1/6/2023 0 Comments Otomata offline![]() ![]() The idea is that we do not need to change the planar shallowĬutting for most insertions. #OTOMATA OFFLINE UPDATE#Z-coordinate is considered (i.e., the next insertion), we update the correspondingĢD shallow cutting. ![]() Points of P that lie below the sweep plane. Insertions, specifically the planar shallow cutting of the xy-projection of the The problem to the problem of maintaining a 2D shallow cutting under only The xy-plane considering the points in non-decreasing z-coordinate. Following we sort the points and sweep a plane parallel to Points and for any integer k can be constructed deterministically in O(n log log n)Īlgorithm Sketch. An optimal k-shallow cutting for 3D dominance ranges on n input Points ci, di, the corners of the planar shallow cutting.Ĭonstruction of 3D Dominance Shallow Cuttings dt−1 ct of alternating vertical line segments ci di = × Īnd horizontal line segments di ci+1 = × (Fig.A planar shallow cutting has the shape of an orthogonal staircase curve Shallow cutting S is optimal when it contains at most cmax nk , qd ) if pi 1 and (ii) any point in Rd with level in P at most k is dominatedīy some vertex of S. Points in Rd, report the input points that are dominated by each query point Version of the offline dominance reporting problem: given n input and query It is well-known that the rectangle enclosure problem is reducible to the 4D Hold in the standard RAM model with w = log n, since we can pre-sort and (In fact, for all time bounds that are Ω(n log n), they All presented algorithms operate in the word-RAM model However their algorithm uses randomization and thus the time bound In improving the running time to the desired bound of O(n log n + k) using linear space. Recently, Chan, Larsen and Pătraşcu succeeded The linear-space algorithm of Gupta, Janardan, Smid and Dasgupta and an alternative implementation by Lagogiannis, Makris and A. Preparata improved the space bound to linear.įurther improvements were discovered in the 1990s. Vaishnavi and Wood first addressed the question presenting an O(n log2 n + k)-time algorithm that uses O(n log2 n) space. R2 ), raising the question whether the same bound could be achieved for rectangle enclosure. O(n log n + k) worst-case time and linear-space algorithm for the related rectangle intersection problem (reporting all k pairs (r1, r2 ) where r1 intersects An early paper by Bentley and Wood presented an Science and Engineering, Hong Kong University of Science and Technology. ![]() The work of this author was done during his visit to the Department of Computer Through Center for Massive Data Algorithmics (MADALGO). Work supported in part by the Danish National Research Foundation grant DNRF84 Geometry with applications to VLSI design, image processing, computer This is a classic problem in the field of compu tational We study the problem of rectangle enclosure: given a set of n axis-aligned rectangles on the plane, report all k pairs (r1, r2 ) of input rectangles where r1Ĭompletely encloses r2. Incorporated, including a linear-time algorithm for merging shallow cuttings and an algorithm for an offline tree point location problem. We first present an improved deterministic construction algorithm that runs in O(n log log n) time in the word-RAM Recently, Afshani and Tsakalidis obtained a deterministic O(n log n)-time algorithm toĬonstruct such cuttings. Shallow cuttings for 3D dominance ranges. Range reporting problem in 4D, and our result leads to the currentlyįastest deterministic algorithm for offline dominance reporting in anyĪ key tool behind Chan et al.’s previous randomized algorithm is The 2D rectangle enclosure problem is related to the offline dominance Larsen and Pătraşcu that attains the same time complexity We achieve the result by derandomizing the algorithm of Chan, Previous deterministic algorithms with O((n log n + k) log log n) running Worst-case time and O(n) space in the word-RAM model. We present the first deterministic algorithm that takes O(n log n + k) We want to report all k enclosing pairs of n input rectangles in 2D. That has been studied since the 1980s: in the rectangle enclosure problem Cheriton School of Computer Science, University of Waterloo, Science and Engineering Department, CUHK, Hong We revisit a classical problem in computational geometry MADALGO, Department of Computer Science, Aarhus University, R. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |