How to Walk a Dog in Parallel: On Parallel Computation of the Discrete Frechet Distance

Abstract

The Fréchet distance (also known as the dog-leash distance) is a well-studied and prevalent distance measure for polygonal curves. In this paper we present parallel algorithms for computing the discrete \frechet\ distance between two curves, in both the PRAM and MPC models. For two polygonal curves of length $n$, we show that the distance between them can be computed in the PRAM model in $O(\log n/ σ)$ rounds and $\tilde{O}(n^{2+σ})$ total work for any $0<σ<1$. In the sublinear MPC model with $n^ρ$ space per machine, it can be computed in $O(\log n / ρ^2)$ rounds and $\tilde{O}(n^{2+ρ})$ total space. For $c$-packed curves, we match our bounds to the sequential algorithms, showing that a $(1+\eps)$-approximation of the distance can be computed in the PRAM model in $O(\log{n}+1/σ^3)$ rounds and $\tilde{O}(n^{1+σ})$ total work, and in the sublinear MPC model in $O(1 /ρ^3)$ rounds and $\tilde{O}(n^{1+ρ})$ total space, for any constant $\eps>0$. As part of our algorithms for $c$-packed curves, we also present parallel algorithms for computing curve simplifications, which may be of independent interest.