On Computing the $k$-Shortcut Fr\'echet Distance

The Fr\'echet distance is a popular measure of dissimilarity for polygonal curves. It is defined as a min-max formulation that considers all direction-preserving continuous bijections of the two curves. Because of its susceptibility to noise, Driemel and Har-Peled introduced the shortcut Fr\'echet distance in 2012, where one is allowed to take shortcuts along one of the curves, similar to the edit distance for sequences. We analyse the parameterized version of this problem, where the number of shortcuts is bounded by a parameter $k$. The corresponding decision problem can be stated as follows: Given two polygonal curves $T$ and $B$ of at most $n$ vertices, a parameter $k$ and a distance threshold $\delta$, is it possible to introduce $k$ shortcuts along $B$ such that the Fr\'echet distance of the resulting curve and the curve $T$ is at most $\delta$? We study this problem for polygonal curves in the plane. We provide a complexity analysis for this problem with the following results: (i) assuming the exponential-time-hypothesis (ETH), there exists no algorithm with running time bounded by $n^{o(k)}$; (ii) there exists a decision algorithm with running time in $O(kn^{2k+2}\log n)$. In contrast, we also show that efficient approximate decider algorithms are possible, even when $k$ is large. We present a $(3+\varepsilon)$-approximate decider algorithm with running time in $O(k n^2 \log^2 n)$ for fixed $\varepsilon$. In addition, we can show that, if $k$ is a constant and the two curves are $c$-packed for some constant $c$, then the approximate decider algorithm runs in near-linear time.