A fast tool for minimum hybridization networks

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 exhaustive-search stage of the algorithm. The third idea is that during the exhaustive-search 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 single-threaded program. Despite this disadvantage, our experimental data shows that FastHN runs substantially faster than the multi-threaded Dendroscope 3 on a PC with multiple cores. Indeed, FastHN can finish within 16 minutes (on average on a Windows-7 (x64) desktop PC with i7-2600 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 in-degree 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 tree-like 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 NP-hard [1–3], 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 [4–9]. 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 T₁ and T₂ with the same leaf set X. In some cases, we may want to enumerate all minimum hybridization networks displaying both T₁ and T₂. Unfortunately, it is not hard to construct two example phylogenetic trees T₁ and T₂ such that there are too many minimum hybridization networks displaying both T₁ and T₂. So, we instead want to enumerate only a representative set of minimum hybridization networks displaying both T₁ and T₂. Here, a hybridization network N represents another hybridization network N’ if for every pair (x,y) of species in X x 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 so-called “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, or http://www.cs.cityu.edu.hk/~lwang/software/FastHN/fastHN.html, 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 3-bit 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 Windows-7 (x64) desktop PC with i7-2600 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 single-threaded, its total elapsed time is usually more than its CPU time. In contrast, Dendroscope 3 is multi-threaded, 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 out-degree 1 resulting from the removal of edge (p,c)), and further re-attaches 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 tree-pairs, 6 (respectively, 22, 33, 11, 21, or 27) tree-pairs have hybridization number 12 (respectively, 13, 14, 15, 16, or 17). Figure 1 summarizes the average running time of the programs for the generated tree-pairs, where each average is taken over those tree-pairs 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 tree-pairs randomly generated by their program.

Figure 1

FastHN vs. Dendroscope 3 on simulated data.

The comparison is done on 120 randomly generated tree-pairs with relatively small hybridization numbers, where each tree has 100 leaves. If the running time of a program for a tree-pair 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 tree-pairs, 3 (respectively, 17, 26, 26, 20, 20, or 6) tree-pairs have hybridization number 11 (respectively, 12, 13, 14, 15, 16, or 17). Moreover, for each h∈{9,10}, there is exactly one generated tree-pair with hybridization number h. Figure 2 summarizes the average running time of the programs for those generated tree-pairs with hybridization number in the range [12 .. 17], where each average is taken over those tree-pairs 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

FastHN vs. Dendroscope 3 on simulated data.

The comparison is done on 120 randomly generated tree-pairs with relatively small hybridization numbers, where each tree has 50 leaves.

To compare the performance of the algorithms for tree-pairs 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 tree-pairs, 4 (respectively, 10, 15, 18, or 12) tree-pairs have hybridization number 21 (respectively, 22, 23, 24, or 25). Moreover, there is exactly one generated tree-pair with hybridization number 20. Figure 3 summarizes the average running time of the two best versions of FastHN for the generated tree-pairs, where each average is taken over those tree-pairs 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 tree-pairs.

Figure 3

Average running time of FastHN on simulated data.

The time is measured on 60 randomly generated tree-pairs 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 1,5-biphosphate 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 tree-pairs, column #taxa shows the number of leaves in an input tree, and column h shows the hybridization number of each tree-pair. Moreover, columns FastHN and Dendroscope show the running times (in seconds) of FastHN and Dendroscope 3, respectively. Furthermore, column FastHN has 8 sub-columns 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 sub-column labeled 000 corresponds to HybridNet.

Table 1 FastHN vs. Dendroscope 3 on 15 tree-pairs in the Poaceae dataset

As can be seen from Table 1, for most of the tree-pairs, there is not much difference in speed between Dendroscope 3 and FastHN with initial cluster reductions. This is because most of the tree-pairs have small hybridization numbers. For the tree-pair (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 tree-pairs 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 exhaustive-search 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(2.4^2dn)-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 exhaustive-search 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 time-consuming (namely, takes O(^{n 2}) 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 exhaustive-search 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 tree-pairs 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 exhaustive-search stage of FastHN.

· During the exhaustive-search 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 in-degree at most 1 and out-degree at most 2.

