We present a number of new results on one of the most extensively studied topics in computational geometry, orthogonal range searching. All our results are in the standard word RAM model:1. We present two data structures for 2-d orthogonal range emptiness. The first achieves O(n lg lg n) space and O(lg lg n) query time, assuming that the n given points are in rank space. This improves the previous results by Alstrup, Brodal, and Rauhe (FOCS'00), with O(n lg ε n) space and O(lg lg n) query time, or with O(n lg lg n) space and O(lg 2 lg n) query time. Our second data structure uses O(n) space and answers queries in O(lg ε n) time. The best previous O(n)-space data structure, due to Nekrich (WADS'07), answers queries in O(lg n/ lg lg n) time.2. We give a data structure for 3-d orthogonal range reporting with O(n lg 1+ε n) space and O(lg lg n + k) query time for points in rank space, for any constant ε > 0. This improves the previous results by Afshani (ESA'08), Karpinski and Nekrich (COCOON'09), and Chan (SODA'11), with O(n lg 3 n) space and O(lg lg n + k) query time, or with O(n lg 1+ε n) space and O(lg 2 lg n + k) query time. Consequently, we obtain improved upper bounds for orthogonal range reporting in all constant dimensions above 3. Our approach also leads to a new data structure for 2-d orthogonal range minimum queries with O(n lg ε n) space and O(lg lg n) query time for points in rank space.3. We give a randomized algorithm for 4-d offline dominance range reporting/emptiness with running time O(n lg n) plus the output size. This resolves two open problems (both appeared in Preparata and Shamos' seminal book): (a) given a set of n axis-aligned rectangles in the plane, we can report all k enclosure pairs (i.e., pairs (r 1 , r 2 ) where rectangle r 1 completely encloses rectangle r 2 ) in O(n lg n + k) expected time; (b) given a set of n points in 4-d, we can find all maximal points (points not dominated by any other points) in O(n lg n) expected time. The most recent previous development on (a) was reported back in SoCG'95 by Gupta, Janardan, Smid, and Dasgupta, whose main result was an O([n lg n + k] lg lg n) algorithm. The best previous result on (b) was an O(n lg n lg lg n) algorithm due to Gabow, Bentley, and Tarjan-from STOC'84! As a consequence, we also obtain the current-record time bound for the maxima problem in all constant dimensions above 4.

A mode of a multiset S is an element a ∈ S of maximum multiplicity; that is, a occurs at least as frequently as any other element in S. Given an array A[1 : n] of n elements, we consider a basic problem: constructing a static data structure that efficiently answers range mode queries on A. Each query consists of an input pair of indices (i, j) for which a mode of A[i : j] must be returned. The best previous data structure with linear space, by Krizanc, Morin, and Smid (ISAAC 2003), requires O( √ n log log n) query time. We improve their result and present an O(n)-space data structure that supports range mode queries in O( p n/ log n) worst-case time. Furthermore, we present strong evidence that a query time significantly below √ n cannot be achieved by purely combinatorial techniques; we show that boolean matrix multiplication of two 1998 ACM Subject Classification E.1 DATA STRUCTURES

In this paper, we consider two fundamental problems in the pointer machine model of computation, namely orthogonal range reporting and rectangle stabbing. Orthogonal range reporting is the problem of storing a set of n points in ddimensional space in a data structure, such that the t points in an axis-aligned query rectangle can be reported efficiently. Rectangle stabbing is the "dual" problem where a set of n axis-aligned rectangles should be stored in a data structure, such that the t rectangles that contain a query point can be reported efficiently. Very recently an optimal O(log n + t) query time pointer machine data structure was developed for the three-dimensional version of the orthogonal range reporting problem. However, in four dimensions the best known query bound of O(log 2 n/ log log n + t) has not been improved for decades.We describe an orthogonal range reporting data structure that is the first structure to achieve significantly less than O(log 2 n + t) query time in four dimensions. More precisely, we develop a structure that uses O(n(log n/ log log n) d ) space and can answer d-dimensional orthogonal range reporting queries (for d ≥ 4) in O(log n(log n/ log log n) d−4+1/(d−2) +t) time. Ignoring log log n factors, this speeds up the best previous query time by a log 1−1/(d−2) n factor. For the rectangle stabbing problem, we show that any data structure that uses nh space must use Ω(log n(log n/ log h) d−2 + t) time * This research was done while the author was a postdoctoral researcher at Dalhousie University, and was supported by Natural Sciences and Engineering Research Council of Canada (NSERC) through the Postdoctoral Fellowships Program (PDF). † Is supported in part by MADALGO and in part by a Google Fellowship in Search and Information Retrieval ‡ Center for Massive Data Algorithmics, a center of the Danish National Research Foundation.Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. to answer a query. This improves the previous results by a log h factor, and is the first lower bound that is optimal for a large range of h, namely for h ≥ log d−2+ε n where ε > 0 is an arbitrarily small constant. By a simple geometric transformation, our result also implies an improved query lower bound for orthogonal range reporting.

We study regular expression membership testing: Given a regular expression of size m and a string of size n, decide whether the string is in the language described by the regular expression. Its classic O(nm) algorithm is one of the big success stories of the 70s, which allowed pattern matching to develop into the standard tool that it is today.Many special cases of pattern matching have been studied that can be solved faster than in quadratic time. However, a systematic study of tractable cases was made possible only recently, with the first conditional lower bounds reported by Backurs and Indyk [FOCS'16]. Restricted to any "type" of homogeneous regular expressions of depth 2 or 3, they either presented a nearlinear time algorithm or a quadratic conditional lower bound, with one exception known as the Word Break problem.In this paper we complete their work as follows:• We present two almost-linear time algorithms that generalize all known almost-linear time algorithms for special cases of regular expression membership testing. * Theorem 3. For any δ > 0, if 4-Clique has no O(n 3+δ ) algorithm, then Word Break has no O(n 1+δ/3 ) algorithm for n = m.We remark that this situation of having matching conditional lower bounds only for combinatorial algorithms is not uncommon, see, e.g., Sliding Window Hamming Distance [6].New Almost-Linear Time Algorithms. We establish two more types for which the membership problem is in almost-linear time. Theorem 4. We design a deterministicÕ(n) + O(m) algorithm for | + •+-membership and an expected time n 1+o(1) + O(m) algorithm for | + •|-membership. These algorithms also work for t-membership for any subsequence t of | + •+ or | + •|, respectively.This generalizes all previously known almost-linear time algorithms for any t-membership problem, as all such types t are proper subsequences of |+•+ or |+•|. Moreover, no further generalization of our algorithms is possible, as shown below.

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