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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)

Abstract

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.)

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Copyright information

© 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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