Why Walking the Dog Takes Time: Frechet Distance Has No Strongly Subquadratic Algorithms Unless SETH Fails

Bringmann, Karl

doi:10.1109/focs.2014.76

Cited by 163 publications

(175 citation statements)

References 36 publications

(61 reference statements)

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“…This leads to an O(mn(d + log 1 ε ))-time algorithm, giving the first approximation algorithm that runs faster than known exact algorithms for the general case. Moreover, as shown by Bringmann [4], our result is tight up to subpolynomial factors, assuming SETH. Finally, we apply our framework to the Euclidean distance (Sect.…”

Section: Introductionmentioning

confidence: 78%

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Computing the Fréchet Distance with a Retractable Leash

Buchin

Leusden

et al. 2013

Lecture Notes in Computer Science

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This is the accepted version of the paper.This version of the publication may differ from the final published version. Abstract All known algorithms for the Fréchet distance between curves proceed in two steps: first, they construct an efficient oracle for the decision version; second, they use this oracle to find the optimum from a finite set of critical values. We present a novel approach that avoids the detour through the decision version. This gives the first quadratic time algorithm for the Fréchet distance between polygonal curves in R d under polyhedral distance functions (e.g., L 1 and L ∞ ). We also get a (1 + ε)-approximation of the Fréchet distance under the Euclidean metric, in quadratic time for any fixed ε > 0. For the exact Euclidean case, our framework currently yields an algorithm with running time O(n 2 log 2 n). However, we conjecture that it may eventually lead to a faster exact algorithm. Permanent repository link Editor in Charge: Kenneth ClarksonA preliminary version appeared as K.

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Section: Introductionmentioning

confidence: 78%

“…4 To construct π , we combine the witness for B * a , the witness π i−1 , and the segment from L * i−1 to x. By assumption, π i−1 enters row j at or to the left of B a .…”

Section: ) Has An ε-Witness That Passes Through B a If And Only If ε mentioning

confidence: 99%

Computing the Fréchet Distance with a Retractable Leash

Buchin

Leusden

et al. 2013

Lecture Notes in Computer Science

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“…Karl Bringmann provided an interesting answer to this question by showing that any algorithm for the Fréchet distance with running time O(n 2−ε ), for some fixed ε > 0, would violate the strong exponential time hypothesis (SETH) [17]. These results were later refined and improved to show that the lower bound holds in basically all settings (with the notable exception of the one-dimensional continuous Fréchet distance, which is still unresolved) [18].…”

Section: Recent Developmentsmentioning

confidence: 99%

Four Soviets Walk the Dog: Improved Bounds for Computing the Fréchet Distance

Buchin

Meulemans

et al. 2017

Discrete Comput Geom

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Given two polygonal curves in the plane, there are many ways to define a notion of similarity between them. One popular measure is the Fréchet distance. Since it was proposed by Alt and Godau in 1992, many variants and extensions have been studied. Nonetheless, even more than 20 years later, the original O(n 2 log n) algorithm by Alt and Godau for computing the Fréchet distance remains the state of the art (here, n denotes the number of edges on each curve). This has led Helmut Alt to conjecture that the associated decision problem is 3SUM-hard. In recent work, Agarwal et al. show how to break the quadratic barrier for the discrete version of the Fréchet distance, where one considers sequences of points instead of polygonal curves. Building on their work, we give a randomized algorithm to compute the Fréchet distance between two polygonal curves in time O(n 2 √ log n(log log n) 3/2 ) on a pointer machine and in time O(n 2 (log log n) 2 ) on a word RAM. Furthermore, we show that there exists an algebraic decision tree for the decision problem of depth O(n 2−ε ), for some ε > 0. We believe that this reveals an intriguing new aspect of this well-studied problem. Finally, we show how to obtain the first subquadratic algorithm for computing the weak Fréchet distance on a word RAM.

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“…The distance is assumed as 600 meters based on the studies in travel behaviour of public transit passengers and the walking speed [21,22,29]. Also, the direction of the trips is defined as the angle between the two trips; if the minimum angle between the two trips is between 0 and 6 degrees, then the two trips will be assumed in the same direction.…”

Section: Spatial Similaritymentioning

confidence: 99%

“…The ones above are the major measures introduced, studied, and compared in the literature. Each measure has its pros and cons that make them suitable for specific applications and networks [22][23][24][25][26][27][28].…”

Section: Spatial Similaritymentioning

confidence: 99%

Spatial-Temporal Similarity Correlation between Public Transit Passengers Using Smart Card Data

Faroqi

Mesbah

Kim

2017

Journal of Advanced Transportation

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The increasing availability of public transit smart card data has enabled several studies to focus on identifying passengers with similar spatial and/or temporal trip characteristics. However, this paper goes one step further by investigating the relationship between passengers' spatial and temporal characteristics. For the first time, this paper investigates the correlation of the spatial similarity with the temporal similarity between public transit passengers by developing spatial similarity and temporal similarity measures for the public transit network with a novel passenger-based perspective. The perspective considers the passengers as agents who can make multiple trips in the network. The spatial similarity measure takes into account direction as well as the distance between the trips of the passengers. The temporal similarity measure considers both the boarding and alighting time in a continuous linear space. The spatial-temporal similarity correlation between passengers is analysed using histograms, Pearson correlation coefficients, and hexagonal binning. Also, relations between the spatial and temporal similarity values with the trip time and length are examined. The proposed methodology is implemented for four-day smart card data including 80,000 passengers in Brisbane, Australia. The results show a nonlinear spatial-temporal similarity correlation among the passengers.

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Why Walking the Dog Takes Time: Frechet Distance Has No Strongly Subquadratic Algorithms Unless SETH Fails

Cited by 163 publications

References 36 publications

Computing the Fréchet Distance with a Retractable Leash

Computing the Fréchet Distance with a Retractable Leash

Four Soviets Walk the Dog: Improved Bounds for Computing the Fréchet Distance

Spatial-Temporal Similarity Correlation between Public Transit Passengers Using Smart Card Data

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