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

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 ExampleComponents

•Each sensor transmitsfor 1 continuous hour.

•Network monitorsfor 3 hours.

•Uniform sensingrange.

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 ExampleSet K-Cover Approach

•When Red is active,23 out of 24 locations are covered.

Sensor Monitoring ExampleActivate Red

•When Green is active,16 out of 24 locations are covered.

Sensor Monitoring ExampleActivate Green

•When Blue is active,18 out of 24 locationsare covered.

Sensor Monitoring ExampleActivate Blue

23 Red

16 Green

18 Blue

47 Total

+

Sensor Monitoring ExampleObjective Function

Compared with naïve simultaneous sensor activation:

24 Total

Given:• Set S of locations.• Sj is the set of locations covered by sensor j.• A collection of subsets.• Positive integer k > 1.

Find:• Partition the sensors into k covers {c1, ...,ck} such that is maximized.

Set K-Cover Problem Formal Definition

Sensors Locations

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 ci.

• 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|Smax|, ½ approximation ratio.

While t < jReceive message that location v is covered by

sensor t in cover ci if Sj covers v.If t = j

Choose ci that has the smallest intersection with Sj.

Assigns self to cover ci.Broadcast this assignment to neighbors.

= Number of elements newly covered by adding .

Greedy Sensor Partition

Areas

Red CoverGreen 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 CoverGreen Cover

Distributed Greedy Algorithm ProofContribution of OPT

Two Observations:1. 2.

Therefore,

Recall, = Number of elements newly covered by adding . = Number of elements newly covered by adding .

Proof Conclusion for Distributed Greedy Algorithm

Centralized 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|Smax|, deterministic approximation ratio 1-1/e.

For j = 1 until n

Assign Sj to cover ci

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| = 200n = 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, Nv).

Proportional Fairness Criteria