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Communication Dans Un Congrès Discrete Mathematics and Theoretical Computer Science Année : 2010

Bounded discrete walks

Résumé

This article tackles the enumeration and asymptotics of directed lattice paths (that are isomorphic to unidimensional paths) of bounded height (walks below one wall, or between two walls, for $\textit{any}$ finite set of jumps). Thus, for any lattice paths, we give the generating functions of bridges ("discrete'' Brownian bridges) and reflected bridges ("discrete'' reflected Brownian bridges) of a given height. It is a new success of the "kernel method'' that the generating functions of such walks have some nice expressions as symmetric functions in terms of the roots of the kernel. These formulae also lead to fast algorithms for computing the $n$-th Taylor coefficients of the corresponding generating functions. For a large class of walks, we give the discrete distribution of the height of bridges, and show the convergence to a Rayleigh limit law. For the family of walks consisting of a $-1$ jump and many positive jumps, we give more precise bounds for the speed of convergence. We end our article with a heuristic application to bioinformatics that has a high speed-up relative to previous work.
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Dates et versions

hal-00542185 , version 1 (01-12-2010)
hal-00542185 , version 2 (20-08-2015)

Licence

Paternité - Pas d'utilisation commerciale - Partage selon les Conditions Initiales

Identifiants

Citer

Cyril Banderier, Pierre Nicodème. Bounded discrete walks. Analysis of Algorithms 2010 (AofA'10), Jun 2010, Wien, Austria. pp.35-48, ⟨10.46298/dmtcs.2792⟩. ⟨hal-00542185v2⟩
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