Results 1 
3 of
3
Persistent Predecessor Search and Orthogonal Point Location on the Word RAM
"... We answer a basic data structuring question (for example, raised by Dietz and Raman back in SODA 1991): can van Emde Boas trees be made persistent, without changing their asymptotic query/update time? We present a (partially) persistent data structure that supports predecessor search in a set of int ..."
Abstract

Cited by 14 (5 self)
 Add to MetaCart
(Show Context)
We answer a basic data structuring question (for example, raised by Dietz and Raman back in SODA 1991): can van Emde Boas trees be made persistent, without changing their asymptotic query/update time? We present a (partially) persistent data structure that supports predecessor search in a set of integers in {1,..., U} under an arbitrary sequence of n insertions and deletions, with O(log log U) expected query time and expected amortized update time, and O(n) space. The query bound is optimal in U for linearspace structures and improves previous nearO((log log U) 2) methods. The same method solves a fundamental problem from computational geometry: point location in orthogonal planar subdivisions (where edges are vertical or horizontal). We obtain the first static data structure achieving O(log log U) worstcase query time and linear space. This result is again optimal in U for linearspace structures and improves the previous O((log log U) 2) method by de Berg, Snoeyink, and van Kreveld (1992). The same result also holds for higherdimensional subdivisions that are orthogonal binary space partitions, and for certain nonorthogonal planar subdivisions such as triangulations without small angles. Many geometric applications follow, including improved query times for orthogonal range reporting for dimensions ≥ 3 on the RAM. Our key technique is an interesting new vanEmdeBoas–style recursion that alternates between two strategies, both quite simple.
Binary Space Partitions  Recent Developments
, 2004
"... A binary space partition tree is a data structure for the representation of a set of objectsin space. It found an increasing number of applications over the last decades. In recent years, intensifying research focused on its combinatorial properties, which affect directly the efficiency of applica ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
A binary space partition tree is a data structure for the representation of a set of objectsin space. It found an increasing number of applications over the last decades. In recent years, intensifying research focused on its combinatorial properties, which affect directly the efficiency of applications. Important advances were made on binary space partitions for disjoint line segments in the plane and for axisaligned objects in higher dimensions. New research directions were also initiated on some realistic polygonal scenes and on kinetic binary space partitions. This paper attempts to give an overview of these results and reiterates some of the most pressing open problems.
The Rectilinear Minimum Bends Path Problem in Three Dimensions
"... Abstract. In this paper we consider the Rectilinear Minimum Bends Path Problem among rectilinear obstacles in three dimensions. The problem is well studied in two dimensions, but is relatively unexplored in higher dimensions. We give an algorithm which solves the problem in worstcase O(βn log n) ti ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. In this paper we consider the Rectilinear Minimum Bends Path Problem among rectilinear obstacles in three dimensions. The problem is well studied in two dimensions, but is relatively unexplored in higher dimensions. We give an algorithm which solves the problem in worstcase O(βn log n) time, where n is the number of corners among all obstacles, and β is the size of a BSP decomposition of the space containing the obstacles. It has been shown that in the worst case β = Θ(n 3/2), giving us an overall worst case time of O(n 5/2 log n). However in many practical circumstances we will have β ≈ O(n). Previously known algorithms have a worstcase running time of O(n 3).