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Bansal et al. Algorithms for Molecular Biology 2010, 5:18 http://www.almob.org/content/5/1/18 BAM ALGORITHMS FOR MOLECULAR BIOLOGY RESEARCHER O RobinsonFoulds Supertrees Mukul S Bansal '2, J Gordon Burleigh3, Oliver Eulenstein2, David FernandezBaca; Introduction Supertree methods provide a formal approach for com bining small phylogenetic trees with incomplete species overlap in order to build comprehensive species phylo genies, or supertrees, that contain all species found in the input trees. Supertree analyses have produced the first familylevel phylogeny of flowering plants [1] and the first phylogeny of nearly all extant mammal species [2]. They have also enabled phylogenetic analyses using largescale genomic data sets in bacteria, across eukar yotes, and within plants [3,4] and have helped elucidate the origin of eukaryotic genomes [5]. Furthermore, supertrees have been used to examine rates and patterns of species diversification [1,2], to test hypotheses regard ing the structure of ecological communities [6], and to examine extinction risk in current species [7]. * Correspondence' fernande@cs iastate edu 2Department of Computer Science, Iowa State University, Ames, A 50011, USA 0 BioMed Central Although supertrees can support largescale evolution ary and ecological analyses, there are still numerous concerns about the performance of existing supertree methods (e.g., [814]). In general, an effective supertree method must accurately estimate phylogenies from large data sets in a reasonable amount of time while retaining much of the phylogenetic information from the input trees. By far the most commonly used supertree method is matrix representation with parsimony (MRP), which works by solving the parsimony problem on a binary matrix representation of the input trees [15,16]. While the parsimony problem is NPhard, MRP can take advantage of fast and effective hillclimbing heuristics implemented in PAUP* or TNT (e.g., [1719]). MRP heuristics often perform well in analyses of both simu lated and empirical data sets (e.g., [2022]); however, there are numerous criticisms of MRP. For example, MRP shows evidence of biases based on the shape and size of input trees [8,11], and MRP supertrees may 2010 Bansal et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommon.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Background: Supertree methods synthesize collections of small phylogenetic trees with incomplete taxon overlap into comprehensive trees, or supertrees, that include all taxa found in the input trees. Supertree methods based on the well established RobinsonFoulds (RF) distance have the potential to build supertrees that retain much information from the input trees. Specifically, the RF supertree problem seeks a binary supertree that minimizes the sum of the RF distances from the supertree to the input trees. Thus, an RF supertree is a supertree that is consistent with the largest number of clusters (or clades) from the input trees. Results: We introduce efficient, local search based, hillclimbing heuristics for the intrinsically hard RF supertree problem on rooted trees. These heuristics use novel nontrivial algorithms for the SPR and TBR local search problems which improve on the time complexity of the best known (naive) solutions by a factor of 0(n) and 0(n2) respectively (where n is the number of taxa, or leaves, in the supertree). We use an implementation of our new algorithms to examine the performance of the RF supertree method and compare it to matrix representation with parsimony (MRP) and the triplet supertree method using four supertree data sets. Not only did our RF heuristic provide fast estimates of RF supertrees in all data sets, but the RF supertrees also retained more of the information from the input trees (based on the RF distance) than the other supertree methods. Conclusions: Our heuristics for the RF supertree problem, based on our new local search algorithms, make it possible for the first time to estimate large supertrees by directly optimizing the RF distance from rooted input trees to the supertrees. This provides a new and fast method to build accurate supertrees. RF supertrees may also be useful for estimating majorityrule() supertrees, which are a generalization of majorityrule consensus trees. Bansal et al. Algorithms for Molecular Biology 2010, 5:18 http://www.almob.org/content/5/1/18 