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Register a new userSublinear data structures for short Frechet queries
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Added by
Ioannis Psarros
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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We study metric data structures for curves in doubling spaces, such as trajectories of moving objects in Euclidean $mathbb{R}^d$, where the distance between two curves is measured using the discrete Frechet distance. We design data structures in an emph{asymmetric} setting where the input is a curve (or a set of $n$ curves) each of complexity $m$ and the queries are with curves of complexity $kll m$. We show that there exist approximate data structures that are independent of the input size $N = d cdot n cdot m$ and we study how to maintain them dynamically if the input is given in the stream. Concretely, we study two types of data structures: (i) distance oracles, where the task is to store a compressed version of the input curve, which can be used to answer queries for the distance of a query curve to the input curve, and (ii) nearest-neighbor data structures, where the task is to preprocess a set of input curves to answer queries for the input curve closest to the query curve. In both cases we are interested in approximation. For curves embedded in Euclidean $mathbb{R}^d$ with constant $d$, our distance oracle uses space in $mathcal{O}((k log(epsilon^{-1}) epsilon^{-d})^k)$ ($epsilon$ is the precision parameter). The oracle performs $(1+epsilon)$-approximate queries in time in $mathcal{O}(k^2)$ and is deterministic. We show how to maintain this distance oracle in the stream using polylogarithmic additional memory. In the stream, we can dynamically answer distance queries to the portion of the stream seen so far in $mathcal{O}(k^4 log^2 m)$ time. We apply our techniques to the second problem, approximate near neighbor (ANN) data structures, and achieve an exponential improvement in the dependency on the complexity of the input curves compared to the state of the art.
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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.
We study geometric set cover problems in dynamic settings, allowing insertions and deletions of points and objects. We present the first dynamic data structure that can maintain an $O(1)$-approximation in sublinear update time for set cover for axis-aligned squares in 2D. More precisely, we obtain randomized update time $O(n^{2/3+delta})$ for an arbitrarily small constant $delta>0$. Previously, a dynamic geometric set cover data structure with sublinear update time was known only for unit squares by Agarwal, Chang, Suri, Xiao, and Xue [SoCG 2020]. If only an approximate size of the solution is needed, then we can also obtain sublinear amortized update time for disks in 2D and halfspaces in 3D. As a byproduct, our techniques for dynamic set cover also yield an optimal randomized $O(nlog n)$-time algorithm for static set cover for 2D disks and 3D halfspaces, improving our earlier $O(nlog n(loglog n)^{O(1)})$ result [SoCG 2020].
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.
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.
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.
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