On characteristic points and approximate decision algorithms for the minimum Hausdorff distance

We investigate {\em approximate decision algorithms} for determining whether the minimum Hausdorff distance between two points sets (or between two sets of nonintersecting line segments) is at most $\varepsilon$.\def\eg{(\varepsilon/\gamma)} An approximate decision algorithm is a standard decision algorithm that answers {\sc yes} or {\sc no} except when $\varepsilon$ is in an {\em indecision interval} where the algorithm is allowed to answer {\sc don't know}. We present algorithms with indecision interval $[\delta-\gamma,\delta+\gamma]$ where $\delta$ is the minimum Hausdorff distance and $\gamma$ can be chosen by the user. In other words, we can make our algorithm as accurate as desired by choosing an appropriate $\gamma$. For two sets of points (or two sets of nonintersecting lines) with respective cardinalities $m$ and $n$ our approximate decision algorithms run in time $O(\eg^2(m+n)\log(mn))$ for Hausdorff distance under translation, and in time $O(\eg^2mn\log(mn))$ for Hausdorff distance under Euclidean motion