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Frchet Distance Between Uncertain Trajectories Computing Expected Value and Upper Bound Kevin Buchin 1 Maarten Lffler 2 Aleksandr Popov 1 Marcel Roeloffzen 1 1 Department of Mathematics and Computer Science Eindhoven University of Technology 2

  1. Fréchet Distance Between Uncertain Trajectories Computing Expected Value and Upper Bound Kevin Buchin 1 Maarten Löffler 2 Aleksandr Popov 1 Marcel Roeloffzen 1 1 Department of Mathematics and Computer Science Eindhoven University of Technology 2 Department of Information and Computing Sciences Utrecht University 16th March 2020

  2. Why (uncertain) trajectories? K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 2 / 15

  3. Why (uncertain) trajectories? K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 2 / 15

  4. Why (uncertain) trajectories? K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 2 / 15

  5. Why (uncertain) trajectories? K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 2 / 15

  6. Why (uncertain) trajectories? K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 2 / 15

  7. Why (uncertain) trajectories? K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 2 / 15

  8. Why (uncertain) trajectories? K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 2 / 15

  9. Uncertain trajectories Measurement uncertainty, regions connected with line segments. K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 3 / 15

  10. Uncertain trajectories Measurement uncertainty, regions connected with line segments. Indecisive point: ℓ options per point ℓ = 3 K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 3 / 15

  11. Uncertain trajectories Measurement uncertainty, regions connected with line segments. Indecisive point: Imprecise point: ℓ options per point Connected region ℓ = 3 circle K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 3 / 15

  12. Similarity measures Discrete Fréchet distance and continuous Fréchet distance: K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 4 / 15

  13. Similarity measures Discrete Fréchet distance and continuous Fréchet distance: K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 4 / 15

  14. Similarity measures Discrete Fréchet distance and continuous Fréchet distance: K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 4 / 15

  15. Similarity measures Discrete Fréchet distance and continuous Fréchet distance: K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 4 / 15

  16. Similarity measures Discrete Fréchet distance and continuous Fréchet distance: K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 4 / 15

  17. Similarity measures Discrete Fréchet distance and continuous Fréchet distance: K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 4 / 15

  18. Similarity measures Discrete Fréchet distance and continuous Fréchet distance: K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 4 / 15

  19. Similarity measures Discrete Fréchet distance and continuous Fréchet distance: K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 4 / 15

  20. Fréchet distance on uncertain trajectories Lower bound? Upper bound? Expected value? K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 5 / 15

  21. Fréchet distance on uncertain trajectories Lower bound? Upper bound? Expected value? K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 5 / 15

  22. Fréchet distance on uncertain trajectories Lower bound? Upper bound? Expected value? K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 5 / 15

  23. Fréchet distance on uncertain trajectories Lower bound? Upper bound? Expected value? K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 5 / 15

  24. Overview imprecise indecisive disks line segments LB Polynomial* Polynomial* Polynomial* discrete FD UB NP-complete NP-complete NP-complete Exp #P-hard — #P-hard Polynomial † NP-complete † LB — FD UB NP-complete NP-complete NP-complete Exp #P-hard — — * Ahn et al., 2012. † Joint work with Fan and Raichel, submitted. K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 6 / 15

  25. Overview imprecise indecisive disks line segments LB Polynomial* Polynomial* Polynomial* discrete FD UB NP-complete NP-complete NP-complete Exp #P-hard — #P-hard Polynomial † NP-complete † LB — FD UB NP-complete NP-complete NP-complete Exp #P-hard — — * Ahn et al., 2012. † Joint work with Fan and Raichel, submitted. K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 6 / 15

  26. Overview imprecise indecisive disks line segments LB Polynomial* Polynomial* Polynomial* discrete FD UB NP-complete NP-complete NP-complete Exp #P-hard — #P-hard Polynomial † NP-complete † LB — FD UB NP-complete NP-complete NP-complete Exp #P-hard — — * Ahn et al., 2012. † Joint work with Fan and Raichel, submitted. Indecisive: Polynomial-time algorithms with time bands for all cases. K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 6 / 15

  27. Hardness results K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 7 / 15

  28. Construction: Indecisive, discrete FD, upper bound ( 0 , 0 . 5 + ε) T Reduction from SAT. Two curves: ( 0 , 0 . 5 ) F ◮ Precise curve: formula structure; ( − 1 , 0 ) ( 0 , 0 ) ( 1 , 0 ) N/A ( − 2 , 0 ) ( 2 , 0 ) ◮ Indecisive curve: ( 0 , − 0 . 5 ) T variable synchronisation synchronisation assignment. ( 0 , − 0 . 5 − ε) F x = 0 K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 8 / 15

  29. Construction: Indecisive, discrete FD, upper bound ( 0 , 0 . 5 + ε) T Reduction from SAT. Two curves: ( 0 , 0 . 5 ) F ◮ Precise curve: formula structure; ( − 1 , 0 ) ( 0 , 0 ) ( 1 , 0 ) N/A ( − 2 , 0 ) ( 2 , 0 ) ◮ Indecisive curve: ( 0 , − 0 . 5 ) T variable synchronisation synchronisation assignment. ( 0 , − 0 . 5 − ε) F x = 0 K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 8 / 15

