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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
Article
  • 35 Downloads

Abstract

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.

Keywords

Partition of integer Monotonic sequence Succinct data structure 

Notes

Acknowledgements

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

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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

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