Zoë Abrams, Ashish Goel, Serge Plotkin Stanford University Set K-Cover Algorithms for Energy Efficient Monitoring in Wireless Sensor Networks

Square field Locations to monitor Sensors scattered across the field Sensor Monitoring Example Components

Each sensor transmits for 1 continuous hour. Network monitors for 3 hours. Uniform sensing range. Sensor Monitoring Example Problem Parameters

Activate covers iteratively in a round robin fashion. Partition sensors into K=3 covers. Covers = {Red, Green, Blue} Sensor Monitoring Example Set K-Cover Approach

When Red is active, 23 out of 24 locations are covered. Sensor Monitoring Example Activate Red

When Green is active, 16 out of 24 locations are covered. Sensor Monitoring Example Activate Green

When Blue is active, 18 out of 24 locations are covered. Sensor Monitoring Example Activate Blue

23 Red 16 Green 18 Blue 47 Total + Sensor Monitoring Example Objective Function Compared with naïve simultaneous sensor activation: 24 Total

Given: Set S of locations. S j is the set of locations covered by sensor j. A collection of subsets. Positive integer k > 1. Find: Partition the sensors into k covers {c 1,...,c k } such that is maximized. Set K-Cover Problem Formal Definition SensorsLocations

Negative Result It is NP-Complete to guarantee better than 15/16 of the optimal coverage. This is due to a reduction from E4 Set Splitting.

Maximize the number of times the least covered location is covered. First Set K-Cover formulation considers fairness criteria (Slijepcevic and Potkonjak [2001]). — Require every locations is in all covers. A few, or even a single location with low coverage can drastically limit the size of k. Fairness Criteria

Sensor Schedules to Conserve Energy D. Tian, and N.D. Georganas [2003]. F. Ye, G. Zhong, S. Lu, and L. Zhang [2002]. T. Yan, T. He, and J.A. Stankovic [2003]. Related Work

Our Contributions Set K-Cover is NP-Complete Randomized Algorithm Distributed Greedy Algorithm Centralized Greedy Algorithm Simulation Results

Randomized Algorithm Each sensor chooses a random number i  {1,...,k} and assigns self to cover c i. Minimal assumptions, simple algorithm, running time O(1). Expected approximation ratio 1 – 1/e.

Fairness of Randomized Algorithm Each location is within expected 1- 1/e of its optimum coverage. Maximizing the minimum covered element. — With high probability (  1 - 1/n), the solution is within O(log n) of optimum.

Distributed Greedy Algorithm Distributed Greedy Algorithm at sensor j Few assumptions, running time nk|S max |, ½ approximation ratio. While t < j Receive message that location v is covered by sensor t in cover c i if S j covers v. If t = j Choose c i that has the smallest intersection with S j. Assigns self to cover c i. Broadcast this assignment to neighbors.

= Number of elements newly covered by adding. Greedy Sensor Partition Areas Red Cover Green Cover Distributed Greedy Algorithm Proof

OPT Sensor Partition = Number of elements newly covered by adding. Iterate back through sensors. = Number of elements newly covered by adding. Greedy Sensor Partition Areas Red Cover Green Cover Distributed Greedy Algorithm Proof Contribution of OPT

Two Observations: Therefore, Recall, = Number of elements newly covered by adding. Proof Conclusion for Distributed Greedy Algorithm

Centralized Greedy Algorithm Derandomization using the method of conditional expectation. Each area is weighted according to how likely it is to be chosen in a future iteration. Many assumptions, running time 2nk|S max |, deterministic approximation ratio 1-1/e. For j = 1 until n Assign S j to cover c i

Objective Function Simulation Results |S| = 1000 and k = 10. Deterministic algorithms perform far above their worst case bounds (consistently more than 72% of OPT).

Network Longevity Simulation Results Maximize k such that the total coverage is more than.8kn. Increase in longevity is proportional to amount of overlap between sensors.

Fairness Simulation Results Number of sensors that cover location v Number of covers that cover location v in solution divided by k k = 10 |S| = 200 n = 100 |E| = 2000

Summary of Results

The End

Location cannot be in more covers than there are sensors that cover it. Location cannot be in more than k covers. Coverage of an area is proportional to to min(k, N v ). Proportional Fairness Criteria