Approximating the Directed Hausdorff Distance

TL;DR

This paper introduces a method to efficiently approximate the directed Hausdorff distance between geometric sets in doubling metric spaces, significantly reducing computation time while maintaining accuracy, and also addresses robustness to outliers.

Contribution

It presents a preprocessing-based approach for fast approximation of Hausdorff distances, including partial variants, in doubling metric spaces, with theoretical guarantees and linear-time algorithms.

Findings

Approximate Hausdorff distance computed in linear time after preprocessing.

Preprocessing time is $O(n ext{log}\Delta)$, reducible to $O(n ext{log} n)$.

Provides a linear-time algorithm for all $k$-partial Hausdorff distances simultaneously.

Abstract

The Hausdorff distance is a metric commonly used to compute the set similarity of geometric sets. For sets containing a total of $n$ points, the exact distance can be computed na\"{i}vely in $O(n^2)$ time. In this paper, we show how to preprocess point sets individually so that the Hausdorff distance of any pair can then be approximated in linear time. We assume that the metric is doubling. The preprocessing time for each set is $O(n\log \Delta)$ where $\Delta$ is the ratio of the largest to smallest pairwise distances of the input. In theory, this can be reduced to $O(n\log n)$ time using a much more complicated algorithm. We compute $(1+\varepsilon)$-approximate Hausdorff distance in $(2 + \frac{1}{\varepsilon})^{O(d)}n$ time in a metric space with doubling dimension $d$. The $k$-partial Hausdorff distance ignores $k$ outliers to increase stability. Additionally, we give a linear-time algorithm to compute directed $k$-partial Hausdorff distance for all values of $k$ at once with no change to the preprocessing.