Fr\'echet Distance Under Translation: Conditional Hardness and an Algorithm via Offline Dynamic Grid Reachability

TL;DR

This paper improves algorithms for computing the discrete Fréchet distance under translation by solving offline dynamic reachability more efficiently and establishes a near-tight lower bound based on the Strong Exponential Time Hypothesis.

Contribution

It introduces an amortized efficient offline dynamic reachability algorithm for directed grid graphs and proves a conditional lower bound for the problem's complexity.

Findings

Improved running time to rac{n^{4.66}}{} for the problem.

Offline reachability can be solved in rac{N^{1/3}}{} amortized time.

Constructing the arrangement of size rac{n^{4}}{} is necessary in the worst case.

Abstract

The discrete Fr\'echet distance is a popular measure for comparing polygonal curves. An important variant is the discrete Fr\'echet distance under translation, which enables detection of similar movement patterns in different spatial domains. For polygonal curves of length $n$ in the plane, the fastest known algorithm runs in time $\tilde{\cal O}(n^{5})$ [Ben Avraham, Kaplan, Sharir '15]. This is achieved by constructing an arrangement of disks of size ${\cal O}(n^{4})$, and then traversing its faces while updating reachability in a directed grid graph of size $N := {\cal O}(n^2)$, which can be done in time $\tilde{\cal O}(\sqrt{N})$ per update [Diks, Sankowski '07]. The contribution of this paper is two-fold. First, although it is an open problem to solve dynamic reachability in directed grid graphs faster than $\tilde{\cal O}(\sqrt{N})$, we improve this part of the algorithm: We observe that an offline variant of dynamic $s$-$t$-reachability in directed grid graphs suffices, and we solve this variant in amortized time $\tilde{\cal O}(N^{1/3})$ per update, resulting in an improved running time of $\tilde{\cal O}(n^{4.66...})$ for the discrete Fr\'echet distance under translation. Second, we provide evidence that constructing the arrangement of size ${\cal O}(n^{4})$ is necessary in the worst case, by proving a conditional lower bound of $n^{4 - o(1)}$ on the running time for the discrete Fr\'echet distance under translation, assuming the Strong Exponential Time Hypothesis.