A Dimension-Reducing Fréchet Simplification Oracle

Let P be a polygonal curve with n vertices in the plane. We construct a data structure of size O(nlog n) suited for simplification queries of the following kind. Given a query line ℓ and an integer k ≥ 1, find a curve Q on ℓ with at most k vertices that minimizes the discrete Fréchet distance to P, among all such curves. Using our data structure, a query can be handled in O(k² log³ n + klog⁴ n) time. More generally, a geometric tree T on n vertices in the plane can be preprocessed into a near-linear-size structure so that, given a pair u, v of its vertices, a line ℓ, and an integer k ≥ 1, one can find a curve Q on ℓ with at most k vertices that minimizes the discrete Fréchet distance to the path from u to v in T, in time O(k² polylog n). For the general dimension-reduction problem, where P is a curve in R^d (d ≥ 3), 0 < ε0 < 1 is a real parameter, and a query specifies a g-flat h (1 ≤ g ≤ d− 1) and an integer k ≥ 1, we construct a data structure of size O(nlog n + f(ε0)n), where f(ε0) = (1 + 1/ε0)^(d−1)/², that allows us to find a curve Q on h with at most k vertices, whose discrete Fréchet distance to P is at most 1 + ε0 times the distance of Q^∗ to P, where Q^∗ is such a curve that minimizes the distance to P. The query handling time is O(f(ε0)k² log² n).