On Hill et al's conjecture for calculating the subtree prune and regraft distance between phylogenies.

BACKGROUND: Recently, Hill et al. 1 implemented a new software package--called SPRIT--which aims at calculating the minimum number of horizontal gene transfer events that is needed to simultaneously explain the evolution of two rooted binary phylogenetic trees on the same set of taxa. To this end, SPRIT computes the closely related so-called rooted subtree prune and regraft distance between two phylogenies. However, calculating this distance is an NP-hard problem and exact algorithms are often only applicable to small- or medium-sized problem instances. Trying to overcome this problem, Hill et al. propose a divide-and-conquer approach to speed up their algorithm and conjecture that this approach can be used to compute the rooted subtree prune and regraft distance exactly.
RESULTS: In this note, we present a counterexample to Hill et al's conjecture and subsequently show that a modified version of their conjecture holds.
CONCLUSION: While Hill et al's conjecture may result in an overestimate of the rooted subtree prune and regraft distance, a slightly more restricted version of their approach gives the desired outcome and can be applied to speed up the exact calculation of this distance between two phylogenies.

Background

In recent years, one of the main research foci in the development of theoretical frameworks that aim at approaching questions in evolutionary biology turns from the reconstruction of phylogenetic trees towards the reconstruction of phylogenetic networks. This has partly been triggered by the exponentially growing amount of available sequence data arising from whole genome sequencing projects and a successive detection of genes whose sequences are chimeras of distinct ancestral gene sequences, and hence, are likely to be the result of reticulation (e.g. horizontal gene transfer or hybridization). Although evolutionary biologists are now mostly acknowledging the existence of species arising from reticulation within certain groups of organisms, the extent to which such events have influenced the evolutionary history for a set of present-day species remains controversially discussed until today. To shed light on this question, Hill et al. [1] recently published a study that is centered around the identification and quantification of horizontal gene transfer. The authors have implemented a new software package--called SPRIT--consisting of a heuristic as well as an exact algorithm, applied it to several data sets of variable size, and compared their results and running times with those obtained from other algorithms that have previously been developed to analyze reticulate evolution.

Algorithmically, SPRIT draws on ideas that are borrowed from work that has been done in the context of the graph-theoretic operation of rooted subtree prune and regraft (rSPR) which is a popular tool to quantify the dissimilarity between two trees. Loosely speaking, an rSPR operation cuts (prunes) a subtree and reattaches (regrafts) it to another part of the tree. A lower bound on the number of reticulation events that is needed to simultaneously explain two phylogenies is the minimum number of rSPR operations that transform one phylogeny into the other [2,3]. This minimum number, which is computed by SPRIT, is referred to as the rSPR distance. However, since the task of calculating this distance is an NP-hard optimization problem, the application of exact algorithms is often restricted to medium-sized data sets.

In trying to overcome this obstacle, thus to speed up SPRIT, Hill et al. propose a divide-and-conquer-type reduction that breaks the problem into several smaller and more tractable subproblems before calculating the rSPR distance for each subproblem separately. Briefly, the authors conjecture that the sum of rSPR distances over all smaller subproblems is equal to the rSPR distance of the original unreduced trees. In this note, we give a counterexample to their conjecture. Nevertheless, we subsequently show that a slightly more restricted version of their conjecture holds and can be used to exactly calculate the rSPR distance between two phylogenies by breaking the problem into smaller subproblems.

The remainder of this paper is organized as follows. The next section contains some mathematical preliminaries that are needed to formally state Hill at al's conjecture. This conjecture is then given in the subsequent section which also contains the aforementioned counterexample. We then show that a modified version of the conjecture holds in the following section. We end this note with a brief conclusion.

Preliminaries

In this section, we give some preliminary definitions that are used throughout this paper. Unless otherwise stated, the notation and terminology follows [4].

Phylogenetic Trees

A rooted binary phylogenetic X-tree is a rooted tree whose root has degree two while all other interior vertices have degree three and whose leaf set is X . The set X is the label set of and is frequently denoted by . Furthermore, let X′ be a subset of X. The minimal rooted subtree of that connects all the leaves in X′ is denoted by (X′) while the restriction of to X′, denoted by |X′, is the rooted binary phylogenetic X′-tree obtained from (X′) by contracting all degree-two vertices apart from the root.

Rooted Subtree Prune and Regraft

Let be a rooted binary phylogenetic X-trees. For the purposes of the upcoming definition, we view the root of as a vertex ρ adjoined to the original root by a pendant edge. Now, let e = {u, v} be any edge of that is not incident with ρ such that u is the vertex on the path from ρ to v. Let be the rooted binary phylogenetic X-tree obtained from by deleting e and reattaching the resulting subtree with root v via a new edge, say f , as follows. Subdivide an edge of the component that contains ρ with a new vertex u′, join u′ and v with f , and contract u. Then has been obtained from by a rooted subtree prune and regraft (rSPR) operation. The rSPR distance between two rooted binary phylogenetic X-trees and is the minimum number of rSPR operations that transform into . We denote this distance by .

