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We show that a variant of the continuous Fréchet distance between polygonal curves can be computed using essentially the same algorithm used to solve the discrete version. The new variant is not necessarily monotone, but this shortcoming can be easily handled via refinement. Combined with a Dijkstra/Prim type algorithm, this leads to a realization of the Fréchet distance (i.e., a morphing) that is locally optimal (aka locally correct), that is both easy to compute, and in practice, takes near linear time on many inputs. The new morphing has the property that the leash is always as short as possible. These matchings/morphings are more natural, and are better than the ones computed by standard algorithms - in particular, they handle noise more graciously. This should make the Fréchet distance more useful for real world applications. We implemented the new algorithm, and various strategies to obtain fast practical performance. We performed extensive experiments with our new algorithm, and released publicly available (and easily installable and usable) Julia and Python packages. In particular, the Julia implementation, for computing the regular Fréchet distance, seems to be {significantly faster} than other currently available implementations. See Table 2.2. Our algorithms can be used to compute the almost-exact Fréchet distance between polygonal curves. Implementations and numerous examples are available here: https://frechet.xyz. 

Given a set P ⊂ ℝ^d of n points, with diameter Δ, and a parameter δ ∈ (0,1), it is known that there is a partition of P into sets P_1, …, P_t, each of size O(1/δ²), such that their convex hulls all intersect a common ball of radius δΔ. We prove that a random partition, with a simple alteration step, yields the desired partition, resulting in a (randomized) linear time algorithm (i.e., O(dn)). We also provide a deterministic algorithm with running time O(dn log n). Previous proofs were either existential (i.e., at least exponential time), or required much bigger sets. In addition, the algorithm and its proof of correctness are significantly simpler than previous work, and the constants are slightly better. We also include a number of applications and extensions using the same central ideas. For example, we provide a linear time algorithm for computing a "fuzzy" centerpoint, and prove a no-dimensional weak ε-net theorem with an improved constant.