Section: Scientific Foundations

Keywords : discrete algorithms, discrete structures, algorithmic complexity, sequence algorithms, string matching, discrete geometry.

Scientific Foundations

Text algorithms

The area of string algorithms (also called text or sequence algorithms) has been very actively developed during last years, as witnessed by the publication of several monographs [30] , [35] , [28] , [29] . While string algorithms remain a natural part of discrete algorithms in general, they form now their own research area, similar to graph algorithms for example. Recent advances in string algorithms have been motivated by their numerous applications, of which the computational biology and the web search are two most salient examples. Our general goal here is to develop new efficient algorithms on words, based on our studies of word combinatorial properties. A direct application of those algorithms is the analysis of biological sequences, that we will discuss in Section  4.1 .

Discrete geometry

While words are general discrete structures, here we are interested in discrete objects having a geometric (planar or spatial) interpretation and studied within the area of Disrete Geometry . Its general goal is to define a theoretical framework to translate to basic notions of the Euclidean geometry (such as distance, length, convexity, ...) as ``faithfully'' as possible. Several approaches exist to pursue this goal  [26] . In our studies, we follow an arithmetical approach, where discrete objects, as straight lines or planes, are defined with arithmetical defnitions. These analytical definitions allow us to represent in a compact way any elementary digital object, to study some objects that are intrinsically discrete (and are not only approximations of continuous objects), and to define infinite discrete objects.

Methods of discrete geometry are mainly applied to geometric and graphical information, in particular to image and document processing and to medical imaging. However, other application areas exist, such as the cristallography for example. In general, this research direction is in fast progress now, as it is witnessed by the international conference Discrete Geometry for Computer Imagery . A technical committee on discrete geometry (TC18) of International Association of Pattern Recognition (IAPR) has been created (http://www.cb.uu.se/~tc18/ )in order to promote this research area.

Discrete probability

Probabilistic models and probabilistic analysis are getting an increasing importance in our studies in general, and in bioinformatics applications in particular (see Section  4.1 ). Our contribution here is of applicative nature, as we develop, study, or use specific probabilistic models in order to solve our bioinformatics problems.