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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. 

In the classical prophet inequality setting, a gambler is given a sequence of n random variables X₁, … , X_n, taken from known distributions, observes their values in adversarial order and selects one of them, immediately after it is being observed, aiming to select a value that is as high as possible. The classical prophet inequality shows a strategy that guarantees a value at least half of the value of an omniscience prophet that always picks the maximum, and this ratio is optimal. Here, we generalize the prophet inequality, allowing the gambler some additional information about the future that is otherwise privy only to the prophet. Specifically, at any point in the process, the gambler is allowed to query an oracle 𝒪. The oracle responds with a single bit answer: YES if the current realization is greater than the remaining realizations, and NO otherwise. We show that the oracle model with m oracle calls is equivalent to the Top-1-of-(m+1) model when the objective is maximizing the probability of selecting the maximum. This equivalence fails to hold when the objective is maximizing the competitive ratio, but we still show that any algorithm for the oracle model implies an equivalent competitive ratio for the Top-1-of-(m+1) model. We resolve the oracle model for any m, giving tight lower and upper bound on the best possible competitive ratio compared to an almighty adversary. As a consequence, we provide new results as well as improvements on known results for the Top-1-of-m model.