Improved Algorithms for Integer Complexity

TL;DR

This paper introduces two efficient algorithms for calculating the integer complexity of numbers, significantly improving speed for both all numbers up to N and individual large numbers, advancing computational methods in number theory.

Contribution

The paper presents the first sublinear-time algorithm for a single integer's complexity and an improved near-optimal algorithm for all integers up to N.

Findings

New $O(N ext{polylog} N)$ algorithm for all $n \\leq N$

First sublinear-time algorithm for single $n$

Improved computational efficiency over previous methods

Abstract

The integer complexity $f(n)$ of a positive integer $n$ is defined as the minimum number of 1's needed to represent $n$, using additions, multiplications and parentheses. We present two simple and faster algorithms for computing the integer complexity: 1) A near-optimal $O(N\mathop{\mathrm{polylog}} N)$-time algorithm for computing the integer complexity of all $n\leq N$, improving the previous $O(N^{1.223})$ one [Cordwell et al., 2017]. 2) The first sublinear-time algorithm for computing the integer complexity of a single $n$, with running time $O(n^{0.6154})$. The previous algorithms for computing a single $f(n)$ require computing all $f(1),\dots,f(n)$.