contain relationships that are not supported by any of the input trees [9,12]. Furthermore, it is unclear if or why minimizing the parsimony score of a matrix repre sentation of input trees is a good optimality criterion or should produce accurate supertrees. Since evolutionary biologists rarely, if ever, know the true relationships for a group of species, it is difficult to assess the accuracy of supertree, or any phylogenetic, methods. One approach to evaluate the accuracy of supertrees is with simulations (e.g., [20,21]). However, simulations inherently simplify the true processes of evolution, and it is unclear how well the performance of a phylogenetic method in simulations corresponds to its performance with empirical data. Perhaps a more useful way to define the accuracy of a supertree method is to quantify the amount of phylogenetic information from the input trees that is retained in the supertree. Ideally, we want the supertree to reflect the input tree topolo gies as much as possible. This suggests that the super tree objective should directly evaluate the similarity of the supertree to the input trees (e.g., [11,23,24]). Numerous metrics exist to measure the similarity of input trees to a supertree, and the RobinsonFoulds (RF) distance metric [25] is among the most widely used. In fact, numerous studies have evaluated the performance of supertree methods, including MRP, by measuring the RF distance between collections of input trees and the resulting supertrees (e.g., [11,20,21]). The RF supertree problem seeks a binary supertree that minimizes the sum of the RF distances between every rooted input tree and the supertree. The intuition behind seeking a binary supertree is that, in this setting, minimizing the RF dis tance is equivalent to maximizing the number of clus ters (or clades) that are shared by the supertree and the input trees. Thus, an RF supertree is a supertree that is consistent with the largest number of clusters from the input trees. Unfortunately, as with MRP, computing RF supertrees is NPhard [26]. In this work, we describe efficient hillclimbing heuristics to estimate RF super trees. These heuristics allow the first largescale esti mates of RF supertrees and comparisons of the accuracy of RF supertrees to other commonly used supertree methods. The RF distance metric between two rooted trees is defined to be a normalized count of the symmetric dif ference between the sets of clusters of the two trees. In the supertree setting, the input trees will often have only a strict subset of the taxa present in the supertree. Thus, a high RF distance between an input tree and a supertree does not necessarily correspond to conflicting evolutionary histories; it can also indicate incomplete phylogenetic information. Consequently, in order to compute the RF distance between an input tree which has only a strict subset of the taxa in the supertree, we first restrict the supertree to only the leaf set of the input tree. This adapted version of the RF distance is not a metric, or even a distance measure (mathemati cally speaking). However, for convenience, we will refer to this adapted version of the RF distance metric using the same name. Previous work Supertree methods are a generalization of consensus methods, in which all the input trees have the same leaf set. The problem of finding an optimal median tree under the RF distance in such a consensus setting is wellstudied. In particular, it is known that the majority rule consensus of the input trees must be a median tree [27], and it can be found in polynomial time. On the other hand, finding the optimum binary median tree, i. e. an RF supertree, in the consensus setting is NPhard [26]. This implies that computing an RF supertree in general is NPhard as well. Our definition of RF distance between two trees where one has only a strict subset of the taxa in the other, cor responds to the distance measure used to define "major ityrule() supertrees" by Cotton and Wilkinson [28]. This definition restricts the larger tree to only the leaf set of the smaller tree before evaluating the RF distance. Majorityrule() supertrees are defined to be the strict consensus of all the optimal median trees under the RF distance. These median trees are defined similarly to RF supertrees, except that RF supertrees must be binary while the median trees can be nonbinary. In general, majorityrule supertrees [28], in both their () and (+) variants, seek to generalize the majorityrule consensus. Indeed, majorityrule supertrees have been shown to have several desirable properties reminiscent of major ityrule consensus trees [29]. Although majorityrule supertrees and RF supertrees are both based on mini mizing RF distance, they represent two different approaches to supertree construction. In particular, the RF supertree method seeks a supertree that is consistent with the largest number of clusters (clades) from the input trees, while majorityrule supertrees do not. Nevertheless, as we discuss later, RF supertrees could be used as a starting point to estimate majorityrule() supertrees. The RF distance between two trees on the same size n leaf set, with leaves labeled by integers {1, ..., n}, can be computed in O(n) time [30]. In fact, an (1 + e)approxi mate value of the RF distance can be computed in sub linear time, with high probability [31]. In the case of unrooted trees, the RF distance metric is sometimes also known as the splits metric (e.g., [32]). The supertree analysis package Clann [23] provides heuristics that operate on unrooted trees and attempt to maximize the number of splits shared between the input Page 2 of 12 Bansal et al. Algorithms for Molecular Biology 2010, 5:18 http://www.almob.org/content/5/1/18 trees and the inferred supertree. This method is called the "maximum splitsfit" method. Local Search We use a heuristic approach for the RF supertree pro blem. Local search is the basis of effective heuristics for many phylogenetic problems. These heuristics iteratively search through the space of possible super trees guided, at each step, by solutions to some local search problem. More formally, in these heuristics, a tree graph (see [32,33]) is defined for the given set of input trees and some fixed tree edit operation. The node set of this tree graph represents the set of all supertrees on the given input trees. An edge is drawn between two nodes exactly if the corresponding trees can be transformed into each other by one tree edit operation. In our setting, the cost of a node in the graph is the RF distance between the supertree repre sented by that node and the given input trees. Given an initial node in the tree graph, the heuristic's task is to find a maximallength path of steepest descent in the cost of its nodes and to return the last node on such a path. This path is found by solving the local search problem at every node along the path. The local search problem is to find a node with the minimum cost in the neighborhood of a given node. The neigh borhood is defined by some tree edit operation, and hence, the time complexity of the local search problem depends on the tree edit operation used. Two of the most extensively used tree edit operations for supertrees are rooted Subtree Prune and Regraft (SPR) [3335] and rooted Tree Bisection and Reconnec tion (TBR) [22,33,34]. The best known (naive) algo rithms for the SPR and TBR local search problems for the RF supertree problem require O(kn ) and O(kn4) time respectively, where k is the number of input trees, and n is the number of leaves in the supertree solution. Our Contribution We describe efficient hillclimbing heuristics for the RF supertree problem. These heuristics are based on novel nontrivial algorithms that can solve the corresponding local search problems for both SPR and TBR in O(kn2) time, yielding speedups of 0(n) and 0(n2) over the best known solutions respectively. These new algorithms are inspired by fast local search algorithms for the gene duplication problem [36,37]. Note that while the super tree itself must be binary, our algorithms work even if the input trees are not. We also examine the perfor mance of the RF supertree method using four published supertree data sets, and compare its performance with MRP and the triplet supertree method [38]. We demon strate that the new algorithms enable RF supertree ana lyses on large data sets and that the RF supertree method outperforms other supertree methods in finding supertrees that are most similar to the input trees based on the RF distance metric. Basic Notation and Preliminaries A tree T is a connected acyclic graph, consisting of a node set V (T) and an edge set E(T). T is rooted if it has exactly one distinguished node called the root which we denote by rt(T). Throughout this work, the term tree refers to a rooted tree. We define < to be the partial order on V (T) where x < y if y is a node on the path between rt(T) and x. The set of minima under < is denoted by L(T) and its elements are called leaves. The set of all nonroot internal nodes of T, denoted by I(T), is defined to be the set V (T)\((T) U {rt(T)}). If {x, y} e E(T) and x dren of y is denoted by ChA(y). T is fully binary if every node has either zero or two children. If two nodes in T have the same parent, they are called siblings. The least common ancestor of a nonempty subset L V (T), denoted as lca(L), is the unique smallest upper bound of L under For each node v e I(T), the cluster CT (v) is defined to be the set of all leaf nodes in Tv; i.e. CT (v) = L(Tv). We denote the set of all clusters of a tree T by t(T). Given a set L L(T), let T' be the minimal rooted subtree of T with leaf set L. We define the leaf induced subtree T [L] of T on leaf set L to be the tree obtained from T' by successively removing each nonroot node of degree two and adjoining its two neighbors. The symmetric differ ence of two sets A and B, denoted by AAB, is the set (A\B) U (B\A). A profile P is a tuple of trees (T1,..., Tk). The RF Supertree Problem Given a profile P, we define a supertree on 'P to be a fully binary tree T where (T*)= U'i (T,) Definition 1 (RF Distance). Given a profile P = (T, ..., Tk) and a supertree T* on 'P, we define the RF distance as follows: 1. For any T,, where 1 < i < k, RF (Ti, P7) = (rT,) A(P [(Ti)]) . 2. RF(P,T*) = Y= RF(T, T*) 3. Let T be the set of supertrees on 'P, then RF(P) = min,= RF(P, T*) . Remark: Traditionally, the value of the RF distance, as computed above, is normalized by multiplying by 1/2. However, this does not affect the definition or Page 3 of 12 Bansal et al. Algorithms for Molecular Biology 2010, 5:18 http://www.almob.org/content/5/1/18 computation of RF supertrees, and therefore, we do not normalize the RF distance. Problem 1 (RF Supertree). Instance: A profile P . Find: A supertree ToptOn P such that RF (P Tot) = RF (P). Recall that the RF Supertree problem is NPhard [27]. Local Search Problems Here we first provide definitions for the rerooting operation (denoted RR) and the TBR[22] and SPR[35] edit operations and then formulate the related local search problems that were motivated in the introduction. Definition 2 (RR operation). Let T be a tree and x e V (T). RR(T, x) is defined to be the tree T, ifx = rt(T) or x e Ch(rt(T)). Otherwise, RR(T, x) is the tree that is obtained from T by (i) suppressing rt(T), and (ii) subdi viding the edge {pa(x), x} by a new root node. We define the following extension: RR(T) = U, v(){RR(T, x)}. For technical reasons, before we can define the TBR operation, we need the following definition. Definition 3 (Planted tree). Given a tree T, the planted tree 02(T) is the tree obtained by adding a root edge {p, rt(T)}, where p 4 V (T), to T. Definition 4 (TBR operation). (See Fig. 1) Let T be a tree, e = (u, v) e E(T), where u = pa(v), and X, Y be the connected components that are obtained by removing edge e from T where v e X and u e Y We define TBRT (v, x, y) for x e X and y e Y to be the tree that is obtained from D(T) by first removing edge e, then repla cing the component X by RR(X, x), and then adjoining a new edge f between x' = rt(RR(X, x)) and Y as follows: 1. Create a new node y' that subdivides the edge (pa (y), y). 2. Adjoin the edge between nodes x' and y'. 3. Suppress the node u, and rename x' as v and y' as u. 4. Contract the root edge. Notation. We define the following: 1. TBRT (v, x) = U y {TBRT (v, x, y)} 2. TBRT (v) = U, x TBRT (v, x) 3. TBRT = U(u, v) e E(T TBRT(v) Definition 5 (SPR operation). Let T be a tree, e = (u, v) e E(T), where u = pa(v), and X, Y be the con nected components that are obtained by removing edge e from T where v e X and u e Y. We define SPRT (v, y), for y Y, to be the tree TBRT (v, v, y). We say that the tree SPRT (v, y) is obtained from T by a subtree prune and regraft (SPR) operation that prunes subtree Tv and regrafts it above node y. Notation. We define the following: 1. SPRT (v) = Uy ySPRT (v, y)} 2. SPRT = U(u, v) e E(T) SPRT(v) Note that an SPR operation for a given tree T can be briefly described through the following four steps: (i) prune some subtree P from T, (ii) add a root edge to the remaining tree S, (iii) regraft P into an edge of the remaining tree S, and (iv) contract the root edge. We now define the relevant