An Efficient Representation of Partitions of Integers

  • Kentaro SumigawaEmail author
  • Kunihiko Sadakane
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10979)


We introduce a data structure for representing a partition of an integer n, which uses \(\mathrm{O}(\sqrt{n})\) bits of space. This is constant multiple of the information theoretic lower bound. Three types of operations \(\mathsf{access}_\mathsf{p},\mathsf{bound}_\mathsf{p},\mathsf{prefixsum}_\mathsf{p}\) are supported in constant time by using the notion of conjugate of a partition. In order to construct this data structure, we also construct a data structure for representing a monotonic sequence, which supports the same operations in constant time and uses \(\mathrm{O}(\min \{\frac{1}{\delta }u\left( \frac{n}{u}\right) ^{\delta },\frac{1}{\delta }n\left( \frac{u}{n}\right) ^\delta \})\) bits of space for any positive constant \(\delta \). (n is the number of terms, and u denotes the size of the universe.)


  1. 1.
    Elias, P.: Efficient storage and retrieval by content and address of static files. J. ACM 21(2), 246–260 (1974)MathSciNetCrossRefGoogle Scholar
  2. 2.
    El-Zein, H., Lewenstein, M., Munro, J.I., Raman, V., Chan, T.M.: On the succinct representation of equivalence classes. Algorithmica 78(3), 1020–1040 (2017)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Fano, R.M.: On the number of bits required to implement an associative memory. Project MAC, Massachusetts Institute of Technology (1971)Google Scholar
  4. 4.
    Fulton, W.: Young Tableaux. Cambridge University Press, Cambridge (2012)Google Scholar
  5. 5.
    Golynski, A., Orlandi, A., Raman, R., Rao, S.S.: Optimal indexes for sparse bit vectors. Algorithmica 69(4), 906–924 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hardy, G.H., Ramanujan, S.: Asymptotic formulae in combinatory analysis. Proc. Lond. Math. Soc. 2(1), 75–115 (1918)CrossRefGoogle Scholar
  7. 7.
    Munro, J.I., Raman, R., Raman, V., Rao, S.S.: Succinct representation of permutation and functions. Theor. Comput. Sci. 438(22), 74–88 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Pibiri, G.E., Venturini, R.: Dynamic Elias-Fano representation. In: 28th Annual Symposium on Combinatorial Pattern Matching (CPM 2017), vol. 78, issue 30, pp. 1–14 (2017)Google Scholar
  9. 9.
    Raman, R., Raman, V., Rao, S.S.: Succinct indexable dictionaries with applications to encoding \(k\)-ary trees and multisets. ACM Trans. Algorithms 3(4) (2007). Article No. 43Google Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematical Informatics, Graduate School of Information Science and TechnologyThe University of TokyoTokyoJapan

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