Abstract
Background
Due to hybridization events in evolution, studying two different genes of a set of species may yield two related but different phylogenetic trees for the set of species. In this case, we want to combine the two phylogenetic trees into a hybridization network with the fewest hybridization events. This leads to three computational problems, namely, the problem of computing the minimum size of a hybridization network, the problem of constructing one minimum hybridization network, and the problem of enumerating a representative set of minimum hybridization networks. The previously best software tools for these problems (namely, Chen and Wang’s HybridNet and Albrecht et al.’s Dendroscope 3) run very slowly for large instances that cannot be reduced to relatively small instances. Indeed, when the minimum size of a hybridization network of two given trees is larger than 23 and the problem for the trees cannot be reduced to relatively smaller independent subproblems, then HybridNet almost always takes longer than 1 day and Dendroscope 3 often fails to complete. Thus, a faster software tool for the problems is in need.
Results
We develop a software tool in ANSI C, named FastHN, for the following problems: Computing the minimum size of a hybridization network, constructing one minimum hybridization network, and enumerating a representative set of minimum hybridization networks. We obtain FastHN by refining HybridNet with three ideas. The first idea is to preprocess the input trees so that the trees become smaller or the problem becomes to solve two or more relatively smaller independent subproblems. The second idea is to use a fast algorithm for computing the rSPR distance of two given phylognetic trees to cut more branches of the search tree in the exhaustivesearch stage of the algorithm. The third idea is that during the exhaustivesearch stage of the algorithm, we find two sibling leaves in one of the two forests (obtained from the given trees by cutting some edges) such that they are as far as possible in the other forest. As the result, FastHN always runs much faster than HybridNet. Unlike Dendroscope 3, FastHN is a singlethreaded program. Despite this disadvantage, our experimental data shows that FastHN runs substantially faster than the multithreaded Dendroscope 3 on a PC with multiple cores. Indeed, FastHN can finish within 16 minutes (on average on a Windows7 (x64) desktop PC with i72600 CPU) even if the minimum size of a hybridization network of two given trees is about 25, the trees each have 100 leaves, and the problem for the input trees cannot be reduced to two or more independent subproblems via cluster reductions. It is also worth mentioning that like HybridNet, FastHN does not use much memory (indeed, the amount of memory is at most quadratic in the input size). In contrast, Dendroscope 3 uses a huge amount of memory. Executables of FastHN for Windows XP (x86), Windows 7 (x64), Linux, and Mac OS are available (see the Results and discussion section for details).
Conclusions
For both biological datasets and simulated datasets, our experimental results show that FastHN runs substantially faster than HybridNet and Dendroscope 3. The superiority of FastHN in speed over the previous tools becomes more significant as the hybridization number becomes larger. In addition, FastHN uses much less memory than Dendroscope 3 and uses the same amount of memory as HybridNet.
Background
Constructing the evolutionary history of a set of species is an important problem in the study of biological evolution. Phylogenetic trees are used in biology to represent the ancestral history of a collection of existing species. This is appropriate for many groups of species. However, there are some groups for which the ancestral history cannot be represented by a tree. This is caused by processes such as hybridization, recombination, and lateral gene transfer. We refer to those processes as reticulation events. For this kind of groups of species, it is more appropriate to represent their ancestral history by rooted acyclic digraphs, where vertices of indegree at least two represent reticulation events.
When studying the evolutionary history of a set of existing species, one can obtain a phylogenetic tree of the set of species with high confidence by looking at a segment of sequences or a set of genes. When looking at another segment of sequences, a different phylogenetic tree can be obtained with high confidence, too. This indicates that reticulation events may occur. Thus, we have the following problem: Given two rooted phylogenetic trees on a set of species that correctly represent the treelike evolution of different parts of their genomes, what is the smallest number of reticulation events needed to explain the evolution of the species under consideration?
The subtree prune and regraft (rSPR) distance and the hybridization number are two important measures for evolutionary tree comparison and hybridization network construction. Since both problems are NPhard [13], it is challenging to develop programs that can give exact solutions when the two given trees are large or have a large rSPR distance or hybridization number. Previously, several software packages have been developed for these problems [49]. This new breakthrough brings us a hope that one can routinely solve these hard problems for two given large trees. However, the previously fastest software packages can still take hours to finish even when the given trees are of moderate sizes. Thus, a faster software tool for these problems is in need.
