Enumerating solutions to p(a) + q(b) = r(c) + s(d)
Mathematics of Computation
On the Power of Random Access Machines
Proceedings of the 6th Colloquium, on Automata, Languages and Programming
How much precision is needed to compare two sums of square roots of integers?
Information Processing Letters
| Hi-index | 0.00 |
Let k and n be positive integers, nk. Define r(n,k) to be the minimum positive value of$$ |\sqrt{a_1} + \cdots + \sqrt{a_k} - \sqrt{b_1} - \cdots -\sqrt{b_k} | $$ where a1, a2,⋯, ak, b1, b2,⋯, bk are positive integers no larger than n. It is important to find a tight bound for r(n,k), in connection to the sum-of-square-roots problem, a famous open problem in computational geometry. The current best lower bound and upper bound are far apart. In this paper, we present an algorithm to find r(n,k) exactly in nk+o(k) time and in n⌈k/2 ⌉+o(k) space. As an example, we are able to compute r(100,7) exactly in a few hours on one PC. The numerical data indicate that the known upper bound seems closer to the truth value of r(n,k).