Querying approximate shortest paths in anisotropic regions Siu-Wing - - PowerPoint PPT Presentation

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Querying approximate shortest paths in anisotropic regions Siu-Wing - - PowerPoint PPT Presentation

Dec 16, 2022 180 likes •800 views

Querying approximate shortest paths in anisotropic regions Siu-Wing Cheng (HKUST) Hyeon-Suk Na (Soongsil University) Antoine Vigneron (INRA Jouy-en-Josas) Yajun Wang (HKUST) The weighted region problem cost = weight length t w=1 w=2 s

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Querying approximate shortest paths in anisotropic regions

Siu-Wing Cheng (HKUST) Hyeon-Suk Na (Soongsil University) Antoine Vigneron (INRA Jouy-en-Josas) Yajun Wang (HKUST)

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t s cost = weight × length w=1 w=2 Shortest path

The weighted region problem

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t s cost = weight × length w=1 w=2 Shortest path

The weighted region problem

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  • Triangulation with n vertices
  • Each face has a weight
  • Find an approximate shortest path with cost at most 1 + ǫ

times the minimum s t

The weighted region problem

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No (strongly polynomial) FPTAS is known

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No (strongly polynomial) FPTAS is known Known results: polynomial in n, 1/ǫ, and other parameters.

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No (strongly polynomial) FPTAS is known Known results: polynomial in n, 1/ǫ, and other parameters.

  • ρ: maximum weight/ minimum weight
  • θ: minimum angle in the triangulation
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No (strongly polynomial) FPTAS is known Known results: polynomial in n, 1/ǫ, and other parameters.

  • ρ: maximum weight/ minimum weight
  • θ: minimum angle in the triangulation

Known results

  • Mitchell and Papadimitriou (1987): O(n8 log(nNρ/ǫ))
  • Aleksandrov, Maheshwari, and Sacks:

O(Cn/√ǫ × polylog(n, 1/ǫ)), C depends on ρ, θ and other parameters.

  • Sun and Reif: O(C′n/ǫ × polylog(n, 1/ǫ)),

C′ depends on θ, other parameters, but not ρ.

  • Cheng et al. (2007): O(n3ρ/ǫ × polylog(n, ρ, 1/ǫ))
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Query problem

s

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Query problem

s Build a data structure

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Query problem

s Build a data structure q Given a query point q, find a (1 + ε)-approximation of the cost of the shortest path from s to q.

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Query problem

s Build a data structure q Given a query point q, find a (1 + ε)-approximation of the cost of the shortest path from s to q.

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Query problem

s Build a data structure q Given a query point q, find a (1 + ε)-approximation of the cost of the shortest path from s to q. Output an approximate shortest path in time linear in its size

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Previous results

Aleksandrov et al.: Space and preprocessing time depend on several parameters including n, ε, ρ and θ.

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Previous results

Aleksandrov et al.: Space and preprocessing time depend on several parameters including n, ε, ρ and θ. Applies to polyhedral surfaces

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Previous results

Aleksandrov et al.: Space and preprocessing time depend on several parameters including n, ε, ρ and θ. Applies to polyhedral surfaces

Our results

Query time: O(log(ρn/ε)) Space: O(ρ2n4/ε2 log(ρn/ε)) Preprocessing time: O(ρ2n4/ε2 log2(ρn/ε))

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Previous results

Aleksandrov et al.: Space and preprocessing time depend on several parameters including n, ε, ρ and θ. Applies to polyhedral surfaces

Our results

Query time: O(log(ρn/ε)) Space: O(ρ2n4/ε2 log(ρn/ε)) Preprocessing time: O(ρ2n4/ε2 log2(ρn/ε)) Applies to anisotropic distance functions

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Convex distance function

Unit ball B: set of points at distance 1 from O O B

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Convex distance function

Unit ball B: set of points at distance 1 from O O x y x + λB dB(x, y) = λ B

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Convex distance function

Unit ball B: set of points at distance 1 from O O x y x + λB dB(x, y) = λ B Shortest path is a straight line segment

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Model

1 1/ρ B One convex distance function per face

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Model

1 1/ρ B One convex distance function per face Speed is in interval [1/ρ, 1]

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Model

1 1/ρ B One convex distance function per face Speed is in interval [1/ρ, 1] Cost of a length ℓ path is in [ℓ, ρℓ].