In general, there may exist two or more minimum hybridization networks displaying two given phylogenetic trees T_{1} and T_{2} with the same leaf set X. In some cases, we may want to enumerate all minimum hybridization networks displaying both T_{1} and T_{2}. Unfortunately, it is not hard to construct two example phylogenetic trees T_{1} and T_{2} such that there are too many minimum hybridization networks displaying both T_{1} and T_{2}. So, we instead want to enumerate only a representative set of minimum hybridization networks displaying both T_{1} and T_{2}. Here, a hybridization network Nrepresents another hybridization network N’ if for every pair (x,y) of species in Xx and y fall into the same connected component of F^{N} if and only if they fall into the same connected component of F’_{N}, where F^{N} (respectively, F’_{N}) is the forest obtained from N (respectively, N’) by removing all the edges entering reticulate nodes. HybridNet[6] and Dendroscope 3[4] are able to enumerate a representative set of of minimum hybridization networks for two given phylogenetic trees. If the problem for the two given trees can be reduced to relatively smaller independent subproblems (by socalled “cluster reductions”), Dendroscope is much faster than HybridNet; otherwise, the two have almost the same speed. Unfortunately, both tools run very slowly when the minimum hybridization number of a hybridization network of two given trees is large (say, larger than 23) and the problem for the trees cannot be reduced to relatively smaller independent subproblems. Thus, a much faster tool is in need.
Results and discussion
We have developed a new tool (called FastHN) for the problem of enumerating a representative set of minimum hybridization networks of two given phylognetic trees. Of course, FastHN can also compute the minimum hybridization number of a hybridization network of two given phylognetic trees and construct a single minimum hybridization network of two given phylognetic trees. FastHN is implemented in ANSI C and is available at http://rnc.r.dendai.ac.jp/∼chen/fastHN.html webcite, or http://www.cs.cityu.edu.hk/∼lwang/software/FastHN/fastHN.html webcite, where one can download executables for Windows XP (x86), Windows 7 (x64), Linux, and Mac OS.
After downloading FastHN, one can run it as follows:
FastHN T1 T2 OPTION HEURISTIC or simply FastHN T1 T2 OPTION
Here, T1 and T2 are two text files each containing a phylogenetic tree in the Newick format (ended with a semicolon). The label of each leaf in an input tree should be a string consisting of letters in {0,1,…,9,a,b,…,z,A,B,…,Z,_,.}. There is no limit on the length of the label of each leaf.
OPTION is a string in the set {HN, MAAF, MAAFs} controlling the output as follows:
· HN: The output is the hybridization number of T1 and T2.
· MAAF: The output is one MAAF of T1 and T2 together with one minimum hybridization network for the MAAF.
· MAAFs: The output is all MAAFs of T1 and T2 together with one minimum hybridization network for each MAAF.
FastHN outputs an MAAF (respectively, MAF) by printing out the leaf sets of the trees in the MAAF (respectively, MAF), while it outputs a hybridization network in its extended Newick format [10]. When OPTION is MAAFs (respectively, MAFs), FastHN outputs the MAAFs (respectively, MAFs) without repetition. We remind the reader that one can view a tree in the Newick format and a network in the extended Newick format by using Dendroscope due to [11].
HEURISTIC is a 3bit binary string specifying the version of FastHN as follows:
· The first bit is 1 if and only if FastHN adopts initial cluster reductions.
· The second bit is 1 if and only if FastHN adopts Heuristic 1.
· The last bit is 1 if and only if FastHN adopts Heuristic 2.
HEURISTIC can be omitted; in that case, it is set to be 111.
To compare the efficiency of FastHN with the previous bests (namely, HybridNet[6] and Dendroscope 3[4]), we have run them on both simulated datasets and biological datasets for the problem of computing all MAAFs of two given phylogenetic trees. The experiment has been performed on a Windows7 (x64) desktop PC with i72600 CPU and 4GB RAM. It is worth mentioning that in our experiments, we have used the total elapsed time (rather than the CPU time) to measure the running time of FastHN. Since FastHN is singlethreaded, its total elapsed time is usually more than its CPU time. In contrast, Dendroscope 3 is multithreaded, its total elapsed time can be less than its CPU time. Because it is not clear how Dendroscope 3 measures its running time, we have pessimistically measured the running time of FastHN using the total elapsed time (in order to do a fair comparison with Dendroscope 3).
Simulated data
To generate simulated datasets, we use a program due to Beiko and Hamilton [12]. To obtain a pair (T, T’) of trees, their program first generates T randomly and then obtains T’ from T by performing a specified number r (say, 20) of random rSPR operations on T. Recall that an rSPR operation on a tree T first removes an edge (p,c) from T, then contracts p (the vertex of outdegree 1 resulting from the removal of edge (p,c)), and further reattaches the subtree rooted at c to an edge (p’,c’) of T (by introducing a new vertex m’, splitting edge (p’,c’) into two edges (p’,m’) and (m’,c’), and adding a new edge (m’,c)). So, the actual rSPR distance of T and T’ is at most r. Moreover, the hybridization number of T and T’ can be r, smaller than r, or larger than r.
