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 log n) and generate a global rounding uniformly at random in linear time under a nondegeneracy assumption and in time O(n log n) in the general case