1. Optimal Sleep-Wakeup Algorithms for Barriers of Wireless Sensors S. Kumar, T. Lai, M. Posner and P. Sinha, BROADNETS ’ 2007

2. Outline Brief Introduction Problem Definition Proposed Algorithms Simulation Results

3. Network Model Omni-directional sensors are used. Assume every sensor knows its location and boundaries location. A WSN provides k- barrier coverage over a region if all crossing paths through this region can be covered k times.

4. Brief Introduction to Wireless Sensor Networks The major concern of wireless sensor network (WSN) is the network lifetime. A widely used technique to extend WSN lifetime is to use sleep-wakeup mechanism. In a WSN, the remaining lifetime of every sensor may be heterogeneous. For example, different traffic load, additional deployment. Usually k-coverage is required. Sensor lifetime and network lifetime.

5. Real Sensor Network To Topology Graph Construct a topology graph G(V,E) according to the real network N: 1. Every vertex in G represents a sensor in the real network. 2. Add two virtual vertices, s and t, which are correspond to the left and right boundaries. 3. There is an edge with capacity 1 between two vertices if their sensing range overlaps. This topology graph is called coverage graph in this paper. A path in G is equivalent to a barrier in the real sensor network. A path is referred to a group of sensors.

6. Problem Definition and Main Contributions Problem definition: 1. Achieve and maintain k-barrier coverage in the WSN. 2. Maximize the network lifetime. 3. Minimize the number of path switches. Path switches: 1. Anytime the sensors consisting the path are turned off then turned on later, it is considered a path switch. 2. Each time a sensor is turned on, some energy consuming activities have to be performed, i.e. route computation, synchronization. Contributions: 1. Propose an algorithm to optimally solve the maximizing network lifetime problem when sensor initial lifetimes are same. This algorithm also minimizes the number of path switches. 2. Propose an algorithm to optimally solve the maximizing network lifetime problem when sensor lifetimes are different. Prove minimizing the number of path switches is NP-Hard for the heterogeneous lifetime case.

7. Homogeneous Sensor Lifetime Case — Stint Algorithm The maximum number of node-disjoint path in a graph can be computed by applying the max-flow algorithm. Lemma: Consider a sensor network N. Let m≥k be the maximum number of node- disjoint paths between the two virtual vertices s and t in the coverage graph G(N). The maximum time for which the network N can provide k-barrier coverage is at most m/k.

8. An example of Stint Algorithm Compute the maximum number of node-disjoint path. m=8, k=3. First select {(1,2,3)}, keep them active for their entire lifetime. Then the remaining 5 paths will be arranged in f=5/gcd(5,3)=5 sets of 3 disjoint paths which is {(4,5,6),(5,6,7),(6,7,8),(7,8,4),( 8,4,5)}. Each of these sets will be active for gcd(5,3)/3=1/3 unit of lifetime. The total lifetime will be 1+5/3=8/3=m/k.

9. Stint Algorithm Can Minimizing Path Switches Compare these two cases: 1. {(1,2,3)}, 1 unit lifetime {(4,5,6),(5,6,7),(6,7,8),(7,8,4),(8,4,5)}, each with 1/3 unit lifetime 2. {(1,2,3),(4,5,6),(7,8,1),(2,3,4),(5,6,7),(8,1,2),(3,4, 5),(6,7,8)}, each with 1/3 unit lifetime. Case 1, path switch number=2; Case 2, path switch number=16. Theorem: If k<m<2k, the minimum number of path switches is k-gcd(m,k). Time complexity is dominated by Max-Flow algorithm which is O(VE 2 ).

10. Heterogeneous Lifetime Case Assume every sensor node can measure the remaining battery level and estimates its lifetime. Real sensor network N to coverage graph G L (N) with lifetime: 1. Every vertex in G L (N) represents a sensor in the real network. 2. Add two virtual vertices, s and t, which are correspond to the left and right boundaries. 3. There is an edge with infinite capacity between two vertices if their sensing range overlaps. 4. Each vertex u has a capacity c(u), equals to its remaining lifetime.

11. Basic Flow and Composite Flow Basic k-Flow of Value a: A basic k- flow of value a in G L (N) is a set of k node-disjoint flows, each have a value a. Composite k-Flow: A set of flows in G L (N) is called a composite k-flow if it can be expressed as a sum of basic k-flows. The maximum composite k-flow can be computed by an optimal algorithm MEM (1996). If the maximum composite k-flow computed by MEM is not node- disjoint, SEM(1993) algorithm will be invoked to optimally decompose the flows in component basic k- flows.

12. An Example of Scheduling The maximum time for which the sensor network N can provide k-barrier coverage is f/k. f is the maximum value of composite k-flow in G L (N). The time complexity of the proposed algorithm is dominated by the SEM algorithm which is O(kV 3 /log(V)). Minimizing path switches is NP-Hard.

13. Simulation Results

14. The End Thank you!