Let F be a rooted forest. The roots (respectively, leaves) of F are those nodes whose in-degrees (respectively, out-degrees) 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 non-root 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 out-degree of every non-leaf 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 non-root 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 non-root 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 out-degree 0 (still called the leaves) is X, each non-leaf node has out-degree 2, there is exactly one node of in-degree 0 (called the root), and each non-root node has in-degree larger than 0. Note that the in-degree of a non-root node in N may be larger than 1. A node of in-degree 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 in-degree 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 X displays 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 T₁ and T₂ on X is a hybridization network N on X such that N displays both T₁ and T₂. A hybridization network of T₁ and T₂ is minimum if its hybridization number is minimized among all hybridization networks of T₁ and T₂. Obviously, if N is a minimum hybridization network of T₁ and T₂, then the in-degree 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 T₁ and T₂ to be the minimum hybridization number of a hybridization network of T₁ and T₂.

We are now ready to define one problem studied in this paper:

Hybridization Network Construction (HNC):

· Input: Two phylogenetic trees T₁ and T₂ on the same set X of species.

· Goal: To construct a minimum hybridization network of T₁ and T₂.

Agreement forests

Throughout this subsection, let T₁ and T₂ be two phylogenetic trees on the same set X of species. If we can apply a sequence of detaching operations on each of T₁ and T₂ so that they become the same forest F, then we refer to F as an agreement forest (AF) of T₁ and T₂. A maximum agreement forest (MAF) of T₁ and T₂ is an agreement forest of T₁ and T₂ whose size is minimized over all agreement forests of T₁ and T₂. The size of an MAF of T₁ and T₂ is called the rSPR distance between T₁ and T₂. The following lemma is shown in [16].

Lemma 1

[16] Given two phylogenetic trees T₁ and T₂, we can compute the rSPR distance between T₁ and T₂ in O(2.4^2dn) time, where n is the number of leaves in T₁ and T₂ and d is the rSPR distance between T₁ and T₂.

Let F be an agreement forest of T₁ and T₂. Obviously, for each i∈{1,2}, the leaves of T _i one-to-one 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 non-leaf nodes of F correspond to distinct non-leaf nodes of T _i . More precisely, a non-leaf node u of F corresponds to the LCA of {_{v 1},…, _{v ℓ} } in T _i , where v₁, …, _{v ℓ} are the leaf descendants of u in F. Again for convenience, we hereafter identify each non-leaf node u of F with the non-leaf node of T _i corresponding to u. With these correspondences, we can use F, T₁, and T₂ to construct a directed graph G _F as follows:

· The nodes of G _F are the roots of F.

· For every two roots _{r 1}and _{r 2}of F, there is an edge from _{r 1}to _{r 2}in G _F if and only if _{r 1}is an ancestor of _{r 2}in T₁ or T₂.

We refer to G _F as the decision graph associated with F. If G _F is acyclic, then F is an acyclic agreement forest (AAF) of T₁ and T₂; otherwise, F is a cyclic agreement forest (CAF) of T₁ and T₂. If F is an AAF of T₁ and T₂ and its size is minimized over all AAFs of T₁ and T₂, then F is a maximum acyclic agreement forest (MAAF) of T₁ and T₂. 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 T₁ and T₂ is equal to the hybridization number of T₁ and T₂[18]. The following lemma is shown in [19]:

Lemma 2

[19] Suppose that C is a cycle of G _F and r₁, …, _{r ℓ} are the nodes of C. Then, each _{r j} ∈{_{r 1},…, _{r ℓ} } has two children u _j and u’ _j in F. Moreover, for every non-root node v of F not contained in {_{u 1}u_1^{\prime },…, _{u ℓ} u_{\ell }^{\prime }}, C remains a cycle in G _F after F is modified by detaching the subtree of F rooted at v.

Let N be a minimum hybridization network of T₁ and T₂. 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 T₁ and T₂ and the size of F(N) is exactly the hybridization number of N. So, each MAAF F’ of T₁ and T₂represents 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 T₁ and T₂, the idea in previous work [6] has been to enumerate all MAAFs of T₁ and T₂ and construct a minimum hybridization network for each enumerated MAAF. Since we can easily use an MAAF of T₁ and T₂ to construct a hybridization network displaying T₁ and T₂[6], the difficulty is in how to enumerate all MAAFs of T₁ and T₂.

We are now ready to define another problem studied in this paper:

Hybridization Network Enumeration (HNE):

· Input: Two phylogenetic trees T₁ and T₂ on the same set X of species.

· Input: Two phylogenetic trees T₁ and T₂ on the same set X of species.

