Multiple Quickselect — Hoare's Find algorithm for several elements

doi:10.1016/0020-0190(95)00150-B

Information Processing Letters

Volume 56, Issue 3, 10 November 1995, Pages 123-129

https://doi.org/10.1016/0020-0190(95)00150-B Get rights and content

Abstract

Hoare's Find algorithm can be used to select p specified order statistics j₁, j₂, …, j_p from a file of n elements simultaneously. We give precise formulæ for both the average number of passes and the average number of comparisons. Averaging again over all possible subsets of p elements, we get results of Lent and Mahmoud as corollaries.

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Cited by (34)

Near-optimal online multiselection in internal and external memory
2016, Journal of Discrete Algorithms
We introduce an online version of the multiselection problem, in which q selection queries are requested on an unsorted array of n elements. We provide the first online algorithm that is 1-competitive with the offline algorithm proposed by Kaligosi et al. [14] in terms of comparison complexity. Our algorithm also supports online search queries efficiently. We then extend our algorithm to the dynamic setting, while retaining online functionality, by supporting arbitrary insertions and deletions on the array. Assuming that the insertion of an element is immediately preceded by a search for that element, our dynamic online algorithm performs an optimal number of comparisons, up to lower order terms and an additive $O (n)$ term.
For the external memory model, we describe the first online multiselection algorithm that is $O (1)$ -competitive. This result improves upon the work of Sibeyn [20] when $q = ω (m^{1 - ϵ})$ for any fixed positive constant ϵ, where m is the number of blocks that can be stored in main memory. We also extend it to support searches, insertions, and deletions of elements efficiently.
The analysis of Range Quickselect and related problems
2011, Theoretical Computer Science
Range Quickselect, a simple modification of the well-known Quickselect algorithm for selection, can be used to efficiently find an element with rank $k$ in a given range $[i . . j]$ , out of $n$ given elements. We study basic cost measures of Range Quickselect by computing exact and asymptotic results for the expected number of passes, comparisons and data moves during the execution of this algorithm.
The key element appearing in the analysis of Range Quickselect is a trivariate recurrence that we solve in full generality. The general solution of the recurrence proves to be very useful, as it allows us to tackle several related problems, besides the analysis that originally motivated us.
In particular, we have been able to carry out a precise analysis of the expected number of moves of the $p$ th element when selecting the $j$ th smallest element with standard Quickselect, where we are able to give both exact and asymptotic results.
Moreover, we can apply our general results to obtain exact and asymptotic results for several parameters in binary search trees, namely the expected number of common ancestors of the nodes with rank $i$ and $j$ , the expected size of the subtree rooted at the least common ancestor of the nodes with rank $i$ and $j$ , and the expected distance between the nodes of ranks $i$ and $j$ .
Distributional analysis of swaps in Quick Select
2010, Theoretical Computer Science
We investigate the number of swaps made by Quick Select (a variant of Quick Sort for finding order statistics) to find an element with a randomly selected rank under realistic partition algorithms such as Lomuto’s or Hoare’s. This kind of grand average provides a smoothing over all individual distributions for specific fixed order statistics. The grand distribution for the number of swaps (when suitably scaled) is a perpetuity (a sum of products of independent mixed continuous random variables supported on the interval $(0, 1)$ ). The tool for this proof is contraction in the Wasserstein metric space, and identifying the limit as the fixed-point solution of a distributional equation. The same methodology carries over when Quick Select is commissioned to find an extremal order statistic (of a relatively small or relatively large rank) and the results are of similar nature. It is one of our purposes to show that analysis under different partition algorithms leads to different results.
On Quickselect, partial sorting and Multiple Quickselect
2006, Information Processing Letters
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. Furthermore 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 an algorithm for sorting a subarray $A [r \dots r + p - 1]$ , given the array $A [1 \dots n]$ , is provided.
Analysis of multiple quickselect variants
2003, Theoretical Computer Science
Multiple Quickselect is an algorithm that uses the idea of Quicksort to search for several order statistics simultaneously. In order to improve the efficiency of Quicksort, one can use the median of 2t+1 randomly chosen elements as pivot element in the partitioning stage. Such a median of 2t+1 partition can also be applied to Multiple Quickselect to reduce the number of comparisons. Here we give an analysis of such Multiple Quickselect variants that use median of 2t+1 partition and describe for these algorithms the asymptotic behaviour of the expected number of required comparisons to find p-order statistics in a data set of size n for n→∞ and fixed p.
Binary search tree recursions with harmonic toll functions
2002, Journal of Computational and Applied Mathematics

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Multiple Quickselect — Hoare's Find algorithm for several elements

Abstract

Concrete Mathematics

Mathematics for the Analysis of Algorithms

Find (Algorithm 65)

Comm. ACM

Analysis of Hoare's Find algorithm with median-of-three partition