1. On the Construction of Data Aggregation Tree with Minimum Energy Cost in Wireless Sensor Networks: NP-Completeness and Approximation Algorithms National Tsing Hua University Tung-Wei Kuo and Ming-Jer Tsai Department of Computer Science Hsinchu 30013, Taiwan, ROC

2. Motivation (1/2) In a wireless sensor network (WSN), a sink collects reports from each sensor periodically. For example: – In a building – Collecting data like 1.temperature, 2.concentration of CO, 3.power consumed by some equipment. 2

3. Motivation (2/2) Sensors are equipped with an AC power plug or sustained power supply. The Octopus X WSN [1] : [1] Octopus wireless sensor network, http://163.13.128.59/. Our goal is to minimize the total energy cost. 3

4. Data aggregation is a way to reduce the number of transmitted packets. – The energy cost is decreased. – It is performed according to the aggregation ratio, q [2]. [2] C. Liu and G. Cao, “Distributed monitoring and aggregation in wireless sensor networks,” in IEEE INFOCOM, 2010. The aggregation ratio, q, is the size of report that can be aggregated into 1 packet. 4 Data aggregation (1/2)

5. Data aggregation (2/2) 2 2 1 1 4 4 3 3 30℃ 31℃ q = 3 An example n(transmitted packets) = 5 sink 29℃ 31℃ 28℃ 32℃ 31℃ 29℃ 31℃ 30℃ 29℃ 32℃ 31℃ 28℃ 5

6. We can simulate this using our model by setting q to large enough (e.g. 4) Data aggregation model: a special case when q = ∞ Simulate n(transmitted packets) of MAX query 2 2 1 1 4 4 3 3 30℃ 31℃ q = 4 sink 29℃ 31℃ 28℃ 32℃ 31℃ 29℃ 31℃32℃ Each node sends exactly one packet 31℃29℃31℃ 30℃ 29℃ 32℃ 31℃ 28℃ 6 Max temperature query

7. Problem definition A static routing tree is considered here. To estimate the energy cost, we consider – Tx, the energy to transmit a packet, and – Rx, the energy to receive a packet. Given the aggregation ratio q, Tx, and Rx: We want to find an optimal tree to minimize the energy cost. 7

8. 31℃ 30℃ 31℃ Why does routing structure matter? 2 2 1 1 4 4 3 3 30℃ 31℃ q = 3 sink 29℃ 31℃ 28℃ 32℃ Tx = 2 Rx = 1 energy cost = (2+1)⨉5 This is a shortest path tree. Let’s see the optimal tree. 29℃ 31℃ 29℃ 31℃ 30℃ 29℃ 32℃ 31℃ 28℃32℃ 31℃ 28℃ Shortest path tree may NOT be an optimal tree. energy cost = (2+1)⨉4 8

9. NP-completeness This problem is NP-complete. Idea of the proof: – Does there exist a tree such that every node sends only one packet? We will design an approximation algorithm. 9

10. Our approximation algorithm Our Algorithm: Shortest path tree. It is a 2-approximation algorithm. Other benefits: 1.Distributed implementation. 2.Only one input: the network topology. 10

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12. A new problem – when relay nodes exist Relay nodes do not generate reports. A feasible routing tree only needs to span all non-relay nodes in this problem. 4 4 3 3 sink 2 2 1 4 4 3 3 2 2 A feasible routing tree 12 A relay node.

13. Steiner tree and shortest path tree: Bad news: bad approximation ratios Good news: perform well on some case q is smallq is large Shortest path tree Steiner tree We want to combine this 2 advantages Inspiration (1/2) 13

14. [3] F. S. Salman, J. Cheriyan, R. Ravi, and S. Subramanian, “Approximating the single-sink link- installation problem in network design,” SIAM J. on Optimization, vol. 11, pp. 595–610, 2000. We want a subgraph such that 1.The path for each non-relay node is short. 2.The number of spanned edges is small. Salman et al. compute a subgraph that has the above properties [3].  But, the subgraph might not be a tree. Inspiration (2/2) 14

15. Our algorithm: A shortest path tree on Salman’s subgraph It is a 7-approximation algorithm. Only one input: the network topology. Our approximation algorithm 15

16. Using the subgraph, Salman et al. design a 7- approximation algorithm for the Capacitated Network Design (CND) problem. The CND problem is similar to ours except that … Difference: the solution may NOT be a tree. A better approximation algorithm (1/3) 16

17. Our algorithm: A shortest path tree on the CND problem’s approximation solution A better approximation algorithm (2/3) For any λ-approximation algorithm of the CND problem, there is a corresponding 2λ- approximation algorithm for our problem. 17

18. When all the report sizes are the same: – We obtain a 5.1-approximation algorithm – It is based on Hassin’s 2.55-CND approximation algorithm [4]. In other case: – We obtain a 7.1-approximation algorithm for our problem. – It is based on Hassin’s 3.55-CND approximation algorithm [4]. A better approximation algorithm (3/3) [4] R. Hassin, R. Ravi, and F. S. Salman, “Approximation algorithms for a capacitated network design problem,” Algorithmica, vol. 38, pp. 417–431, 2004. 18

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20. Simulation Simulation Settings: – 100 sensors are randomly placed in a 100*100 field – Transmission range = 20 – Tx = 2, Rx = 1 – Report size = 1 (uniform report size), or 1~5 (non-uniform report size) – Aggregation ratio = 2, 4, 6, …, 50 for uniform report size, and 2, 4, 6, …, 100 for non-uniform report size The result is obtained by averaging data of 30 different networks. 20

21. Simulation We will compute a lower bound (LB). LB = the maximum of 2 other lower bounds 1.The optimal value if fractional packets are allowed (min cost flow problem) E.g. report size = 5, aggregation ratio = 10 → transmit 0.5 packet, instead of 1 packet 2.Minimum number of spanned edges (Steiner tree problem) We use a 2-approximation algorithm to compute Steiner tree [5]. [5] L. Kou, G. Markowsky, and L. Berman, “A fast algorithm for steiner trees,” Acta Informatica, vol. 15, pp. 141–145, 1981. 21

22. 300 500 Simulation -without relay node 22 5 101520 25 30 35404550 Aggregation Ratio Energy Cost 200 400 600 700 800 900 1000 Lower Bound: Uniform Report Size

23. 300 500 Simulation -without relay node 23 5 101520 25 30 35404550 Aggregation Ratio Energy Cost 200 400 600 700 800 900 1000 Lower Bound: Uniform Report Size Lower Bound: Non-Uniform Report Size

24. 300 500 Simulation -without relay node 24 5 101520 25 30 35404550 Aggregation Ratio Energy Cost 200 400 600 700 800 900 1000 Shortest Path Tree: Uniform Report Size Lower Bound: Uniform Report Size Lower Bound: Non-Uniform Report Size Shortest Path Tree: Uniform Report Size

25. 300 500 Simulation -without relay node 25 5 101520 25 30 35404550 Aggregation Ratio Energy Cost 200 400 600 700 800 900 1000 Shortest Path Tree: Uniform Report Size Shortest Path Tree: Non-Uniform Report Size Lower Bound: Uniform Report Size Lower Bound: Non-Uniform Report Size Shortest Path Tree: Non-Uniform Report Size

26. 300 500 300 500 Simulation -without relay node 26 5 101520 25 30 35404550 Aggregation Ratio Energy Cost 200 400 600 700 800 900 1000 Shortest Path Tree: Uniform Report Size Shortest Path Tree: Non-Uniform Report Size Lower Bound: Uniform Report Size Lower Bound: Non-Uniform Report Size The ratios are less than 2

27. 300 500 Simulation -without relay node 27 5 101520 25 30 35404550 Aggregation Ratio Energy Cost 200 400 600 700 800 900 1000 Shortest Path Tree: Uniform Report Size Shortest Path Tree: Non-Uniform Report Size Lower Bound: Uniform Report Size Lower Bound: Non-Uniform Report Size The performances are close to the optimums when the aggregation ratio is large

28. Simulation -without relay node 28 5 101520 25 30 35404550 Aggregation Ratio Energy Cost 200 300 400 500 600 700 800 900 1000 Arbitrary Spanning Tree: Uniform Report Size Shortest Path Tree: Uniform Report Size Shortest Path Tree: Non-Uniform Report Size Lower Bound: Uniform Report Size Lower Bound: Non-Uniform Report Size Arbitrary Spanning Tree: Uniform Report Size

29. Arbitrary Spanning Tree: Non-Uniform Report Size Simulation -without relay node 29 5 101520 25 30 35404550 Aggregation Ratio Energy Cost 200 300 400 500 600 700 800 900 1000 Arbitrary Spanning Tree: Uniform Report Size Arbitrary Spanning Tree: Non-Uniform Report Size Shortest Path Tree: Uniform Report Size Shortest Path Tree: Non-Uniform Report Size Lower Bound: Uniform Report Size Lower Bound: Non-Uniform Report Size Arbitrary Spanning Tree: Non-Uniform Report Size

