LvD: A New Algorithm for Computing the Likelihood of a Phylogeny

TL;DR

This work introduces LvD (Likelihood via Decomposition), a novel algorithm for computing phylogenetic likelihoods by building a decomposition tree of connected components, achieving a worst-case update and parallel time of $O(\log n)$, in contrast to the traditional $O(n)$. By defining partial likelihoods on clades and segments and merging them via five merger types, LvD enables efficient updates and potential parallelization while maintaining compatibility with standard nucleotide models. Empirical and simulated results show appreciable speed-ups (often 20–30%, up to several-fold in unbalanced trees), with gains sensitive to tree balance and data characteristics. The approach offers memory efficiency, a clear path to parallel speedups, and substantial practical impact for large-scale phylogenetic likelihood computations, though gains are less pronounced on already balanced trees and require careful implementation for best results.

Abstract

There are few, if any, algorithms in statistical phylogenetics which are used more heavily than Felsenstein's 1973 pruning method for computing the likelihood of a tree. We present LvD, (Likelihood via Decomposition), an alternative to Felsenstein's algorithm based on a different decomposition of the underlying phylogeny. It works for all standard nucleotide models. The new algorithm allows updates of the likelihood calculation in worst case $O(\log n)$ time with $n$ taxa, as opposed to worst case $O(n)$ time for existing methods. In practice this leads to appreciable improvements in likelihood calculations, the extent of speed-up depending on how balanced or unbalanced the trees are. We explore implications for parallel computing, and show that the approach allows likelihoods to be computed in $O(\log n)$ parallel time per site, compared to (worst case) $O(n)$ time. We implemented and applied the algorithm to large numbers of simulated and empirical data sets and showed that these theoretical advances lead to a significant practical speed-up, although the extent of the improvement depends on how balanced the phylogenies already are.

Figures (13)

• Figure 1: Terminology and notation for phylogenies. The root is denoted $\rho$. For each node $v$, the state at that node is denoted $\mathbf{x}_v$ and the set of taxa which are descendants of $v$ is denoted by $C_v$.
• Figure 2: Illustration of the three rules used to evaluate the partial likelihoods for nodes in the phylogeny, using Felsenstein's algorithm: (a.) leaves; (b.) internal nodes; (c.) the root.
• Figure 3: Examples of components: two clades ($S_1$ and $S_2$) and two segments ($S_3$ and $S_4$) in a tree. Boundary nodes are depicted using hollow circles and squares: hollow circles indicate component roots and hollow squares indicate component buds. Note that the root of $S_2$ coincides with the bud of $S_3$.
• Figure 4: The five ways that two components ($S_1$ and $S_2$) can be merged to give a single component ($S$).
• Figure 5: A decomposition tree. (a) The original phylogeny (b) Decomposition tree (with five levels). (c) - (g) Components corresponding to each level in the decomposition tree. (c) Level one in the decomposition tree corresponds to single-edge components. Subsequent levels (d)--(g) are formed by merging components from the previous level. Note that components 7 and 10 in (c) are not merged in (d) as this would create a component with more than two boundaries.