· Goal: To enumerate all MAAFs of T₁ and T₂ and construct a minimum hybridization network of T₁ and T₂ from each MAAF of T₁ and T₂.

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 T₁ and T₂.

Extending Whidden et al.’s Algorithm

Throughout this subsection, let T₁ and T₂ 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 T₁ and T₂, 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 T₁ and T₂ of size k and stop immediately once such an AF is found. To find an AF of T₁ and T₂ of size k, we start by setting _{F 1}=_{T 1} and _{F 2}=_{T 2} and associating a label set{x} to each leaf x of F₁ and F₂. We then repeatedly modify F₁ and F₂ (until either |_{F 1}|>k or F₁ becomes a forest without edges) as follows. We find two arbitrary sibling leaves u and v in F₂. If u and v are also siblings in F₁, then we modify F₁ and F₂ separately by merging the identical subtrees of F₁ and F₂ 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 F₁, then we distinguish three cases as follows. Case 1: u and v are in different component trees of F₁. In this case, in order to transform F₁ and F₂ into an AF of T₁ and T₂, 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 F₁ and either (1) u and the parent of v are siblings in F₁ or (2) v and the parent of u are siblings in F₁. In this case, if (1) (respectively, (2)) holds, then we modify F₁ by detaching the subtree rooted at the sibling of v (respectively, u). Case 3: u and v are in the same component tree of F₁ and neither (1) nor (2) in Case 2 holds. In this case, in order to transform F₁ and F₂ into an AF of T₁ and T₂, 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 F₁ by detaching the subtrees rooted at those non-root nodes w such that the parent of w appears on the (not necessarily directed) path between u and v in F₁ but w does not.

By the above three cases, we always have the following:

· |_{F 1}|≥|_{F 2}|.

· All component trees of F₂ except at most one have no edges.

· For each component tree _{Γ 2}of F₂ without edges, F₁ has a component tree _{Γ 1}without edges such that the label sets associated with the unique leaves of _{Γ 1}and _{Γ 2}are identical.

Once |_{F 1}| becomes larger than k, we know that F₁ and F₂ have no AF of size k. On the other hand, once F₁ becomes a forest without edges, we can use the label sets L(v) of the leaves v of F₁ to obtain an AF of T₁ and T₂ of size |_{F 1}| by modifying T₁ as follows. For each leaf v of F₁ such that L(v) does not contain the dummy leaf, detach the subtree of T₁ 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(2.4^2kn) time:

rSPR Distance Checking (rSPRDC):

· Input:(_{T 1},_{T 2},k,_{F 1},_{F 2}), where T₁ and T₂ are two phylogenetic trees on the same set X of species, k is an integer, F₁ (respectively, F₂) is a rooted forest obtained from T₁ (respectively, T₂) by performing zero or more detaching operations, and every component tree of F₂ except at most one is identical to a component tree of F₁.

· Goal: To decide if performing k more detaching operations on F₁ leads to an AF of T₁ and T₂.

Finally, if we want to enumerate all MAFs of T₁ and T₂, then we need to modify Whidden et al.’s algorithm as follows. First, we do not distinguish Cases 2 and 3 because modifying F₁ as in Case 2 may lose some MAF of T₁ and T₂. Moreover, whenever an AF of T₁ and T₂ of size k is found, we do not stop immediately and instead continue to find other AFs of T₁ and T₂ of size k. The resulting algorithm runs more slowly, namely, in O(^3kn) time.

Speeding up HybridNet

Throughout this subsection, let T₁ and T₂ be two phylogenetic trees on the same set X of species. We first sketch how HybridNet enumerates all MAAFs of T₁ and T₂, 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 T₁. If F≠^F″and for every set Y\in \mathcal{ℒ}\left(F\right), there is a set Y^{″}\in \mathcal{ℒ}\left(F^{″}\right) with Y\subseteq Y^{″}, then we say that F is finer than F’ and F’ is coarser than F.

To enumerate all MAAFs of T₁ and T₂, the idea behind HybridNet is to design an algorithm for the following problem:

Generalized Agreement Forest (GAF)

· Input:(_{T 1},_{T 2},k,_{F 1}), where T₁ and T₂ are two phylogenetic trees on the same set X of species, k is an integer, and F₁ is a rooted forest obtained from T₁ by performing zero or more detaching operations.

· Goal: To find a sequence of AFs of T₁ and T₂ including all AFs F of T₁ and T₂ such that (1) F can be obtained by performing at most k detaching operations on F₁ (or equivalently, at most |_{F 1}| + k detaching operations on T₂) and (2) no AF of T₁ and T₂ is finer than F₁ and coarser than F.

