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Non uniform random generation of generalized Motzkin paths

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Abstract

We consider in this paper the class M k n of generalized Motzkin paths of length n, that is, lattice paths using steps (1,1), (1,−1), (k,0), where k is a fixed positive integer, starting at the origin (0,0), running above the x-axis, and ending at (n,0). The area is the region bounded by the path and the x-axis. We first establish a bijection between the area of paths in M k n and some lattice paths of length n+1. Then, by using a rejection technique, we obtain a linear algorithm with an average time complexity (k mod 2 +1)(n+1).

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Authors and Affiliations

  1. LaCIM, Département d'informatique, Un. Québec à Montréal, CP 8888 Succ. Centre-ville, Montréal (QC), Canada, H3C 3P8

    Srečko Brlek

  2. Dipartimento di Sistemi e Informatica, Un. di Firenze, Firenze, Italy

    Elisa Pergola

  3. LaBRI, Université Bordeaux 1, Bordeaux, France

    Olivier Roques

Authors
  1. Srečko Brlek
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  2. Elisa Pergola
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  3. Olivier Roques
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Correspondence to Srečko Brlek.

Additional information

With the support of NSERC (Canada), G.N.C.S. – Istituto Nazionale di Alta Matematica (Italia), MIUR Project: Linguaggi Formalie Automi: Teoriae Applicazioni

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Brlek, S., Pergola, E. & Roques, O. Non uniform random generation of generalized Motzkin paths. Acta Informatica 42, 603–616 (2006). https://doi.org/10.1007/s00236-006-0008-x

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  • DOI: https://doi.org/10.1007/s00236-006-0008-x

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