The fast marching algorithm computes an approximate solution to the eikonal equation in O(N log N) time, where the factor log N is due to the administration of a priority queue. Recently, Yatziv, Bartesaghi and Sapiro have suggested to use an untidy priority queue, reducing the overall complexity to O(N) at the price of a small error in the computed solution. In this paper, we give an explicit estimate of the error introduced, which is based on a discrete comparison principle. This estimates implies in particular that the choice of an accuracy level that is independent of the speed function F results in the complexity bound O(Fmax /Fmin N). A numerical experiment illustrates this robustness problem for large ratios Fmax /Fmin .