IWOCA 2018: Combinatorial Algorithms pp 361-373

# An Efficient Representation of Partitions of Integers

• Kentaro Sumigawa
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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