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, r2) where rectangle r 1 completely encloses rectangle r2) 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. Orthogonal range searching, geometric data structures, dominance, maxima, word RAM 38] for just a fraction of the vast amount of publications on orthogonal range searching. Recent papers [2, 3] have made progress on the pointer machine model and I/O model. In this paper, we study orthogonal range searching in the standard word RAM model, which is arguably the most natural and realistic model of computation to consider in internal memory. We obtain the best RAM upper bounds known to date for a number of problems, including: 2-d orthogonal range emptiness, 3-d orthogonal range reporting, and offline 4-d dominance range reporting.
When d = 2, Ref. REF preprocessed a data structure with O(n log log n) space and only O(log log n) query time; for larger d, the query time O(log log n) can be kept with a slight overhead on its space complexity.
Orthogonal range searching on the RAM, revisited
SoCG '11
Mathematics Computer Science