  30. Construction: Indecisive, discrete FD, upper bound ( 0 , 0 . 5 + ε) T Reduction from SAT. Two curves: ( 0 , 0 . 5 ) F ◮ Precise curve: formula structure; ( − 1 , 0 ) ( 0 , 0 ) ( 1 , 0 ) N/A ( − 2 , 0 ) ( 2 , 0 ) ◮ Indecisive curve: ( 0 , − 0 . 5 ) T variable synchronisation synchronisation assignment. ( 0 , − 0 . 5 − ε) F x = 0 K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 8 / 15

  31. Construction: Indecisive, discrete FD, upper bound ( 0 , 0 . 5 + ε) T Reduction from SAT. Two curves: ( 0 , 0 . 5 ) F ◮ Precise curve: formula structure; ( − 1 , 0 ) ( 0 , 0 ) ( 1 , 0 ) N/A ( − 2 , 0 ) ( 2 , 0 ) ◮ Indecisive curve: ( 0 , − 0 . 5 ) T variable synchronisation synchronisation assignment. ( 0 , − 0 . 5 − ε) F x = 0 K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 8 / 15

  32. Construction: example ( 0 , 0 . 5 + ε) ( 0 , 0 . 5 ) ( − 1 , 0 ) ( 0 , 0 ) ( 1 , 0 ) ( − 2 , 0 ) ( 2 , 0 ) ( 0 , − 0 . 5 ) ( 0 , − 0 . 5 − ε) C = x 1 ∨ x 2 Realisation: x 1 = T, x 2 = F, x 3 = F. K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 9 / 15

  33. Construction: example ( 0 , 0 . 5 + ε) ( 0 , 0 . 5 ) ( − 1 , 0 ) ( 0 , 0 ) ( 1 , 0 ) ( − 2 , 0 ) ( 2 , 0 ) ( 0 , − 0 . 5 ) ( 0 , − 0 . 5 − ε) C = x 1 ∨ x 2 Realisation: x 1 = T, x 2 = F, x 3 = F. K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 9 / 15

  34. Construction: example ( 0 , 0 . 5 + ε) ( 0 , 0 . 5 ) ( − 1 , 0 ) ( 0 , 0 ) ( 1 , 0 ) ( − 2 , 0 ) ( 2 , 0 ) ( 0 , − 0 . 5 ) ( 0 , − 0 . 5 − ε) C = x 1 ∨ x 2 Realisation: x 1 = T, x 2 = F, x 3 = F. K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 9 / 15

  35. Construction: example ( 0 , 0 . 5 + ε) ( 0 , 0 . 5 ) ( − 1 , 0 ) ( 0 , 0 ) ( 1 , 0 ) ( − 2 , 0 ) ( 2 , 0 ) ( 0 , − 0 . 5 ) ( 0 , − 0 . 5 − ε) C = x 1 ∨ x 2 Realisation: x 1 = T, x 2 = F, x 3 = F. K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 9 / 15

  36. Construction: Indecisive, discrete FD, upper bound C 1 C 2 C 3 C 4 C 5 1 + ε 1 1 1 1 1 1 1 1 1 V ( 0 , 0 ) ( 0 , 0 ) K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 10 / 15

  37. Construction: Indecisive, discrete FD, upper bound C 1 C 2 C 3 C 4 C 5 1 + ε 1 1 1 1 1 1 1 1 1 V ( 0 , 0 ) ( 0 , 0 ) C 4 = F K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 10 / 15

  38. Construction: Indecisive, discrete FD, upper bound C 1 C 2 C 3 C 4 C 5 1 + ε 1 1 1 1 1 1 1 1 1 V ( 0 , 0 ) ( 0 , 0 ) C i = T for all i K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 10 / 15

  39. Hardness results imprecise indecisive disks line segments UB NP-complete NP-complete NP-complete discrete FD Exp #P-hard — #P-hard UB NP-complete NP-complete NP-complete FD Exp #P-hard — — K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 11 / 15

  40. Algorithms using time bands K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 12 / 15

  41. Time bands: idea K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 13 / 15

  42. Time bands: idea K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 13 / 15

  43. Time bands: idea w = 1 K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 13 / 15

  44. Time bands: results Time band width w , indecisive trajectories with ℓ options, length n : K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 14 / 15

  45. Time bands: results Time band width w , indecisive trajectories with ℓ options, length n : ◮ UB dFD precise + indecisive: Θ ( 4 w ℓ n √ w ) ; K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 14 / 15

  46. Time bands: results Time band width w , indecisive trajectories with ℓ options, length n : ◮ UB dFD precise + indecisive: Θ ( 4 w ℓ n √ w ) ; ◮ UB dFD 2x indecisive: Θ ( 4 w ℓ 2 w n √ w ) ; K. Buchin, M. Löffler, A. Popov, M. Roeloffzen 14 / 15

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