905 research outputs found
Fréchet Distance for Uncertain Curves
In this article, we study a wide range of variants for computing the (discrete and continuous) FrĂ©chet distance between uncertain curves. An uncertain curve is a sequence of uncertainty regions, where each region is a disk, a line segment, or a set of points. A realisation of a curve is a polyline connecting one point from each region. Given an uncertain curve and a second (certain or uncertain) curve, we seek to compute the lower and upper bound FrĂ©chet distance, which are the minimum and maximum FrĂ©chet distance for any realisations of the curves. We prove that both problems are NP-hard for the FrĂ©chet distance in several uncertainty models, and that the upper bound problem remains hard for the discrete FrĂ©chet distance. In contrast, the lower bound (discrete [5] and continuous) FrĂ©chet distance can be computed in polynomial time in some models. Furthermore, we show that computing the expected (discrete and continuous) FrĂ©chet distance is #P-hard in some models.On the positive side, we present an FPTAS in constant dimension for the lower bound problem when Î/ÎŽis polynomially bounded, where ÎŽis the FrĂ©chet distance and Îbounds the diameter of the regions. We also show a near-linear-time 3-approximation for the decision problem on roughly ÎŽ-separated convex regions. Finally, we study the setting with Sakoe-Chiba time bands, where we restrict the alignment between the curves, and give polynomial-time algorithms for the upper bound and expected discrete and continuous FrĂ©chet distance for uncertainty modelled as point sets.</p
Computing the Fréchet distance with shortcuts is NP-hard
We study the shortcut Fréchet distance, a natural variant of the Fréchet distance that allows us to take shortcuts from and to any point along one of the curves. We show that, surprisingly, the problem of computing the shortcut Fréchet distance exactly is NP-hard. Furthermore, we give a 3-approximation algorithm for the decision version of the problem
Approximability of the Discrete {Fr\'echet} Distance
<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 -dimensional point sequences of length ) 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 . Moreover, we design an -approximation algorithm that runs in time , for any . Hence, an -approximation of the Fréchet distance can be computed in strongly subquadratic time, for any \varepsilon > 0.</p
Computing the Fréchet Distance with a Retractable Leash
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 (Formula presented.) under polyhedral distance functions (e.g., (Formula presented.) and (Formula presented.)). We also get a (Formula presented.)-approximation of the Fréchet distance under the Euclidean metric, in quadratic time for any fixed (Formula presented.). For the exact Euclidean case, our framework currently yields an algorithm with running time (Formula presented.). However, we conjecture that it may eventually lead to a faster exact algorithm
Computing the Similarity Between Moving Curves
In this paper we study similarity measures for moving curves which can, for
example, model changing coastlines or retreating glacier termini. Points on a
moving curve have two parameters, namely the position along the curve as well
as time. We therefore focus on similarity measures for surfaces, specifically
the Fr\'echet distance between surfaces. While the Fr\'echet distance between
surfaces is not even known to be computable, we show for variants arising in
the context of moving curves that they are polynomial-time solvable or
NP-complete depending on the restrictions imposed on how the moving curves are
matched. We achieve the polynomial-time solutions by a novel approach for
computing a surface in the so-called free-space diagram based on max-flow
min-cut duality
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