5 research outputs found

    Approximating (k,ℓ)(k,\ell)-center clustering for curves

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    The Euclidean kk-center problem is a classical problem that has been extensively studied in computer science. Given a set G\mathcal{G} of nn points in Euclidean space, the problem is to determine a set C\mathcal{C} of kk centers (not necessarily part of G\mathcal{G}) such that the maximum distance between a point in G\mathcal{G} and its nearest neighbor in C\mathcal{C} is minimized. In this paper we study the corresponding (k,ℓ)(k,\ell)-center problem for polygonal curves under the Fr\'echet distance, that is, given a set G\mathcal{G} of nn polygonal curves in Rd\mathbb{R}^d, each of complexity mm, determine a set C\mathcal{C} of kk polygonal curves in Rd\mathbb{R}^d, each of complexity ℓ\ell, such that the maximum Fr\'echet distance of a curve in G\mathcal{G} to its closest curve in C\mathcal{C} is minimized. In this paper, we substantially extend and improve the known approximation bounds for curves in dimension 22 and higher. We show that, if ℓ\ell is part of the input, then there is no polynomial-time approximation scheme unless P=NP\mathsf{P}=\mathsf{NP}. Our constructions yield different bounds for one and two-dimensional curves and the discrete and continuous Fr\'echet distance. In the case of the discrete Fr\'echet distance on two-dimensional curves, we show hardness of approximation within a factor close to 2.5982.598. This result also holds when k=1k=1, and the NP\mathsf{NP}-hardness extends to the case that ℓ=∞\ell=\infty, i.e., for the problem of computing the minimum-enclosing ball under the Fr\'echet distance. Finally, we observe that a careful adaptation of Gonzalez' algorithm in combination with a curve simplification yields a 33-approximation in any dimension, provided that an optimal simplification can be computed exactly. We conclude that our approximation bounds are close to being tight.Comment: 24 pages; results on minimum-enclosing ball added, additional author added, general revisio

    Approximability of the Discrete {Fr\'echet} Distance

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    <p>The Fréchet distance is a popular and widespread distance measure for point sequences and for curves. About two years ago, Agarwal et al. [SIAM J. Comput. 2014] presented a new (mildly) subquadratic algorithm for the discrete version of the problem. This spawned a flurry of activity that has led to several new algorithms and lower bounds.</p><p>In this paper, we study the approximability of the discrete Fréchet distance. Building on a recent result by Bringmann [FOCS 2014], we present a new conditional lower bound showing that strongly subquadratic algorithms for the discrete Fréchet distance are unlikely to exist, even in the one-dimensional case and even if the solution may be approximated up to a factor of 1.399.</p><p>This raises the question of how well we can approximate the Fréchet distance (of two given dd-dimensional point sequences of length nn) in strongly subquadratic time. Previously, no general results were known. We present the first such algorithm by analysing the approximation ratio of a simple, linear-time greedy algorithm to be 2Θ(n)2^{\Theta(n)}. Moreover, we design an α\alpha-approximation algorithm that runs in time O(nlog⁥n+n2/α)O(n\log n + n^2/\alpha), for any α∈[1,n]\alpha\in [1, n]. Hence, an nΔn^\varepsilon-approximation of the Fréchet distance can be computed in strongly subquadratic time, for any \varepsilon > 0.</p

    LIPIcs, Volume 258, SoCG 2023, Complete Volume

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    LIPIcs, Volume 258, SoCG 2023, Complete Volum
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