  When performing an analysis on a collection of molecular sequences, it can be
convenient to reduce the number of sequences under consideration while
maintaining some characteristic of a larger collection of sequences. For
example, one may wish to select a subset of high-quality sequences that
represent the diversity of a larger collection of sequences. One may also wish
to specialize a large database of characterized "reference sequences" to a
smaller subset that is as close as possible on average to a collection of
"query sequences" of interest. Such a representative subset can be useful
whenever one wishes to find a set of reference sequences that is appropriate to
use for comparative analysis of environmentally-derived sequences, such as for
selecting "reference tree" sequences for phylogenetic placement of metagenomic
reads. In this paper we formalize these problems in terms of the minimization
of the Average Distance to the Closest Leaf (ADCL) and investigate algorithms
to perform the relevant minimization. We show that the greedy algorithm is not
effective, show that a variant of the Partitioning Among Medoids (PAM)
heuristic gets stuck in local minima, and develop an exact dynamic programming
approach. Using this exact program we note that the performance of PAM appears
to be good for simulated trees, and is faster than the exact algorithm for
small trees. On the other hand, the exact program gives solutions for all
numbers of leaves less than or equal to the given desired number of leaves,
while PAM only gives a solution for the pre-specified number of leaves. Via
application to real data, we show that the ADCL criterion chooses chimeric
sequences less often than random subsets, while the maximization of
phylogenetic diversity chooses them more often than random. These algorithms
have been implemented in publicly available software.
