Results 1 
9 of
9
Randomized KDimensional Binary Search Trees
, 1998
"... This paper introduces randomized Kdimensional binary search trees (randomized Kdtrees), a variant of Kdimensional binary trees. This data structure allows the efficient maintenance of multidimensional records for any sequence of insertions and deletions; and thus, is fully dynamic. We show that ..."
Abstract

Cited by 14 (3 self)
 Add to MetaCart
This paper introduces randomized Kdimensional binary search trees (randomized Kdtrees), a variant of Kdimensional binary trees. This data structure allows the efficient maintenance of multidimensional records for any sequence of insertions and deletions; and thus, is fully dynamic. We show that several types of associative queries are efficiently supported by randomized Kdtrees. In particular, a randomized Kdtree with n records answers exact match queries in expected O(log n) time. Partial match queries are answered in expected O(n 1\Gammaf (s=K) ) time, when s out of K attributes are specified, with 0 ! f(s=K) ! 1 a real valued function of s=K). Nearest neighbor queries are answered online in expected O(log n) time. Our randomized algorithms guarantee that their expected time bounds hold irrespective of the order and number of insertions and deletions. Keywords: Randomized Algorithms, Multidimensional Data Structures, Kdtrees, Associative Queries, Multidimensional Diction...
Analysis of Range Search for Random KD Trees
 Acta Informatica
, 1999
"... . We analyze the expected time complexity of range searching with kd trees in all dimensions when the data points are uniformly distributed in the unit hypercube. The partial match results of Flajolet and Puech are reproved using elementary probabilistic methods. In addition, we give asymptotic exp ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
. We analyze the expected time complexity of range searching with kd trees in all dimensions when the data points are uniformly distributed in the unit hypercube. The partial match results of Flajolet and Puech are reproved using elementary probabilistic methods. In addition, we give asymptotic expected time analysis for orthogonal and convex range search, as well as nearest neighbor search. We disprove a conjecture by Bentley that nearest neighbor search for a given random point in the kd tree can be done in O(1) expected time. Keywords and phrases. kd trees, partial match query, range search, expected time, probabilistic analysis of algorithms, data structures, nearest neoghbor search. Research of the authors was sponsored by NSERC grant A3456. The third author received a DGAPAUNAM Scholarship. x1. Introduction The kd tree, or kdimensional binary search tree, was proposed by Bentley in 1975. It is a binary tree in which each record contains k keys, right and left pointers to ...
Limit laws for partial match queries in quadtrees
 ANN. APPL. PROBAB
, 2001
"... It is proved that in an idealized uniform probabilistic model the cost of a partial match query in a multidimensional quadtree after normalization converges in distribution. The limiting distribution is given as a fixed point of a random affine operator. Also a firstorder asymptoticexpansion for th ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
It is proved that in an idealized uniform probabilistic model the cost of a partial match query in a multidimensional quadtree after normalization converges in distribution. The limiting distribution is given as a fixed point of a random affine operator. Also a firstorder asymptoticexpansion for the variance of the cost is derived and results on exponential moments are given. The analysis is based on the contraction method.
Partial match queries in random kd trees
 SIAM Journal on Computing
, 2005
"... Abstract. We solve the open problem of characterizing the leading constant in the asymptotic approximation to the expected cost used for random partial match queries in random kd trees. Our approach is new and of some generality; in particular, it is applicable to many problems involving differenti ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. We solve the open problem of characterizing the leading constant in the asymptotic approximation to the expected cost used for random partial match queries in random kd trees. Our approach is new and of some generality; in particular, it is applicable to many problems involving differential equations (or difference equations) with polynomial coefficients. Key words. kd trees, partialmatch queries, differential equations, averagecase analysis of algorithms, method of linear operators, asymptotic analysis. AMS subject classifications. 68W40 68P05 68P10 68U05 1. Introduction. Multidimensional
unknown title
, 2002
"... www.academicpress.com On the average performance of orthogonal range search in multidimensional data structures ✩ ..."
Abstract
 Add to MetaCart
(Show Context)
www.academicpress.com On the average performance of orthogonal range search in multidimensional data structures ✩
Phase changes in random point quadtrees
, 2005
"... Dedicated to the memory of ChingZong Wei (1949–2004) We show that a wide class of linear cost measures (such as the number of leaves) in random ddimensional point quadtrees undergo a change in limit laws: if the dimension d D 1;:::;8, then the limit law is normal; if d 9 then there is no convergenc ..."
Abstract
 Add to MetaCart
(Show Context)
Dedicated to the memory of ChingZong Wei (1949–2004) We show that a wide class of linear cost measures (such as the number of leaves) in random ddimensional point quadtrees undergo a change in limit laws: if the dimension d D 1;:::;8, then the limit law is normal; if d 9 then there is no convergence to a fixed limit law. Stronger approximation results such as convergence rates and local limit theorems are also derived for the number of leaves, additional phase changes being unveiled. Our approach is new and very general, and also applicable to other classes of search trees. A brief discussion of Devroye’s gridtrees (covering mary search trees and quadtrees as special cases) is given. We also propose an efficient numerical procedure for computing the constants involved to high precision. Contents