Culminating paths

TL;DR

This paper studies culminating paths in Z^2, providing enumeration formulas, asymptotic analysis, and efficient algorithms for their random generation, with applications in bioinformatics and physics.

Contribution

It introduces a detailed enumeration and asymptotic analysis of culminating paths, and develops efficient algorithms for their uniform random generation.

Findings

Closed-form generating functions for culminating paths.

Asymptotic behavior of path counts for different parameter regimes.

Linear-time algorithms for uniform random generation when a >= b.

Abstract

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, we only determine the exponential growth of the number of culminating paths. Finally, we study the uniform random generation of culminating paths via various methods. The rejection approach, coupled with a symmetry argument, gives an algorithm that is linear when a>= b, with no precomputation stage nor non-linear storage required. The choice of the best algorithm is not as clear when a<b. An elementary recursive approach yields a linear algorithm after a precomputation stage involving O(n^3) arithmetic operations, but we also present some alternatives that may be more efficient in practise.