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

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

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, Our goal is to minimize the total energy cost. 3

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, The aggregation ratio, q, is the size of report that can be aggregated into 1 packet. 4 Data aggregation (1/2)

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

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 ℃ 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

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

31℃ 30℃ 31℃ Why does routing structure matter? ℃ 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

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

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

11

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 A feasible routing tree 12 A relay node.

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

[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, 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

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

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

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

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,

19

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

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,

Simulation -without relay node Aggregation Ratio Energy Cost Lower Bound: Uniform Report Size

Simulation -without relay node Aggregation Ratio Energy Cost Lower Bound: Uniform Report Size Lower Bound: Non-Uniform Report Size

Simulation -without relay node Aggregation Ratio Energy Cost Shortest Path Tree: Uniform Report Size Lower Bound: Uniform Report Size Lower Bound: Non-Uniform Report Size Shortest Path Tree: Uniform Report Size

Simulation -without relay node Aggregation Ratio Energy Cost 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

Simulation -without relay node Aggregation Ratio Energy Cost 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

Simulation -without relay node Aggregation Ratio Energy Cost 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

Simulation -without relay node Aggregation Ratio Energy Cost 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

Arbitrary Spanning Tree: Non-Uniform Report Size Simulation -without relay node Aggregation Ratio Energy Cost 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

Simulation -without relay node Aggregation Ratio Energy Cost 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

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

500 Simulation-with relay node uniform report size Aggregation Ratio Energy Cost Lower Bound

500 Simulation-with relay node uniform report size Aggregation Ratio Energy Cost Algorithm 1 Lower Bound

500 Simulation-with relay node uniform report size Aggregation Ratio Energy Cost Algorithm 1 Lower Bound Algorithm 2

500 Simulation-with relay node uniform report size Aggregation Ratio Energy Cost Algorithm 1 Algorithm 2 The ratios are less than 2 Lower Bound

500 Simulation-with relay node uniform report size Aggregation Ratio Energy Cost Algorithm 1 Algorithm 2 Hassin’s Algorithm Lower Bound

500 Simulation-with relay node uniform report size Aggregation Ratio Energy Cost Algorithm 1 Algorithm 2 Hassin’s Algorithm The performances are close Lower Bound

500 Simulation-with relay node uniform report size Aggregation Ratio Energy Cost Algorithm 1 Algorithm 2 Hassin’s Algorithm Shortest Path Tree Lower Bound

500 Simulation-with relay node uniform report size Aggregation Ratio Energy Cost Algorithm 1 Algorithm 2 Hassin’s Algorithm Shortest Path Tree Steiner Tree Lower Bound

500 Simulation-with relay node uniform report size Aggregation Ratio Energy Cost 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

500 Simulation-with relay node uniform report size Aggregation Ratio Energy Cost 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

500 Simulation-with relay node uniform report size Aggregation Ratio Energy Cost Algorithm 1 Algorithm 2 Hassin’s Algorithm Shortest Path Tree Steiner Tree Arbitrary Spanning Tree Lower Bound

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

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

Thank You 45