TY - JOUR

T1 - The rectangle enclosure and point-dominance problems revisited

AU - Gupta, Prosenjit

AU - Janardan, Ravi

AU - Smid, Michiel

AU - Dasgupta, Bhaskar

N1 - Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

PY - 1997

Y1 - 1997

N2 - We consider the problem of reporting the pairwise enclosures in a set of n axesparallel rectangles in IR2, which is equivalent to reporting dominance pairs in a set of n points in IR4. Over a decade ago, Lee and Preparata7 gave an O(n log2 n + k)-time and O(n)-space algorithm for these problems, where k is the number of reported pairs. Since that time, the question of whether there is a faster algorithm has remained an intriguing open problem. In this paper, we give an algorithm which uses O(n + k) space and runs in O (n log n log log n + k log log n) time. Thus, although our result is not a strict improvement over the Lee-Preparata algorithm for the full range of k, it is, nevertheless, the first result since Ref. (6) to make any progress on this long-standing open problem. Our algorithm is based on the divide-and-conquer paradigm. The heart of the algorithm is the solution to a red-blue dominance reporting problem (the "merge" step). We give a novel solution for this problem which is based on the iterative application of a sequence of non-trivial sweep routines. This solution technique should be of independent interest. We also present another algorithm whose bounds match the bounds given in Fief. (6), but which is simpler. Finally, we consider the special case where the rectangles have a small number, α, of different aspect ratios, which is often the case in practice. For this problem, we give an algorithm which runs in O(αn log n + k) time and uses O(n) space.

AB - We consider the problem of reporting the pairwise enclosures in a set of n axesparallel rectangles in IR2, which is equivalent to reporting dominance pairs in a set of n points in IR4. Over a decade ago, Lee and Preparata7 gave an O(n log2 n + k)-time and O(n)-space algorithm for these problems, where k is the number of reported pairs. Since that time, the question of whether there is a faster algorithm has remained an intriguing open problem. In this paper, we give an algorithm which uses O(n + k) space and runs in O (n log n log log n + k log log n) time. Thus, although our result is not a strict improvement over the Lee-Preparata algorithm for the full range of k, it is, nevertheless, the first result since Ref. (6) to make any progress on this long-standing open problem. Our algorithm is based on the divide-and-conquer paradigm. The heart of the algorithm is the solution to a red-blue dominance reporting problem (the "merge" step). We give a novel solution for this problem which is based on the iterative application of a sequence of non-trivial sweep routines. This solution technique should be of independent interest. We also present another algorithm whose bounds match the bounds given in Fief. (6), but which is simpler. Finally, we consider the special case where the rectangles have a small number, α, of different aspect ratios, which is often the case in practice. For this problem, we give an algorithm which runs in O(αn log n + k) time and uses O(n) space.

KW - Dominance

KW - Output-sensitive algorithm

KW - Rectangle enclosure

KW - Space sweep

KW - Van Emde Boas tree

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U2 - 10.1142/S0218195997000260

DO - 10.1142/S0218195997000260

M3 - Article

AN - SCOPUS:0031493807

SN - 0218-1959

VL - 7

SP - 437

EP - 455

JO - International Journal of Computational Geometry and Applications

JF - International Journal of Computational Geometry and Applications

IS - 5

ER -