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Dynamic Trees and Dynamic Point Location
 In Proc. 23rd Annu. ACM Sympos. Theory Comput
, 1991
"... This paper describes new methods for maintaining a pointlocation data structure for a dynamicallychanging monotone subdivision S. The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graphtheoretic planar dual of S. Queries are answered by using ..."
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Cited by 46 (11 self)
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This paper describes new methods for maintaining a pointlocation data structure for a dynamicallychanging monotone subdivision S. The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graphtheoretic planar dual of S. Queries are answered by using a centroid decomposition of the dual tree to drive searches in the primal tree. These trees are maintained via the linkcut trees structure of Sleator and Tarjan, leading to a scheme that achieves vertex insertion/deletion in O(log n) time, insertion/deletion of kedge monotone chains in O(log n + k) time, and answers queries in O(log 2 n) time, with O(n) space, where n is the current size of subdivision S. The techniques described also allow for the dual operations expand and contract to be implemented in O(log n) time, leading to an improved method for spatial pointlocation in a 3dimensional convex subdivision. In addition, the interlacedtree approach is applied to online pointlo...
Methods for Achieving Fast Query Times in Point Location Data Structures
, 1997
"... Given a collection S of n line segments in the plane, the planar point location problem is to construct a data structure that can efficiently determine for a given query point p the first segment(s) in S intersected by vertical rays emanating out from p. It is well known that linearspace data struc ..."
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Cited by 20 (1 self)
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Given a collection S of n line segments in the plane, the planar point location problem is to construct a data structure that can efficiently determine for a given query point p the first segment(s) in S intersected by vertical rays emanating out from p. It is well known that linearspace data structures can be constructed so as to achieve O(log n) query times. But applications, such as those common in geographic information systems, motivate a reexamination of this problem with the goal of improving query times further while also simplifying the methods needed to achieve such query times. In this paper we perform such a reexamination, focusing on the issues that arise in three different classes of pointlocation query sequences: ffl sequences that are reasonably uniform spatially and temporally (in which case the constant factors in the query times become critical), ffl sequences that are nonuniform spatially or temporally (in which case one desires data structures that adapt to s...
Optimal Finger Search Trees in the Pointer Machine
, 2002
"... We develop a new finger search tree with worst case constant update time in the Pointer Machine (PM) model of computation. This was a major problem in the field of Data Structures and was tantalizingly open for over twenty years, while many attempts by researchers were made to solve it. The result c ..."
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Cited by 10 (3 self)
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We develop a new finger search tree with worst case constant update time in the Pointer Machine (PM) model of computation. This was a major problem in the field of Data Structures and was tantalizingly open for over twenty years, while many attempts by researchers were made to solve it. The result comes as a consequence of the innovative mechanism that guides the rebalancing operations, combined with incremental multiple splitting and fusion techniques over nodes.
Multiple Templates Access of Trees in Parallel Memory Systems
 Proc. of Intern. Parallel Processing Symp. (IPPS
, 1998
"... We study the problem of mapping the N nodes of a data structure on M memory modules so that they can be accessed in parallel by templates i.e. distinct sets of nodes. In literature several algorithms are available for arrays (accessed by rows, columns, diagonals and subarrays) and trees (accessed ..."
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Cited by 6 (4 self)
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We study the problem of mapping the N nodes of a data structure on M memory modules so that they can be accessed in parallel by templates i.e. distinct sets of nodes. In literature several algorithms are available for arrays (accessed by rows, columns, diagonals and subarrays) and trees (accessed by subtrees, roottoleaf paths, levels, etc.). Although some mapping algorithms for arrays allow conflictfree access to several templates at once (for example rows and columns), no mapping algorithm is known for efficiently accessing subtree, path and level templates in complete binary trees. In our paper, we, first, prove that any mapping algorithm that is conflictfree for tree/level template has \Omega\Gamma M= log M ) conflicts when access is done according to path template and vice versa. Therefore, no mapping algorithm can be found that is conflictfree on both path and tree (or path and level) templates. Our main result is an algorithm for mapping complete binary trees wi...
Threedimensional layers of maxima
 Algorithmica
"... Abstract. We present an O(n log n)time algorithm to solve the threedimensional layersofmaxima 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 i ..."
