Sets of medians in the non-geodesic pseudometric space of unsigned genomes with breakpoints

Abstract

BACKGROUND: The breakpoint median in the set Sn of permutations on n terms is known to have some unusual behavior, especially if the input genomes are maximally different to each other. The mathematical study of the set of medians is complicated by the facts that breakpoint distance is not a metric but a pseudo-metric, and that it does not define a geodesic space. RESULTS: We introduce the notion of partial geodesic, or geodesic patch between two permutations, and show that if two permutations are medians, then every permutation on a geodesic patch between them is also a median. We also prove the conjecture that the input permutations themselves are medians.