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Towards a Realistic Analysis of Some Popular Sorting Algorithms

Published online by Cambridge University Press:  11 December 2014

J. CLÉMENT ,
T. H. NGUYEN THI  and
B. VALLÉE
J. CLÉMENT
Affiliation:
Université de Caen/ENSICAEN/CNRS – GREYC – Caen, France (e-mails: Julien.Clement@unicaen.fr, thu-hien.nguyen-thi@unicaen.fr, brigitte.vallee@unicaen.fr)
T. H. NGUYEN THI
Affiliation:
Université de Caen/ENSICAEN/CNRS – GREYC – Caen, France (e-mails: Julien.Clement@unicaen.fr, thu-hien.nguyen-thi@unicaen.fr, brigitte.vallee@unicaen.fr)
B. VALLÉE
Affiliation:
Université de Caen/ENSICAEN/CNRS – GREYC – Caen, France (e-mails: Julien.Clement@unicaen.fr, thu-hien.nguyen-thi@unicaen.fr, brigitte.vallee@unicaen.fr)
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Abstract

We describe a general framework for realistic analysis of sorting algorithms, and we apply it to the average-case analysis of three basic sorting algorithms (QuickSort, InsertionSort, BubbleSort). Usually the analysis deals with the mean number of key comparisons, but here we view keys as words produced by the same source, which are compared via their symbols in lexicographic order. The ‘realistic’ cost of the algorithm is now the total number of symbol comparisons performed by the algorithm, and, in this context, the average-case analysis aims to provide estimates for the mean number of symbol comparisons used by the algorithm. For sorting algorithms, and with respect to key comparisons, the average-case complexity of QuickSort is asymptotic to 2n log n, InsertionSort to n2/4 and BubbleSort to n2/2. With respect to symbol comparisons, we prove that their average-case complexity becomes Θ (n log2n), Θ(n2), Θ (n2 log n). In these three cases, we describe the dominant constants which exhibit the probabilistic behaviour of the source (namely entropy and coincidence) with respect to the algorithm.

Keywords

Primary 68W3268P1068P0568W4068Q2568Q8768Q17Secondary 68W0594A1594A1797I8030D1030E1530E20
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Special Issue 1: Honouring the Memory of Philippe Flajolet - Part 3 , January 2015 , pp. 104 - 144
Copyright
Copyright © Cambridge University Press 2014 

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