On computing the (exact) Fréchet distance with a frog

2025-12-08

AI Summary

This paper addresses three fundamental limitations of existing “frog-leap” methods for computing the continuous Fréchet distance: (i) lack of guaranteed accuracy, (ii) asymptotic incorporation of monotonicity events, and (iii) heuristic curve simplification. We propose the first exact algorithm with provable convergence guarantees. Methodologically, we construct a dynamic graph representation of the discrete free-space diagram, integrating recursive edge partitioning with explicit modeling of monotonicity events to ensure exact convergence in polynomial time. Furthermore, we devise a near-optimal lossless curve simplification scheme, replacing prior heuristic approaches. Experiments show that our algorithm significantly outperforms existing methods in worst-case scenarios and achieves state-of-the-art performance on several instances. However, it remains slower than the practical SoCG’19 algorithm overall, exposing an inherent efficiency bottleneck of the frog-leap paradigm. Our implementation is publicly available.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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 value 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, 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.

Problem

• Computes exact Fréchet distance for polygonal curves
• Ensures polynomial-time convergence to exact value
• Provides near-optimal lossless curve simplifications

Innovation

• Exact Frechet distance via polynomial-time convergence
• Near-optimal lossless simplification for efficiency
• Open-source C++ implementation with empirical evaluation