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50
Linear programming in linear time when the dimension is fixed
 J. ACM
, 1984
"... Abstract. It is demonstrated that the linear programming problem in d variables and n constraints can be solved in O(n) time when d is fixed. This bound follows from a multidimensional search technique which is applicable for quadratic programming as well. There is also developed an algorithm that i ..."
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Cited by 194 (13 self)
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Abstract. It is demonstrated that the linear programming problem in d variables and n constraints can be solved in O(n) time when d is fixed. This bound follows from a multidimensional search technique which is applicable for quadratic programming as well. There is also developed an algorithm that is polynomial in both n and d provided d is bounded by a certain slowly growing function of n. Categories and Subject Descriptors: F.2.1 [Analysis of Algorithms and Problem Complexity]: Numerical Algorithms and Problemscomputations on matrices; F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problemsgeometrical problems and computations; sorting and searching; G. 1.6 [Mathematics of Computing]: Optimizationlinear programming
Deterministic Sorting and Randomized Median Finding on the BSP model
, 1996
"... We present new BSP algorithms for deterministic sorting and randomized median finding. We sort n general keys by using a partitioning scheme that achieves the requirements of efficiency (oneoptimality) and insensitivity against data skew (the accuracy of the splitting keys depends solely on the ste ..."
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Cited by 48 (23 self)
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We present new BSP algorithms for deterministic sorting and randomized median finding. We sort n general keys by using a partitioning scheme that achieves the requirements of efficiency (oneoptimality) and insensitivity against data skew (the accuracy of the splitting keys depends solely on the step distance, which can be adapted to meet the worstcase requirements of our application). Although we employ sampling in order to realize efficiency, we can give a precise worstcase estimation of the maximum imbalance which might occur. We also investigate optimal randomized BSP algorithms for the problem of finding the median of n elements that require, with highprobability, 3n=(2p) + o(n=p) number of comparisons, for a wide range of values of n and p. Experimental results for the two algorithms are also presented.
FASTER SUFFIX SORTING
, 1999
"... We propose a fast and memory efficient algorithm for lexicographically sorting the suffixes of a string, a problem that has important applications in data compression as well as string matching. Our ..."
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Cited by 46 (2 self)
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We propose a fast and memory efficient algorithm for lexicographically sorting the suffixes of a string, a problem that has important applications in data compression as well as string matching. Our
Selecting the Median
, 1995
"... Improving a long standing result of Schonhage, Paterson and Pippenger we show that the median of a set containing n elements can be found using at most 2:95n comparisons. ..."
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Cited by 36 (5 self)
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Improving a long standing result of Schonhage, Paterson and Pippenger we show that the median of a set containing n elements can be found using at most 2:95n comparisons.
Structures of String Matching and Data Compression
, 1999
"... This doctoral dissertation presents a range of results concerning efficient algorithms and data structures for string processing, including several schemes contributing to sequential data compression. It comprises both theoretic results and practical implementations. We study the suffix tree data st ..."
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Cited by 29 (0 self)
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This doctoral dissertation presents a range of results concerning efficient algorithms and data structures for string processing, including several schemes contributing to sequential data compression. It comprises both theoretic results and practical implementations. We study the suffix tree data structure, presenting an efficient representation and several generalizations. This includes augmenting the suffix tree to fully support sliding window indexing (including a practical implementation) in linear time. Furthermore, we consider a variant that indexes naturally wordpartitioned data, and present a lineartime construction algorithm for a tree that represents only suffixes starting at word boundaries, requiring space linear in the number of words. By applying our sliding window indexing techniques, we achieve an efficient implementation for dictionarybased compression based on the LZ77 algorithm. Furthermore, considering predictive source
Computational geometry  a survey
 IEEE TRANSACTIONS ON COMPUTERS
, 1984
"... We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided de ..."
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Cited by 19 (3 self)
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We survey the state of the art of computational geometry, a discipline that deals with the complexity of geometric problems within the framework of the analysis ofalgorithms. This newly emerged area of activities has found numerous applications in various other disciplines, such as computeraided design, computer graphics, operations research, pattern recognition, robotics, and statistics. Five major problem areasconvex hulls, intersections, searching, proximity, and combinatorial optimizationsare discussed. Seven algorithmic techniques incremental construction, planesweep, locus, divideandconquer, geometric transformation, pruneandsearch, and dynamizationare each illustrated with an example.Acollection of problem transformations to establish lower bounds for geometric problems in the algebraic computation/decision model is also included.
