Presentation is loading. Please wait.

Presentation is loading. Please wait.

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

Similar presentations


Presentation on theme: "Approximations for Min Connected Sensor Cover Ding-Zhu Du University of Texas at Dallas."— Presentation transcript:

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

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

3 Have you watched movie Twister? sensor Bucket of sensors tornado

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

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

6 Buildings Where are all the sensors?

7 Transportation systems, etc Where are all the sensors?

8 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

9 Applications of Senor Web

10 observer An example of sensor web

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

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

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

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

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

16 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

17 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

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

19 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.

20 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

21 1 2 Min-CSCMin-CTCGST

22 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.

23 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.

24 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.

25 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

26 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

27 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

28 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.

29 What Is Link Radius? Communication disk Sensing disk

30 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’].

31 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

32 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

33 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)

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

35 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.

36 THANK YOU!


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

Similar presentations


Ads by Google