Computing Sequences with Addition Chains

TL;DR: It is shown that the sequence 2^0, 2^1, 2 \cdots, 2^{n - 1} ,2^ n - 1\} can be computed with $n + 2.13\sqrt n + \log n$ additions, and that the lower bound result is applied to show that the addition-sequence problem is NP-complete.

Abstract: Given a sequence $n_1 , \cdots ,n_m $ of positive integers, what is the smallest number of additions needed to compute all m integers starting with 1? This generalization of the addition chain ($m = 1$) problem will be called the addition-sequence problem. We show that the sequence $\{ 2^0 ,2^1 , \cdots ,2^{n - 1} ,2^n - 1\} $ can be computed with $n + 2.13\sqrt n + \log n$ additions, and that $n + \sqrt n - 2$ is a lower bound. This lower bound result is applied to show that the addition-sequence problem is NP-complete.