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Cited by 5 (0 self)
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Abstract. We present an O(n log n)time algorithm to solve the threedimensional layersofmaxima 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
Finger Search Trees
, 2005
"... One of the most studied problems in computer science is the problem of maintaining a sorted sequence of elements to facilitate efficient searches. The prominent solution to the problem is to organize the sorted sequence as a balanced search tree, enabling insertions, deletions and searches in logari ..."
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Cited by 5 (0 self)
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One of the most studied problems in computer science is the problem of maintaining a sorted sequence of elements to facilitate efficient searches. The prominent solution to the problem is to organize the sorted sequence as a balanced search tree, enabling insertions, deletions and searches in logarithmic time. Many different search trees have been developed and studied intensively in the literature. A discussion of balanced binary search trees can e.g. be found in [4]. This chapter is devoted to finger search trees which are search trees supporting fingers, i.e. pointers, to elements in the search trees and supporting efficient updates and searches in the vicinity of the fingers. If the sorted sequence is a static set of n elements then a simple and space efficient representation is a sorted array. Searches can be performed by binary search using 1+⌊log n⌋ comparisons (we throughout this chapter let log x denote log 2 max{2, x}). A finger search starting at a particular element of the array can be performed by an exponential search by inspecting elements at distance 2 i − 1 from the finger for increasing i followed by a binary search in a range of 2 ⌊log d ⌋ − 1 elements, where d is the rank difference in the sequence between the finger and the search element. In Figure 11.1 is shown an exponential search for the element 42 starting at 5. In the example d = 20. An exponential search requires
Toward a Universal Mapping Algorithm for Accessing Trees in Parallel Memory Systems
, 1998
"... We study the problem of mapping the N nodes of a complete tary tree on M memory modules so that they can be accessed in parallel by templates, i.e. distinct sets of nodes. Typical templates for accessing trees are subtrees, roottoleaf paths, or levels which will be referred to as elementary templ ..."
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Cited by 4 (3 self)
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We study the problem of mapping the N nodes of a complete tary tree on M memory modules so that they can be accessed in parallel by templates, i.e. distinct sets of nodes. Typical templates for accessing trees are subtrees, roottoleaf paths, or levels which will be referred to as elementary templates. In this paper, we first propose a new mapping algorithm for accessing both paths and subtrees of size M with an optimal number of conflicts (i.e., only one conflict) when the number of memory modules is limited to M . We also propose another mapping algorithm for a composite template, say V (as versatile), such that its size is not fixed and an instance of V is composed of any combination of c instances of elementary templates. The number of conflicts for accessing an Snode instance of template V is O ` S p M log M + c ' and the memory load is 1 + o(1) where load is defined as the ratio between the maximum and minimum number of data items mapped onto each memory module. 1. In...
Versatile Access to Parallel Memory Systems
 Proc. of 1998 Wkshp. on Distributed Data and Structures
"... In this paper, we present a survey of results about the problem of mapping the N items of a data structure on M memory modules so that items can be accessed in parallel by templates i.e. distinct sets of nodes. In particular, we present some results that allow to access several different templates a ..."
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Cited by 2 (2 self)
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In this paper, we present a survey of results about the problem of mapping the N items of a data structure on M memory modules so that items can be accessed in parallel by templates i.e. distinct sets of nodes. In particular, we present some results that allow to access several different templates at once, i.e. we focus on versatile mapping algorithms (for a comprehensive survey of other related results see [14] in this volume). In particular, we present some of the algorithms in literature for accessing arrays (by rows, columns, diagonals and subarrays) and trees (accessed by subtrees, roottoleaf paths, levels and compositions thereof). 1 Introduction In this paper we present a survey of results related to the problem of mapping a data structure to M memory modules when it is known how the access is required i.e. templates of memory access are known. The problem is to minimize the conflicts on the memory modules, i.e. simultaneous access to the same module to retrieve different dat...
Algorithmica DOI 10.1007/s0045301296364 Improved Bounds for Finger Search on a RAM
, 2011
"... Abstract We present a new finger search tree with O(log log d) expected search time in the Random Access Machine (RAM) model of computation for a large class of input distributions. The parameter d represents the number of elements (distance) between the search element and an element pointed to by a ..."
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Abstract We present a new finger search tree with O(log log d) expected search time in the Random Access Machine (RAM) model of computation for a large class of input distributions. The parameter d represents the number of elements (distance) between the search element and an element pointed to by a finger, in a finger search tree that stores n elements. Our data structure improves upon a previous result by Anders