local search problems based on the TBR and SPR operations. Problem 2 (TBRScoring (TBRS)). Given instance ( P T), where P is the profile (Ti, ..., Tk) and T is a supertree on 'P, find a tree T* e TBRT such that RF(P, T*) = minT'eTBR, RF(P, T'). Problem 3 (TBRRestricted Scoring (TBRRS)). Given instance (P T, v), where P is the profile (T, ..., Tk), T is a supertree on 'P, and v is a nonroot node in V(T), find a tree T e TBRT (v) such that RF(P,T*) = minT'eTBR(v) RF(P,T') . The problems SPRScoring (SPRS) and SPR Restricted Scoring (SPRRS) are defined analogously to the problems TBRS and TBRRS respectively. S S' v y vre V /f T'\ /T" AT' a b c x d( d a b cx Figure 1 TBR Operation. Example depicting a TBR operation which transforms tree S into tree S' TBRs(v, x, y). Page 4 of 12 Bansal et al. Algorithms for Molecular Biology 2010, 5:18 http://www.almob.org/content/5/1/18 Throughout the remainder of this manuscript, k is the number of trees in the profile P T denotes a supertree on 'P, and n is the number of leaves in T. The follow ing observation follows from Definition 4. Observation 1. The TBRS problem on instance (P , T) can be solved by solving the TBRRS problem IE(T) times. We show how to solve the TBRS problem on the instance (P, T) in O(kn2) time. Since SPRT c TBRT this also implies an O(kn2) solution for the SPRS pro blem. This gives a speedup of 0(n2) and 0(n) over the best known (naive) algorithms for the TBRS and SPRS problems respectively. In particular, we first show that any instance of the TBRRS problem can be decomposed into an instance of an SPRRS problem, and an instance of a Rooting pro blem (defined in the next section). We show how to solve both these problems in O(kn) time, yielding an O (kn) time solution for the TBRRS problem. This imme diately implies an O(kn2) time algorithm for the TBRS problem (see Observation 1). Note that the size of the set TBRT is 0(n3). Thus, for each tree in the input profile the time complexity of computing and enumerating the RF distances of all trees in TBRT is i(n3). However, to solve the TBRS problem one only needs to find a tree with the minimum RF dis tance. This lets us solve the TBRS problem in time that is sublinear in the size of TBRT. In fact, after the initial O(kn2) preprocessing step, our algorithm can output the RF distance of any tree in TBRT in 0(1) time. Structural Properties Throughout this section, we limit our attention to one tree S from the profile P. We show how to solve the TBRRS problem for the instance ((S), T, v) for some nonroot node v e V (T) in O(n) time. Based on this solution, it is straightforward to solve the TBRRS pro blem on the instance (p, T, v) within O(kn) time as well. For clarity, we will also assume that (S) = (T). In general, if (S) c (T) then we can simply set T to be T [L(S)]. This takes O(n) time and, consequently, does not affect the time complexity of our algorithm. Our algorithm makes use of the LCA mapping from S to T. This mapping is defined as follows. Definition 6 (LCA Mapping). Given two trees T' and T such that (T) (T), the LCA mapping &T', T: V (T) V(T) is the mapping &; T (u) = IcaT (( T )). Notation. We define a boolean functionfr: I(S) > {0, 1} such thatfT (u) = 1 if there exists a node v e I(T) such that Cs (u) = C, (v), andfT (u) = 0 otherwise. Thus, fT (u) = 1 if and only if the cluster Cs (u) exists in the tree T as well. Additionally, we define _FT = {u e I(S):fT (u) = 0}; that is, rtT is the set of all nodes u e I(S) such that the cluster Cs (u) does not exist in the tree T. The following lemma associates the value RF(S, T) with the cardinality of the set SFT. Lemma 1. RF(S, T) = I(T) II(S) + 2.4T . Proof. Let gT denote the set {u e I(S):fT (u) = 1}. By the definition of RF (S, T), we must have RF(S, T) = II (T) + II(S) 2.1 gT . And hence, since  T I +  Tr = I(S), we get RF(S, T) = I(T) II(S) + 2. 