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s t

Exemple of a non-polygonal shortest path

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s t

Exemple of a non-polygonal shortest path

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s t

Exemple of a non-polygonal shortest path

There exists a rectifiable (=finite length) shortest path

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s t

Exemple of a non-polygonal shortest path

There exists a rectifiable (=finite length) shortest path There exists a (1 + ε)-approximate shortest path with O(ρn2/ε) edges.

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s t

Static (non query) algorithm

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s t Circle with radius ρ d(s, t) and center s

Static (non query) algorithm

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s t Circle with radius ρ d(s, t) and center s Steiner points with uniform spacing δ = 1/poly(ρ, n, 1/ε)

Static (non query) algorithm

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s t Circle with radius ρ d(s, t) and center s Steiner points with uniform spacing δ = 1/poly(ρ, n, 1/ε)

Static (non query) algorithm

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Observation

s q δ

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Observation

s q shortest path δ

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Observation

s q shortest path approximate shortest path δ

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Observation

s q shortest path approximate shortest path δ

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Observation

s q shortest path approximate shortest path δ additive error at most δ

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Observation

(1 + ε) approximation when q is in an annulus s q t radius ratio=poly(ρ, n, 1/ε)

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Answering queries within the annulus

s Additively weighted Voronoi diagram

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Answering queries within the annulus

s Weight= cost of approximate shortest path Additively weighted Voronoi diagram

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Answering queries within the annulus

s Weight= cost of approximate shortest path Additively weighted Voronoi diagram q

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Answering queries within the annulus

s Weight= cost of approximate shortest path Additively weighted Voronoi diagram q

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Answering queries within the annulus

s Weight= cost of approximate shortest path Additively weighted Voronoi diagram q

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Answering queries within the annulus

s Weight= cost of approximate shortest path Additively weighted Voronoi diagram q

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Overview

For each vertex vi, we construct a data structure for the annulus with radius ri = d(s, vi)/2ρ and Ri =poly(ρ, n, 1/ε)d(s, vi)

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Overview

For each vertex vi, we construct a data structure for the annulus with radius ri = d(s, vi)/2ρ and Ri =poly(ρ, n, 1/ε)d(s, vi) s vi ri Ri q

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Overview

For each vertex vi, we construct a data structure for the annulus with radius ri = d(s, vi)/2ρ and Ri =poly(ρ, n, 1/ε)d(s, vi) s vi ri Ri q We handle the neighborhood of s through scaling

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Overview

For each vertex vi, we construct a data structure for the annulus with radius ri = d(s, vi)/2ρ and Ri =poly(ρ, n, 1/ε)d(s, vi) s vi ri Ri q We handle the neighborhood of s through scaling We handle the empty space (if any) through scaling and perturbation

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Close range queries s

v1 q

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Close range queries s

v1 q q′

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Close range queries s

v1 q q′

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Close range queries s

v1 q q′

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Between two annuli s

q

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Between two annuli s

q

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Between two annuli s

q q′ Perturbation ϕ maps black points → blue points and q → q′

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Between two annuli s

q q′ Find approx shortest path in data structure for the blue subdivision

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Between two annuli s

q q′ Find approx shortest path in data structure for the blue subdivision Clip the path

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Between two annuli s

q q′ Find approx shortest path in data structure for the blue subdivision Clip the path Apply ϕ−1

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Between two annuli s

q q′ Find approx shortest path in data structure for the blue subdivision Clip the path Apply ϕ−1 Go straight to s

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Conclusion

We can also handle obstacles

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Conclusion

We can also handle obstacles 2 point queries?

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Conclusion

We can also handle obstacles 2 point queries?