A General Technique for Searching in Implicit Sets via Function Inversion
Boris S. Aronov, Jean Cardinal, Justin Dallant, John Iacono · Society for Industrial and Applied Mathematics eBooks · 2024
Given a function f from the set [N] to a d-dimensional integer grid, we consider data structures that allow efficient orthogonal range searching queries in the image of f, without explicitly storing it. We show that, if f is of the form [N] → [2w]d for some w = polylog(N) and is computable in constant time, then, for any 0 < α < 1, we can obtain a data structure using Õ(N1-α/3) words of space such that, for a given d-dimensional axis-aligned box B, we can search for some x ∈ [N] such that f (x) ∈ B in time Õ(Nα). This result is obtained simply by combining integer range searching with the Fiat-Naor function inversion scheme, which was already used in data-structure problems previously. We further obtain