b7b7 Find a group Steiner tree in communication network. Min- Coverage & Connectivity b2b2 b6b6 b3b3 b4b4 b1b1 b5b5 b7b7 S 1 S2 S 1 S3 S 1 S2 S 3 S 2 S3 S 2 S4 S 3 S4 * G i contains all sensors covering b i. S 2 S3 S4

1 2 Min-CSCMin-CTCGST 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(log 2 n log log n log k) from optimal.

C onnected S ensor C over with Target Area C onnected Sensor C over with T arget Points Connect output of Min-TC into Min-CTC. It can be done in Ο(r) - approximation. 1 Algorithm 2 2 Min-CSCMin-CTCMin-TC Refer to Lidong Wu’s paper [INFOCOM 2013’].

There exists a polynomial-time (1 + ε)- approximation for MIN-TC. Green is an opt (CTC), Red is an approx (TC). # < (1+ε) · opt (TC), < (1+ε) · opt (CTC) Step 2 Target Coverage

Byrka et al. [6] showed there exists a polynomial-time1.39- approximation of for Network Steiner Minimum Tree. Green is an opt (CTC), Red is an approx (TC). Step 2 Network Steiner Tree Let S′ ⊆ S be a (1 + ε )-approximation for MIN-TC. Assign weight one to every edge of G. Interconnect sensors in S′ to compute a Steiner tree T as network Steiner minimum tree. All sensors on the tree form an approx for min CTC. # nodes % approx for min CTC = # edges +1 % approx for Network ST < 1.39 · opt (Network ST) +1 < 1.39 · ??? · opt (CTC) + 1

Step 2 Network Steiner Tree Green is an opt (CTC). Red is an approx (TC). Each orange line has distance < r. opt (Network ST) < opt (CTC) -1 + r · # = opt (CTC) · O( r ) Note: # < (1+ε) · opt (CTC)

Outline I. Introduction II. Two Approximations III. Final Remarks

Future Works Ο(log 3 n 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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