We first use Beiko and Hamilton’s program to generate 120 pairs of trees each of which has 100 leaves. The first (respectively, second) 60 pairs are generated by setting r = 14 (respectively, r = 17). It turns out that among the 120 generated treepairs, 6 (respectively, 22, 33, 11, 21, or 27) treepairs have hybridization number 12 (respectively, 13, 14, 15, 16, or 17). Figure 1 summarizes the average running time of the programs for the generated treepairs, where each average is taken over those treepairs with the same hybridization number. As can be seen from the figure, FastHN with Heuristic 1 and/or Heuristic 2 is much faster than HybridNet and Dendroscope 3. This difference in speed becomes more significant as the hybridization number becomes larger. Moreover, Heuristic 1 contributes the most to the saving of running time. Indeed, when Heuristic 1 is used, both Heuristic 2 and initial cluster reductions do not help much. It is worth noting that Beiko and Hamilton’s program tends to create a pair of trees without a relatively large common clusters. This is why initial cluster reductions do not help much for treepairs randomly generated by their program.
Figure 1. FastHNvs. Dendroscope 3 on simulated data.
The comparison is done on 120 randomly generated treepairs with relatively small hybridization numbers, where each tree has 100 leaves. If the running time of a program for a treepair is more than 600 seconds, then it has been rounded down to 600 seconds. Among the 120 pairs, Dendroscope 3 takes more than 600 seconds for 28 pairs, FastHN without Heuristic 1, 2, or initial cluster reductions takes more than 600 seconds for 20 pairs, FastHN with only initial cluster reductions takes more than 600 seconds for 18 pairs, and FastHN with Heuristic 1 or 2 takes less than 60 seconds for every pair.
To test how the number of leaves in an input tree influences the running time of the algorithms, we next use Beiko and Hamilton’s program to generate 120 pairs of trees each of which has 50 leaves. The first (respectively, second) 60 pairs are generated by setting r = 14 (respectively, r = 17). It turns out that among the 120 generated treepairs, 3 (respectively, 17, 26, 26, 20, 20, or 6) treepairs have hybridization number 11 (respectively, 12, 13, 14, 15, 16, or 17). Moreover, for each h∈{9,10}, there is exactly one generated treepair with hybridization number h. Figure 2 summarizes the average running time of the programs for those generated treepairs with hybridization number in the range [12 .. 17], where each average is taken over those treepairs with the same hybridization number. As can be seen from the figure, the superiority of FastHN over HybridNet and Dendroscope 3 remains the same (as in Figure 1) if Heuristic 1 or 2 is used. Moreover, Heuristic 1 contributes the most to the saving of running time.
Figure 2. FastHNvs. Dendroscope 3 on simulated data.
The comparison is done on 120 randomly generated treepairs with relatively small hybridization numbers, where each tree has 50 leaves.
To compare the performance of the algorithms for treepairs with relatively large hybridization numbers, we further use Beiko and Hamilton’s program to generate 60 pairs of trees by setting r = 25, where each tree has 100 leaves. It turns out that among the 60 generated treepairs, 4 (respectively, 10, 15, 18, or 12) treepairs have hybridization number 21 (respectively, 22, 23, 24, or 25). Moreover, there is exactly one generated treepair with hybridization number 20. Figure 3 summarizes the average running time of the two best versions of FastHN for the generated treepairs, where each average is taken over those treepairs with the same hybridization number. As can be seen from the figure, both versions take less than 16 minutes (on average) even when the hybridization number is as large as 25, while FastHN with both Heuristics 1 and 2 is the faster version. In contrast, Dendroscope 3 fails to complete for each of the 60 treepairs.
Figure 3. Average running time ofFastHNon simulated data.
The time is measured on 60 randomly generated treepairs with relatively large hybridization numbers, where each tree has 100 leaves. (Note:Dendroscope 3 fails to complete for each of the 60 pairs.)
Biological data
We use the Poaceae dataset from the Grass Phylogeny Working Group [13]). The dataset contains sequences for six loci: internal transcribed spacer of ribosomal DNA (ITS); NADH dehydrogenase, subunit F (ndhF); phytochrome B (phyB); ribulose 1,5biphosphate carboxylase/oxygenase, large subunit (rbcL); RNA polymerase II, subunit ^{β′} (rpoC2); and granule bound starch synthase I (waxy). The Poaceae dataset was previously analyzed by [14], who generated the inferred rooted binary trees for these loci. See Table 1 for the experimental results. In this table, column pair shows the treepairs, column #taxashows the number of leaves in an input tree, and column h shows the hybridization number of each treepair. Moreover, columns FastHN and Dendroscope show the running times (in seconds) of FastHN and Dendroscope 3, respectively. Furthermore, column FastHN has 8 subcolumns each labeled by 3 bits, where the first (respectively, middle, or the last) bit is 1 if and only if initial cluster reductions (respectively, Heuristic 1, or Heuristic 2) are adopted. In particular, the subcolumn labeled 000 corresponds to HybridNet.
