Products of Euclidean metrics and applications to proximity questions among curves

TL;DR

This paper introduces efficient data structures for approximate nearest neighbor search among discretized curves in Euclidean space, generalizing popular distance measures like Fréchet and Dynamic Time Warping, with improved query times.

Contribution

It presents the first data structures and algorithms for ANN on discretized curves using Euclidean metric products, achieving arbitrarily good approximation with practical efficiency.

Findings

Supports a wide range of p in Euclidean products

Achieves arbitrarily good approximation factors

Improves query time complexity for bounded curve lengths

Abstract

The problem of Approximate Nearest Neighbor (ANN) search is fundamental in computer science and has benefited from significant progress in the past couple of decades. However, most work has been devoted to pointsets whereas complex shapes have not been sufficiently treated. Here, we focus on distance functions between discretized curves in Euclidean space: they appear in a wide range of applications, from road segments to time-series in general dimension. For $\ell_p$-products of Euclidean metrics, for any $p$, we design simple and efficient data structures for ANN, based on randomized projections, which are of independent interest. They serve to solve proximity problems under a notion of distance between discretized curves, which generalizes both discrete Fr\'echet and Dynamic Time Warping distances. These are the most popular and practical approaches to comparing such curves. We offer the first data structures and query algorithms for ANN with arbitrarily good approximation factor, at the expense of increasing space usage and preprocessing time over existing methods. Query time complexity is comparable or significantly improved by our algorithms, our algorithm is especially efficient when the length of the curves is bounded.