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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 • 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 a given value of f, and • data structures for selection and ranking queries on geometric quantities computed from tuples of points in d-space. These results unify and generalize previously known results on 3SUM-indexing and string searching, and are widely applicable as a black box to a variety of problems. In particular, we give a data structure for a generalized version of gapped string indexing, and show how to preprocess a set of points on an integer grid in order to efficiently compute (in sublinear time), for points contained in a given axis-aligned box, their Theil-Sen estimator, the kth largest area triangle, or the induced hyperplane that is the kth furthest from the origin. 

We study the relationship between memory accesses, bank conflicts, thread multiplicity (also known as over-subscription) and instruction-level parallelism in comparison-based sort- ing algorithms for Graphics Processing Units (GPUs). We experimentally validate a proposed formula that relates these parameters with asymptotic analysis of the number of mem- ory accesses by an algorithm. Using this formula we analyze and compare several GPU sorting algorithms, identifying key performance bottlenecks in each one of them. Based on this analysis we propose a GPU-efficient multiway merge- sort algorithm, GPU-MMS, which minimizes or eliminates these bottlenecks and balances various limiting factors for specific hardware. We realize an implementation of GPU-MMS and compare it to sorting algorithm implementations in state-of-the-art GPU libraries on three GPU architectures. Despite these library implementations being highly optimized, we find that GPU-MMS outperforms them by an average of 21% for random integer inputs and 14% for random key-value pairs.