Table 1. FastHN vs. Dendroscope 3 on 15 treepairs in the Poaceae dataset
As can be seen from Table 1, for most of the treepairs, there is not much difference in speed between Dendroscope 3 and FastHN with initial cluster reductions. This is because most of the treepairs have small hybridization numbers. For the treepair (ndhf, ITS), FastHN with cluster reductions runs substantially faster than FastHN without cluster reductions. This is because the problem for this pair can be reduced to two treepairs of roughly equal sizes by initial cluster reductions in the preprocessing stage of the algorithm.
Discussion
Roughly speaking, FastHN consists of two stages, namely, the preprocessing stage and the exhaustivesearch stage. In the preprocessing stage, FastHN performs only subtree reductions and cluster reductions. Indeed, other kinds of reductions are also known. One of them is chain reduction [15]. Performing chain reductions on the input trees results in trees whose nodes are weighted. Unfortunately, it seems that Whidden et al.’s O(2.4^{2d}n)time algorithm for computing the rSPR distance d of two given phylognetic trees with n leaves does not work when the trees are weighted. This is why FastHN does not perform chain reductions.
In the exhaustivesearch stage, FastHN also performs subtree reductions whenever possible, but does not perform cluster reductions. The main reason of not performing cluster reductions in the 2nd stage is that performing a cluster reduction is too timeconsuming (namely, takes O(^{n2}) time, where n is the number of leaves in the trees).
When running FastHN, one can decide whether to adopt initial cluster reductions, Heuristic 1, or Heuristic 2. If two input trees have relatively large common clusters, performing initial cluster reductions on them lead to solving independent and significantly smaller subproblems. So, we should always choose to adopt initial cluster reductions. Moreover, as can be seen from our simulated results, we should always choose to adopt Heuristic 1 because it enables the algorithm to save a lot of time by cutting more branches of the search tree in the exhaustivesearch stage. Our simulated results also show that adopting both Heuristics 1 and 2 makes FastHN run faster (on average) than adopting only Heuristic 1. Thus, in general, we should choose to adopt Heuristic 2 as well. However, in our experiments, we have found some treepairs for which FastHN with Heuristic 1 but without Heuristic 2 runs significantly faster than FastHN with both Heuristics 1 and 2. Hence, as long as Heuristic 1 is adopted, there is still room to decide whether to adopt Heuristic 2 as well.
Conclusions
Our experiments show that FastHN runs substantially faster than the previously best tools (namely, HybridNet and Dendroscope 3). The fast speed of FastHN originates from two key new ideas (which have not been used to solve the problems before, as far as we know):
· We use a fast algorithm for computing the rSPR distance of two given phylognetic trees to cut more branches of the search tree during the exhaustivesearch stage of FastHN.
· During the exhaustivesearch stage of FastHN, we always try to find a pair of sibling leaves in one of the two forests (obtained from the given trees by cutting some edges) such that the two leaves is as far apart as possible in the other forest.
Methods
Throughout this section, a rooted forest always means a directed acyclic graph in which every node has indegree at most 1 and outdegree at most 2.
Let F be a rooted forest. The roots (respectively, leaves) of F are those nodes whose indegrees (respectively, outdegrees) are 0. The size of F, denoted by —F—, is the number of roots in F minus 1. A node v of F is unifurcate if it has only one child in F. If a root v of F is unifurcate, then contracting v in F is the operation that modifies F by deleting v. If a nonroot node v of F is unifurcate, then contracting v in F is the operation that modifies F by first adding an edge from the parent of v to the child of v and then deleting v.
For convenience, we view each node u of F as an ancestor and descendant of u itself. A node u is lower than another node v≠u in F if u is a descendant of v in F. The lowest common ancestor (LCA) of a set U of nodes in F is the lowest node v in F such that for every node u∈U, v is an ancestor of u in F. For a node v of F, the subtree of F rooted at v is the subgraph of F whose nodes are the descendants of v in F and whose edges are those edges connecting two descendants of v in F. If v is a root of F, then the subtree of F rooted at v is a component tree of F. F is a rooted tree if it has only one root.
A rooted binary forest is a rooted forest in which the outdegree of every nonleaf node is 2. Let F be a rooted binary forest. F is a rooted binary tree if it has only one root. If v is a nonroot node of F with parent p and sibling u, then detaching the subtree of F rooted at v is the operation that modifies F by first deleting the edge (p,v) and then contracting p. A detaching operation on F is the operation of detaching the subtree of F rooted at a nonroot node.
Hybridization networks and phylogenetic trees
Let X be a set of existing species. A hybridization network on X is a directed acyclic graph N in which the set of nodes of outdegree 0 (still called the leaves) is X, each nonleaf node has outdegree 2, there is exactly one node of indegree 0 (called the root), and each nonroot node has indegree larger than 0. Note that the indegree of a nonroot node in N may be larger than 1. A node of indegree larger than 1 in N is called a reticulation node of N. Intuitively speaking, a reticulation node corresponds to a reticulation event. The hybridization number of a reticulation node in N is its indegree in N minus one. The hybridization number of N is the total hybridization number of reticulation nodes in N.
