Frchet Distance Between Uncertain Trajectories Computing Expected - - PowerPoint PPT Presentation

▶
fr chet distance between uncertain trajectories
SMART_READER_LITE
LIVE PREVIEW

Frchet Distance Between Uncertain Trajectories Computing Expected - - PowerPoint PPT Presentation

Jan 07, 2024 103 likes •609 views

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

slide-1
SLIDE 1

Fréchet Distance Between Uncertain Trajectories

Computing Expected Value and Upper Bound

Kevin Buchin1 Maarten Löffler2 Aleksandr Popov1 Marcel Roeloffzen1

1Department of Mathematics and Computer Science

Eindhoven University of Technology

2Department of Information and Computing Sciences

Utrecht University

16th March 2020

slide-2
SLIDE 2

Why (uncertain) trajectories?

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

2 / 15

slide-3
SLIDE 3

Why (uncertain) trajectories?

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

2 / 15

slide-4
SLIDE 4

Why (uncertain) trajectories?

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

2 / 15

slide-5
SLIDE 5

Why (uncertain) trajectories?

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

2 / 15

slide-6
SLIDE 6

Why (uncertain) trajectories?

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

2 / 15

slide-7
SLIDE 7

Why (uncertain) trajectories?

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

2 / 15

slide-8
SLIDE 8

Why (uncertain) trajectories?

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

2 / 15

slide-9
SLIDE 9

Uncertain trajectories

Measurement uncertainty, regions connected with line segments.

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

3 / 15

slide-10
SLIDE 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

slide-11
SLIDE 11

Uncertain trajectories

Measurement uncertainty, regions connected with line segments. Indecisive point: ℓ options per point ℓ = 3 Imprecise point: Connected region circle

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

3 / 15

slide-12
SLIDE 12

Similarity measures

Discrete Fréchet distance and continuous Fréchet distance:

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

4 / 15

slide-13
SLIDE 13

Similarity measures

Discrete Fréchet distance and continuous Fréchet distance:

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

4 / 15

slide-14
SLIDE 14

Similarity measures

Discrete Fréchet distance and continuous Fréchet distance:

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

4 / 15

slide-15
SLIDE 15

Similarity measures

Discrete Fréchet distance and continuous Fréchet distance:

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

4 / 15

slide-16
SLIDE 16

Similarity measures

Discrete Fréchet distance and continuous Fréchet distance:

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

4 / 15

slide-17
SLIDE 17

Similarity measures

Discrete Fréchet distance and continuous Fréchet distance:

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

4 / 15

slide-18
SLIDE 18

Similarity measures

Discrete Fréchet distance and continuous Fréchet distance:

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

4 / 15

slide-19
SLIDE 19

Similarity measures

Discrete Fréchet distance and continuous Fréchet distance:

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

4 / 15

slide-20
SLIDE 20

Fréchet distance on uncertain trajectories

Lower bound? Upper bound? Expected value?

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

5 / 15

slide-21
SLIDE 21

Fréchet distance on uncertain trajectories

Lower bound? Upper bound? Expected value?

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

5 / 15

slide-22
SLIDE 22

Fréchet distance on uncertain trajectories

Lower bound? Upper bound? Expected value?

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

5 / 15

slide-23
SLIDE 23

Fréchet distance on uncertain trajectories

Lower bound? Upper bound? Expected value?

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

5 / 15

slide-24
SLIDE 24

Overview

indecisive imprecise disks line segments discrete FD LB Polynomial* Polynomial* Polynomial* UB NP-complete NP-complete NP-complete Exp #P-hard — #P-hard FD LB Polynomial† — NP-complete† 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

slide-25
SLIDE 25

Overview

indecisive imprecise disks line segments discrete FD LB Polynomial* Polynomial* Polynomial* UB NP-complete NP-complete NP-complete Exp #P-hard — #P-hard FD LB Polynomial† — NP-complete† 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

slide-26
SLIDE 26

Overview

indecisive imprecise disks line segments discrete FD LB Polynomial* Polynomial* Polynomial* UB NP-complete NP-complete NP-complete Exp #P-hard — #P-hard FD LB Polynomial† — NP-complete† 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

slide-27
SLIDE 27

Hardness results

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

7 / 15

slide-28
SLIDE 28

Construction: Indecisive, discrete FD, upper bound

Reduction from SAT. Two curves: ◮ Precise curve: formula structure; ◮ Indecisive curve: variable assignment. (0, 0) N/A (0, 0.5) F (0, −0.5) T (0, 0.5 + ε) T (0, −0.5 − ε) F x = 0 (1, 0) (2, 0) synchronisation (−1, 0) (−2, 0) synchronisation

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

8 / 15

slide-29
SLIDE 29

Construction: Indecisive, discrete FD, upper bound

Reduction from SAT. Two curves: ◮ Precise curve: formula structure; ◮ Indecisive curve: variable assignment. (0, 0) N/A (0, 0.5) F (0, −0.5) T (0, 0.5 + ε) T (0, −0.5 − ε) F x = 0 (1, 0) (2, 0) synchronisation (−1, 0) (−2, 0) synchronisation

