Exact Algorithms for Partial Curve Matching via the Fréchet Distance

Abstract

Curve matching is a fundamental problem that occurs in many applications. In this paper, we study the problem of measuring partial similarity between curves. Specifically, given two curves, we wish to maximize the total length of subcurves that are close to each other, where closeness is measured by the Fréchet distance, a common distance measure for curves. The resulting maximal length is called the partial Fréchet similarity between the two input curves. Given two polygonal curves P and Q in IR d of size m and n, respectively, we present the first exact algorithm that runs in polynomial time to compute F δ (P, Q), the partial Fréchet similarity between P and Q, under the L1 and L∞ norms. Specifically, we formulate the problem of computing F δ (P, Q) as a longest path problem, and solve it in O(mn(m + n) log(mn)) time, under the L1 or L∞ norm, using a "shortest-path map" type decomposition. To the best of our knowledge, this is the first paper to study this natural definition of partial curve similarity in the continuous setting (with all points in the curve considered), and present a polynomial-time exact algorithm for it.

Finding long and similar parts of trajectories

“…In fact, in the discrete case, this measure coincides with DTW under the Euclidean distance. A different solution to partial similarity under the Fréchet distance is given by Buchin et al [5]. The Fréchet distance as similarity measure for trajectories has been further developed by Buchin et al [3].…”

“…Figure 2 shows a set of six trajectories whose clusters we may want to determine. Two clusters are easily visible, but it is not directly clear if the dashed trajectory, τ 3 , fits better with {τ 1 , τ 2 } or with {τ 4 , τ 5 , τ 6 }. Based on shape and distance similarity, it fits slightly better in the first group, but if we consider time-dependent similarity like the average distance at corresponding times (defined formally later), it fits better in the second group, basically because the speeds are more similar.…”

Detecting Commuting Patterns by Clustering Subtrajectories

“…This subvariant can be seen as a partial matching problem of curves. Partial matching problems for two curves using the Fréchet distance that have been considered are matching a curve to a subcurve of a different trajectory [4] and matching two curves where several subcurves may contribute to the matching [12].…”

Four Soviets Walk the Dog: Improved Bounds for Computing the Fréchet Distance

“…They argued that the Fréchet distance is better suited as a similarity measure, and they described an O(n 2 log n) time algorithm to compute it on a real RAM or pointer machine. 1 Since Alt and Godau's seminal paper, there has been a wealth of research in various directions, such as extensions to higher dimensions [7,23,26,28,33,46], approximation algorithms [9,10,37], the geodesic and the homotopic Fréchet distance [29,34,38,48], and much more [2,22,25,35,36,51,54,55]. Most known approximation algorithms make further assumptions on the curves, and only an O(n 2 )-time approximation algorithm is known for arbitrary polygonal curves [24].…”