Improved Algorithms for Integer Complexity
SIAM Symposium on Simplicity in Algorithms(2023)
摘要
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 $f(1),\dots,f(n)$.
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关键词
integer complexity,improved algorithms
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