Approximating the packedness of polygonal curves

Gudmundsson, Joachim; Yuan, Sha; Wong, Sampson

doi:10.48550/arxiv.2009.07789

Cited by 3 publications

(8 citation statements)

References 16 publications

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“…The congestion of a square with respect to a curve π is the total length of π inside it, divided by its side length. Following [GSW20], we reduce the problem of approximating the congestion of a curve to that of computing the congestion of O(n) squares. Then we build a quadtree whose cells approximate these squares, so that it suffices to compute the congestion of the cells.…”

Section: Sketch Of Algorithmmentioning

confidence: 99%

“…We are interested in approximating c(S). To this end, we follow Gudmundsson et al [GSW20], who reduced the problem to querying the lengths of intersections between the curve and some squares. While Gudmundsson et al state their result for a curve, it holds for any set of segments.…”

Section: Preliminariesmentioning

confidence: 99%

“…Lemma 2.4 (Lemma 12 in [GSW20]). Given a set S of n segments in the plane, and a parameter ε ∈ (0, 1), one can compute, in…”

Section: Preliminariesmentioning

confidence: 99%

“…More recently, Gudmundsson et al [GSW20] gave a (6 + ε)-approximation in O (n 4/3 /ε 4 ) time. As part of their algorithm, they showed that one can quickly find a set of O(n) squares whose congestions yield a constant approximation to the congestion of the curve.…”

Section: Introductionmentioning

confidence: 99%

“…If we replace a point by a short segment, then deciding the packedness for sufficiently small squares centered at these points is equivalent to solving the Hopcroft problem. For this reason, Gudmunsson et al [GSW20] deemed it unlikely that their approach, "or a similar approach, can lead to a considerably faster algorithm" for computing congestion.…”

Section: Introductionmentioning

confidence: 99%

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Har-Peled¹,

Zhou²

2021

Preprint

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The congestion of a curve is a measure of how much it zigzags around locally. More precisely, a curve π is c-packed if the length of the curve lying inside any ball is at most c times the radius of the ball, and its congestion is the maximum c for which π is c-packed. This paper presents a randomized (288 + ε)-approximation algorithm for computing the congestion of a curve (or any set of segments in constant dimension). It runs in O(n log 2 n) time and succeeds with high probability. Although the approximation factor is large, the running time improves over the previous fastest constant approximation algorithm [GSW20], which took O (n 4/3 ) time. We carefully combine new ideas with known techniques to obtain our new, near-linear time algorithm.

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Section: Sketch Of Algorithmmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

“…Lemma 2.4 (Lemma 12 in [GSW20]). Given a set S of n segments in the plane, and a parameter ε ∈ (0, 1), one can compute, in…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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How Packed Is It, Really?

Har-Peled¹,

Zhou²

2021

Preprint

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Approximating the packedness of polygonal curves

Gudmundsson

Sha

Wong

2023

Computational Geometry

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Faster Approximate Covering of Subcurves under the Fréchet Distance

Frederik¹,

Jacobus²,

Driemel³

2022

Preprint

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Subtrajectory clustering is an important variant of the trajectory clustering problem, where the start and endpoints of trajectory patterns within the collected trajectory data are not known in advance. We study this problem in the form of a set cover problem for a given polygonal curve: find the smallest number k of representative curves such that any point on the input curve is contained in a subcurve that has Fréchet distance at most a given ∆ to a representative curve. We focus on the case where the representative curves are line segments and approach this NP-hard problem with classical techniques from the area of geometric set cover: we use a variant of the multiplicative weights update method which was first suggested by Brönniman and Goodrich for set cover instances with small VC-dimension. We obtain a bicriteria-approximation algorithm that computes a set of O(k log(k)) line segments that cover a given polygonal curve of n vertices under Fréchet distance at most O(∆). We show that the algorithm runs in O(k 2 n + kn 3 ) time in expectation and uses O(kn + n 3 ) space. For two dimensional input curves that are c-packed, we bound the expected running time by O(k 2 c 2 n) and the space by O(kn + c 2 n). In R d the dependency on n instead is quadratic. In addition, we present a variant of the algorithm that uses implicit weight updates on the candidate set and thereby achieves near-linear running time in n without any assumptions on the input curve, while keeping the same approximation bounds. This comes at the expense of a small (polylogarithmic) dependency on the relative arclength.

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Approximating the packedness of polygonal curves

Cited by 3 publications

References 16 publications

How Packed Is It, Really?

How Packed Is It, Really?

Approximating the packedness of polygonal curves

Faster Approximate Covering of Subcurves under the Fréchet Distance

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