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Lexicographic Unranking of Combinations Revisited

CNRS, Laboratoire de Paris 6—lip6umr 7606, Sorbonne Université, F-75005 Paris, France
Author to whom correspondence should be addressed.
Academic Editor: Henning Fernau
Algorithms 2021, 14(3), 97;
Received: 30 November 2020 / Revised: 15 March 2021 / Accepted: 16 March 2021 / Published: 19 March 2021
(This article belongs to the Special Issue Selected Algorithmic Papers From CSR 2020)
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. View Full-Text
Keywords: unranking algorithm; combinatorial generation; combination; lexicographic order; complexity analysis unranking algorithm; combinatorial generation; combination; lexicographic order; complexity analysis
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MDPI and ACS Style

Genitrini, A.; Pépin, M. Lexicographic Unranking of Combinations Revisited. Algorithms 2021, 14, 97.

AMA Style

Genitrini A, Pépin M. Lexicographic Unranking of Combinations Revisited. Algorithms. 2021; 14(3):97.

Chicago/Turabian Style

Genitrini, Antoine, and Martin Pépin. 2021. "Lexicographic Unranking of Combinations Revisited" Algorithms 14, no. 3: 97.

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