Fast computation of approximate weak common intervals in multiple indeterminate strings

In ongoing work to define a principled method for syntenic block discovery and structuring, work based on homology-derived constraints and a generalization of common intervals, we faced a fundamental computational problem: how to determine quickly, among a set of indeterminate strings (strings whose elements consist of subsets of characters), contiguous intervals that would share a vast majority of their elements, but allow for sharing subsets of characters subsumed by others, and also for certain elements to be missing from certain genomes. An algorithm for this problem in the special case of determinate strings (where each element is a single character of the alphabet, i.e., "normal" strings) was described by Doerr et al., but its running time would explode if generalized to indeterminate strings. In this paper, we describe an algorithm for computing these special common intervals in time close to that of the simpler algorithm of Doerr et al. and show that can compute these intervals in just a couple of hours for large collections (tens to hundreds) of bacterial genomes.