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Quickselect and Dickman function
 Combinatorics, Probability and Computing
, 2000
"... We show that the limiting distribution of the number of comparisons used by Hoare's quickselect algorithm when given a random permutation of n elements for finding the mth smallest element, where m = o(n), is the Dickman function. The limiting distribution of the number of exchanges is also derived ..."
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We show that the limiting distribution of the number of comparisons used by Hoare's quickselect algorithm when given a random permutation of n elements for finding the mth smallest element, where m = o(n), is the Dickman function. The limiting distribution of the number of exchanges is also derived. 1 Quickselect Quickselect is one of the simplest and e#cient algorithms in practice for finding specified order statistics in a given sequence. It was invented by Hoare [19] and uses the usual partitioning procedure of quicksort: choose first a partitioning key, say x; regroup the given sequence into two parts corresponding to elements whose values are less than and larger than x, respectively; then decide, according to the size of the smaller subgroup, which part to continue recursively or to stop if x is the desired order statistics; see Figure 1 for an illustration in terms of binary search trees. For more details, see Guibas [15] and Mahmoud [26]. This algorithm , although ine#cient in the worst case, has linear mean when given a sequence of n independent and identically distributed continuous random variables, or equivalently, when given a random permutation of n elements, where, here and throughout this paper, all n! permutations are equally likely. Let C n,m denote the number of comparisons used by quickselect for finding the mth smallest element in a random permutation, where the first partitioning stage uses n 1 comparisons. Knuth [23] was the first to show, by some di#erencing argument, that E(C n,m ) = 2 (n + 3 + (n + 1)H n (m + 2)Hm (n + 3 m)H n+1m ) , n, where Hm = 1#k#m k 1 . A more transparent asymptotic approximation is E(C n,m ) (#), (#) := 2 #), # Part of the work of this author was done while he was visiting School of C...
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.
SPANNING TREE SIZE IN RANDOM BINARY SEARCH TREES
, 2004
"... This paper deals with the size of the spanning tree of p randomly chosen nodes in a binary search tree. It is shown via generating functions methods, that for fixed p, the (normalized) spanning tree size converges in law to the Normal distribution. The special case p = 2 reproves the recent result ( ..."
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Cited by 10 (3 self)
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This paper deals with the size of the spanning tree of p randomly chosen nodes in a binary search tree. It is shown via generating functions methods, that for fixed p, the (normalized) spanning tree size converges in law to the Normal distribution. The special case p = 2 reproves the recent result (obtained by the contraction method by Mahmoud and Neininger [Ann. Appl. Probab. 13 (2003) 253–276]), that the distribution of distances in random binary search trees has a Gaussian limit law. In the proof we use the fact that the spanning tree size is closely related to the number of passes in Multiple Quickselect. This parameter, in particular, its first two moments, was studied earlier by Panholzer and Prodinger [Random Structures Algorithms 13 (1998) 189–209]. Here we show also that this normalized parameter has for fixed porder statistics a Gaussian limit law. For p = 1 this gives the wellknown result that the depth of a randomly selected node in a random binary search tree converges in law to the
Generalized reciprocity laws for sums of harmonic numbers
, 2005
"... We present summation identities for generalized harmonic numbers, which generalize reciprocity laws discovered when studying the algorithm quickselect. Furthermore, we demonstrate how the computer algebra package Sigma can be used in order to find/prove such identities. We also discuss alternating h ..."
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We present summation identities for generalized harmonic numbers, which generalize reciprocity laws discovered when studying the algorithm quickselect. Furthermore, we demonstrate how the computer algebra package Sigma can be used in order to find/prove such identities. We also discuss alternating harmonic sums, as well as limiting relations. 1.
Addendum: On quickselect, partial sorting and multiple quickselect, 2006. Available at: http://www.dmg.tuwien.ac.at/kuba
"... Abstract. We present explicit solutions of a class of recurrences related to the Quickselect algorithm. Thus we are immediately able to solve recurrences arising at the partial sorting problem, which are contained in this class. We show how the partial sorting problem is connected to the Multiple Qu ..."
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Abstract. We present explicit solutions of a class of recurrences related to the Quickselect algorithm. Thus we are immediately able to solve recurrences arising at the partial sorting problem, which are contained in this class. We show how the partial sorting problem is connected to the Multiple Quickselect algorithm and present a method for the calculation of solutions for a class of recurrences related to the Multiple Quickselect algorithm. Further an analysis of the partial sorting problem for the ranks r,..., r+p−1 given the array A[1,..., n] is provided. 1.
The Analysis of Multiple Quickselect
"... Quickselect is a version of Quicksort that makes it possible to find efficiently any element in an unsorted file given its rank. "Multiple Quickselect" is designed to find simultaneously a collection of elements, also specified by their ranks. This talk shows how to analyse Multiple Quickselect when ..."
