Near-medians that avoid the corners; a combinatorial probability approach

Abstract

BackgroundThe breakpoint median for a set of k ≥ 3 random genomes tends to approach (any) one of these genomes ("corners") as genome length increases, although there are diminishing proportion of medians equidistant from all k ("medians in the middle"). Algorithms are likely to miss the latter, and this has consequences for the general case where input genomes share some or many gene adjacencies, where the tendency for the median to be closer to one input genome may be an artifact of the corner tendency.ResultsWe present a simple sampling procedure for constructing a "near median" that represents a compromise among k random genomes and that has only a slightly greater breakpoint distance to all of them than the median does. We generalize to the realistic case where genomes share varying proportions of gene adjacencies. We present a supplementary sampling scheme that brings the constructed genome even closer to median status.ConclusionsOur approach is of particular use in the phylogenetic context where medians are repeatedly calculated at ancestral nodes, and where the corner effect prevents different parts of the phylogeny from communicating with each other.

Compromise or optimize? The breakpoint anti-median

“…These anti-medians contrast with arbitrary random genomes whose normalized sums of scores to g 1 , g 2 , and g 3 approach 3. At the other extreme, they also contrast with the “near medians” [9] completed by maximum matching algorithms, whose scores are less than those of the randomly completed samples constructed here.…”

“…We also assume n is large so that for an arbitrary proportion θ , the (1) difference between θ n and the nearest integer to θ n may be neglected. The probabilistic justification behind these assumptions is discussed in [9]. …”

“…A sample is “randomly completed” to form a genome with n genes by the addition of 1− ψ − ω 1,2 − ω 1,3 − ω 2,3 adjacencies constructed by randomly pairing gene ends that are not in any of the adjacencies in the sample or in the overlaps. In other words, to focus on the purely statistical consequences of the sampling procedure we thus do not consider the increment in the number of adjacencies obtainable in individual instances by the ad hoc matching algorithms developed in [9]. The random completion process does not add to the number of adjacencies in the sample in common with one, two or three of g 1 , g 2 and g 3 .…”

“…The “near median” was proposed to get around these difficulties [9]. For k random genomes, the same proportion of gene adjacencies is sampled from each one, in such a way that the union of the samples is compatible – an “end” of a gene is adjacent to no more than one other gene end.…”

“…In the original paper [9], only the following, highly symmetrical cases were studied for k =3: three purely random genomes, three genomes all with common adjacencies forming a proportion ψ of their adjacencies, and three genomes all with a proportion ψ of common adjacencies and additional proportions ω 1,2 , ω 1,3 , ω 2,3 of adjacencies in their pairwise intersections. We only investigated the maximum θ such that the same proportion θ could be sampled from the three input genomes.…”