Calculating the Unrooted Subtree Prune-and-Regraft Distance

The subtree prune-and-regraft (SPR) distance metric is a fundamen-tal way of comparing evolutionary trees. It has wide-ranging applications,such as to study lateral genetic transfer, viral recombination, and Markovchain Monte Carlo phylogenetic inference. Although the rooted versionof SPR distance can be computed relatively efficiently between rootedtrees using fixed-parameter-tractable maximum agreement forest (MAF)algorithms, no MAF formulation is known for the unrooted case. Cor-respondingly, previous algorithms are unable to compute unrooted SPRdistances larger than 7.In this paper, we substantially advance understanding of and com-putational algorithms for the unrooted SPR distance. First we identifyfour properties of optimal SPR paths, each of which suggests that noMAF formulation exists in the unrooted case. Then we introduce thereplug distance, a new lower bound on the unrooted SPR distance thatis amenable to MAF methods, and give an efficient fixed-parameter al-gorithm for calculating it. Finally, we develop a “progressive A*” searchalgorithm using multiple heuristics, including the TBR and replug dis-tances, to exactly compute the unrooted SPR distance. Our algorithmis nearly two orders of magnitude faster than previous methods on smalltrees, and allows computation of unrooted SPR distances as large as 14on trees with 50 leaves.