on the construction of data aggregation tree with minimum energy cost in wireless sensor networks:...
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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)
2143
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?
2143
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
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 SizeLower Bound: Non-Uniform Report Size
Lower Bound: Non-Uniform Report Size Lower Bound: Non-Uniform Report Size
300
500
Simulation
-without relay node
24
5 10 15 20 25 30 35 40 45 50Aggregation Ratio
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
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 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
Simulation
-without relay node
26
5 10 15 20 25 30 35 40 45 50Aggregation Ratio
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
The ratios are less than 2
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 Shortest Path Tree: Uniform Report SizeShortest Path Tree: Non-Uniform Report SizeLower Bound: Uniform Report SizeLower Bound: Non-Uniform Report Size
The performances are close to the optimums when the aggregation ratio is large
Simulation
-without relay node
28
5 10 15 20 25 30 35 40 45 50Aggregation Ratio
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
Simulation
-without relay node
29
5 10 15 20 25 30 35 40 45 50Aggregation Ratio
Energy
Cost
200300400500600700800900
1000Arbitrary Spanning Tree: Uniform Report SizeArbitrary Spanning Tree: Non-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: Non-Uniform Report Size
Simulation
-without relay node
30
5 10 15 20 25 30 35 40 45 50Aggregation Ratio
Energy
Cost
200300400500600700800900
1000Arbitrary Spanning Tree: Uniform Report SizeArbitrary Spanning Tree: Non-Uniform Report SizeThe ratios are big
Shortest Path Tree: Uniform Report SizeShortest Path Tree: Non-Uniform Report SizeLower Bound: Uniform Report SizeLower Bound: Non-Uniform Report Size
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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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500Simulation-with relay node
uniform report size
5 10 15 20 25 30 35 40 45Aggregation Ratio
Energy
Cost
200250300350400450
Algorithm 1
Algorithm 1Algorithm 1
Lower Bound
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500Simulation-with relay node
uniform report size
5 10 15 20 25 30 35 40 45Aggregation Ratio
Energy
Cost
200250300350400450
Algorithm 1 Lower Bound Algorithm 2
Algorithm 2Algorithm 2
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500Simulation-with relay node
uniform report size
5 10 15 20 25 30 35 40 45Aggregation Ratio
Energy
Cost
200250300350400450
Algorithm 1Algorithm 2
The ratios are less than 2
Lower Bound
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500Simulation-with relay node
uniform report size
5 10 15 20 25 30 35 40 45Aggregation Ratio
Energy
Cost
200250300350400450
Algorithm 1Algorithm 2Hassin’s Algorithm
Hassin’s AlgorithmHassin’s Algorithm
Lower Bound
37
500Simulation-with relay node
uniform report size
5 10 15 20 25 30 35 40 45Aggregation Ratio
Energy
Cost
200250300350400450
Algorithm 1Algorithm 2Hassin’s Algorithm
The performances are close
Lower Bound
38
500Simulation-with relay node
uniform report size
5 10 15 20 25 30 35 40 45Aggregation Ratio
Energy
Cost
200250300350400450
Algorithm 1Algorithm 2Hassin’s AlgorithmShortest Path Tree
Shortest Path TreeShortest Path Tree
Lower Bound
39
500Simulation-with relay node
uniform report size
5 10 15 20 25 30 35 40 45Aggregation Ratio
Energy
Cost
200250300350400450
Algorithm 1Algorithm 2Hassin’s AlgorithmShortest Path Tree Steiner TreeSteiner TreeSteiner Tree
Lower Bound
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500Simulation-with relay node
uniform report size
5 10 15 20 25 30 35 40 45Aggregation Ratio
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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500Simulation-with relay node
uniform report size
5 10 15 20 25 30 35 40 45Aggregation Ratio
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
42
500Simulation-with relay node
uniform report size
5 10 15 20 25 30 35 40 45Aggregation Ratio
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
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