Julien Clément

Dynamical Sources in Information Theory : A General Analysis of Trie Structures

Digital trees, also known as tries, are a general purpose flexible data structure that implements dictionaries built on sets of words. An analysis is given of three major representations of tries in the form of array-tries, list tries, and bst-tries (“ternary search tries”). The size and the search costs of the corresponding representations are analysed precisely in the average case, while a complete distributional analysis of the height of tries is given. The unifying data model used is that of dynamical sources and it encompasses classical models like those of memoryless sources with independent symbols, of finite Markov chains, and of nonuniform densities. The probabilistic behaviour of the main parameters, namely, size, path length, or height, appears to be determined by two intrinsic characteristics of the source: the entropy and the probability of letter coincidence. These characteristics are themselves related in a natural way to spectral properties of specific transfer operators of the Ruelle type.

REVERSE ENGINEERING PREFIX TABLES

The Prefix table of a string reports for each position the maximal length of its prefixes starting here. The Prefix table and its dual Suffix table are basic tools used in the design of the most efficient string-matching and pattern extraction algorithms. These tables can be computed in linear time independently of the alphabet size. We give an algorithmic characterisation of a Prefix table (it can be adapted to a Suffix table). Namely, the algorithm tests if an integer table of size

The Number of Symbol Comparisons in QuickSort and QuickSelect

n

The standard factorization of Lyndon words: an average point of view

is the Prefix table of some word and, if successful, it constructs the lexicographically smallest string having it as a Prefix table. We show that the alphabet of the string can be bounded to

Assessing the significance of sets of words

\log_2 n

Parsing with a finite dictionary

letters. The overall algorithm runs in

Optimal prefix codes for some families of two-dimensional geometric distributions

O(n)

Counting occurrences for a finite set of words: Combinatorial methods

time.

Towards a Realistic Analysis of the QuickSelect Algorithm

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 log2 n ) [rather than O (n logn ), classically], whereas that of QuickSelect remains O (n ). Explicit expressions for the implied constants are provided by our combinatorial---analytic methods.

Towards a Realistic Analysis of Some Popular Sorting Algorithms

A non-empty word w is a Lyndon word if and only if it is strictly smaller for the lexicographical order than any of its proper suffixes. Such a word w is either a letter or admits a standard factorization uv where v is its smallest proper suffix. For any Lyndon word v, we show that the set of Lyndon words having v as right factor of the standard factorization is regular and compute explicitly the associated generating function. Next, considering the Lyndon words of length n over a twoletter alphabet, we establish that, for the uniform distribution, the average length of the right factor v of the standard factorization is asymptotically 3n/4.