Fast RSK Correspondence by Doubling Search

Author Alexander Tiskin

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Alexander Tiskin
  • Department of Mathematics and Computer Science, St. Petersburg State University, Russia


I thank Nikolay Vasilyev, Vasilii Duzhin and Artem Kuzmin for advice and fruitful discussions.

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Alexander Tiskin. Fast RSK Correspondence by Doubling Search. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 86:1-86:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


The Robinson-Schensted-Knuth (RSK) correspondence is a fundamental concept in combinatorics and representation theory. It is defined as a certain bijection between permutations and pairs of Young tableaux of a given order. We consider the RSK correspondence as an algorithmic problem, along with the closely related k-chain problem. We give a simple, direct description of the symmetric RSK algorithm, which is implied by the k-chain algorithms of Viennot and of Felsner and Wernisch. We also show how the doubling search of Bentley and Yao can be used as a subroutine by the symmetric RSK algorithm, replacing the default binary search. Surprisingly, such a straightforward replacement improves the asymptotic worst-case running time for the RSK correspondence that has been best known since 1998. A similar improvement also holds for the average running time of RSK on uniformly random permutations.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Permutations and combinations
  • combinatorics of permutations
  • Robinson-Schensted-Knuth correspondence
  • k-chains
  • RSK algorithm


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  1. Jon Louis Bentley and Andrew Chi-Chih Yao. An almost optimal algorithm for unbounded searching. Information Processing Letters, 5(3):82-87, 1976. URL:
  2. Sergei Bespamyatnikh and Michael Segal. Enumerating longest increasing subsequences and patience sorting. Information Processing Letters, 76:7-11, 2000. URL:
  3. Henrik Blunck and Jan Vahrenhold. In-Place Algorithms for Computing (Layers of) Maxima. Algorithmica, 57:1-21, 2010. URL:
  4. Adam L. Buchsbaum and Michael T. Goodrich. Three-Dimensional Layers of Maxima. Algorithmica, 39:275-286, 2004. URL:
  5. Jeff Calder, Selim Esedoḡlu, and Alfred O. Hero. A PDE-based Approach to Nondominated Sorting. SIAM Journal on Numerical Analysis, 53:82-104, 2015. URL:
  6. E W Dijkstra. Some beautiful arguments using mathematical induction. Acta Informatica, 13(1):1-8, 1980. URL:
  7. V. S. Duzhin and N. N. Vasilyev. Asymptotic behavior of normalized dimensions of standard and strict Young diagrams: growth and oscillations. Journal of Knot Theory and Its Ramifications, 25(12):1642002, 2016. URL:
  8. P Erdös and G Szekeres. A combinatorial problem in geometry. Compositio Mathematica, 2:463-470, 1935. Google Scholar
  9. Stefan Felsner and Lorenz Wernisch. Maximum k-Chains in Planar Point Sets: Combinatorial Structure and Algorithms. SIAM Journal on Computing, 28:192-209, 1998. URL:
  10. Michael L Fredman. On computing the length of longest increasing subsequences. Discrete Mathematics, 11:29-35, 1975. URL:
  11. Curtis Greene. An Extension of Schensted’s Theorem. Advances in Mathematics, 14:254-265, 1974. Google Scholar
  12. Dan Gusfield. Algorithms on Strings, Trees, and Sequences. Cambridge University Press, 1997. URL:
  13. D E Knuth. Permutations, matrices, and generalized Young tableaux. Pacific Journal of Mathematics, 34(3):709-727, 1970. Google Scholar
  14. S. N. Majumdar and S. Nechaev. Exact asymptotic results for the Bernoulli matching model of sequence alignment. Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 72(2):020901, 2005. URL:
  15. G de B Robinson. On the representations of the symmetric group. American Journal of Mathematics, 60:745-760, 1938. URL:
  16. D. Romik. The Number of Steps in the Robinson-Schensted Algorithm. Functional Analysis and Its Applications, 39(2):152-155, 2005. URL:
  17. Dan Romik. The Surprising Mathematics of Longest Increasing Subsequences. Cambridge University Press, Cambridge, 2014. URL:
  18. C. Schensted. Longest Increasing and Decreasing Subsequences. Canadian Journal of Mathematics, 13:179-191, 1961. URL:
  19. M. P. Schützenberger. Quelques remarques sur une construction de Schensted. Mathematica Scandinavica, 12:117-128, 1963. URL:
  20. Richard P. Stanley. Algebraic Combinatorics. Undergraduate Texts in Mathematics. Springer, New York, NY, 2013. URL:
  21. Hugh Thomas and Alexander Yong. Longest increasing subsequences, Plancherel-type measure and the Hecke insertion algorithm. Advances in Applied Mathematics, 46(1-4):610-642, 2011. URL:
  22. N. N. Vasiliev and V. S. Duzhin. A Study of the Growth of the Maximum and Typical Normalized Dimensions of Strict Young Diagrams. Journal of Mathematical Sciences, 216(1):53-64, 2016. URL:
  23. G. Viennot. Une forme geometrique de la correspondance de Robinson-Schensted. In Combinatoire et Représentation du Groupe Symétrique, volume 579 of Lecture Notes in Mathematics, pages 29-58. Springer, 1977. URL:
  24. G. Viennot. Chain and antichain families, grids and Young tableaux. Annals of Discrete Mathematics, 23:409-463, 1984. URL:
  25. X Viennot. Growth diagrams and edge local rules. In Proceedings of GASCom, 2018. Google Scholar