Biased finger trees and three-dimensional layers of maxima

Atallah, Mikhail J.; Goodrich, Michael T.; Ramaiyer, Kumar

doi:10.1145/177424.177601

Cited by 9 publications

(15 citation statements)

References 43 publications

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“…The previous algorithm [2] was based on the use of two new data structures. The first was a dynamic extension of the point-location structure of Preparata [14] to work in the context of staircase subdivisions, and the second was an extension of the biased search tree data structure of Bent, Sleator, and Tarjan [3] to support finger searches and updates.…”

Section: Relation To the Prior Claimmentioning

confidence: 99%

“…We sketch this result in Section 2. In addition, Atallah, Goodrich, and Ramaiyer [2] claim an O(n log n)-time algorithm, but their presentation appears to have several problems. A simple, linear-time reduction from sorting gives an Ω(n log n)-time lower bound in the comparison model.…”

Section: Introductionmentioning

confidence: 99%

“…Before we present our algorithm, however, we briefly outline how it relates to the prior algorithm claimed to run in O(n log n) time [2].…”

Section: Introductionmentioning

confidence: 99%

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Three-Dimensional Layers of Maxima

Buchsbaum

Goodrich

2002

Algorithms — ESA 2002

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Abstract. We present an O(n log n)-time algorithm to solve the threedimensional layers-of-maxima problem, an improvement over the prior O(n log n log log n)-time solution. A previous claimed O(n log n)-time solution due to Atallah, Goodrich, and Ramaiyer [SCG'94] has technical flaws. Our algorithm is based on a common framework underlying previous work, but to implement it we devise a new data structure to solve a special case of dynamic planar point location in a staircase subdivision. Our data structure itself relies on a new extension to dynamic fractional cascading that allows vertices of high degree in the control graph.

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Section: Relation To the Prior Claimmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Three-Dimensional Layers of Maxima

Buchsbaum

Goodrich

2002

Algorithms — ESA 2002

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“…A good example are the three templates presented previously for complete binary trees: levels can have as many nodes as (N + 1)/2 while root-to-leaf paths all have the same size, i.e., log(N + 1). In an effort to standardize the efficiency measure for multiple templates access, we model the access similarly to the so-called finger updates [AGR94,GMPR77,HM82,K81]. What is given as input to access a template is a pointer (or finger) to a leader node and, from that node, an M-size instance of the template is retrieved.…”

Section: Introductionmentioning

confidence: 99%

Multiple Templates Access of Trees in Parallel Memory Systems

Auletta

Vivo

Scarano

1998

Journal of Parallel and Distributed Computing

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“…We sketch this result in Section 2. Atallah et al [2] claim an O(n log n)-time algorithm, but their presentation appears to have several problems. A simple, linear-time reduction from sorting gives an (n log n)-time lower bound in the comparison model.…”

mentioning

confidence: 99%

Three-Dimensional Layers of Maxima

Buchsbaum

Goodrich

2004

Algorithmica

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We present an O(n log n)-time algorithm to solve the three-dimensional layers-of-maxima problem. This is an improvement over the prior O(n log n log log n)-time solution. A previous claimed O(n log n)-time solution due to Atallah et al.[2] has technical flaws. Our algorithm is based on a common framework underlying previous work, but to implement it we devise a new data structure to solve a special case of dynamic planar point location in a staircase subdivision. Our data structure itself relies on a new extension to dynamic fractional cascading that allows vertices of high degree in the control graph.Key Words. Fractional cascading, Layers of maxima, Planar point location. Introduction.A point p ∈ d dominates another point q ∈ d if each coordinate of p exceeds that of q. Given a set S of n points in d , the maximum points are those that are not dominated by any point in S. The maxima set problem, of finding all the maximum points in S, is a classic problem in computational geometry, dating back to the early days of the discipline [11]. Interestingly, the algorithm presented in the original paper by Kung et al. [11] is still the most efficient known method for solving this problem; it runs in O(n log n) time when d = 2 or 3 and O(n log d−2 n) time when d ≥ 4. This problem has subsequently been studied in many other contexts, including solutions for parallel computation models [9], for point sets subject to insertions and deletions [10], and for moving points [7].The layers-of-maxima problem iterates this discovery: after finding the maximum points, remove them from S and find the maximum points in the remaining set, iterating until S becomes empty. The iteration index in which a point is a maximum is defined to be its layer. More formally, for p ∈ S, layer( p) = 1 if p is a maximum point; otherwise, layer( p) = 1 + max{layer(q) : q dominates p}. The layers-of-maxima problem, which is related to the convex layers problem [4], is to determine layer( p) for each p ∈ S, given S.With some effort [1], the three-dimensional layers-of-maxima problem can be solved in time O(n log n log log n) using techniques from dynamic fractional cascading [13]. We sketch this result in Section 2. Atallah et al. [2] claim an O(n log n)-time algorithm, but their presentation appears to have several problems. A simple, linear-time reduction from sorting gives an (n log n)-time lower bound in the comparison model.

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Biased finger trees and three-dimensional layers of maxima

Cited by 9 publications

References 43 publications

Three-Dimensional Layers of Maxima

Three-Dimensional Layers of Maxima

Multiple Templates Access of Trees in Parallel Memory Systems

Three-Dimensional Layers of Maxima

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