eng
episciences.org
Discrete Mathematics & Theoretical Computer Science
1365-8050
2007-06-05
Vol. 10 no. 2
10.46298/dmtcs.438
438
journal article
Culminating paths
Mireille Bousquet-MÃ©lou
Yann Ponty
Let a and b be two positive integers. A culminating path is a path of Z^2 that starts from (0,0), consists of steps (1,a) and (1,-b), stays above the x-axis and ends at the highest ordinate it ever reaches. These paths were first encountered in bioinformatics, in the analysis of similarity search algorithms. They are also related to certain models of Lorentzian gravity in theoretical physics. We first show that the language on a two letter alphabet that naturally encodes culminating paths is not context-free. Then, we focus on the enumeration of culminating paths. A step by step approach, combined with the kernel method, provides a closed form expression for the generating fucntion of culminating paths ending at a (generic) height k. In the case a=b, we derive from this expression the asymptotic behaviour of the number of culminating paths of length n. When a>b, we obtain the asymptotic behaviour by a simpler argument. When a= b, with no precomputation stage nor non-linear storage required. The choice of the best algorithm is not as clear when a
https://dmtcs.episciences.org/438/pdf
Lattice paths
enumeration
random generation
05A15, 05A16 , 60C05
[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]