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Register a new userLocally Correct Frechet Matchings
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Added by
Kevin Buchin
Publication date
2012
fields
Informatics Engineering
and research's language is
English
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The Frechet distance is a metric to compare two curves, which is based on monotonous matchings between these curves. We call a matching that results in the Frechet distance a Frechet matching. There are often many different Frechet matchings and not all of these capture the similarity between the curves well. We propose to restrict the set of Frechet matchings to natural matchings and to this end introduce locally correct Frechet matchings. We prove that at least one such matching exists for two polygonal curves and give an O(N^3 log N) algorithm to compute it, where N is the total number of edges in both curves. We also present an O(N^2) algorithm to compute a locally correct discrete Frechet matching.
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In this paper we study a wide range of variants for computing the (discrete and continuous) Frechet distance between uncertain curves. We define an uncertain curve as 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 Frechet distance, which are the minimum and maximum Frechet distance for any realisations of the curves. We prove that both the upper and lower bound problems are NP-hard for the continuous Frechet distance in several uncertainty models, and that the upper bound problem remains hard for the discrete Frechet distance. In contrast, the lower bound (discrete and continuous) Frechet distance can be computed in polynomial time. Furthermore, we show that computing the expected discrete Frechet distance is #P-hard when the uncertainty regions are modelled as point sets or line segments. The construction also extends to show #P-hardness for computing the continuous Frechet distance when regions are modelled as point sets. On the positive side, we argue that in any constant dimension there is a FPTAS for the lower bound problem when $Delta / delta$ is polynomially bounded, where $delta$ is the Frechet distance and $Delta$ bounds the diameter of the regions. We then argue there is a near-linear-time 3-approximation for the decision problem when the regions are convex and roughly $delta$-separated. Finally, we also study the setting with Sakoe--Chiba time bands, where we restrict the alignment between the two curves, and give polynomial-time algorithms for upper bound and expected discrete and continuous Frechet distance for uncertainty regions modelled as point sets.
This paper studies the $r$-range search problem for curves under the continuous Frechet distance: given a dataset $S$ of $n$ polygonal curves and a threshold $r>0$, 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.
The Frechet distance is a popular similarity measure between curves. For some applications, it is desirable to match the curves under translation before computing the Frechet distance between them. This variant is called the Translation Invariant Frechet distance, and algorithms to compute it are well studied. The query version, finding an optimal placement in the plane for a query segment where the Frechet distance becomes minimized, is much less well understood. We study Translation Invariant Frechet distance queries in a restricted setting of horizontal query segments. More specifically, we preprocess a trajectory in $mathcal O(n^2 log^2 n) $ time and $mathcal O(n^{3/2})$ space, such that for any subtrajectory and any horizontal query segment we can compute their Translation Invariant Frechet distance in $mathcal O(text{polylog } n)$ time. We hope this will be a step towards answering Translation Invariant Frechet queries between arbitrary trajectories.
A matching is compatible to two or more labeled point sets of size $n$ with labels ${1,dots,n}$ if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled convex sets of $n$ points there exists a compatible matching with $lfloor sqrt {2n}rfloor$ edges. More generally, for any $ell$ labeled point sets we construct compatible matchings of size $Omega(n^{1/ell})$. As a corresponding upper bound, we use probabilistic arguments to show that for any $ell$ given sets of $n$ points there exists a labeling of each set such that the largest compatible matching has ${mathcal{O}}(n^{2/({ell}+1)})$ edges. Finally, we show that $Theta(log n)$ copies of any set of $n$ points are necessary and sufficient for the existence of a labeling such that any compatible matching consists only of a single edge.
The Frechet distance is a popular distance measure for curves which naturally lends itself to fundamental computational tasks, such as clustering, nearest-neighbor searching, and spherical range searching in the corresponding metric space. However, its inherent complexity poses considerable computational challenges in practice. To address this problem we study distortion of the probabilistic embedding that results from projecting the curves to a randomly chosen line. Such an embedding could be used in combination with, e.g. locality-sensitive hashing. We show that in the worst case and under reasonable assumptions, the discrete Frechet distance between two polygonal curves of complexity $t$ in $mathbb{R}^d$, where $dinlbrace 2,3,4,5rbrace$, degrades by a factor linear in $t$ with constant probability. We show upper and lower bounds on the distortion. We also evaluate our findings empirically on a benchmark data set. The preliminary experimental results stand in stark contrast with our lower bounds. They indicate that highly distorted projections happen very rarely in practice, and only for strongly conditioned input curves. Keywords: Frechet distance, metric embeddings, random projections
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