Finding Nested Common Intervals Efficiently

In this article, we study the problem of efficiently finding gene clusters formalized by nested common intervals between two genomes represented either as permutations or as sequences. Considering permutations, we give several algorithms whose running time depends on the size of the actual output rather than the output in the worst case. Indeed, we first provide a straightforward cubic time algorithm for finding all nested common intervals. We reduce this complexity by providing a quadratic time algorithm computing an irredundant output. We then show, by providing a third algorithm, that finding only the maximal nested common intervals can be done in linear time. Finally, we prove that finding approximate nested common intervals is fixed parameter tractable. Considering sequences, we provide solutions (modifications of previously defined algorithms and a new algorithm) for different variants of the problem, depending on the treatment one wants to apply to duplicated genes. This includes a polynomial-time algorithm for a variant implying a matching of the genes in the cluster, a setting that for other problems often leads to hardness.