1. Set K-Cover Algorithms for Energy Efficient Monitoring in Wireless Sensor Networks
Dileep Kumar
HMCL
1st June, 2014

2. Cooking example Salad: 25m prep, 0m cooking Chicken noodle:
10m prep, 40 min cooking
Rice pudding:
15 mins prep, 20m cooking

3. In what order should Martha make the dishes?
Martha can work on preparing one dish at a time, however once something is cooking, she can prepare another dish.
How quickly can she get all the dishes ready?
She starts at 5pm, and her guests will arrive at 6pm….

4. First try Prep time Cook time (10,40) (25,0) (15,20) 5:00pm 5:25pm

5. Second try (15,20) (10,40) (25,0) 5:00pm 5:10pm 5:25pm 5:50pm 5:50pm
First work on dishes with shortest preparation time?

6. This rule may not work all the time
Suppose the required times are:
Bulgur (5,10) Lentils (10, 60) Lamb (15, 75)
Shortest prep time order: start at 5pm, and finish lamb at 6:45pm
Longest cooking time first: food ready at 6:30pm.

7. What if she had to make several dishes?
For 3 dishes, there are only 6 possible orders. SCR,SRC,RSC,RCS,CSR,CRS.
The number of possible orderings of 10 dishes is 3,628,800.
For 15 dishes the number of possible orderings is 1,307,674,368,000!

8. Key Idea Order dishes in longest cooking time order.
Chicken noodle soup goes first (40 mins of cook time), next is the Rice pudding (20 mins of cook time), followed by the Salad (0 mins of cook time).
This is the best ordering. In other words, no other order can take less time.

9. Where to fill gas?
Suppose you want to go on a road trip across the US. You start from New York City and would like to drive to San Francisco.
You have :
roadmap
gas station locations and their gas prices
Want to:
minimize travel cost
The Gas Station Problem (Khuller, Malekian,Mestre), Eur. Symp. of Algorithms

10. Finding gas prices online

11. Structure of the Optimal Solution
Two vertices s & t
A fixed path v1 v2 v3 vn s t
2.99$
2.98$
2.97$
1.00$

12. Key Property Suppose the optimal sequence of stops is u1,u2,…,u.
If c(ui) < c(ui+1)  Fill up the whole tank
If c(ui) > c(ui+1)  Just fill enough to reach ui+1. ui
ui+1
c(ui)
c(ui+1)

15. Lifetime = 5 C1={(s1,r2)(s2,r2)} C2={(s1,r2)(s3,r2)}

16. Lifetime = 6 C1={(s1,r1)(s2,r2)} C2={(s1,r2)(s3,r1)}
C5={(s1,r1)(s2,r1)(s3,r1)}
C6={(s4,r2)}
Lifetime = 6

17. Distributed Greedy Algorithm
Preprocessing Phase
Each sensor determine the interest area
Sensor’s neighbors
Partition Phase
Few assumptions, running time nk|Smax|,
½ approximation ratio.

18. Distributed Greedy Algorithm
Distributed Greedy Algorithm at sensor j
While t < j
Receive 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.

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

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

21. Proof Conclusion for Distributed Greedy Algorithm
Recall,
= Number of elements newly covered by adding .
Two Observations: 1. 2.
Therefore,

22. Centralized Greedy Algorithm
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.
Initialize C={c1:=empty,….Ck:=empty}
For j = 1 until n
Assign Sj to cover ci

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

24. Network Longevity Simulation Results

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

26. End