A phylogenetic tree on X is a rooted binary tree whose leaf set is X. A hybridization network N on Xdisplays a phylogenetic tree T on X if N has a subgraph M such that M is a rooted tree, the root of M has exactly two children in M, and modifying M by contracting its unifurcate nodes yields T. A hybridization network of two phylogenetic trees T_{1} and T_{2} on X is a hybridization network N on X such that N displays both T_{1} and T_{2}. A hybridization network of T_{1} and T_{2} is minimum if its hybridization number is minimized among all hybridization networks of T_{1} and T_{2}. Obviously, if N is a minimum hybridization network of T_{1} and T_{2}, then the indegree of every reticulation node in N is exactly 2 and hence the hybridization number of N is equal to the number of reticulation nodes in N. For convenience, we define the hybridization number of T_{1} and T_{2} to be the minimum hybridization number of a hybridization network of T_{1} and T_{2}.
We are now ready to define one problem studied in this paper:
Hybridization Network Construction (HNC):
· Input: Two phylogenetic trees T_{1} and T_{2} on the same set X of species.
· Goal: To construct a minimum hybridization network of T_{1} and T_{2}.
Agreement forests
Throughout this subsection, let T_{1} and T_{2} be two phylogenetic trees on the same set X of species. If we can apply a sequence of detaching operations on each of T_{1} and T_{2} so that they become the same forest F, then we refer to F as an agreement forest (AF) of T_{1} and T_{2}. A maximum agreement forest (MAF) of T_{1} and T_{2} is an agreement forest of T_{1} and T_{2} whose size is minimized over all agreement forests of T_{1} and T_{2}. The size of an MAF of T_{1} and T_{2} is called the rSPR distance between T_{1} and T_{2}. The following lemma is shown in [16].
Lemma 1
[16] Given two phylogenetic trees T_{1} and T_{2}, we can compute the rSPR distance between T_{1} and T_{2} in O(2.4^{2d}n) time, where n is the number of leaves in T_{1} and T_{2} and d is the rSPR distance between T_{1} and T_{2}.
Let F be an agreement forest of T_{1} and T_{2}. Obviously, for each i∈{1,2}, the leaves of T_{i} onetoone correspond to the leaves of F. For convenience, we hereafter identify each leaf v of F with the leaf of T_{i} corresponding to v. Similarly, for each i∈{1,2}, the nonleaf nodes of F correspond to distinct nonleaf nodes of T_{i}. More precisely, a nonleaf node u of F corresponds to the LCA of {_{v1},…,_{vℓ}} in T_{i}, where v_{1}, …, _{vℓ}are the leaf descendants of u in F. Again for convenience, we hereafter identify each nonleaf node u of F with the nonleaf node of T_{i} corresponding to u. With these correspondences, we can use F, T_{1}, and T_{2} to construct a directed graph G_{F} as follows:
· The nodes of G_{F} are the roots of F.
· For every two roots _{r1}and _{r2}of F, there is an edge from _{r1}to _{r2}in G_{F} if and only if _{r1}is an ancestor of _{r2}in T_{1} or T_{2}.
We refer to G_{F} as the decision graph associated withF. If G_{F} is acyclic, then F is an acyclic agreement forest (AAF) of T_{1} and T_{2}; otherwise, F is a cyclic agreement forest (CAF) of T_{1} and T_{2}. If F is an AAF of T_{1} and T_{2} and its size is minimized over all AAFs of T_{1} and T_{2}, then F is a maximum acyclic agreement forest (MAAF) of T_{1} and T_{2}. Note that our definition of an AAF is the same as those in [15,17] but is different from that in [16]. Moreover, it is known that the size of an MAAF of T_{1} and T_{2} is equal to the hybridization number of T_{1} and T_{2}[18]. The following lemma is shown in [19]:
Lemma 2
[19] Suppose that C is a cycle of G_{F} and r_{1}, …, _{rℓ}are the nodes of C. Then, each _{rj}∈{_{r1},…,_{rℓ}} has two children u_{j} and u’_{j} in F. Moreover, for every nonroot node v of F not contained in {_{u1}
Let N be a minimum hybridization network of T_{1} and T_{2}. Suppose that we modify N to obtain a forest F(N) by first removing all edges entering reticulation nodes, then removing those nodes v such that neither v nor its descendants are in X, and further contracting all unifurcate nodes. Obviously, F(N) is an AAF of T_{1} and T_{2} and the size of F(N) is exactly the hybridization number of N. So, each MAAF F’ of T_{1} and T_{2}represents the set of all minimum hybridization networks N such that F(N) is the same as F’. Thus, to enumerate a representative set of minimum hybridization networks of T_{1} and T_{2}, the idea in previous work [6] has been to enumerate all MAAFs of T_{1} and T_{2} and construct a minimum hybridization network for each enumerated MAAF. Since we can easily use an MAAF of T_{1} and T_{2} to construct a hybridization network displaying T_{1} and T_{2}[6], the difficulty is in how to enumerate all MAAFs of T_{1} and T_{2}.
