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Given a function $f$ from the set $[N]$ to a $d$-dimensional integer grid, weconsider data structures that allow efficient orthogonal range searchingqueries in the image of $f$, without explicitly storing it.  We show that, if $f$ is of the form $[N]\to [2^{w}]^d$ for some$w=\mathrm{polylog} (N)$ and is computable in constant time, then, for any$0<\alpha <1$, we can obtain a data structure using $\tilde {O}(N^{1-\alpha /3})$ words of space such that, for a given $d$-dimensional axis-aligned box$B$, we can search for some $x\in [N]$ such that $f(x) \in B$ in time$\tilde{O}(N^{\alpha})$.  This result is obtained simply by combining integer range searching with theFiat-Naor function inversion scheme, which was already used in data-structureproblems previously.  We further obtain  - data structures for range counting and reporting, predecessor, selection,ranking queries, and combinations thereof, on the set $f([N])$,  - data structures for preimage size and preimage selection queries for agiven value of $f$, and  - data structures for selection and ranking queries on geometric quantitiescomputed from tuples of points in $d$-space.  These results unify and generalize previously known results on 3SUM-indexingand string searching, and are widely applicable as a black box to a variety ofproblems.  In particular, we give a data structure for a generalized version of gappedstring indexing, and show how to preprocess a set of points on an integer gridin order to efficiently compute (in sublinear time), for points contained in agiven axis-aligned box, their Theil-Sen estimator, the $k$th largest areatriangle, or the induced hyperplane that is the $k$th furthest from the origin.

A General Technique for Searching in Implicit Sets via Function Inversion