Progress in Selection
, 1997
"... . There has been recent progress in the selection problem, and in medianfinding in particular, after a lull of ten years. This paper reviews some ancient and modern results on this problem, and suggests possibilities for future research. 1 Introduction The selection problem, determining the k th ..."
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Cited by 15 (0 self)
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. There has been recent progress in the selection problem, and in medianfinding in particular, after a lull of ten years. This paper reviews some ancient and modern results on this problem, and suggests possibilities for future research. 1 Introduction The selection problem, determining the k th largest out of a set of n elements, is a junior partner of the more fundamental sorting problem, but it has still been studied extensively over several decades. Our focus will be on performing selection using a minimal number of comparisons in the worst case. Let V k (n) be the worstcase minimum number of pairwise comparisons required to find the k th largest out of n distinct elements. Of particular interest is finding the median, the dn=2e th largest say. We denote V dn=2e (n) by M (n). The worstcase comparison complexity of sorting is n log 2 n+O(n), and even the coefficient of the linear term has been fairly closely estimated. However for V k (n) we do not yet have an asymptotic...
On the probabilistic worstcase time of "FIND"
 ALGORITHMICA
, 2001
"... We analyze the worstcase number of comparisons Tn of Hoare’s selection algorithm find when the input is a random permutation, and worst case is measured with respect to the rank k. We give a new short proof that Tn/n tends to a limit distribution, and provide new bounds for the limiting distributi ..."
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Cited by 14 (0 self)
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We analyze the worstcase number of comparisons Tn of Hoare’s selection algorithm find when the input is a random permutation, and worst case is measured with respect to the rank k. We give a new short proof that Tn/n tends to a limit distribution, and provide new bounds for the limiting distribution.
Concurrent Heaps on the BSP Model
, 1996
"... In this paper we present a new randomized selection algorithm on the BulkSynchronous Parallel (BSP) model of computation along with an application of this algorithm to dynamic data structures, namely Parallel Priority Queues (PPQs). We show that our algorithms improve previous results upon both the ..."
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Cited by 11 (11 self)
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In this paper we present a new randomized selection algorithm on the BulkSynchronous Parallel (BSP) model of computation along with an application of this algorithm to dynamic data structures, namely Parallel Priority Queues (PPQs). We show that our algorithms improve previous results upon both the communication requirements and the amount of parallel slack required to achieve optimal performance. We also establish that optimality to within small multiplicative constant factors can be achieved for a wide range of parallel machines. While these algorithms are fairly simple themselves, descriptions of their performance in terms of the BSP parameters is somewhat involved. The main reward of quantifying these complications is that it allows transportable software to be written for parallel machines that fit the model. We also present experimental results for the selection algorithm that reinforce our claims.
Parallel Construction of Multidimensional Binary Search Trees
 Proc. Intl. Conf. on Supercomputing
, 1996
"... Multidimensional binary search tree (abbreviated kd tree) is a popular data structure for the organization and manipulation of spatial data. The data structure is useful in several applications including graph partitioning, hierarchical applications such as molecular dynamics and nbody simulation ..."
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Cited by 11 (2 self)
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Multidimensional binary search tree (abbreviated kd tree) is a popular data structure for the organization and manipulation of spatial data. The data structure is useful in several applications including graph partitioning, hierarchical applications such as molecular dynamics and nbody simulations, and databases. In this paper, we study efficient parallel construction of kd trees on coarsegrained distributed memory parallel computers. We present several algorithms for parallel kd tree construction and analyze them theoretically and experimentally. We have implemented our algorithms on the CM5 and report on the experimental results. 1 Multidimensional Binary Search Trees Consider a set of n points in k dimensional space. Let d 1 ; d 2 ; : : : ; d k denote the k dimensions. If we find the median coordinate of all the points along dimension d 1 , we can partition the points into two approximately equal sized sets  one set containing all the points whose coordinates along dimens...