On clustering to minimize the sum of radii
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Finding the smallest gap between sums of square roots
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
On Clustering to Minimize the Sum of Radii
SIAM Journal on Computing
On the Sum of Square Roots of Polynomials and Related Problems
ACM Transactions on Computation Theory (TOCT)
| Hi-index | 0.89 |
In this paper, we investigate the problem of the minimum nonzero difference between two sums of square roots of integers. Let r(n, k) be the minimum positive value of |Σi=1k √ai-Σi=1k√bi| where ai and bi are integers not larger than integer n. We prove by an explicit construction that r(n,k) = O(n-2k+3/2) for fixed k and any n. Our result implies that in order to compare two sums of k square roots of integers with at most d digits per integer, one might need precision of as many as (2k - 3/2)d digits. We also prove that this bound is optimal for a wide range of integers, i.e., r(n, k) = Θ(n-2k+3/2) for fixed k and for those integers in the form of ai = (2k-1 2i)2(n' + 2i) and bi = (2k-1 2i+1)2 (n' + 2i + 1), where n' is any integer satisfied the form and i is any integer in [0, k-1]. We finally show that for k = 2 and any n, this bound is also optimal, i.e., r(n, 2) = Θ(n-7/2).