Makoto Yamashita (Tokyo Institute of Technology) I-Lin Wang (National Cheng Kung University)

1. 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) ISMP 2012 (TU Berlin, Berlin, Germerny)

2. Outline

3. SNL(Sensor Network Localization Problem) • We want to infer locationsfrom distance information • System of Equation

4. 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. 1AX8, 1003 atoms

5. 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 heuristics • Middle accuracy & Middle computation cost

6. Our Approach • Combination of heuristics • Shortest path • Gradient method • Connectivity consistency

7. Trilateration • Three anchors determine the location uniquely.

8. Shortest Path • Propagation from anchors • Moredistance information⇒Shortest Path • Rough estimate ⇒Gradient method

9. Minimization of difference • Instead of solving the systemminimize • Effective for noisy distance input input true noise(e.g.:20%～30%)

10. Gradient Method • Repeatuntil Shortest Path result

11. Connectivity Consistency • Given distance is usually less than radio range. Attraction Adjustment Repulsion

12. Framework of our heuristics • Select initial anchors • Estimate roughly with Shortest Path • Apply Gradient Methodwith estimate distance • Apply Gradient Method with original distance • Adjust sensors by Connectivity Consistency • Go to Step 4 until there is no significant improvement

13. Numerical Experiments • Effect of Shortest Path & Consistency Adjustment • SNLsa (Shortest Path & Consistency Adjustment)vs. SNLa (Consistency Adjustment)vs. SNLs (Shortest Path) • Comparison with SDP relaxation • SNLsa vs. SFSDP (Kim et at, 2009) Sparse Full SDP relaxation

14. 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)

15. SNLsa vs. SNLa vs. SNLs 1000sensors • The accuracy of SNLa is poor⇒Shortest Path is effective 500sensors

16. SNLsa vs. SNLs vs. SNLa (2) • For middle radioranges, Consistency Adjustment works well. 200 sensors

17. SNLsa vs. SFSDP 1000sensors • For large radioranges, SNLsa is faster. 500sensors

18. 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.

19. 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.

20. Densest Subset • Collect all solutions for each sensor. • Generate a graph by connecting each other. • Find the densest subsetvia discrete optimization.(Nagano et al, 2011) • Take the average ofthe densest subset.

21. Numerical Results of Densest Subset • Protein 1HOE (2D projection) • Protein 1KDH (2D projection) RMSD(Densest Average) = 0.227412 RMSD(Densest Average) = 1.248964 Ignoring deviations, we obtain reasonable solution.

22. 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. 謝謝聆聽, Thank you very much for your attention.