A Dimension-Reducing Fréchet Simplification Oracle

Abstract

Let P be a polygonal curve with n vertices in the plane. We construct a data structure of size O(n log 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 + k log⁴n) 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).