In the supplementary material of [19], an O(^3kn)-time algorithm for solving GAF is detailed. The algorithm differs from Whidden et al.’s algorithm for enumerating all MAFs of T₁ and T₂ only in that we start with F₁ (as it is given) and _{F 2}=_{T 2}(instead of starting with _{F 1}=_{T 1}and _{F 2}=_{T 2}) and then repeatedly modify F₁ and F₂ until either |_{F 1}|>k + _{k 0}or F₁ becomes a forest without edges, where k₀ is the original size of F₁. 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 F₁ and F₂, we call the algorithm for rSPRDC in Lemma 1 on input (_{T 1},_{T 2},k−|_{F 1}| + _{k 0},_{F 1},_{F 2}) to check if performing k−|_{F 1}| + _{k 0}more detaching operations on F₁ leads to an AF of T₁ and T₂.

As the result, if we know that performing k−|_{F 1}| + _{k 0}more detaching operations on F₁ does not lead to an AF of T₁ and T₂, then no more choice of modifying F₁ and F₂ is necessary; otherwise, we proceed to make multiple choices of modifying F₁ and F₂ the same as before but with the following difference:

· Heuristic 2: Instead of selecting two arbitrary sibling leaves u and v in F₂ (cf. the Extending Whidden et al.’s Algorithm section), we select two sibling leaves u and v in F₂ such that they are as far apart as possible in F₁.

The intuition behind Heuristic 2 is that if u and v are far apart in F₁, then either u and v fall into two different connected components of F₁ 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 F₁ and we can detach a lot of subtrees from F₁ in Case 3.

Finally, to enumerate all MAAFs of T₁ and T₂, we initialize k = 0 and then proceed as follows.

• 1. Simulate the sped-up algorithm for GAF on input (_{T 1},_{T 2},k,_{T 1}). During the simulation, whenever an AF F of T₁ and T₂ 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 T₁ and T₂ such that F’ can be obtained from F by performing k−|F| detaching operations on F.

• 2. If at least one AAF of T₁ and T₂ 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 G_F″ 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″|=k and G _F’ is still cyclic. Once F’ becomes acyclic, we output it. The total time taken by Step 11b is O(²(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 worst-case 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 T₁ and T₂. 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 T₁ and T₂ be two phylogenetic trees on the same set X of species. As mentioned before, we can easily use an MAAF of T₁ and T₂ to construct a hybridization network displaying T₁ and T₂[6]. So, we only explain how to enumerate all MAAFs of T₁ and T₂.

In the last subsection, we have explained how to speed up HybridNet so that it can enumerate all MAAFs of T₁ and T₂ within shorter time. Indeed, we can make HybridNet even faster. The idea is to preprocess T₁ and T₂ 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 T₁ and T₂ until neither of them is available.

Subtree reduction

Suppose that T₁ has a non-leaf node v₁ and T₂ has a non-leaf node v₂ such that the subtree of T₁ rooted at v₁ is identical to the subtree of T₂ rooted at v₂. Then, we modify T₁ (respectively, T₂) by merging the subtree of T₁ (respectively, T₂) rooted at v₁ (respectively, v₂) 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 T₁ and T₂.

Cluster reduction

Suppose that subtree reductions on T₁ and T₂ are not available but T₁ has a non-leaf node T₁ and T₂ has a non-leaf node T₂ such that the subtree of T₁ rooted at T₁ has the same leaf set as the subtree of T₂ rooted at T₂. Then, we split T₁ (respectively, T₂) 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 T₁ (respectively, T₂) rooted at T₁ (respectively, T₂), while T”₁ (respectively, T”₂) is obtained by merging the subtree T₁ (respectively, T₂) rooted at T₁ (respectively, T₂) 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 T₁ and T₂ 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 sped-up HybridNet (as described in the last subsection) on T₁ and T₂; otherwise, we have obtained two or more subproblems. Suppose that we have h subproblems and the i th subproblem (1≤i≤h) is to enumerate all MAAFs of two trees _{T 1,i}and _{T 2,i}. Then, for each 1≤i≤h, we run the sped-up HybridNet to enumerate the set _i of MAAFs of _{T 1,i} and _{T 2,i}. Finally, we output the Cartesan product ₁×⋯× _h .

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.