National Tsing Hua University

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

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

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

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

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

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

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.

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

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.

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

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. sink sink 2 2 A relay node. 1 A feasible routing tree 3 3 4 4

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

14. Inspiration(2/2) • We want a subgraph such that • The path for each non-relay node is short. • 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. [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.

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

16. A better approximation algorithm (1/3) • 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.

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

18. A better approximation algorithm(3/3) • 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]. [4] R. Hassin, R. Ravi, and F. S. Salman, “Approximation algorithms for a capacitated network design problem,” Algorithmica, vol. 38, pp. 417–431, 2004.

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.

21. Simulation • We will compute a lower bound (LB). • LB = the maximum of 2 other lower bounds • 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 • 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 steinertrees,” ActaInformatica, vol. 15, pp. 141–145, 1981.

22. Simulation -without relay node 1000 Lower Bound: Uniform Report Size Lower Bound: Uniform Report Size Lower Bound: Uniform Report Size 900 800 Energy Cost 700 600 500 400 300 200 30 10 15 20 35 40 45 50 25 5 Aggregation Ratio

23. Simulation -without relay node 1000 Lower Bound: Uniform Report Size Lower Bound: Non-Uniform Report Size Lower Bound: Non-Uniform Report Size Lower Bound: Non-Uniform Report Size 900 800 Energy Cost 700 600 500 400 300 200 30 10 15 20 35 40 45 50 25 5 Aggregation Ratio

24. Simulation -without relay node 1000 Lower Bound: Uniform Report Size Lower Bound: Non-Uniform Report Size 900 Shortest Path Tree: Uniform Report Size Shortest Path Tree: Uniform Report Size Shortest Path Tree: Uniform Report Size 800 Energy Cost 700 600 500 400 300 200 30 10 15 20 35 40 45 50 25 5 Aggregation Ratio

25. Simulation -without relay node Shortest Path Tree: Non-Uniform Report Size 1000 Lower Bound: Uniform Report Size Lower Bound: Non-Uniform Report Size 900 Shortest Path Tree: Uniform Report Size Shortest Path Tree: Non-Uniform Report Size Shortest Path Tree: Non-Uniform Report Size 800 Energy Cost 700 600 500 400 300 200 30 10 15 20 35 40 45 50 25 5 Aggregation Ratio

26. Simulation -without relay node 1000 Lower Bound: Uniform Report Size Lower Bound: Non-Uniform Report Size 900 Shortest Path Tree: Uniform Report Size Shortest Path Tree: Non-Uniform Report Size 800 Energy Cost 700 600 500 500 400 300 300 The ratios are less than 2 200 30 10 15 20 35 40 45 50 25 5 Aggregation Ratio

27. Simulation -without relay node 1000 Lower Bound: Uniform Report Size Lower Bound: Non-Uniform Report Size 900 Shortest Path Tree: Uniform Report Size Shortest Path Tree: Non-Uniform Report Size 800 Energy Cost 700 The performances are close to the optimums when the aggregation ratio is large 600 500 400 300 200 30 10 15 20 35 40 45 50 25 5 Aggregation Ratio

28. Simulation -without relay node 1000 Lower Bound: Uniform Report Size Lower Bound: Non-Uniform Report Size 900 Shortest Path Tree: Uniform Report Size Shortest Path Tree: Non-Uniform Report Size 800 Arbitrary Spanning Tree: Uniform Report Size Arbitrary Spanning Tree: Uniform Report Size Arbitrary Spanning Tree: Uniform Report Size Energy Cost 700 600 500 400 300 200 30 10 15 20 35 40 45 50 25 5 Aggregation Ratio

29. Simulation -without relay node 1000 Lower Bound: Uniform Report Size Lower Bound: Non-Uniform Report Size 900 Shortest Path Tree: Uniform Report Size Shortest Path Tree: Non-Uniform Report Size 800 Arbitrary Spanning Tree: Uniform Report Size Arbitrary Spanning Tree: Non-Uniform Report Size Arbitrary Spanning Tree: Non-Uniform Report Size Arbitrary Spanning Tree: Non-Uniform Report Size Energy Cost 700 600 500 400 300 200 30 10 15 20 35 40 45 50 25 5 Aggregation Ratio

