A parameterized algorithm for (1,2)-exemplar breakpoint distance

The exemplar breakpoint distance problem is motivated by finding conserved sets of genes between two genomes. It asks to find respective exemplars in two genomes to minimize the breakpoint distance between them. If one genome has no repeated gene (called trivial genome) and the other has genes repeating at most twice, it is referred to as the (1,2)-exemplar breakpoint distance problem, EBD(1, 2) for short. Whether there exists a fixed parameter algorithm for this problem has been open for many years. In this paper, we propose the first fixed parameter algorithm for EBD(1,2). Using a dynamic programming approach, we design an algorithm to solve EBD(1,2) with running time O(4
^s
n
²
) and space O(4
^s
n), respectively, where n is the number of gene families, and s is the maximum number of genes (minus 1) located between two copies of a gene family in the second genome. Our algorithm can also be used to compute the maximum adjacencies between two genomes. Simulations on real data have verified the effectiveness of our algorithm. The algorithm has been implemented in C++. The software package is available upon request.