Agreement Forests

Let and be two rooted binary phylogenetic X-trees. Again, to make the following work, regard the roots of and as a vertex ρ adjoined to the original root by a pendant edge. An agreement forest for and is a partition of such that and the following properties are satisfied:

(i) for all i ∈ {ρ, 1, ..., k}, we have , and

(ii) the trees in and are vertex-disjoint subtrees of and , respectively.

Throughout the remainder of this note, we will interchangeably refer to and as an agreement forest for and . A maximum-agreement forest for and is an agreement forest for and with the smallest number of elements over all agreement forests for and . Note that a maximum-agreement forest for and is not necessarily unique.

Bordewich and Semple [5] established the following characterization which directly relates the rSPR distance to the number of elements in a maximum-agreement forest and is crucial to many algorithms that exactly compute the rSPR distance between two rooted binary phylogenetic trees.

Theorem 1. Let and be two rooted binary phylogenetic X-trees, and let be a maximum-agreement forest for and . Then

Clusters

Let be a rooted binary phylogenetic X-tree, and let A be a subset of X with |A| ≥ 2. We say that A is a cluster of if there is a vertex v in whose set of descendants is precisely A. We denote this cluster by .

We next consider several different types of clusters that will play an important role in the remainder of this paper. Let and be two rooted binary phylogenetic X-trees, and let A be a cluster that is common to and ; that is there exists a vertex v in and a vertex in such that . Furthermore, let u (resp. u′) be the parent vertex of v (resp. ) in (resp. ), and let w (resp. w′) be the child vertex of u (resp. u′) with (resp. ). If no proper subset of A is a common cluster of and , we refer to A as a minimal cluster. Moreover, A is a solvable cluster if A is minimal and . Lastly, we say that A is a subtree-like cluster if A is a solvable cluster and . Roughly speaking, the condition is satisfied if the subtree with root w in is identical to the subtree with root w′ in . We refer to as the common subtree associated with A and note that it can exclusively consist of an isolated vertex. For example, A = {1, 2, ..., 6} is a solvable cluster of the two rooted binary phylogenetic X-trees and that are shown in Figure 1 since = {1, 2, ..., 12}. However, as , it follows that A is not a subtree-like cluster of and .

Figure 1

Two rooted binary phylogenetic . Note that and have an additional vertex ρ adjoined to the original root by a pendant edge.

Now, let Θ ∈ {minimal, solvable, subtree-like}. We next describe algorithmically how to obtain a sequence of tree pairs--which is important to mathematically state Hill et al's conjecture--by decomposing two rooted binary phylogenetic X-trees and into smaller subtrees. As previously, view the roots of and as a vertex ρ adjoined to the original root by a pendant edge, and regard ρ as part of the label set; that is . Setting i to be 1, let Abe a common Θ cluster of and with . Let denote the rooted binary phylogenetic tree (viewing the root of as a vertex ρadjoined to the original root by a pendant edge) and reset to be the tree obtained from by replacing with a new vertex a. Analogously, let denote the rooted binary phylogenetic tree (viewing the root of as a vertex ρadjoined to the original root by a pendant edge) and reset to be the tree obtained from by replacing with a new vertex a. If and contain a Θ cluster Awith , stop or increment i by 1 and repeat this process; otherwise, stop. Eventually, we obtain a sequence

of pairs of rooted binary phylogenetic trees, where and denote the two trees after the replacement of and with a vertex a. We call this sequence a cluster sequence of and with respect to a specific cluster type Θ. An example of a cluster sequence with respect to Θ = solvable for the two rooted binary phylogenetic trees depicted in Figure 1 is shown in Figure 2.

Figure 2

A cluster sequence with respect to Θ = . Details on how the tree pairs have been obtained are given in the text.

Hill et al's Conjecture and a Counterexample

We begin this section by formally stating Hill et al's conjecture which was introduced in [1].

Conjecture 2. Let and be two rooted binary phylogenetic X-trees. Let be a cluster sequence for and with respect to Θ = solvable. Then

Next, we detail a counterexample to the above conjecture which is based on the two rooted binary phylogenetic X-trees and that are shown in Figure 1. A maximum-agreement forest for and contains 5 elements and is shown in the top of Figure 3. By Theorem 1, this implies that . Now, consider the cluster sequence with respect to Θ = solvable for and that contains three tree pairs and is depicted in Figure 2. The first tree pair () consists of the restricted subtrees of and whose leaf set is the solvable cluster A1 = {1, 2, ..., 6} of and ; thus and . Similarly, the second tree pair () consists of the restricted subtrees of the two trees that have been obtained from and by replacing and , respectively, with a single leaf a1 whose leaf set is the solvable cluster A2 = {7, 8, ..., 12}. Lastly, the third tree pair () can be regarded as being obtained from and by replacing and with a leaf a1 and replacing and with a leaf a2. For each tree pair () of the cluster sequence shown in Figure 2, a maximum-agreement forest with i ∈ {1, 2, ρ} is depicted in the bottom part of Figure 3. Note that each forest is the unique maximum-agreement forest for and Now, by Equation 1, we have

Figure 3

Maximum-agreement forests. Top: A maximum-agreement forest for and depicted in Figure 1. Bottom: A maximum-agreement forest for each tree pair and shown in Figure 2.

which is strictly greater than ; thus showing that Conjecture 2 does not hold.