30. Simulation -without relay node 30 5 101520 25 30 35404550 Aggregation Ratio Energy Cost 200 300 400 500 600 700 800 900 1000 Arbitrary Spanning Tree: Uniform Report Size Arbitrary Spanning Tree: Non-Uniform Report Size The ratios are big Shortest Path Tree: Uniform Report Size Shortest Path Tree: Non-Uniform Report Size Lower Bound: Uniform Report Size Lower Bound: Non-Uniform Report Size

31. Simulation-with relay node uniform report size Two approximation algorithms here: 1.A 7-approxmiation algorithm based on Salman’s approximation algorithm. (Algorithm 1) 2.A 5.1-approxmiation algorithm based on Hassin’s approximation algorithm. (Algorithm 2) We also compare to the performance of Hassin’s algorithm directly, i.e. a non-tree routing structure. 31

32. 500 Simulation-with relay node uniform report size 32 5 101520 25 30 354045 Aggregation Ratio Energy Cost 200 250 300 350 400 450 Lower Bound

33. 500 Simulation-with relay node uniform report size 33 5 101520 25 30 354045 Aggregation Ratio Energy Cost 200 250 300 350 400 450 Algorithm 1 Lower Bound

34. 500 Simulation-with relay node uniform report size 34 5 101520 25 30 354045 Aggregation Ratio Energy Cost 200 250 300 350 400 450 Algorithm 1 Lower Bound Algorithm 2

35. 500 Simulation-with relay node uniform report size 35 5 101520 25 30 354045 Aggregation Ratio Energy Cost 200 250 300 350 400 450 Algorithm 1 Algorithm 2 The ratios are less than 2 Lower Bound

36. 500 Simulation-with relay node uniform report size 36 5 101520 25 30 354045 Aggregation Ratio Energy Cost 200 250 300 350 400 450 Algorithm 1 Algorithm 2 Hassin’s Algorithm Lower Bound

37. 500 Simulation-with relay node uniform report size 37 5 101520 25 30 354045 Aggregation Ratio Energy Cost 200 250 300 350 400 450 Algorithm 1 Algorithm 2 Hassin’s Algorithm The performances are close Lower Bound

38. 500 Simulation-with relay node uniform report size 38 5 101520 25 30 354045 Aggregation Ratio Energy Cost 200 250 300 350 400 450 Algorithm 1 Algorithm 2 Hassin’s Algorithm Shortest Path Tree Lower Bound

39. 500 Simulation-with relay node uniform report size 39 5 101520 25 30 354045 Aggregation Ratio Energy Cost 200 250 300 350 400 450 Algorithm 1 Algorithm 2 Hassin’s Algorithm Shortest Path Tree Steiner Tree Lower Bound

40. 500 Simulation-with relay node uniform report size 40 5 101520 25 30 354045 Aggregation Ratio Energy Cost 200 250 300 350 400 450 Algorithm 1 Algorithm 2 Hassin’s Algorithm Shortest Path Tree Steiner Tree When the aggregation ratio is small, shortest path tree performs better Lower Bound

41. 500 Simulation-with relay node uniform report size 41 5 101520 25 30 354045 Aggregation Ratio Energy Cost 200 250 300 350 400 450 Algorithm 1 Algorithm 2 Hassin’s Algorithm Shortest Path Tree Steiner Tree When the aggregation ratio is large, Steiner tree is better Both of them perform well on average case Lower Bound

42. 500 Simulation-with relay node uniform report size 42 5 101520 25 30 354045 Aggregation Ratio Energy Cost 200 250 300 350 400 450 Algorithm 1 Algorithm 2 Hassin’s Algorithm Shortest Path Tree Steiner Tree Arbitrary Spanning Tree Lower Bound

43. Simulation-with relay node uniform report size The result is similar to the previous one. Non- 43

44. Conclusion We prove the problem of constructing a data aggregation tree with minimum energy cost is NP- complete and provide a 2-approximation algorithm. For the problem with relay nodes, we prove it is NP-complete and provide a 7-approximation algorithm. We show any λ-approximation algorithm of the CND problem can be used to obtain a 2λ- approximation algorithm of our problem. 44

45. Thank You 45