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

Shablya, Yuriy; Kruchinin, Dmitry; Kruchinin, Vladimir

doi:10.3390/math8060962

Cited by 9 publications

(13 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since the factoradic (with 8 components) of 2021 is (0, 1, 2, 0, 4, 4, 2, 0), the permutation (of size 8) of rank 2021 is (0, 3,6,7,1,5,4,2). To obtain this permutation, we read the factoradic from right to left, and extract iteratively from the list (0, 1, .…”

Section: Link Between the Factorial Number System And Permutationsmentioning

confidence: 99%

“…For example, when n = 5 and k = 3, the number 8 is represented as ( 1 1 ) + ( 32 ) + ( 43 ), thus the combinadic of 8 is (1,3,4). In Table 4, we present for various values of u the combinadic of u and the combination of rank u for n = 6 and k = 2.…”

Section: Unranking Through Combinadicsmentioning

confidence: 99%

“…This generation strategy of a single element is classically called an unranking method. It appears as the fundamental problem in combinatorial generation such as in [1] and in optimization [2]. It is also used as a basic building block in scheduling problems [3], as well as, e.g., in software testing [4].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Lexicographic Unranking of Combinations Revisited

Genitrini¹,

Pépin²

2021

Algorithms

View full text Add to dashboard Cite

In the context of combinatorial sampling, the so-called “unranking method” can be seen as a link between a total order over the objects and an effective way to construct an object of given rank. The most classical order used in this context is the lexicographic order, which corresponds to the familiar word ordering in the dictionary. In this article, we propose a comparative study of four algorithms dedicated to the lexicographic unranking of combinations, including three algorithms that were introduced decades ago. We start the paper with the introduction of our new algorithm using a new strategy of computations based on the classical factorial numeral system (or factoradics). Then, we present, in a high level, the three other algorithms. For each case, we analyze its time complexity on average, within a uniform framework, and describe its strengths and weaknesses. For about 20 years, such algorithms have been implemented using big integer arithmetic rather than bounded integer arithmetic which makes the cost of computing some coefficients higher than previously stated. We propose improvements for all implementations, which take this fact into account, and we give a detailed complexity analysis, which is validated by an experimental analysis. Finally, we show that, even if the algorithms are based on different strategies, all are doing very similar computations. Lastly, we extend our approach to the unranking of other classical combinatorial objects such as families counted by multinomial coefficients and k-permutations.

show abstract

Section: Link Between the Factorial Number System And Permutationsmentioning

confidence: 99%

Section: Unranking Through Combinadicsmentioning

confidence: 99%

See 1 more Smart Citation

Lexicographic Unranking of Combinations Revisited

Genitrini¹,

Pépin²

2021

Algorithms

View full text Add to dashboard Cite

show abstract

“…There are several basic general methods for developing combinatorial generation algorithms, such as backtracking [1], the ECO-method [15], Flajolet's method [16], and Kruchinin's method [17]. This article discusses the application of the latter method, which is based on the representation of combinatorial sets in the form of an AND/OR tree structure.…”

Section: Methodsmentioning

confidence: 99%

Combinatorial Generation Algorithms for Some Lattice Paths Using the Method Based on AND/OR Trees

Shablya

2023

Algorithms

Self Cite

View full text Add to dashboard Cite

Methods of combinatorial generation make it possible to develop algorithms for generating objects from a set of discrete structures with given parameters and properties. In this article, we demonstrate the possibilities of using the method based on AND/OR trees to obtain combinatorial generation algorithms for combinatorial sets of several well-known lattice paths (North-East lattice paths, Dyck paths, Delannoy paths, Schroder paths, and Motzkin paths). For each considered combinatorial set of lattice paths, we construct the corresponding AND/OR tree structure where the number of its variants is equal to the number of objects in the combinatorial set. Applying the constructed AND/OR tree structures, we have developed new algorithms for ranking and unranking their variants. The performed computational experiments have confirmed the obtained theoretical estimation of asymptotic computational complexity for the developed ranking and unranking algorithms.

show abstract

“…Shablya et al [12] look for new combinatorial generation algorithms. They give basic general methods and investigate one of them based on AND/OR trees.…”

mentioning

confidence: 99%