Matching Curves to Imprecise Point Sets using Fr\'echet Distance

Abstract

Let $P$ be a polygonal curve in $\mathbb{R}^d$ of length $n$, and $S$ be a point-set of size $k$. The Curve/Point Set Matching problem consists of finding a polygonal curve $Q$ on $S$ such that the Fr\'echet distance from $P$ is less than a given $\varepsilon$. We consider eight variations of the problem based on the distance metric used and the omittability or repeatability of the points. We provide closure to a recent series of complexity results for the case where $S$ consists of precise points. More importantly, we formulate a more realistic version of the problem that takes into account measurement errors. This new problem is posed as the matching of a given curve to a set of imprecise points. We prove that all three variations of the problem that are in P when $S$ consists of precise points become NP-complete when $S$ consists of imprecise points. We also discuss approximation results.