On computing the (exact) Fr\'echet distance with a frog

TL;DR

This paper improves the frog-based approach for computing the exact continuous Fréchet distance between polygonal curves, addressing its limitations and providing an open-source implementation with extensive empirical evaluation.

Contribution

It develops techniques to guarantee exactness, polynomial-time convergence, and near-optimal simplifications, and compares the method's practical performance with existing approaches.

Findings

Exact computation increases median runtime but can be faster in worst-case scenarios.

The frog-based approach is often outperformed by the method of Bringmann et al. in practice.

The new techniques provide theoretical guarantees and practical improvements over previous frog-based methods.

Abstract

The continuous Frechet distance between two polygonal curves is classically computed by exploring their free space diagram. Recently, Har-Peled, Raichel, and Robson [SoCG'25] proposed a radically different approach: instead of directly traversing the continuous free space, they approximate the distance by computing paths in a discrete graph derived from the discrete free space, recursively bisecting edges until the discrete distance converges to the continuous Frechet distance. They implement this so-called frog-based technique and report substantial practical speedups over the state of the art. We revisit the frog-based approach and address three of its limitations. First, the method does not compute the Frechet distance exactly. Second, the recursive bisection procedure only introduces the monotonicity events required to realise the Frechet distance asymptotically, that is, only in the limit. Third, the applied simplification technique is heuristic. Motivated by theoretical considerations, we develop new techniques that guarantee exactness, polynomial-time convergence, and near-optimal lossless simplifications. We provide an open-source C++ implementation of our variant. Our primary contribution is an extensive empirical evaluation. As expected, exact computation introduces overhead and increases the median running time. Yet, our method is often faster in the worst case, the slowest ten percent of instances, or even on average due to its convergence guarantees. More surprisingly, in our experiments, the implementation of Bringmann, Kuennemann, and Nusser [SoCG'19] consistently outperforms all frog-based approaches in practice. This appears to contrast published claims of the efficiency of the frog-based techniques. These results thereby provide nuanced perspective on frogs: highlighting both the theoretical appeal, but also the practical limitations.