Global roundings of sequences

For a given sequence a = (a1, . . . , an) of numbers, a global rounding is an integer sequence b = (b1, . . . , bn) such that the rounding error |∑i∈I (ai − bi )| is less than one in all intervals I ⊆ {1, . . . , n}. We give a simple characterization of the set of global roundings of a. This allows to compute optimal roundings in time O(n logn) and generate a global rounding uniformly at random in linear time under a non-degeneracy assumption and in time O(n logn) in the general case.  2004 Elsevier B.V. All rights reserved.