Abstract: In this paper, we extend an algorithmic approach to constructing ordered restriction maps from images of a population of individual DNA molecules (clones) digested by restriction enzymes. The original algorithm was capable of producing high-resolution, high-accuracy maps rapidly and in a scalable manner given a certain class of data errors, including contamination, sizing errors, false and missing restriction sites and unknown orientation. Here we extend this set of errors to include possibly broken molecules where the amount of breakage is not known beforehand, which is necessary for handling larger clones. In an earlier paper~\cite{optmapII}, we had shown that the problem of making maps from molecules with end fragments missing as the only source of error is NP-complete. We also show how to handle multiple reliability levels in the input data when calling restriction sites, where the actual reliability levels are not known and must be infered from the data.