# The $k$-Fr\'echet distance

## TL;DR

The paper introduces the $k$-Fréchet distance, a new measure for comparing polygonal curves that captures piecewise similarities, and explores its computational complexity and algorithms for approximation and fixed-parameter tractability.

## Contribution

It defines the $k$-Fréchet distance, analyzes its NP-completeness, and provides algorithms for special cases, approximation, and fixed-parameter tractability.

## Key findings

- Deciding the $k$-Fréchet distance is NP-complete.
- Polynomial-time algorithm exists for $k=2$.
- An approximation algorithm achieves at most twice the optimal number of subcurves.

## Abstract

We introduce a new distance measure for comparing polygonal chains: the $k$-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 $k$ denotes the number of subcurves into which we divide the input curves. The $k$-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 $k$-Fr\'echet distance: besides an exponential-time algorithm for the general case, we give a polynomial-time algorithm for $k=2$, 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 $k$ (the number of allowed subcurves) and $z$ (the number of segments of one curve that intersects the $\varepsilon$-neighborhood of a point on the other curve).