A fast algorithm to generate open meandric systems and meanders

Abstract

An open meandric system is a planar configuration of acyclic curves crossing an infinite horizontal line in the plane such that the curves may extend in both horizontal directions. We present a fast, recursive algorithm to exhaustively generate open meandric systems with n crossings. We then illustrate how to modify the algorithm to generate unidirectional open meandric systems (the curves extend only to the right) and non-isomorphic open meandric systems where equivalence is taken under horizontal reflection. Each algorithm can be modified to generate systems with exactly k curves. In the unidirectional case when k=1, we can apply a minor modification along with some additional optimization steps to yield the first fast and simple algorithm to generate open meanders.

Gray codes for reflectable languages

“…For example, Figure 5 Enumeration sequences for open meandric systems were studied by Bacher [1] and a fast algorithm for generating open meandric systems appears in [2]. The latter paper uses the alphabet Σ = {u,d,o,c} to represent the four different types of crossings: u=up, d=down, o=open, c=close.…”

“…Observe that any OMS of order n can always be extended into an OMS of order n + 1 by appending either an o, d, or u. Only when adding a c to an existing OMS is it possible that that a previously open curve can become closed [2]. Thus, the language of all OMSs is reflectable by setting x i =d and y i =u.…”

“…The algorithm in [2] to generate all OMSs runs in constant amortized time due to the introduction of data structures that can test when it is possible to append a c to an existing OMS in constant time. Therefore, it is possible to directly apply our generic Gray code algorithm to convert their lexicographic algorithm into a Gray code algorithm.…”

Stamp Foldings, Semi-Meanders, and Open Meanders: Fast Generation Algorithms

“…Franz and Earnshaw's [3] algorithm also appears to have an asymptotic running time that is greater than the number of meanders being generated (no implementation details or analysis is provided). The fastest known algorithm is given by Bobier and Sawada [1]. Although a rigorous analysis is not provided, the implementation of the algorithm is very simple.…”

“…Then in Section 3.2, by maintaining the wind-factor for semi-meanders we obtain an efficient algorithm to generate open-meanders. This algorithm is analyzed and compared experimentally with the previously fastest known algorithm to exhaustively list open-meanders [1]. The paper concludes with a summary in Section 4.…”