We consider the problem of performing predecessor searches in a bounded universe while achieving query times that depend on the distribution of queries. We obtain several data structures with various properties: in particular, we give data structures that achieve expected query times logarithmic in the entropy of the distribution of queries but with space bounded in terms of universe size, as well as data structures that use only linear space but with query times that are higher (but still sublinear) functions of the entropy. For these structures, the distribution is assumed to be known. We also consider individual query times on universe elements with general weights, as well as the case when the distribution is not known in advance.
using @math space @cite_0 . By paying an additional @math factor in the first half of this bound, the space can be improved to @math @cite_0 . P a tra s cu and Thorup later effectively settled this line of research with a set of time-space trade-offs @cite_8 .
@cite_0 @cite_8
2949155877 2537252502 2508919161 2000931246
Karpinski and Nekrich (2008) introduced the problem of parameterized range majority, which asks to preprocess a string of length @math such that, given the endpoints of a range, one can quickly find all the distinct elements whose relative frequencies in that range are more than a threshold @math . Subsequent authors have reduced their time and space bounds such that, when @math is given at preprocessing time, we need either @math space and optimal @math query time or linear space and @math query time, where @math is the alphabet size. In this paper we give the first linear-space solution with optimal @math query time. For the case when @math is given at query time, we significantly improve previous bounds, achieving either @math space and optimal @math query time or compressed space and @math query time. Along the way, we consider the complementary problem of parameterized range minority that was recently introduced by (2012), who achieved linear space and @math query time even for variable @math . We improve their solution to use either nearly optimally compressed space with no slowdown, or optimally compressed space with nearly no slowdown. Some of our intermediate results, such as density-sensitive query time for one-dimensional range counting, may be of independent interest. Given a set Z of n positive integers and a target value t, the S ubset S um problem asks whether any subset of Z sums to t. A textbook pseudopolynomial time algorithm by Bellman from 1957 solves S ubset S um in time O(nt). This has been improved to O(n max Z) by Pisinger [J. Algorithms'99] and recently to [EQUATION] by Koiliaris and Xu [SODA'17]. Here we present a simple and elegant randomized algorithm running in time O(n+t). This improves upon a classic algorithm and is likely to be near-optimal, since it matches conditional lower bounds from S et C over and k-C lique . We then use our new algorithm and additional tricks to improve the best known polynomial space solution from time O(n3t) and space O(n2) to time O(nt) and space O(n log t), assuming the Extended Riemann Hypothesis. Unconditionally, we obtain time O(nt1+e) and space O(nte) for any constant e > 0. [See the paper for the full abstract.] We show tight upper and lower bounds for time-space trade-offs for the @math -Approximate Near Neighbor Search problem. For the @math -dimensional Euclidean space and @math -point datasets, we develop a data structure with space @math and query time @math for every @math such that: This is the first data structure that achieves sublinear query time and near-linear space for every approximation factor @math , improving upon [Kapralov, PODS 2015]. The data structure is a culmination of a long line of work on the problem for all space regimes; it builds on Spherical Locality-Sensitive Filtering [Becker, Ducas, Gama, Laarhoven, SODA 2016] and data-dependent hashing [Andoni, Indyk, Nguyen, Razenshteyn, SODA 2014] [Andoni, Razenshteyn, STOC 2015]. Our matching lower bounds are of two types: conditional and unconditional. First, we prove tightness of the whole above trade-off in a restricted model of computation, which captures all known hashing-based approaches. We then show unconditional cell-probe lower bounds for one and two probes that match the above trade-off for @math , improving upon the best known lower bounds from [Panigrahy, Talwar, Wieder, FOCS 2010]. In particular, this is the first space lower bound (for any static data structure) for two probes which is not polynomially smaller than the one-probe bound. To show the result for two probes, we establish and exploit a connection to locally-decodable codes. Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253