## Three-dimensional layers of maxima

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Venue: | Algorithmica |

Citations: | 7 - 0 self |

### BibTeX

@ARTICLE{Buchsbaum_three-dimensionallayers,

author = {Adam L. Buchsbaum and Michael T. Goodrich},

title = {Three-dimensional layers of maxima},

journal = {Algorithmica},

year = {},

volume = {39},

pages = {2004}

}

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### Abstract

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. 1

### Citations

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Citation Context ...sional Layers of Maxima 265 Straightforward implementations of T do not improve upon the results of Section 2. For example, if we construct T as a balanced, binary search tree and use red-black trees =-=[8]-=- to implement the entry and exit lists, then Q(N) =O(log 2 N) as before: O(log n) timeateachlevelofT to perform a stabbing test using Lemma 2. R(N) =O(log n), and I(N) =D(N) =O(log 2 N) (argue as with... |

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Citation Context ...e that are not dominated by any point in S. Themaxima set problem, to find 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, Luccio, and Preparata [11] remains the most efficient known solution to this problem; it runs in O(n log n) timewhend =2or3and O... |

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Citation Context ... V }. LetN = |V | + |E| + � v∈V |C(v)|. A generalized path in G is a sequence (v1,...,vp) of vertices such that for each 1 <j≤ p, there exists an edge {vi,vj} ∈E for some 1 ≤ i<j. Chazelle and Guibas =-=[5]-=- consider the problem of traversing a generalized path of G, ateach vertex v determining the predecessor in C(v) (or at each finding the successor) of a query value x ∈ℜ + ,giventhatx∈R(e) for each ed... |

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Citation Context ...es determines layer(p). R(n) =I(n) = D(n) =O(log n), so the algorithm runs in O(n log 2 n) total time and O(n) space. We can improve the running time to O(n log n log log n) byusingvanEmde Boas trees =-=[16]-=- instead of ordered lists, but the space becomes O(n 2 ). We can reduce the space back to O(n) withthesameO(n log n log log n) time bound using dynamic fractional cascading [13]. We omit the details o... |

61 | On the convex layers of a planar set
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(Show Context)
Citation Context ...ts layer. More formally, for p ∈ S, layer(p) =1ifp 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) foreachp ∈ S, givenS. 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 fraction... |

50 |
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Citation Context ... 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. The second structure was used as an auxiliary structure in the first, and both were analyzed by detailed case analyses. Unfortunately, there are crucial cases ... |

44 |
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Citation Context ...09 and NSF Grant CCR-0098068. R. Möhring and R. Raman (Eds.): ESA 2002, LNCS 2461, pp. 257–269, 2002. c○ Springer-Verlag Berlin Heidelberg 2002s258 Adam L. Buchsbaum and Michael T. Goodrich cascading =-=[13]-=-. 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-tim... |

33 |
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Citation Context ... n) time[2]. 1.1 Relation to the Prior Claim 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. The... |

26 |
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Citation Context ... entry and exit lists reduces the running time to O(n log n log log n), but as each slice’s entry and exit list requires O(n) space, the total space increases to O(n 2 ). Using Willard’s q-fast tries =-=[17]-=- yields time O(n log 1.5 n)andspaceO(n log n). In general, however, if we set the arity of T to some d and use a dictionary that on N items admits predecessor and successor queries, insertions, and de... |

23 | Dynamic fractional cascading. Algorithmica - Mehlhorn, Näher - 1990 |

22 |
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Citation Context ...log N +logN). Their results assume that d = O(1). Raman [15] notes that this result is easily extended to yield O(p(log d+loglogN)+ log N) query time in the case of arbitrary d and with Dietz goes on =-=[6, 15]-=- to maketheupdatetimeworstcaseO(log d +loglogN). The above extension of dynamic fractional cascading to arbitrary d uses an idea of Chazelle and Guibas that replaces each vertex of G by a uniform star... |

18 | Constant Time Algorithms for Computational Geometry on the Recon Mesh
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Citation Context ...s problem; it runs in O(n log n) timewhend =2or3and O(n log d−2 n)timewhend ≥ 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 fin... |

15 |
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Citation Context ...date is known, i.e., given a pointer to the successor (or predecessor) of the element to be inserted or deleted. The query time becomes O(p log log N +logN). Their results assume that d = O(1). Raman =-=[15]-=- notes that this result is easily extended to yield O(p(log d+loglogN)+ log N) query time in the case of arbitrary d and with Dietz goes on [6, 15] to maketheupdatetimeworstcaseO(log d +loglogN). The ... |

14 | Bounded ordered dictionaries in O(log log n) time and O(n) space - Mehlhorn, Naher - 1990 |

13 |
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(Show Context)
Citation Context ...log d−2 n)timewhend ≥ 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 unti... |

9 | Biased finger trees and threedimensional layers of maxima
- Atallah, Goodrich, et al.
- 1994
(Show Context)
Citation Context ...257–269, 2002. c○ Springer-Verlag Berlin Heidelberg 2002s258 Adam L. Buchsbaum and Michael T. Goodrich cascading [13]. 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.... |

6 |
Bounded Ordered Dictionaries in O(log log N
- Mehlhorn, Näher
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(Show Context)
Citation Context ...If d = O( √ log n) andD(N) = O(log log n), this would yield an O(n log n)-time algorithm. Again we could use van Emde Boas trees [16], but the space would still be O(n 2 ). Using Mehlhorn and Näher’s =-=[12]-=- modification reduces the space to O(n), but the time becomes expected. We next show how to apply dynamic fractional cascading with this method to achieve O(n log n) time (worst case) and O(n log n/ l... |

4 |
An optimal algorithm for the maxima set problem for data in motion
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(Show Context)
Citation Context ...s 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 iter... |

3 |
Bounded ordered dictionaries in O(log logN) time and O(n) space
- Mehlhorn, Näher
- 1990
(Show Context)
Citation Context .... If d = O( √ logn) and D(N) = O(log log n), this would yield an O(n logn)-time algorithm. Again we could use van Emde Boas trees [16], but the space would still be O(n2). Using Mehlhorn and Näher’s =-=[12]-=- modification reduces the space to O(n), but the time becomes expected. We next show how to apply dynamic fractional cascading with this method to achieve O(n logn) time (worst case) and O(n log n/ lo... |