The Number of Symbol Comparisons in QuickSort and QuickSelect

Abstract

We revisit the classical QuickSort and QuickSelect algorithms, under a complexity model that fully takes into account the elementary comparisons between symbols composing the records to be processed. Our probabilistic models belong to a broad category of information sources that encompasses memoryless (i.e., independent-symbols) and Markov sources, as well as many unbounded-correlation sources. We establish that, under our conditions, the average-case complexity of QuickSort is O(n log 2 n) [rather than O(n log n), classically], whereas that of QuickSelect remains O(n). Explicit expressions for the implied constants are provided by our combinatorial-analytic methods.Introduction. Every student of a basic algorithms course is taught that, on average, the complexity of Quicksort is O(n log n), that of binary search is O(log n), and that of radix-exchange sort is O(n log n); see for instance [13,16]. Such statements are based on specific assumptions-that the comparison of data items (for the first two) and the comparison of symbols (for the third one) have unit cost-and they have the obvious merit of offering an easy-to-grasp picture of the complexity landscape. However, as noted by Sedgewick, these simplifying assumptions suffer from limitations: they do not make possible a precise assessment of the relative merits of algorithms and data structures that resort to different methods (e.g., comparison-based versus radix-based sorting) in a way that would satisfy the requirements of either information theory or algorithms engineering. Indeed, computation is not reduced to its simplest terms, namely, the manipulation of totally elementary symbols, such as bits, bytes, characters. Furthermore, such simplified analyses say little about a great many application contexts, in databases or natural language processing, for instance, where information is highly "non-atomic", in the sense that it does not plainly reduce to a single machine word.First, we observe that, for commonly used data models, the mean costs S n and K n of any algorithm under the symbol-comparison and the key-comparison model, respectively, are connected by the universal relation S n = K n · O(log n).(This results from the fact that at most O(log n) symbols suffice, with high probability, to distinguish n keys; cf. the analysis of the height of digital trees,

Towards a Realistic Analysis of the QuickSelect Algorithm

“…However, this cost involves various constants that depend on the source S (and possibly on the real α); these are precisely described in Theorem 2 and displayed in Figure 2. Here, for the QuickSelect algorithms, we prove all the results which were only stated in the extended abstract [26], and we exhibit the probabilistic features of the source which play a role in the analysis: each algorithm of interest is related to a particular constant depending on the source; this constant describes the interplay between the algorithm and the source and explains how the efficiency of the algorithm depends on the source.…”

“…The surprise is that there are cases where this upper bound is tight, as in QuickSort; others where both costs are of the same order, as in QuickSelect. In previous works [26,3], we have already shown that the expected cost of QuickSort is Θpn log 2 nq, not Θpn log nq, when all elementary operations-symbol comparisons-are taken into account. By contrast, we prove here that the cost of QuickSelect turns out to be Θpnq, in both the old and the new world, albeit, of course, with different implied constants.…”

Data-Specific Analysis of String Sorting

“…This immediately implies the following: Corollary 1.1. All the expected running time bounds proved for quicksort in [5] and [14] carry over to mergesort with the constant for the leading term improved by a factor of 2 log 2 ≈ 1.39. Theorem 1.3.…”

“…There are several reasons. Firstly, we can reprove some of the results in [5,14,4], arguably in a much simpler way. Secondly, such tries turn up in our analysis of mergesort.…”

“…6.3]. The second was to assume that the n strings are generated by some random process and to analyze a general, key comparison based algorithm, in particular Quicksort, for its average performance with respect to symbol comparison complexity [5,14].…”