Method for Developing Combinatorial Generation Algorithms Based on AND/OR Trees and Its Application

• Main Authors: Yuriy Shablya, Dmitry Kruchinin, Vladimir Kruchinin
• Format: Article
• Language: English
• Published: MDPI AG 2020-06-01
• Series: Mathematics
• Subjects: combinatorial generation; method; algorithm; AND/OR tree; Euler–Catalan’s triangle; labeled Dyck path
• Online Access: https://www.mdpi.com/2227-7390/8/6/962

In this paper, we study the problem of developing new combinatorial generation algorithms. The main purpose of our research is to derive and improve general methods for developing combinatorial generation algorithms. We present basic general methods for solving this task and consider one of these methods, which is based on AND/OR trees. This method is extended by using the mathematical apparatus of the theory of generating functions since it is one of the basic approaches in combinatorics (we propose to use the method of compositae for obtaining explicit expression of the coefficients of generating functions). As a result, we also apply this method and develop new ranking and unranking algorithms for the following combinatorial sets: permutations, permutations with ascents, combinations, Dyck paths with return steps, labeled Dyck paths with ascents on return steps. For each of them, we construct an AND/OR tree structure, find a bijection between the elements of the combinatorial set and the set of variants of the AND/OR tree, and develop algorithms for ranking and unranking the variants of the AND/OR tree.