30. Simulation -without relay node 1000 Lower Bound: Uniform Report Size Lower Bound: Non-Uniform Report Size 900 Shortest Path Tree: Uniform Report Size Shortest Path Tree: Non-Uniform Report Size 800 Arbitrary Spanning Tree: Uniform Report Size Arbitrary Spanning Tree: Non-Uniform Report Size Energy Cost 700 The ratios are big 600 500 400 300 200 30 10 15 20 35 40 45 50 25 5 Aggregation Ratio

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

32. Simulation-with relay nodeuniform report size 500 Lower Bound Lower Bound Lower Bound 450 400 Energy Cost 350 300 250 200 30 10 15 20 35 40 45 25 5 Aggregation Ratio

33. Simulation-with relay nodeuniform report size 500 Lower Bound Algorithm 1 Algorithm 1 Algorithm 1 450 400 Energy Cost 350 300 250 200 30 10 15 20 35 40 45 25 5 Aggregation Ratio

34. Simulation-with relay nodeuniform report size 500 Lower Bound Algorithm 1 Algorithm 2 Algorithm 2 Algorithm 2 450 400 Energy Cost 350 300 250 200 30 10 15 20 35 40 45 25 5 Aggregation Ratio

35. Simulation-with relay nodeuniform report size 500 Lower Bound Algorithm 1 Algorithm 2 450 400 Energy Cost 350 The ratios are less than 2 300 250 200 30 10 15 20 35 40 45 25 5 Aggregation Ratio

36. Simulation-with relay nodeuniform report size 500 Hassin’s Algorithm Lower Bound Algorithm 1 Algorithm 2 450 Hassin’s Algorithm Hassin’s Algorithm 400 Energy Cost 350 300 250 200 30 10 15 20 35 40 45 25 5 Aggregation Ratio

37. Simulation-with relay nodeuniform report size 500 Lower Bound Algorithm 1 Algorithm 2 450 Hassin’s Algorithm 400 Energy Cost 350 300 The performances are close 250 200 30 10 15 20 35 40 45 25 5 Aggregation Ratio

38. Simulation-with relay nodeuniform report size 500 Lower Bound Algorithm 1 Algorithm 2 450 Hassin’s Algorithm Shortest Path Tree Shortest Path Tree Shortest Path Tree 400 Energy Cost 350 300 250 200 30 10 15 20 35 40 45 25 5 Aggregation Ratio

39. Simulation-with relay nodeuniform report size 500 Lower Bound Algorithm 1 Algorithm 2 450 Hassin’s Algorithm Shortest Path Tree Steiner Tree Steiner Tree Steiner Tree 400 Energy Cost 350 300 250 200 30 10 15 20 35 40 45 25 5 Aggregation Ratio

40. Simulation-with relay nodeuniform report size 500 Lower Bound Algorithm 1 Algorithm 2 450 Hassin’s Algorithm Shortest Path Tree Steiner Tree 400 Energy Cost When the aggregation ratio is small, shortest path tree performs better 350 300 250 200 30 10 15 20 35 40 45 25 5 Aggregation Ratio

41. Simulation-with relay nodeuniform report size 500 Lower Bound Algorithm 1 Algorithm 2 450 Hassin’s Algorithm Shortest Path Tree Steiner Tree 400 Energy Cost When the aggregation ratio is large, Steiner tree is better Both of them perform well on average case 350 300 250 200 30 10 15 20 35 40 45 25 5 Aggregation Ratio

42. Simulation-with relay nodeuniform report size 500 Lower Bound Algorithm 1 Algorithm 2 450 Hassin’s Algorithm Shortest Path Tree Steiner Tree 400 Energy Cost Arbitrary Spanning Tree Arbitrary Spanning Tree Arbitrary Spanning Tree 350 300 250 200 30 10 15 20 35 40 45 25 5 Aggregation Ratio

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

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.

45. Thank You