We are now ready to define another problem studied in this paper:
Hybridization Network Enumeration (HNE):
· Input: Two phylogenetic trees T_{1} and T_{2} on the same set X of species.
· Input: Two phylogenetic trees T_{1} and T_{2} on the same set X of species.
· Goal: To enumerate all MAAFs of T_{1} and T_{2} and construct a minimum hybridization network of T_{1} and T_{2} from each MAAF of T_{1} and T_{2}.
Basically, HNE is the problem of enumerating a representative set of minimum hybridization networks of two given phylogenetic trees. As in previous studies [5,6,8], when we consider HNC and HNE, we always assume that each given phylogenetic tree has been modified by first introducing a new root and a dummy leaf and then letting the old root and the dummy leaf be the children of the new root.
The following lemma is shown in [19]:
Lemma 3
[19] The dummy leaf alone does not form a component tree of an MAAF of T_{1} and T_{2}.
Extending Whidden et al.’s Algorithm
Throughout this subsection, let T_{1} and T_{2} be two phylogenetic trees on the same set X of species. We sketch the fastest known algorithm (due to Whidden et al.[16]) for computing an MAF of T_{1} and T_{2}, and then state a slight extension of the algorithm that will be used in our algoirthm for HNE.
The basic idea behind Whidden et al.’s algorithm is as follows. For k = 0, 1, 2, …(in this order), we try to find an AF of T_{1} and T_{2} of size k and stop immediately once such an AF is found. To find an AF of T_{1} and T_{2} of size k, we start by setting _{F1}=_{T1} and _{F2}=_{T2} and associating a label set{x} to each leaf x of F_{1} and F_{2}. We then repeatedly modify F_{1} and F_{2} (until either _{F1}>k or F_{1} becomes a forest without edges) as follows. We find two arbitrary sibling leaves u and v in F_{2}. If u and v are also siblings in F_{1}, then we modify F_{1} and F_{2} separately by merging the identical subtrees of F_{1} and F_{2} rooted at the parent of u and v each into a single leaf whose label set is the union of the label sets of u and v. On the other hand, if u and v are not siblings in F_{1}, then we distinguish three cases as follows. Case 1:u and v are in different component trees of F_{1}. In this case, in order to transform F_{1} and F_{2} into an AF of T_{1} and T_{2}, we have two choices to modify them, namely, by either detaching the subtree rooted at u or detaching the subtree rooted at v. Case 2:u and v are in the same component tree of F_{1} and either (1) u and the parent of v are siblings in F_{1} or (2) v and the parent of u are siblings in F_{1}. In this case, if (1) (respectively, (2)) holds, then we modify F_{1} by detaching the subtree rooted at the sibling of v (respectively, u). Case 3:u and v are in the same component tree of F_{1} and neither (1) nor (2) in Case 2 holds. In this case, in order to transform F_{1} and F_{2} into an AF of T_{1} and T_{2}, we have three choices to modify them. The first two choices are the same as those in Case 1. In the third choice, we modify F_{1} by detaching the subtrees rooted at those nonroot nodes w such that the parent of w appears on the (not necessarily directed) path between u and v in F_{1} but w does not.
By the above three cases, we always have the following:
· _{F1}≥_{F2}.
· All component trees of F_{2} except at most one have no edges.
· For each component tree _{Γ2}of F_{2} without edges, F_{1} has a component tree _{Γ1}without edges such that the label sets associated with the unique leaves of _{Γ1}and _{Γ2}are identical.
Once _{F1} becomes larger than k, we know that F_{1} and F_{2} have no AF of size k. On the other hand, once F_{1} becomes a forest without edges, we can use the label sets L(v) of the leaves v of F_{1} to obtain an AF of T_{1} and T_{2} of size _{F1} by modifying T_{1} as follows. For each leaf v of F_{1} such that L(v) does not contain the dummy leaf, detach the subtree of T_{1} rooted at the LCA of the leaves in L(v).
Now, we are now ready to make a key observation in this paper. By (b) and (c) in the above, Whidden et al.’s MAF algorithm can actually be used to solve the following slightly more general problem in O(2.4^{2k}n) time:
rSPR Distance Checking (rSPRDC):
· Input:(_{T1},_{T2},k,_{F1},_{F2}), where T_{1} and T_{2} are two phylogenetic trees on the same set X of species, k is an integer, F_{1} (respectively, F_{2}) is a rooted forest obtained from T_{1} (respectively, T_{2}) by performing zero or more detaching operations, and every component tree of F_{2} except at most one is identical to a component tree of F_{1}.
