Incomplete Directed Perfect Phylogeny in Linear Time

TL;DR

This paper presents a simpler and faster algorithm for the Incomplete Directed Perfect Phylogeny problem, achieving optimal linear time complexity by leveraging problem-specific properties and avoiding complex data structures.

Contribution

The authors develop a new, simpler $ ilde{O}(nm)$-time algorithm for IDPP and further improve it to an optimal $O(nm)$-time solution by exploiting problem-specific insights.

Findings

Achieved a simpler $ ilde{O}(nm)$-time algorithm for IDPP.

Developed an asymptotically faster $O(nm)$-time algorithm.

Demonstrated the importance of problem-specific properties in algorithm design.

Abstract

Reconstructing the evolutionary history of a set of species is a central task in computational biology. In real data, it is often the case that some information is missing: the Incomplete Directed Perfect Phylogeny (IDPP) problem asks, given a collection of species described by a set of binary characters with some unknown states, to complete the missing states in such a way that the result can be explained with a perfect directed phylogeny. Pe'er et al. proposed a solution that takes $\tilde{O}(nm)$ time for $n$ species and $m$ characters. Their algorithm relies on pre-existing dynamic connectivity data structures: a computational study recently conducted by Fern{\'a}ndez-Baca and Liu showed that, in this context, complex data structures perform worse than simpler ones with worse asymptotic bounds. This gives us the motivation to look into the particular properties of the dynamic connectivity problem in this setting, so as to avoid the use of sophisticated data structures as a blackbox. Not only are we successful in doing so, and give a much simpler $\tilde{O}(nm)$-time algorithm for the IDPP problem; our insights into the specific structure of the problem lead to an asymptotically faster algorithm, that runs in optimal $O(nm)$ time.