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Abstract:
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.
