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An approach based on shortest path and connectivity consistency for sensor network localization problems
Makoto Yamashita (Tokyo Institute of Technology)I-Lin Wang (National Cheng Kung University)Zih-Cin Lin (National Cheng Kung University)
2012/08/22 ISMP 2012 (TU Berlin, Berlin, Germerny)
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Outline
Sensor Network Localization1• Mathematical Formulation
Framework of our approach2• Shortest Path• Gradient Method• Connectivity Consistency
Numerical Results3Multiple Start4
• Starting Point Selection• Combination of Location Results
Conclusion and Future works5
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SNL(Sensor Network Localization Problem)
We want to infer locationsfrom distance information
System of Equation
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Protein Structure
We can use distances between atoms measured by NOE effect.
We want to infer whole structure.
Structure determines chemical property of protein.
There are many other applications.
2012/08/221AX8, 1003 atoms
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Existing Methods
Multidimensional Scaling[Merit] Low computation cost[Demerit] All distances are necessary
SDP relaxation (Biswas & Ye 2004)[Merit] High accuracy[Demerit] High compuation cost
We combine some heuristicsMiddle accuracy & Middle computation cost
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Our Approach
Combination of heuristicsShortest pathGradient methodConnectivity consistency
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Trilateration
Three anchors determine the location uniquely.
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1a
2a
3a
1d
2d
3d
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Shortest Path
Propagation from anchors
Moredistance information⇒Shortest Path
Rough estimate Gradient method⇒
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Minimization of difference
Instead of solving the system
minimize
Effective for noisy distance input
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input true noise(e.g.:20% ~ 30%)
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Gradient Method
Repeat
until
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Connectivity Consistency
Given distance is usually less than radio range.
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Repulsion
Attraction
Adjustment
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Framework of our heuristics
1. Select initial anchors2. Estimate roughly with Shortest Path3. Apply Gradient Method with estimate distance4. Apply Gradient Method with original distance5. Adjust sensors by Connectivity Consistency6. Go to Step 4
until there is no significant improvement
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Numerical Experiments
Effect of Shortest Path & Consistency AdjustmentSNLsa (Shortest Path & Consistency Adjustment)
vs. SNLa (Consistency Adjustment)vs. SNLs (Shortest Path)
Comparison with SDP relaxationSNLsa vs. SFSDP (Kim et at, 2009)
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Sparse Full SDP relaxation
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Test Instances & RMSD
[0,1]x[0,1] space in 2D exact distance (zero noise) #sensors = 200, 500, 1000 #anchors = #sensors/10 radiorange = 0.05, 0.10, 0.15, 0.20, 0.25, 0.30 average of 100 randomly generated instances
Evaluate RMSD (Root Mean Square Deviation)
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SNLsa vs. SNLa vs. SNLs
The accuracy of SNLa is poor⇒Shortest Path is effective
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0.3 0.25 0.2 0.15 0.1 0.051.00E-071.00E-061.00E-051.00E-041.00E-031.00E-021.00E-01
1.00E+001.00E+01
0
5
10
15
20
25
et_SNLsaet_SNLaet_SNLsrmsd_SNLsarmsd_SNLarmsd_SNLs
radiorange
RMSD
time(s)
0.3 0.25 0.2 0.15 0.1 0.051.00E-071.00E-061.00E-051.00E-041.00E-031.00E-021.00E-01
1.00E+001.00E+01
0
2
4
6
8
10
12
14
et_SNLsaet_SNLaet_SNLsrmsd_SNLsarmsd_SNLarmsd_SNLs
radiorange
RMSD
time(s)
1000sensors
500sensors
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SNLsa vs. SNLs vs. SNLa (2)
For middle radioranges, Consistency Adjustment works well.
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0.3 0.25 0.2 0.15 0.1 0.051.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
1.00E+01
0
0.5
1
1.5
2
2.5
3
3.5
et_SNLsaet_SNLaet_SNLsrmsd_SNLsarmsd_SNLarmsd_SNLs
radiorange
RMSD
time(s)
200 sensors
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SNLsa vs. SFSDP
For large radioranges, SNLsa is faster.
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0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.11.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
1.00E+01
0.00
2.00
4.00
6.00
8.00
10.00
12.00
et_SFSDPet_SNLsarmsd_SFSDPrmsd_SNLsa
radiorange
RMSD
time(s)
0.3 0.2 0.1 0.3 0.2 0.1 0.3 0.2 0.11.00E-07
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
1.00E+01
0.00
2.00
4.00
6.00
8.00
10.00
12.00
et_SFSDPet_SNLsarmsd_SFSDPrmsd_SNLsa
radiorange
RMSD
time(s)
1000sensors
500sensors
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Multiple Start
For the starting anchors, there are many candidates.
We list-up cliques of size 4 and select better cliques.(e.g.: large volume of triangle or tetrahedra.)
Each starting anchors generates different locations.
We want reasonable result from multiple solutions.
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Different Solutions
For similar solutions, we can take their average.
Lack of edges often make the instance harder.
We need different approach to select locations.
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Densest Subset
1. Collect all solutions for each sensor.
2. Generate a graph by connecting each other.
3. Find the densest subsetvia discrete optimization.(Nagano et al, 2011)
4. Take the average ofthe densest subset.
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Numerical Results of Densest Subset Protein 1HOE (2D projection)
Protein 1KDH (2D projection)
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Clique 1 2 3 4 5 6
RMSD 0.223434 0.223498 0.223498 0.223498 0.223498 0.223498
7 8 9 10 11 12
0.223498 1.416665 4.620514 6.158431 6.189689 7.732091
RMSD(Densest Average) = 0.227412
Clique 1 2 3 4 5 6
RMSD 1.356912 1.356923 1.356971 1.356971 1.356971 1.356971
7 8 9 10 11 12
1.356971 1.356971 1.356971 4.178180 10.286665 12.354961
RMSD(Densest Average) = 1.248964
Ignoring deviations, we obtain reasonable solution.
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Conclusion and Future Works
Shortest Path & Consistency Adjustment works well for randomly generated instances.
Combination of multiple starts generatesreasonable solutions.
We should discuss multiple sensor types.We should introduce chemical property of proteins.
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謝謝聆聽 , Thank you very much for your attention.