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 $d$-dimensional point sequences of length $n$) 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 $2^{\Theta(n)}$. Moreover, we design an $\alpha$-approximation algorithm that runs in time $O(n\log n + n^2/\alpha)$, for any $\alpha\in [1, n]$. Hence, an $n^\varepsilon$-approximation of the Fréchet distance can be computed in strongly subquadratic time, for any \varepsilon > 0.</p

Fast Frechet Distance Between Curves With Long Edges

Computing the Fr\'echet distance between two polygonal curves takes roughly quadratic time. In this paper, we show that for a special class of curves the Fr\'echet distance computations become easier. Let $P$ and $Q$ be two polygonal curves in $\mathbb{R}^d$ with $n$ and $m$ vertices, respectively. We prove four results for the case when all edges of both curves are long compared to the Fr\'echet distance between them: (1) a linear-time algorithm for deciding the Fr\'echet distance between two curves, (2) an algorithm that computes the Fr\'echet distance in $O((n+m)\log (n+m))$ time, (3) a linear-time $\sqrt{d}$-approximation algorithm, and (4) a data structure that supports $O(m\log^2 n)$-time decision queries, where $m$ is the number of vertices of the query curve and $n$ the number of vertices of the preprocessed curve

A fast implementation of near neighbors queries for Fr\'echet distance (GIS Cup)

This paper describes an implementation of fast near-neighbours queries (also known as range searching) with respect to the Fr\'echet distance. The algorithm is designed to be efficient on practical data such as GPS trajectories. Our approach is to use a quadtree data structure to enumerate all curves in the database that have similar start and endpoints as the query curve. On these curves we run positive and negative filters to narrow the set of potential results. Only for those trajectories where these heuristics fail, we compute the Fr\'echet distance exactly, by running a novel recursive variant of the classic free-space diagram algorithm. Our implementation won the ACM SIGSPATIAL GIS Cup 2017.Comment: ACM SIGSPATIAL'17 invited paper. 9 page

On the General Chain Pair Simplification Problem

The Chain Pair Simplification problem (CPS) was posed by Bereg et al. who were motivated by the problem of efficiently computing and visualizing the structural resemblance between a pair of protein backbones. In this problem, given two polygonal chains of lengths n and m, the goal is to simplify both of them simultaneously, so that the lengths of the resulting simplifications as well as the discrete Frechet distance between them are bounded. When the vertices of the simplifications are arbitrary (i.e., not necessarily from the original chains), the problem is called General CPS (GCPS). In this paper we consider for the first time the complexity of GCPS under both the discrete Frechet distance (GCPS-3F) and the Hausdorff distance (GCPS-2H). (In the former version, the quality of the two simplifications is measured by the discrete Fr\u27echet distance, and in the latter version it is measured by the Hausdorff distance.) We prove that GCPS-3F is polynomially solvable, by presenting an widetilde-O((n+m)^6 min{n,m}) time algorithm for the corresponding minimization problem. We also present an O((n+m)^4) 2-approximation algorithm for the problem. On the other hand, we show that GCPS-2H is NP-complete, and present an approximation algorithm for the problem

FRESH: Fréchet similarity with hashing

This paper studies the r-range search problem for curves under the continuous Fréchet distance: given a dataset S of n polygonal curves and a threshold >0 , construct a data structure that, for any query curve q, efficiently returns all entries in S with distance at most r from q. We propose FRESH, an approximate and randomized approach for r-range search, that leverages on a locality sensitive hashing scheme for detecting candidate near neighbors of the query curve, and on a subsequent pruning step based on a cascade of curve simplifications. We experimentally compare FRESH to exact and deterministic solutions, and we show that high performance can be reached by suitably relaxing precision and recall

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