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Optimal Sampling Strategies in Quicksort and Quickselect
 PROC. OF THE 25TH INTERNATIONAL COLLOQUIUM (ICALP98), VOLUME 1443 OF LNCS
, 1998
"... It is well known that the performance of quicksort can be substantially improved by selecting the median of a sample of three elements as the pivot of each partitioning stage. This variant is easily generalized to samples of size s = 2k + 1. For large samples the partitions are better as the median ..."
Abstract

Cited by 28 (4 self)
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It is well known that the performance of quicksort can be substantially improved by selecting the median of a sample of three elements as the pivot of each partitioning stage. This variant is easily generalized to samples of size s = 2k + 1. For large samples the partitions are better as the median of the sample makes a more accurate estimate of the median of the array to be sorted, but the amount of additional comparisons and exchanges to find the median of the sample also increases. We show that the optimal sample size to minimize the average total cost of quicksort (which includes both comparisons and exchanges) is s = a \Delta p n + o( p n ). We also give a closed expression for the constant factor a, which depends on the medianfinding algorithm and the costs of elementary comparisons and exchanges. The result above holds in most situations, unless the cost of an exchange exceeds by far the cost of a comparison. In that particular case, it is better to select not the median of...
Partial match queries in relaxed multidimensional search trees
 Algorithmica
, 1998
"... Abstract. Partial match queries arise frequently in the context of large databases, where each record contains a distinct multidimensional key, that is, the key of each record is a Ktuple of values. The components of a key are called the coordinates or attributes of the key. In a partial match quer ..."
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Cited by 7 (2 self)
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Abstract. Partial match queries arise frequently in the context of large databases, where each record contains a distinct multidimensional key, that is, the key of each record is a Ktuple of values. The components of a key are called the coordinates or attributes of the key. In a partial match query we specify the value of s attributes, 0 < s < K, and leave the remaining K − s attributes unspecified. The goal is to retrieve all the records in the database that match the specified attributes. In this paper we present several results about the average performance and variance of partial matches in relaxed Kdimensional trees (search trees and digital tries). These data structures are variants of the well known K dtrees and K dtries. In relaxed trees the sequence of attributes used to guide a query is explicitly stored at the nodes of the tree and randomly generated and, in general, will be different for different search paths. In the standard variants, the sequence of attributes that guides a query examines the attributes in a cyclic fashion, fixed and identical for all search paths. We show that the probabilistic analysis of the relaxed multidimensional trees is very similar to that of standard K dtrees and K dtries, and also to the analysis of quadtrees. In fact, besides the average cost and variance of partial match in relaxed K dtrees and K dtries, we also obtain the variance of partial matches in twodimensional quadtrees. We also compute the average cost of partial matches in other relaxed multidimensional digital tries, namely, relaxed K dPatricia and relaxed K ddigital search trees.
Optimal Surface Reconstruction from Digitized Point Data using CI Methods
 University of Dortmund
, 1997
"... In many scientific and technological endeavors, a threedimensional solid must be reconstructed from digitized point data. This paper presents three solutions to the problem of reconstructing smooth surfaces using triangular tiles. The presented algorithms differ in their strategic approach. Here, t ..."
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Cited by 4 (2 self)
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In many scientific and technological endeavors, a threedimensional solid must be reconstructed from digitized point data. This paper presents three solutions to the problem of reconstructing smooth surfaces using triangular tiles. The presented algorithms differ in their strategic approach. Here, two deterministic algorithms and one nondeterministic CI (computational intelligence) strategy will be described. In order to compare triangulations, two quality criteria will be introduced. 1.1 Motivation Practical applications of tactile or optical scanning methods generate huge sets of many thousands of weakly structured threedimensional point coordinates. These are usually very hard to handle using common CAD systems. CAD systems are expecting smooth B'ezier surfaces, NURBS (uniform rational Bsplines)orBsplines rather than a large set of discrete points. Due to approximation theoretical and practical reasons, it is not useful to span a complete surface with only one spline. In orde...