· Goal: To decide if performing k more detaching operations on F_{1} leads to an AF of T_{1} and T_{2}.
Finally, if we want to enumerate all MAFs of T_{1} and T_{2}, then we need to modify Whidden et al.’s algorithm as follows. First, we do not distinguish Cases 2 and 3 because modifying F_{1} as in Case 2 may lose some MAF of T_{1} and T_{2}. Moreover, whenever an AF of T_{1} and T_{2} of size k is found, we do not stop immediately and instead continue to find other AFs of T_{1} and T_{2} of size k. The resulting algorithm runs more slowly, namely, in O(^{3k}n) time.
Speeding up HybridNet
Throughout this subsection, let T_{1} and T_{2} be two phylogenetic trees on the same set X of species. We first sketch how HybridNet enumerates all MAAFs of T_{1} and T_{2}, and then explain how to speed it up.
First, we need several definitions. For a rooted forest F, we use (F) to denote the family of the leaf sets of the component trees of F. Let F and F’ be two forests each obtained by performing zero or more detaching operations on T_{1}. If F≠^{F″}and for every set
To enumerate all MAAFs of T_{1} and T_{2}, the idea behind HybridNet is to design an algorithm for the following problem:
Generalized Agreement Forest (GAF)
· Input:(_{T1},_{T2},k,_{F1}), where T_{1} and T_{2} are two phylogenetic trees on the same set X of species, k is an integer, and F_{1} is a rooted forest obtained from T_{1} by performing zero or more detaching operations.
· Goal: To find a sequence of AFs of T_{1} and T_{2} including all AFs F of T_{1} and T_{2} such that (1) F can be obtained by performing at most k detaching operations on F_{1} (or equivalently, at most _{F1} + kdetaching operations on T_{2}) and (2) no AF of T_{1} and T_{2} is finer than F_{1} and coarser than F.
In the supplementary material of [19], an O(^{3k}n)time algorithm for solving GAF is detailed. The algorithm differs from Whidden et al.’s algorithm for enumerating all MAFs of T_{1} and T_{2} only in that we start with F_{1} (as it is given) and _{F2}=_{T2}(instead of starting with _{F1}=_{T1}and _{F2}=_{T2}) and then repeatedly modify F_{1} and F_{2} until either _{F1}>k + _{k0}or F_{1} becomes a forest without edges, where k_{0} is the original size of F_{1}. Now, we are now ready to make two other key observations in this paper. To speed up Chen and Wang’s algorithm for solving GAF, we modify it as follows:
· Heuristic 1: Every time before we start to make multiple choices of modifying F_{1} and F_{2}, we call the algorithm for rSPRDC in Lemma 1 on input (_{T1},_{T2},k−_{F1} + _{k0},_{F1},_{F2}) to check if performing k−_{F1} + _{k0}more detaching operations on F_{1} leads to an AF of T_{1} and T_{2}.
As the result, if we know that performing k−_{F1} + _{k0}more detaching operations on F_{1} does not lead to an AF of T_{1} and T_{2}, then no more choice of modifying F_{1} and F_{2} is necessary; otherwise, we proceed to make multiple choices of modifying F_{1} and F_{2} the same as before but with the following difference:
· Heuristic 2: Instead of selecting two arbitrary sibling leaves u and v in F_{2} (cf. the Extending Whidden et al.’s Algorithm section), we select two sibling leaves u and v in F_{2} such that they are as far apart as possible in F_{1}.
The intuition behind Heuristic 2 is that if u and v are far apart in F_{1}, then either u and v fall into two different connected components of F_{1} so that we do not have to try Case 3 in the Extending Whidden et al.’s Algorithm section, or u and v fall into the same connected component of F_{1} and we can detach a lot of subtrees from F_{1} in Case 3.
Finally, to enumerate all MAAFs of T_{1} and T_{2}, we initialize k = 0 and then proceed as follows.
1. Simulate the spedup algorithm for GAF on input (_{T1},_{T2},k,_{T1}). During the simulation, whenever an AF F of T_{1} and T_{2} is enumerated, perform one of the following steps depending on whether F is acyclic or not:
(a) If F is acyclic, output it.
(b) If F is cyclic, then output all AAFs F’ of T_{1} and T_{2} such that F’ can be obtained from F by performing k−F detaching operations on F.
2. If at least one AAF of T_{1} and T_{2} was outputted in Step 11a or 11b, then stop; otherwise, increase k by 1 and go to Step 1.
