National Tsing Hua University

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

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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.

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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.

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• 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.

Data aggregation (1/2)

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Data aggregation (2/2)

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30℃

31℃

q = 3An example

n(transmitted packets) = 5

sink

29℃

31℃28℃32℃

31℃ 29℃

31℃30℃29℃32℃31℃ 28℃

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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

2143

30℃

31℃

q = 4sink

29℃

31℃28℃32℃

31℃ 29℃

31℃ 32℃

Each node sends exactly one packet31℃ 29℃

31℃30℃29℃32℃31℃28℃Max temperature query

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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.

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31℃30℃

31℃

Why does routing structure matter?

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30℃

31℃

q = 3sink

29℃

31℃28℃32℃

Tx = 2Rx = 1energy cost = (2+1)⨉5This is a shortest path tree.Let’s see the optimal tree.29℃

29℃31℃

29℃31℃30℃

29℃

29℃32℃31℃ 28℃32℃31℃28℃

Shortest path tree may NOT be an optimal tree.energy cost = (2+1)⨉4

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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.

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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.

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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.

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sink21

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sink2A feasible routing tree A relay node.

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• Steiner tree and shortest path tree:• Bad news: bad approximation ratios• Good news: perform well on some caseq is small q is largeShortest path treeSteiner tree

We want to combine this 2 advantages

Inspiration(1/2)

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 that1. 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)

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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

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• 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)

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Our algorithm:A shortest path tree on the CND problem’s approximation solutionA better approximation algorithm

(2/3)

For any λ-approximation algorithm of the CND problem, there is a corresponding 2λ-approximation algorithm for our problem.

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• 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.

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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.

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Simulation• We will compute a lower bound (LB).• LB = the maximum of 2 other lower bounds1. 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 packet2. 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.

300

500

Simulation-without relay node

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5 10 15 20 25 30 35 40 45 50Aggregation Ratio

Energy

Cost

200

400

600700800900

1000 Lower Bound: Uniform Report Size

Lower Bound: Uniform Report SizeLower Bound: Uniform Report Size

300

500


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Energy

Cost

200

400

600700800900

1000 Lower Bound: Uniform Report SizeLower Bound: Non-Uniform Report Size

Lower Bound: Non-Uniform Report Size Lower Bound: Non-Uniform Report Size

300

500


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Energy

Cost

200

400

600700800900

1000 Shortest Path Tree: Uniform Report SizeLower Bound: Uniform Report SizeLower Bound: Non-Uniform Report Size

Shortest Path Tree: Uniform Report Size Shortest Path Tree: Uniform Report Size

300

500


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Energy

Cost

200

400

600700800900

1000 Shortest Path Tree: Uniform Report SizeShortest Path Tree: Non-Uniform Report SizeLower Bound: Uniform Report SizeLower Bound: Non-Uniform Report Size

Shortest Path Tree: Non-Uniform Report SizeShortest Path Tree: Non-Uniform Report Size

300500300

500


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Energy

Cost

200

400

600700800900


The ratios are less than 2

300

500


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Energy

Cost

200

400

600700800900


The performances are close to the optimums when the aggregation ratio is large


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Energy

Cost

200300400500600700800900

1000Arbitrary Spanning Tree: Uniform Report Size

Shortest Path Tree: Uniform Report SizeShortest Path Tree: Non-Uniform Report SizeLower Bound: Uniform Report SizeLower Bound: Non-Uniform Report Size

Arbitrary Spanning Tree: Uniform Report SizeArbitrary Spanning Tree: Uniform Report Size

Arbitrary Spanning Tree: Non-Uniform Report Size


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Energy

Cost

200300400500600700800900

1000Arbitrary Spanning Tree: Uniform Report SizeArbitrary Spanning Tree: Non-Uniform Report Size


Arbitrary Spanning Tree: Non-Uniform Report Size


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Energy

Cost

200300400500600700800900

1000Arbitrary Spanning Tree: Uniform Report SizeArbitrary Spanning Tree: Non-Uniform Report SizeThe ratios are big


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Simulation-with relay nodeuniform 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.

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500Simulation-with relay node

uniform report size

5 10 15 20 25 30 35 40 45Aggregation Ratio

Energy

Cost

200250300350400450

Lower Bound Lower Bound

Lower Bound

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uniform report size


Energy

Cost

200250300350400450

Algorithm 1

Algorithm 1Algorithm 1

Lower Bound

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uniform report size


Energy

Cost

200250300350400450

Algorithm 1 Lower Bound Algorithm 2


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uniform report size


Energy

Cost

200250300350400450


The ratios are less than 2

Lower Bound

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uniform report size


Energy

Cost

200250300350400450

Algorithm 1Algorithm 2Hassin’s Algorithm

Hassin’s AlgorithmHassin’s Algorithm

Lower Bound

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uniform report size


Energy

Cost

200250300350400450

Algorithm 1Algorithm 2Hassin’s Algorithm

The performances are close

Lower Bound

38


uniform report size


Energy

Cost

200250300350400450

Algorithm 1Algorithm 2Hassin’s AlgorithmShortest Path Tree

Shortest Path TreeShortest Path Tree

Lower Bound

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uniform report size


Energy

Cost

200250300350400450

Algorithm 1Algorithm 2Hassin’s AlgorithmShortest Path Tree Steiner TreeSteiner TreeSteiner Tree

Lower Bound

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uniform report size


Energy

Cost

200250300350400450

Algorithm 1Algorithm 2Hassin’s AlgorithmShortest Path Tree Steiner TreeWhen the aggregation ratio is small, shortest path tree performs better

Lower Bound

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uniform report size


Energy

Cost

200250300350400450

Algorithm 1Algorithm 2Hassin’s AlgorithmShortest Path Tree Steiner TreeWhen the aggregation ratio is large, Steiner tree is betterBoth of them perform well on average case

Lower Bound

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uniform report size


Energy

Cost

200250300350400450

Algorithm 1Algorithm 2Hassin’s AlgorithmShortest Path Tree Steiner TreeArbitrary Spanning TreeArbitrary Spanning TreeArbitrary Spanning Tree

Lower Bound

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Simulation-with relay nodeuniform report size

The result is similar to the previous one.

Non-

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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.

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Thank You

National Tsing Hua University

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