Dynamical sources in information theory: Fundamental intervals and word prefixes

Vallée, Brigitte

doi:10.1007/bf02679622

Cited by 53 publications

(77 citation statements)

References 20 publications

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“…They satisfy, for s < 0, the relations Π(s) ≤ Λ(s) ≤ Π(s + 1). The series Λ(s) always has a singularity at s = −1, and Π(s) always has a singularity at s = 0, The regularity properties of the source can be expressed in terms of Λ near s = −1, and Π near 0, as already shown in [19]. An important subclass is formed by dynamical sources, which are closely related to dynamical systems on the interval; see Figure 4 and [19].…”

Section: Definitionmentioning

confidence: 83%

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The Number of Symbol Comparisons in QuickSort and QuickSelect

Vallée¹,

Clément²,

Fill³

et al. 2009

Automata, Languages and Programming

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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,

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Section: Definitionmentioning

confidence: 83%

“…Our approach otherwise relies on methods from analytic combinatorics [10], as well as on information theory and the theory of dynamical systems. It relates to earlier analyses of digital trees ("tries") by Clément, Flajolet, and Vallée [3,19].…”

mentioning

confidence: 88%

The Number of Symbol Comparisons in QuickSort and QuickSelect

Vallée¹,

Clément²,

Fill³

et al. 2009

Automata, Languages and Programming

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“…More general sources, such as dynamical sources [13], are also considered in the literature, but they are beyond the scope of this article.…”

Section: Distributions On Wordsmentioning

confidence: 99%

A Probabilistic Analysis of the Reduction Ratio in the Suffix-Array IS-Algorithm

Nicaud¹

2015

Combinatorial Pattern Matching

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“…Such sources are most likely constituting the widest class of models, which can be subjected to a complete analytic treatment. To a large extent, Vallée's ideas [59] evolved from the realization that methods, originally developed for the purpose of analysing continued fraction algorithms [60], could be of a much wider scope.…”

Section: Dynamical Sourcesmentioning

confidence: 99%

The Ubiquitous Digital Tree

Flajolet

2006

Stacs 2006

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Abstract. The digital tree also known as trie made its first appearance as a general-purpose data structure in the late 1950's. Its principle is a recursive partitioning based on successive bits or digits of data items. Under various guises, it has then surfaced in the management of very large data bases, in the design of efficient communication protocols, in quantitative data mining, in the leader election problem of distributed computing, in data compression, as well as in some corners of computational geometry. The algorithms are invariably very simple, easy to implement, and in a number of cases surprisingly efficient. The corresponding quantitative analyses pose challenging mathematical problems and have triggered a flurry of research works. Generating functions and symbolic methods, singularity analysis, the saddle-point method, transfer operators of dynamical systems theory, and the Mellin transform have all been found to have a bearing on the probabilistic behaviour of trie algorithms. We offer here a perspective on the rich algorithmic, analytic, and probabilistic aspects of tries, culminating with a connection between a sorting problem and the Riemann hypothesis.

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Dynamical sources in information theory: Fundamental intervals and word prefixes

Cited by 53 publications

References 20 publications

The Number of Symbol Comparisons in QuickSort and QuickSelect

The Number of Symbol Comparisons in QuickSort and QuickSelect

A Probabilistic Analysis of the Reduction Ratio in the Suffix-Array IS-Algorithm

The Ubiquitous Digital Tree

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