A Reduction Algorithm for Computing the Hybridization Number of Two Trees

Abstract

Hybridization is an important evolutionary process for many groups of species. Thus, conflicting signals in a data set may not be the result of sampling or modeling errors, but due to the fact that hybridization has played a significant role in the evolutionary history of the species under consideration. Assuming that the initial set of gene trees is correct, a basic problem for biologists is to compute this minimum number of hybridization events to explain this set. In this paper, we describe a new reduction-based algorithm for computing the minimum number, when the initial data set consists of two trees. Although the two-tree problem is NP-hard, our algorithm always gives the exact solution and runs efficiently on many real biological problems. Previous algorithms for the two-tree problem either solve a restricted version of the problem or give an answer with no guarantee of the closeness to the exact solution. We illustrate our algorithm on a grass data set. This new algorithm is freely available for application at either http://www.bi.uni-duesseldorf.de/~linz or http://www.math.canterbury.ac.nz/~cas83.

Efficiently Calculating Evolutionary Tree Measures Using SAT

“…Simulated Data Set: 50-taxa trees were generated under the Yule-Harding distribution to be the "species tree" and then for each distance and each species tree, 10 "gene trees" of that distance were generated. In both graphs, @ is RIATA-HGT [20], • is the SAT Descent using Walksat [25], and + is the exact algorithm HybridNumber [7]. Due to the similarity in results to HybridNumber, the results for SAT Ascent solution are omitted.…”

“…The biological data set, from the analysis of HybridNumber [7] and described more fully there, is from the Poaceae (Grass) family. Hybridization is a well-recognized occurrence in grasses [12], making this an excellent test data set.…”

“…For each loci, a tree was built using the fastDNAmL program [21] by Heiko Schmidt [23]. As in [7], we looked at pairs of trees, reduced to their common taxa. In all, we have 15 pairs of trees.…”

“…Wu [28] encodes the rSPR problem into an integer linear programming instance, achieving good results for the rSPR problem only. To find exact answers for hybridization numbers, Linz et al [7] used clever combinatorial characterizations to yield an exhaustive search that does well for surprisingly large values.…”

“…# of Hybrid SAT RIATA SAT Descent [23] taxa Number [7] Exact -HGT [20] w [24] a [8] n [8] We compare the exact solver, HybridNumber [7], the fast heuristic, RIATA-HGT [20], and our program using the SAT encodings. The data for HybridNumber in the third column is from [7].…”