Remarks on the implementation of the fast marching method

The fast marching algorithm computes an approximate solution to the eikonal equation in time, where the factor log N is due to the administration of a priority queue. Recently, Yatziv et al. (2006 J. Comput. Phys., 212, 393–399) have suggested using an untidy priority queue, reducing the overall complexity to 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 estimate implies, in particular, that the choice of an accuracy level that is independent of the speed function F results in the complexity bound being . A numerical experiment illustrates this robustness problem for large ratios F max /F min.