1. Achieving Minimum Coverage Breach under Bandwidth Constraints in Wireless Sensor Networks Maggie X. Cheng, Lu Ruan and Weili Wu Dept. of Comput. Sci, Missouri Univ. IEEE INFOCOM 2005

2. Outline Introduction Minimum Breach Problem in Sensor Networks Problem Definition Complexity Classification of the Minimum Breach Problem Approximation Algorithm Integer Programming Formulation of the Minimum Breach Problem Heuristic 1: RELAXATION Heuristic 2: MINBREACH Simulation Study Conclusion and Extensions

3. Introduction Stochastically deployed sensor network. Oscillated between active modes and inactive modes. Divided into mutually exclusive subsets without consideration on subset sizes. Disjoint set covers C 1 = { s 1, s 2 } C 2 = { s 3, s 4 } [Ref 1] Energy-Efficient Target Coverage in Wireless Sensor Networks, infocom 2004Ref 1 [Ref 2] Maximal_Lifetime_Scheduling__in_sensor_surveillance_networksRef 2

4. Why are Bandwidth Constraints? Each Active Sensor will send the sensory data directly to the base station. “ Bandwidth ” is the total number of time slots. The total number of sensors simultaneously sending to the base station must be restricted by the bandwidth. …… Bandwidth Constraints Slot 1Slot 2Slot W

5. Problem Definition(1/2) Object has equal chance of being detected from all direction. If sensors lie within the area boundary, the object is considered covered.

6. Problem Definition(2/2) Breach: If a target is not covered by any active sensor, it is called “ breach ”. PROBLEM: MINIMUM BREACH Given a collection S of sensors, a collection A of targets, and the sensor-target coverage map. QUESTION: Can we divide S into disjoint subsets such that the overall breach is at most B and each subset has at most W sensors in it?

7. Complexity Classification of the Minimum Breach Problem — MINIMUM BREACH Prove that MINIMUM BREACH problem is NP-complete. 1. Divide the sensors into two disjoint subsets to minimize the overall breach. — MINIMUM 2SET BREACH problem 2. MINIMUM SET SPLITTING (NP-Complete)  Given a collection C of subsets of a finite set S  Is there a partition of S into two subsets S 1 and S 2 such that the cardinality of the subsets in C that are not entirely contained in either S 1 or S 2 (splitted) is at least |C|-B.

8. Integer Programming Formulation of the Minimum Breach Problem

10. Heuristic I: RELAXATION First Step: the Integer Programming problem is relaxed to a Linear Programming problem, and an optimal solution for LP is computed. Second Step: a greedy algorithm is employed to find an integer solution based on the optimal solution obtained at the first set. Third Step: the solution from problem is used to construct the subsets.

12. Heuristic II: MINBREACH x1x2x3...xnx1x2x3...xn x = = y k,i. x k,i

13. Heuristic II: MINBREACH I 1 : denote the rows in the upper part {0,1,-1} I 2 : denote the rows in the lower part {0,1} Jx: represent the columns that correspond to the {x k,i } in the original (IP) Jy: represent the columns that correspond to the {y k,j } in the original (IP)

14. Simulation Study As the number of targets increase from 10 to 100, the number of sensors also increase from 10 to 100, and bandwidth increase from 2 to 20. It verifies that higher f leads to higher breach rate.

16. Conclusion and Extensions To improve sensor coverage, deploying more sensors must be accompanied by increasing bandwidth, otherwise, the coverage may be decreased as a result. To minimize the maximal breach is also an NP-complete problem that requires efficient approximation algorithm.

17. [back]back ~  (n 3 ) where n= N(N+M)/W

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