The Review of Socionetwork Strategies

, Volume 13, Issue 2, pp 237–252 | Cite as

Storing Partitions of Integers in Sublinear Space

  • Kentaro Sumigawa
  • Kunihiko SadakaneEmail author


In this study, we introduce a data structure representing a partition of an integer n, which uses \(\mathrm{O}(\sqrt{n})\) bits of space. This is a constant multiple of the information theoretic lower bound. Three types of operations \({\textsf {access}}_{{\textsf {p}}}\), \({\textsf {bound}}_{{\textsf {p}}}\), \({\textsf {prefixsum}}_{{\textsf {p}}}\) are supported in constant time using the notion of conjugate of a partition. To construct this data structure, we establish a data structure 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 \(\delta \) where n is the number of terms, and u denotes the size of the universe.


Partition of integer Monotonic sequence Succinct data structure 



This work was supported by JST CREST Grant Number JPMJCR1402, Japan.


  1. 1.
    Elias, P. (1974). Efficient storage and retrieval by content and address of static files. Journal of ACM, 21(2), 246–260.CrossRefGoogle Scholar
  2. 2.
    El-Zein, H., Lewenstein, M., Munro, J. I., Raman, V., & Chan, T. M. (2017). On the succinct representation of equivalence classes. Algorithmica, 78(3), 1020–1040.CrossRefGoogle Scholar
  3. 3.
    Fano, R. M. (1971). On the number of bits required to implement an associative memory. Massachusetts Institute of Technology, Project MAC.Google Scholar
  4. 4.
    Fulton, W. (2012). Young Tableaux. Cambridge: Cambridge University Press.Google Scholar
  5. 5.
    Golynski, A., Orlandi, A., Raman, R., & Rao, S. S. (2014). Optimal Indexes for Sparse Bit Vectors. Algorithmica, 69(4), 906–924.CrossRefGoogle Scholar
  6. 6.
    Hardy, G. H., & Ramanujan, S. (1918). Asymptotic formulae in combinatory analysis. Proceedings of the London Mathematical Society, 2(1), 75–115.CrossRefGoogle Scholar
  7. 7.
    Munro, J. I., Raman, R., Raman, V., & Rao, S. S. (2012). Succinct representation of permutation ans functions. Theoretical Computer Science, 438(22), 74–88.CrossRefGoogle Scholar
  8. 8.
    Pibiri, G. E., & Venturini, R. (2017). Dynamic Elias-Fano representation. In: 28th Annual symposium on combinatorial pattern matching (CPM 2017), 78(30), 1–14.Google Scholar
  9. 9.
    Raman, R., Raman, V., & Rao, S. S. (2007). Succinct indexable dictionaries with applications to encoding \(k\)-ary trees and multisets. ACM Transactions on Algorithms, 3(4), 43.CrossRefGoogle Scholar

Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

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

Personalised recommendations