A Fast and Exact Algorithm for the Exemplar Breakpoint Distance

Abstract

A fundamental problem in comparative genomics is to compute the distance between two genomes. For two genomes without duplicate genes, we can easily compute a variety of distance measures in linear time, but the problem is NP-hard under most models when genomes contain duplicate genes. Sankoff proposed the use of exemplars to tackle the problem of duplicate genes and gene families: each gene family is represented by a single gene (the exemplar for that family), chosen so as to optimize some metric. Unfortunately, choosing exemplars is itself an NP-hard problem. In this article, we propose a very fast and exact algorithm to compute the exemplar breakpoint distance, based on new insights in the underlying structure of genome rearrangements and exemplars. We evaluate the performance of our algorithm on simulation data and compare its performance to the best effort to date (a divide-and-conquer approach), showing that our algorithm runs much faster and scales much better. We also devise a new algorithm for the intermediate breakpoint distance problem, which can then be applied to assign orthologs. We compare our algorithm with the state-of-the-art method MSOAR by assigning orthologs among five well annotated mammalian genomes, showing that our algorithm runs much faster and is slightly more accurate than MSOAR.

On Computing Breakpoint Distances for Genomes with Duplicate Genes

“…For each pair of species, we compare the running time for the four algorithms (I-A1, I-A0, M-A1 and M-A0). We also run another two algorithms for comparison, namely, the exact algorithm to solve E-BDP described in [12] (referred to as E-A1), and MSOAR [16], which uses heuristics to compute a matching such that the inversion distance induced by this matching is minimized. The results are shown in Table 3.…”

“…We described a fast and exact algorithm for E-BDP in previous work [12]. In this section, we describe fast and exact algorithms for I-BDP and M-BDP.…”

“…Our ILP formulations use the same ideas we used for the exact algorithm to solve E-BDP in [12]. They are similar to those proposed in [14], but have fewer variables.…”

“…Nguyen et al gave a much faster divide-and-conquer approach [11], while Angibaud et al [9] gave an exact algorithm by formulating it as an integer linear program (ILP). In [12], we gave an exact solution through combining ILP and novel preprocessing algorithms and constraint-generating algorithms, which runs very fast (in a few seconds) and scales up to the largest genomes analyzed on both simulations and biological datasets. For M-BDP, Blin et al [8] had defined (but not tested) a branch-andbound algorithm, while Swenson et al [13] gave an approximation algorithm.…”

“…Our results show that our algorithms easily scale beyond the size of mammalian genomes: in all of our testing, almost all instances took a few seconds and none more that 70 s. They also demonstrate that our new algorithm to reduce the number of variables is a crucial contributor to this speed, especially for the more challenging I-BDP. Thus, with our previous (and equally fast) algorithm for E-BDP [12], we now have fast and exact algorithms to compute the breakpoint distance under all three formulations.…”