SETH says: weak Fréchet distance is faster, but only if it is continuous and in one dimension

Kevin Buchin; Tim Ophelders; Bettina Speckmann

doi:10.1137/1.9781611975482.179

SETH says: weak Fréchet distance is faster, but only if it is continuous and in one dimension

Kevin Buchin, Tim Ophelders, Bettina Speckmann

Algorithms, Geometry and Applications

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

15 Citations (Scopus)

1 Downloads (Pure)

Abstract

We show by reduction from the Orthogonal Vectors problem that algorithms with strongly subquadratic running time cannot approximate the Fréchet distance between curves better than a factor 3 unless SETH fails. We show that similar reductions cannot achieve a lower bound with a factor better than 3. Our lower bound holds for the continuous, the discrete, and the weak discrete Fréchet distance even for curves in one dimension. Interestingly, the continuous weak Fréchet distance behaves differently. Our lower bound still holds for curves in two dimensions and higher. However, for curves in one dimension, we provide an exact algorithm to compute the weak Fréchet distance in linear time.

Original language	English
Title of host publication	Proc. 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)
Place of Publication	New York
Publisher	Association for Computing Machinery, Inc
Pages	2887-2899
Number of pages	13
ISBN (Electronic)	978-1-61197-548-2
DOIs	https://doi.org/10.1137/1.9781611975482.179
Publication status	Published - 1 Jan 2019
Event	30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019 - San Diego, United States Duration: 6 Jan 2019 → 9 Jan 2019

Conference

Conference	30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019
Country/Territory	United States
City	San Diego
Period	6/01/19 → 9/01/19

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10.1137/1.9781611975482.179

https://arxiv.org/pdf/1807.08699.pdf

Cite this

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title = "SETH says: weak Fr{\'e}chet distance is faster, but only if it is continuous and in one dimension",

abstract = "We show by reduction from the Orthogonal Vectors problem that algorithms with strongly subquadratic running time cannot approximate the Fr{\'e}chet distance between curves better than a factor 3 unless SETH fails. We show that similar reductions cannot achieve a lower bound with a factor better than 3. Our lower bound holds for the continuous, the discrete, and the weak discrete Fr{\'e}chet distance even for curves in one dimension. Interestingly, the continuous weak Fr{\'e}chet distance behaves differently. Our lower bound still holds for curves in two dimensions and higher. However, for curves in one dimension, we provide an exact algorithm to compute the weak Fr{\'e}chet distance in linear time.",

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booktitle = "Proc. 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)",

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Buchin, K , Ophelders, T & Speckmann, B 2019, SETH says: weak Fréchet distance is faster, but only if it is continuous and in one dimension. in Proc. 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). Association for Computing Machinery, Inc, New York, pp. 2887-2899, 30th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2019, San Diego, United States, 6/01/19. https://doi.org/10.1137/1.9781611975482.179

SETH says: weak Fréchet distance is faster, but only if it is continuous and in one dimension. / Buchin, Kevin ; Ophelders, Tim ; Speckmann, Bettina.

Proc. 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). New York : Association for Computing Machinery, Inc, 2019. p. 2887-2899.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

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N2 - We show by reduction from the Orthogonal Vectors problem that algorithms with strongly subquadratic running time cannot approximate the Fréchet distance between curves better than a factor 3 unless SETH fails. We show that similar reductions cannot achieve a lower bound with a factor better than 3. Our lower bound holds for the continuous, the discrete, and the weak discrete Fréchet distance even for curves in one dimension. Interestingly, the continuous weak Fréchet distance behaves differently. Our lower bound still holds for curves in two dimensions and higher. However, for curves in one dimension, we provide an exact algorithm to compute the weak Fréchet distance in linear time.

AB - We show by reduction from the Orthogonal Vectors problem that algorithms with strongly subquadratic running time cannot approximate the Fréchet distance between curves better than a factor 3 unless SETH fails. We show that similar reductions cannot achieve a lower bound with a factor better than 3. Our lower bound holds for the continuous, the discrete, and the weak discrete Fréchet distance even for curves in one dimension. Interestingly, the continuous weak Fréchet distance behaves differently. Our lower bound still holds for curves in two dimensions and higher. However, for curves in one dimension, we provide an exact algorithm to compute the weak Fréchet distance in linear time.

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DO - 10.1137/1.9781611975482.179

M3 - Conference contribution

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BT - Proc. 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)

PB - Association for Computing Machinery, Inc

CY - New York

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Y2 - 6 January 2019 through 9 January 2019

ER -

SETH says: weak Fréchet distance is faster, but only if it is continuous and in one dimension

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