Using Subtree-Like Clusters to Prove Hill et al's Conjecture

In this section, we show that Conjecture 2 holds, if we consider a subtree-like cluster instead of a solvable cluster in each iteration of computing a cluster sequence for two rooted binary phylogenetic trees. We first prove the result for a cluster sequence of size two and then see that this result generalizes to cluster sequences of greater size.

Lemma 3. Let and be two rooted binary phylogenetic X-trees. Let (), () be a cluster sequence for and with respect to Θ = subtree-like. Then

Proof. Let A1 be the subtree-like cluster of and . We start by making an observation that is crucial for what follows. By the definition of a subtree-like cluster, there exists a common subtree, say , that is associated with A1 in and . Clearly, is also a common subtree of and . Furthermore, as has been obtained from by replacing with a single vertex a1 and as has been obtained from by replacing with a single vertex a1, it is easily checked that |() is a common subtree of and .

We now show that

Let be a maximum-agreement forest for and , and let be a maximum-agreement forest for and . By the observation prior to this paragraph, it follows from Proposition 3.2 of [5] that is a subset of an element, say , in . Furthermore, let be the label set of with ρ1 ∈ . As is an agreement forest for and and as is such a forest for and , it follows that

is an agreement forest for and . As - {a1} always contains an element, note that is never the empty set. Thus and, by Theorem 1, we have

This establishes Equation 2.

We now turn to the second part of this proof and show that

Let be a maximum-agreement forest for and . The remainder of this part splits into two cases. First, assume that there exists an element in , say , such that and . Note that is the only label set with the described properties, as otherwise, is not an agreement forest for and . Let , and let . Since is an agreement forest for and ,

is such a forest for and and

is an agreement forest for and . Second, assume that no such element exists. Hence, every element in is either a subset of A1 or a subset of . Furthermore, as A1 is a subtree-like cluster of and whose associated common subtree is , it again follows from Proposition 3.2 of [5], that is a subset of an element, say , in . Now, as is an agreement forest for and , it follows that

is an agreement forest for and and

is such a forest for and . Regardless of whether or not exists, we have , and therefore,

This establishes Equation 3, and combining Equations 2 and 3 completes the proof of this lemma.

The next theorem directly follows from repeated applications of Lemma 3.

Theorem 4. Let and be two rooted binary phylogenetic X-trees. Let and with respect to Θ = subtree-like. Then

Conclusion

In this paper, we have shown that Hill et al's conjecture [1] and the underlying divide-and-conquer approach cannot be used to calculate the rSPR distance between two phylogenies exactly. To provide some intuition why this conjecture fails, consider the following. Let be a cluster sequence with respect to Θ = solvable for two rooted binary phylogenetic trees and . Calculating a maximum-agreement forest for each tree pair (), taking their union, and, for each i ∈; {1, 2, ..., t}, joining the element containing awith the element containing ρcan potentially result in a set, say , which contains an element that is a subset of {a1, a2, ..., a, ρ1, ρ2, ..., ρ}. In the case of our counterexample,

contains one such element. Trivially, this element is not part of any agreement forest for and while - {{a1, a2, ρ1, ρ2}} is precisely a maximum-agreement forest for and . Consequently, a divide-and-conquer approach that exactly calculates needs to take into account the number of elements in that are subsets of {a1, a2, ..., a, ρ1, ρ2, ..., ρ}; otherwise, the result may be an overestimate of the exact solution. Alternatively, one can approach the problem by finding a strategy which guarantees that no element in is a subset of {a1, a2, ..., a, ρ1, ρ2, ..., ρ}. This is the underlying idea of Theorem 4 which uses a slightly more restricted version of Hill et al's conjecture and finally gives the desired outcome. Hence, decomposing and into a cluster sequence with respect to Θ = subtree-like can be used to speed up the exact calculation of .

However, for practical problem instances, it may be unlikely to find many subtree-like clusters. For example, the two phylogenies shown in Figure 1 do not have any common subtree-like cluster. This is due to the restricted definition of such a cluster which requires that a vertex whose set of descendants is a common cluster of two rooted binary phylogenetic X-trees and has the same parent vertex than a common subtree of and . To lessen this problem, an alternative approach--that has recently been published by Linz and Semple [6]--can be applied. This paper describes a more general divide-and-conquer approach that exactly computes the rSPR distance between and for when a cluster sequence with respect to Θ = minimal for and is given. Loosely speaking, the authors calculate a so-called minimum-weight partition of X ∪ {ρ} ∪ {a1, a2, ..., a, ρ1, ρ2, ..., ρ} such that contains an agreement forest (not necessarily a maximum-agreement forest) for each tree pair (). To compute , it has been shown that applying a 'bottom-up' approach which locally works on subtrees of each tree pair () guarantees that the number of elements in that are subsets of {a1, a2, ..., a, ρ1, ρ2, ..., ρ} is maximized while || is minimized.