Combinatorial Generation Algorithms for Directed Lattice Paths

Yuriy Shablya (), Arsen Merinov and Dmitry Kruchinin
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Yuriy Shablya: Laboratory of Algorithms and Technologies for Discrete Structures Research, Tomsk State University of Control Systems and Radioelectronics, 634050 Tomsk, Russia
Arsen Merinov: Laboratory of Algorithms and Technologies for Discrete Structures Research, Tomsk State University of Control Systems and Radioelectronics, 634050 Tomsk, Russia
Dmitry Kruchinin: Laboratory of Algorithms and Technologies for Discrete Structures Research, Tomsk State University of Control Systems and Radioelectronics, 634050 Tomsk, Russia

Mathematics, 2024, vol. 12, issue 8, 1-18

Abstract: Graphs are a powerful tool for solving various mathematical problems. One such task is the representation of discrete structures. Combinatorial generation methods make it possible to obtain algorithms that can create discrete structures with specified properties. This article is devoted to issues related to the construction of such combinatorial generation algorithms for a wide class of directed lattice paths. The main method used is based on the representation of a given combinatorial set in the form of an AND/OR tree structure. To apply this method, it is necessary to have an expression for the cardinality function of a combinatorial set that satisfies certain requirements. As the main result, we have found recurrence relations for enumerating simple directed lattice paths that satisfy the requirements of the applied method and have constructed the corresponding AND/OR tree structure. Applying the constructed AND/OR tree structure, we have developed new algorithms for ranking and unranking simple directed lattice paths. Additionally, the obtained results were generalized to the case of directed lattice paths.

Keywords: directed lattice path; AND/OR tree; bijection; recurrence; combinatorial generation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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