1 I. E Lemma 2. For any u e I(S),fT (u) = 1 if and only if SCs (u) = I CT (aS, T (u)) . Proof If Cs (u) = I C (s, r(u)) then we must have Cs (u) = CT (s, T (u)) and, consequently, fT (u) = 1. In the other direction, if I Cs (u) # I CT ( &s, T (u)), then we must have Cs (u) c CT (&s, T (u)) and, conse quently, fr (u) = 0. E The LCA mapping from S to T can be computed in O (n) time [39], and consequently, by Lemmas 1 and 2, we can compute the RF distance between S and T in O(n) time as well (other O(n)time algorithms for calculating the RF distance are presented in [30,31]). Moreover, Lemma 1 implies that in order to find a tree T' e TBRT (v) such that RF(P, T*) = minT'eTBRR() RF(P, T'), it is sufficient to find a tree T* e TBRT (v) for which  T. I= minTETBR (v)I TT' I Remark: An implicit assumption here is that the leaves of both trees are labeled by integers {1, ..., n}. If the leaf labels are arbitrary, then we require an additional O(kn log n)time preprocessing step to relabel the leaves of the trees in the given profile. Note, however, that this additional step does not add to the overall time com plexity of solving the TBRS or SPRS problems. We now show that the TBRRS problem can be solved by solving two smaller problems separately and combin ing their solutions. As before, we limit our attention to one tree S from the profile P Given the TBRRS instance ((S), T, v), we define a bipartition {X, 5 } of I(S), where X = {u e I(S): &s, Au) e V(T,)}. Lemma 3. If u e X, then fT'(u) = fT (u) for all T' e TBRT (v, v). If u e j and y denotes the sibling of v, then fTr(u) = fT (u), where T' = TBRT(v, x, y) for any x e V (Tv). Proof. Consider the case when u e X. Let T' be any tree in TBRT(v, v) and let node y e V (T) be such that T' = TBR(v, v, y). Thus, for any node w e V (Tv), the subtrees Tv and T' must be identical. Since u e X, we must have &s, T (u) e Tv and, consequently, TMsT(u) = TM,7(u). Lemma 2 now implies thatfT, (u) = fT (u). Now consider the case when u L k. Node y denotes the sibling of v in tree T and let T' = TBR(v, x, y), for some x e V (Tv). Thus, for any node w e V(T)\V(Tv), we must have T(W) = T' (w). Moreover, the leaf sets of the two subtrees rooted at the children of w in T must be identical to the leaf sets of the two subtrees Page 5 of 12 Bansal et al. Algorithms for Molecular Biology 2010, 5:18 http://www.almob.org/content/5/1/18 rooted at the children of w in T': This implies that if &s, (u) = w, then s, T' (u) = w as well. By Lemma 2 we must therefore havefr' (u) =fT (u). 1 Lemma 3 implies that a tree in TBRT (v) with smallest RF distance can be obtained by optimizing the rooting for the pruned subtree, and optimizing the regraft location separately. This allows us to obtain a tree in TBRT (v) with smallest RF distance by evaluating only O(n) trees. Con trast this with the naive approach to finding a tree in TBRT (v) with smallest total distance, which is to evaluate all trees obtained by rerooting the pruned subtree in all possible ways, and, for each rerooting, regrafting the sub tree in all possible locations. Since there are O(n) ways to reroot the pruned subtree, and O(n) ways to regraft, this would require evaluating O(n2) trees. It is interesting to note that this ability to decompose the TBRRS problem into two simpler problems is not unique to the context of RF supertrees alone. For example, it has been observed that a similar decomposition can be achieved in the con text of the gene duplication problem [37]. Thus, to solve the TBRRS problem, we must find (i) a rerooting T' of the subtree Tv for which ,T' is mini mized, and (ii) a regraft location y for Tv which mini mizes r, y ) . Observe that the problem in part (ii) is simply the SPRRS problem on the input instance ((S), T, v). For part (i), consider the following problem statement. Problem 4 (Rooting). Given instance (P, T, v), where P is the profile (Ti, ..., Tk), T is a supertree on P and v is a nonroot node in V (T), find a node x e V (Tv) for which RF (P, TBRT (v, x, y)) is minimum, where y denotes the sibling of v in T. Note that the problem in part (i) is the Rooting pro blem on the input instance ((S), T, v). We show how to solve both the Rooting and the SPRRS problems in O (n) time on instance ((S), T, v). As seen above, based on Lemma 3, this immediately implies that the TBRRS problem for a profile consisting of a single tree can be solved in O(n) time. To solve the TBRRS problem on instance (P, T, v), we simply solve the Rooting and SPRRS problems separately on the input instance (P, T, v), which takes O(kn) time (see Theorems 3 and 4). We thus have the following two theorems. Theorem 1. The TBRRS problem can be solved in 0 (kn) time. Theorem 2. The TBRS problem can be solved in 0 (kn2) time. Solving the Rooting Problem To solve the Rooting problem on instance ((S), T, v), we rely on an efficient algorithm for computing the value of fT' (u) for any T' e RR(Tv) and any u e I(S). This algo rithm relies on the following five lemmas. