Quartets solved by 327

Incomplete Characters

The modern revolution in genome sequencing has produced a huge amount of genetic data for a wide variety of species. One ultimate goal of possessing all this information is to be able to construct complete phylogenies via direct genome analysis.

For example, say that we have a gene shared by a number of taxa. We could create a character based on whether species are known to possess the gene or not, and then use a huge character table to construct our desired phylogeny. However, the present bottleneck with such a method is that it assumes that we already possess complete genome information for all possible species. The race is on to sequence as many species genomes as possible; for instance, the Genome 10K Project aims to sequence 10,000 species genomes over the next decade. Yet for the time being, possessing a complete genomic picture of all Earth's species remains a dream.

As a result of these practical limitations, we need to be able to work with partial characters, which divide taxa into three separate groups: those possessing the character, those not possessing the character, and those for which we do not yet have conclusive information.

Problem

A partial split of a set $S$ of $n$ taxa models a partial character and is denoted by $A \mid B$, where $A$ and $B$ are still the two disjoint subsets of taxa divided by the character. Unlike in the case of splits, we do not necessarily require that $A \cup B = S$; $(A \cup B)^{\textrm{c}}$ corresponds to those taxa for which we lack conclusive evidence regarding the character.

We can assemble a collection of partial characters into a generalized partial character table $C$ in which the symbol $x$ is placed in $C_{i, j}$ if we do not have conclusive evidence regarding the $j$th taxon with respect to the $i$th partial character.

A quartet is a partial split $A \mid B$ in which both $A$ and $B$ contain precisely two elements. For the sake of simplicity, we often will consider quartets instead of partial characters. We say that a quartet $A \mid B$ is inferred from a partial split $C \mid D$ if $A \subseteq C$ and $B \subseteq D$ (or equivalently $A \subseteq D$ and $B \subseteq C$). For example, $\{1, 3\} \mid \{2, 4\}$ and $\{3, 5\} \mid \{2, 4\}$ can be inferred from $\{1, 3, 5\} \mid \{2, 4\}$.

Given: A partial character table $C$.

Return: The collection of all quartets that can be inferred from the splits corresponding to the underlying characters of $C$.

Sample Dataset

cat dog elephant ostrich mouse rabbit robot
01xxx00
x11xx00
111x00x

Sample Output

{elephant, dog} {rabbit, robot}
{cat, dog} {mouse, rabbit}
{mouse, rabbit} {cat, elephant}
{dog, elephant} {mouse, rabbit}