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Quickselect is a version of Quicksort that makes it possible to find efficiently any element in an unsorted file given its rank. "Multiple Quickselect" is designed to find simultaneously a collection of elements, also specified by their ranks. This talk shows how to analyse Multiple Quickselect when the underlying permutation is random and the collection of ranks is a random psubset (p fixed). The analysis provides a nice illustration of the use of multivariate generating functions. 1. Algorithms The principle of Quicksort is of course extremely well known. Given an array T [1 : : : n] of numbers to be sorted, choose a "pivot", say T [1], then partition (by a linear scan) the original array into the two subarrays, T! and T? , formed by elements that are respectively smaller and larger than the pivot, and proceed recursively. The algorithm was first proposed by Hoare in 1962. Its characteristics are fairly well understood: the algorithms sorts in place (it uses O(1) auxiliary memory i...
Averagecase analysis of moves in Quick Select
"... We investigate the average number of moves made by Quick Select (a variant of Quick Sort for finding order statistics) to find an element with a randomly selected rank. This kind of grand average provides smoothing over all individual cases of a specific fixed order statistic. The variance of the nu ..."
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We investigate the average number of moves made by Quick Select (a variant of Quick Sort for finding order statistics) to find an element with a randomly selected rank. This kind of grand average provides smoothing over all individual cases of a specific fixed order statistic. The variance of the number of moves involves intricate dependencies, and we only give reasonably tight bounds. 1 Quick Select. Quick Sort is a well known fast algorithm for data sorting. It was invented by Hoare (see [3]); see also [5] and [7]. Quick Sort is the default sorting scheme in some operating systems, such as UNIX. The algorithm is twosided: It puts a pivot element in its correct position, and arranges the data in two groups relative to that pivot. Elements below the pivot go in one group, the rest are placed in the other group. The two groups are then handled recursively. The onesided version (Quick Select) of the algorithm is popular for identifying order statistics. This onesided version of Quick Sort to search for order statistics is also known as Hoare’s Find algorithm, which was first given in [2]. To find a certain order statistic, such as the first quartile, Quick Select proceeds with the partition stage, just as in Quick Sort, to place a pivot in its correct position, and creates the two groups on the two sides of the pivot. But then, the algorithm decides if the pivot is the sought order statistic or not. If it is, the algorithm terminates (announcing the pivot to be the sought order statistic), and if not it recursively pursues only a group on one side where the order statistic resides.
Online Sorting via Searching and Selection
, 2009
"... In this paper, we present a framework based on a simple data structure and parameterized algorithms for the problems of finding items in an unsorted list of linearly ordered items based on their rank (selection) or value (search). As a sideeffect of answering these online selection and search queri ..."
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In this paper, we present a framework based on a simple data structure and parameterized algorithms for the problems of finding items in an unsorted list of linearly ordered items based on their rank (selection) or value (search). As a sideeffect of answering these online selection and search queries, we progressively sort the list. Our algorithms are based on Hoare’s Quickselect, and are parameterized based on the pivot selection method. For example, if we choose the pivot as the last item in a subinterval, our framework yields algorithms that will answer q ≤ n unique selection and/or search queries in a total of O(nlog q) average time. After q = Ω(n) queries the list is sorted. Each repeated selection query takes constant time, and each repeated search query takes O(log n) time. The two query types can be interleaved freely. By plugging different pivot selection methods into our framework, these results can, for example, become randomized expected time or deterministic worstcase time. We extend the algorithms and data structures in our framework to obtain results that are cacheoblivious I/O efficient and/or dynamic and/or compressed. Our methods are easy to implement, and we show they perform well in practice. 1
JOINT DISTRIBUTIONS FOR MOVEMENTS OF ELEMENTS IN SATTOLO’S AND THE FISHERYATES ALGORITHM
"... Abstract. Sattolo’s algorithm creates a random cyclic permutation by interchanging pairs of elements in an appropriate manner; the FisherYates algorithm produces random (not necessarily cyclic) permutations in a very similar way. The distributions of the movements of the elements in these two algor ..."
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Abstract. Sattolo’s algorithm creates a random cyclic permutation by interchanging pairs of elements in an appropriate manner; the FisherYates algorithm produces random (not necessarily cyclic) permutations in a very similar way. The distributions of the movements of the elements in these two algorithms have already been treated quite extensively in past works. In this paper, we are interested in the joint distribution of two elements j and k; we are able to compute the bivariate generating functions explicitly, although it is quite involved. From it, moments and limiting distributions can be deduced. Furthermore, we compute the probability that elements i and j ever change places in both algorithms. 1.
Limit Laws for the Randić Index of Random Binary Tree Models
, 2006
"... Abstract. We investigate the Randić index of random binary trees under two standard probability models: the one induced by random permutations and the Catalan (uniform). In both cases the mean and variance are computed by recurrence methods and shown to be asymptotically linear in the size of the tr ..."
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Abstract. We investigate the Randić index of random binary trees under two standard probability models: the one induced by random permutations and the Catalan (uniform). In both cases the mean and variance are computed by recurrence methods and shown to be asymptotically linear in the size of the tree. The recursive nature of binary search trees lends itself in a natural way to application of the contraction method, by which a limit distribution (for a suitably normalized version of the index) is shown to be Gaussian. The Randić index (suitably normalized) is also shown to be normally distributed in binary Catalan trees, but the methodology we use for this derivation is singularity analysis of formal generating functions.