Improved Algorithms for Integer Complexity

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 $O(N\mathrm{polylog}N)$-time algorithm for computing the integer complexity of all $n\le N$, 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 $f(1),\dots ,f(n)$.

include the code and add more experiments