Note that Step 11b is nontrivial. As described in the supplementary material of [19], Lemma 2 is very helpful for this purpose. More specifically, we first find a cycle C in _{GF″} in O(^{F″}^{2}) time. By Lemma 2, in order to make F’ acyclic, we have to choose one node r of C and modify F’ by detaching the subtree of F’ rooted at an (arbitrary) child of r. Note that since r is a root of F’, detaching the subtree of F’ rooted at a child of r is achieved by simply deleting r from F’ and is hence independent of the choice of the child. Moreover, if the parent r’ of the dummy leaf in F’ is a node of C, then by Lemma 3, we can exclude r’ from consideration when choosing r. So, we have at most ^{F″}≤k−1 ways to break C. After modifying F’ in this way, we again construct G_{F’} and test if it is acyclic. If it is acyclic, then we can output F’; otherwise, we again find a cycle C in G_{F’} and use it to modify F’ as before. We repeat modifying F’ in this way, until either F’ becomes acyclic, or ^{F″}=kand G_{F’} is still cyclic. Once F’ becomes acyclic, we output it. The total time taken by Step 11b is O(^{k2}^{(k−1)k−F″}), because we make a total number of at most O(^{(k−1)k−F″}) choices for breaking cycles.
Experiments show that Heuristics 1 and 2 help us speed up the algorithm substantially. However, the two heuristics may not help in the worst case. That is, we are unable to prove that the two heuristics improve the worstcase time complexity of the algorithm which is O(^{3d}X + ^{3d}^{(k−1)k−d + 2}) (as shown in [19]), where d is the size of an MAF of T_{1} and T_{2}. We note that k and d are usually quite close.
The new algorithm for HNE
In this subsection, we only design an algorithm for HNE. Note that it is trivial to obtain a faster algorithm for HNC by modifying the algorithm for HNE so that it stops immediately once an MAAF is found.
Throughout this subsection, let T_{1} and T_{2} be two phylogenetic trees on the same set X of species. As mentioned before, we can easily use an MAAF of T_{1} and T_{2} to construct a hybridization network displaying T_{1} and T_{2}[6]. So, we only explain how to enumerate all MAAFs of T_{1} and T_{2}.
In the last subsection, we have explained how to speed up HybridNet so that it can enumerate all MAAFs of T_{1} and T_{2} within shorter time. Indeed, we can make HybridNet even faster. The idea is to preprocess T_{1} and T_{2} so that the given trees become smaller or the problem becomes to solve two or more smaller independent subproblems. More specifically, we perform the following two reductions on T_{1} and T_{2} until neither of them is available.
Subtree reduction
Suppose that T_{1} has a nonleaf node v_{1} and T_{2} has a nonleaf node v_{2} such that the subtree of T_{1} rooted at v_{1} is identical to the subtree of T_{2} rooted at v_{2}. Then, we modify T_{1} (respectively, T_{2}) by merging the subtree of T_{1} (respectively, T_{2}) rooted at v_{1} (respectively, v_{2}) into a single leaf whose label set is the union of the label sets of the merged leaves. It is known [2] that this reduction preserves the MAAFs of T_{1} and T_{2}.
Cluster reduction
Suppose that subtree reductions on T_{1} and T_{2} are not available but T_{1} has a nonleaf node T_{1} and T_{2} has a nonleaf node T_{2} such that the subtree of T_{1} rooted at T_{1} has the same leaf set as the subtree of T_{2} rooted at T_{2}. Then, we split T_{1} (respectively, T_{2}) into two trees T’_{1} and T”_{1} (respectively, T’_{2} and T”_{2}) as follows. T’_{1} (respectively, T’_{2}) is simply the subtree of T_{1} (respectively, T_{2}) rooted at T_{1} (respectively, T_{2}), while T”_{1} (respectively, T”_{2}) is obtained by merging the subtree T_{1} (respectively, T_{2}) rooted at T_{1} (respectively, T_{2}) into a single leaf whose label set is the union of the label sets of the merged leaves. It is known [20] that the set of MAAFs of T_{1} and T_{2} is the Cartesian product of the set of MAAFs of T’_{1} and T’_{2} and the set of MAAFs of T”_{1} and T”_{2}.
After the preprocessing stage, if no cluster reduction has been performed in the preprocessing stage, then we run the spedup HybridNet (as described in the last subsection) on T_{1} and T_{2}; otherwise, we have obtained two or more subproblems. Suppose that we have h subproblems and the ith subproblem (1≤i≤h) is to enumerate all MAAFs of two trees _{T1,i}and _{T2,i}. Then, for each 1≤i≤h, we run the spedup HybridNet to enumerate the set _{i}of MAAFs of _{T1,i} and _{T2,i}. Finally, we output the Cartesan product _{1}×⋯×_{h}.
Competing interests
The authors declare that they have no competing interests.
Author’s contributions
ZZC was in charge of algorithm design, algorithm implementation, design of experiments, and manuscript preparation. LW participated in algorithm design and manuscript editing. SY was involved in algorithm implementation and testing. All authors read and approved the final manuscript.
Acknowledgements
We would like to thank the reviewers for their valuable suggestions and comments. ZZC was supported in part by the GrantinAid for Scientific Research of the Ministry of Education, Science, Sports and Culture of Japan, under Grant No. 24500023. LW is supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No. CityU 121608].
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