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

Finding a minimum stretch of a function

Given a piecewise monotone function f : R ! R and a real value Tmin, we develop an algorithm that finds an interval of length at least Tmin for which the average value of f is minimized. The run-time of the algorithm is linear in the number of monotone pieces of f if certain operations are available in constant time for f. We use this algorithm to solve a basic problem arising in the analysis of trajectories: Finding the most similar subtrajectories of two given trajectories, provided that the duration is at least Tmin. Since the precise solution requires complex operations, we also give a simple (1+")approximation algorithm in which these operations are not needed

Finding long and similar parts of trajectories

A natural time-dependent similarity measure for two trajectories is their average distance at corresponding times. We give algorithms for computing the most similar subtrajectories under this measure, assuming the two trajectories are given as two polygonal, possibly self-intersecting lines. When a minimum duration is specified for the subtrajectories, and they must start at exactly corresponding times in the input trajectories, we give a linear-time algorithm for computing the starting time and duration of the most similar subtrajectories. The algorithm is based on a result of independent interest: We present a linear-time algorithm to find, for a piece-wise monotone function, an interval of at least a given length that has minimum average value. When the two subtrajectories can start at different times in the two input trajectories, it appears difficult to give an exact algorithm for the most similar subtrajectories problem, even if the duration of the desired two subtrajectories is fixed to some length. We show that the problem can be solved approximately, and with a performance guarantee. More precisely, we present (1 + e)-approximation algorithms for computing the most similar subtrajectories of two input trajectories for the case where the duration is specified, and also for the case where only a minimum on the duration is specified

Model-based segmentation and classification of trajectories (Extended abstract)

We present efficient algorithms for segmenting and classifying a trajectory based on a parameterized movement model like the Brownian bridge movement model. Segmentation is the problem of subdividing a trajectory into parts such that each art is homogeneous in its movement characteristics. We formalize this using the likelihood of the model parameter. We consider the case where a discrete set of m parameter values is given and present an algorithm to compute an optimal segmentation with respect to an information criterion in O(nm) time for a trajectory with n sampling points. Classification is the problem of assigning trajectories to classes. We present an algorithm for discrete classification given a set of trajectories. Our algorithm computes the optimal classification with respect to an information criterion in O(m^2 + mk(log m + log k)) time for m parameter values and k trajectories, assuming bitonic likelihood functions

Four Soviets walk the dog, with an application to Alt's conjecture

Given two polygonal curves in the plane, there are many ways to define a notion of similarity between them. One measure that is extremely popular is the Fréchet distance. Since it has been proposed by Alt and Godau in 1992, many variants and extensions have been studied. Nonetheless, even more than 20 years later, the original O(n^2 log n) algorithm by Alt and Godau for computing the Fréchet distance remains the state of the art (here n denotes the number of vertices on each curve). This has led Helmut Alt to conjecture that the associated decision problem is 3SUM-hard.In recent work, Agarwal et al. show how to break the quadratic barrier for the discrete version of the Fréchet distance, where one considers sequences of points instead of polygonal curves. Building on their work, we give a randomized algorithm to compute the Fréchet distance between two polygonal curves in time O(n^2 \sqrt log n (log log n)^{3/2}) on a pointer machine and in time O(n^2 (log log n)^2) on a word RAM. Furthermore, we show that there exists an algebraic decision tree for the decision problem of depth O(n^{2¿}), for some ¿ > 0. This provides evidence that the decision problem may not be 3SUM-hard after all and reveals an intriguing new aspect of this well-studied problem

Area-preserving C-oriented schematization

We define an edge-move operation for polygons and prove that every simple non-convex polygon P has a non-conflicting pair of complementary edge-moves that reduces the number of edges of P while preserving its area. We use this result to generate area-preserving C-oriented schematizations of polygons

On the number of regular edge labelings

We prove that any irreducible triangulation on n vertices has O (4:6807n ) regular edge labeling,s and that there are irreducible triangulations on n vertices with (3:0426n ) regular edge labelings. Our upper bound relies on a novel application of Shearer's entropy lemma. As an example of the wider applicability of this technique, we also improve the upper bound on the number of 2-orientations of a quadrangulation to O (1:87n ). Keywords: Counting; Regular edge labeling; Shearer's entropy lemm

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

Delaunay triangulations on the word RAM: towards a practical worst-case optimal algorithm

The Delaunay triangulation of n points in the plane can be constructed in o(n log n) time when the coordinates of the points are integers from a restricted range. However, algorithms that are known to achieve such running times had not been implemented so far. We explore ways to obtain a practical algorithm for Delaunay triangulations in the plane that runs in linear time for small integers. For this, we first implement and evaluate variants of an algorithm, BrioDC, that is known to achieve this bound. We find that our implementations of these algorithms are competitive with fast existing algorithms. Secondly, we implement and evaluate variants of an algorithm, BRIO, that runs fast in experiments. Our variants aim to avoid bad worst-case behavior and our squarified orders indeed provide faster point location