Abstract
AbstractWe consider the problem of computing the Fréchet distance between two curves for which the exact locations of the vertices are unknown. Each vertex may be placed in a given uncertainty region for that vertex, and the objective is to place vertices so as to minimise the Fréchet distance. This problem was recently shown to be NP-hard in 2D, and it is unclear how to compute an optimal vertex placement at all.We give a polynomial-time algorithm for 1D curves with intervals as uncertainty regions. In contrast, we show that the problem is NP-hard in 1D in the case that vertices are placed to maximise the Fréchet distance.We also study the weak Fréchet distance between uncertain curves. While finding the optimal placement of vertices seems more difficult than for the regular Fréchet distance—and indeed we can easily prove that the problem is NP-hard in 2D—the optimal placement of vertices in 1D can be computed in polynomial time. Finally, we investigate the discrete weak Fréchet distance, for which, somewhat surprisingly, the problem is NP-hard already in 1D.KeywordsCurvesUncertaintyFréchet distance1DHardnessWeak Fréchet distance
Highlights
The Fréchet distance is a popular distance measure for curves
We study the weak Fréchet distance between uncertain curves
The Fréchet distance corresponds to the minimum leash length needed with which the person and the dog can walk from start to end on their respective curve
Summary
The Fréchet distance is a popular distance measure for curves. Its computational complexity has drawn considerable attention in computational geometry [2, 5, 7, 8, 11, 17, 21]. Each curve is given by a sequence of uncertainty regions; we minimise the Fréchet distance over all possible choices of locations in the regions This is called the lower bound problem for the Fréchet distance between uncertain curves. Ahn et al [4] show a polynomial-time algorithm that decides whether the lower bound discrete Fréchet distance is below a certain threshold, for two curves with uncertainty regions modelled as circles in constant dimension. As has been recently shown, the decision problem for the continuous Fréchet distance is NP-hard already in two dimensions with vertical line segments as uncertainty regions and one precise and one uncertain curve [13]; it is not even clear how to compute the lower bound at all with any uncertainty model that is not discrete.
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