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

8 / 15

slide-30
SLIDE 30

Construction: Indecisive, discrete FD, upper bound

Reduction from SAT. Two curves: ◮ Precise curve: formula structure; ◮ Indecisive curve: variable assignment. (0, 0) N/A (0, 0.5) F (0, −0.5) T (0, 0.5 + ε) T (0, −0.5 − ε) F x = 0 (1, 0) (2, 0) synchronisation (−1, 0) (−2, 0) synchronisation

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

8 / 15

slide-31
SLIDE 31

Construction: Indecisive, discrete FD, upper bound

Reduction from SAT. Two curves: ◮ Precise curve: formula structure; ◮ Indecisive curve: variable assignment. (0, 0) N/A (0, 0.5) F (0, −0.5) T (0, 0.5 + ε) T (0, −0.5 − ε) F x = 0 (1, 0) (2, 0) synchronisation (−1, 0) (−2, 0) synchronisation

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

8 / 15

slide-32
SLIDE 32

Construction: example

(0, 0) (0, 0.5) (0, −0.5) (1, 0) (−1, 0) (0, 0.5 + ε) (0, −0.5 − ε) (2, 0) (−2, 0) C = x1 ∨ x2 Realisation: x1 = T, x2 = F, x3 = F.

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

9 / 15

slide-33
SLIDE 33

Construction: example

(0, 0) (0, 0.5) (0, −0.5) (1, 0) (−1, 0) (0, 0.5 + ε) (0, −0.5 − ε) (2, 0) (−2, 0) C = x1 ∨ x2 Realisation: x1 = T, x2 = F, x3 = F.

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

9 / 15

slide-34
SLIDE 34

Construction: example

(0, 0) (0, 0.5) (0, −0.5) (1, 0) (−1, 0) (0, 0.5 + ε) (0, −0.5 − ε) (2, 0) (−2, 0) C = x1 ∨ x2 Realisation: x1 = T, x2 = F, x3 = F.

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

9 / 15

slide-35
SLIDE 35

Construction: example

(0, 0) (0, 0.5) (0, −0.5) (1, 0) (−1, 0) (0, 0.5 + ε) (0, −0.5 − ε) (2, 0) (−2, 0) C = x1 ∨ x2 Realisation: x1 = T, x2 = F, x3 = F.

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

9 / 15

slide-36
SLIDE 36

Construction: Indecisive, discrete FD, upper bound

C1 C2 C3 C4 C5 V (0, 0) (0, 0) 1 1 1 1 1 1 1 1 + ε 1 1

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

10 / 15

slide-37
SLIDE 37

Construction: Indecisive, discrete FD, upper bound

C1 C2 C3 C4 C5 V (0, 0) (0, 0) 1 1 1 1 1 1 1 1 + ε 1 1 C4 = F

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

10 / 15

slide-38
SLIDE 38

Construction: Indecisive, discrete FD, upper bound

C1 C2 C3 C4 C5 V (0, 0) (0, 0) 1 1 1 1 1 1 1 1 + ε 1 1 Ci = T for all i

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

10 / 15

slide-39
SLIDE 39

Hardness results

indecisive imprecise disks line segments discrete FD UB NP-complete NP-complete NP-complete Exp #P-hard — #P-hard FD UB NP-complete NP-complete NP-complete Exp #P-hard — —

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

11 / 15

slide-40
SLIDE 40

Algorithms using time bands

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

12 / 15

slide-41
SLIDE 41

Time bands: idea

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

13 / 15

slide-42
SLIDE 42

Time bands: idea

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

13 / 15

slide-43
SLIDE 43

Time bands: idea

w = 1

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

13 / 15

slide-44
SLIDE 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

slide-45
SLIDE 45

Time bands: results

Time band width w, indecisive trajectories with ℓ options, length n: ◮ UB dFD precise + indecisive: Θ(4wℓn√w);

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

14 / 15

slide-46
SLIDE 46

Time bands: results

Time band width w, indecisive trajectories with ℓ options, length n: ◮ UB dFD precise + indecisive: Θ(4wℓn√w); ◮ UB dFD 2x indecisive: Θ(4wℓ2wn√w);

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

14 / 15

slide-47
SLIDE 47

Time bands: results

Time band width w, indecisive trajectories with ℓ options, length n: ◮ UB dFD precise + indecisive: Θ(4wℓn√w); ◮ UB dFD 2x indecisive: Θ(4wℓ2wn√w); ◮ Exp dFD 2x indecisive: Θ(4wℓ2wn2w2);

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

14 / 15

slide-48
SLIDE 48

Time bands: results

Time band width w, indecisive trajectories with ℓ options, length n: ◮ UB dFD precise + indecisive: Θ(4wℓn√w); ◮ UB dFD 2x indecisive: Θ(4wℓ2wn√w); ◮ Exp dFD 2x indecisive: Θ(4wℓ2wn2w2); ◮ UB/Exp FD: polynomial in n.

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

14 / 15

slide-49
SLIDE 49

Summary

Hardness results: indecisive imprecise disks line segments discrete FD UB NP-complete NP-complete NP-complete Exp #P-hard — #P-hard FD UB NP-complete NP-complete NP-complete Exp #P-hard — — Time bands: Polynomial-time algorithms for constant-width bands for UB and Exp discrete FD and FD on indecisive trajectories.

  • K. Buchin, M. Löffler, A. Popov, M. Roeloffzen

15 / 15