Frechet Distance Revisited and Extended

Given two simplicial complexes in R^d, and start and end vertices in each complex, we show how to compute curves (in each complex) between these vertices, such that the Fr'echet distance between these curves is minimized. As a polygonal curve is a complex, this generalizes the regular notion of weak Fr'echet distance between curves. We also generalize the algorithm to handle an input of k simplicial complexes. Using this new algorithm we can solve a slew of new problems, from computing a mean curve for a given collection of curves, to various motion planning problems. Additionally, we show that for the mean curve problem, when the k input curves are c-packed, one can (1+epsilon)-approximate the mean curve in near linear time, for fixed k and epsilon. Additionally, we present an algorithm for computing the strong Fr'echet distance between two curves, which is simpler than previous algorithms, and avoids using parametric search.

💡 Research Summary

The paper revisits the classic Fréchet distance problem and extends it far beyond the traditional setting of two polygonal curves. The authors consider two (or more) simplicial complexes in ℝ^d, each equipped with a start and an end vertex, and ask for a pair of paths—one inside each complex—such that the weak Fréchet distance between the two paths is minimized. The key technical contribution is to replace the usual parametric space with the product complex C₁×C₂ (or C₁×…×C_k for k complexes). In this product space each cell is the Cartesian product of a cell from each input complex, yielding a polyhedral cell whose feasible region (points whose Euclidean distance is at most δ) is convex. By exploiting this convexity the decision problem reduces to checking connectivity of the free‑space within each cell, which can be represented as a graph whose edges correspond to admissible transitions between adjacent cells. The optimization problem then becomes a bottleneck‑shortest‑path problem on this graph, solvable in O(n²) time for two complexes (n denotes total combinatorial complexity).

The method generalizes to k complexes with only a linear factor increase: the total running time becomes O(n·k). This is a dramatic improvement over the naïve O(n^k) approach that would arise from a direct extension of the classic algorithm.

Several concrete applications are derived from this framework. First, the authors handle “thick” paths: given two simple polygons representing corridors of non‑zero width, the algorithm finds the pair of interior curves that are closest under Fréchet distance. Second, they address a wiring problem in 3‑D meshes, computing an optimal wire that stays inside a given model while staying as close as possible to a rough guide curve. Third, they formulate multi‑agent motion planning where each agent moves on its own complex; the goal is to minimize the maximum cost (or a weighted sum) incurred by any configuration along the synchronized motion.

A particularly notable application is the mean‑curve problem: given k input curves, find a new curve that minimizes the maximum Fréchet distance to each input. When the input curves are c‑packed (i.e., their total length within any ball of radius r is O(cr)), the authors show that a (1+ε)‑approximate mean curve can be computed in near‑linear time Õ(c·n/ε) for fixed k and ε. The algorithm samples each input curve, builds the product complex, and solves the bottleneck‑path problem, thereby avoiding the exponential dependence on k that appears in earlier exact algorithms.

Finally, the paper presents a new randomized algorithm for the strong (monotone) Fréchet distance between two polygonal curves (or two directed acyclic graph (DAG) complexes). Instead of the classic parametric search, the algorithm uses a linear‑time selection routine to sample candidate distances, tests each candidate by constructing the free‑space in the product complex, and repeats this process O(log n) times. The overall running time is O(n² log n), matching the best known deterministic bound but with a much simpler implementation.

In summary, by introducing the product complex as the central geometric object, the authors unify a wide range of Fréchet‑distance‑related problems—weak and strong variants, multi‑curve extensions, and approximation under packing assumptions—under a single algorithmic paradigm that runs in quadratic (or near‑linear under packing) time and avoids the heavy machinery of parametric search. This work opens the door to efficient motion‑planning and shape‑analysis tools that were previously infeasible due to computational complexity.