The $k$-Fr\'echet distance
Abstract
We introduce a new distance measure for comparing polygonal chains: the -Fr\'echet distance. As the name implies, it is closely related to the well-studied Fr\'echet distance but detects similarities between curves that resemble each other only piecewise. The parameter denotes the number of subcurves into which we divide the input curves. The -Fr\'echet distance provides a nice transition between (weak) Fr\'echet distance and Hausdorff distance. However, we show that deciding this distance measure turns out to be NP-complete, which is interesting since both (weak) Fr\'echet and Hausdorff distance are computable in polynomial time. Nevertheless, we give several possibilities to deal with the hardness of the -Fr\'echet distance: besides an exponential-time algorithm for the general case, we give a polynomial-time algorithm for , i.e., we ask that we subdivide our input curves into two subcurves each. We also present an approximation algorithm that outputs a number of subcurves of at most twice the optimal size. Finally, we give an FPT algorithm using parameters (the number of allowed subcurves) and (the number of segments of one curve that intersects the -neighborhood of a point on the other curve).
Cite
@article{arxiv.1903.02353,
title = {The $k$-Fr\'echet distance},
author = {Hugo A Akitaya and Maike Buchin and Leonie Ryvkin and Jérôme Urhausen},
journal= {arXiv preprint arXiv:1903.02353},
year = {2019}
}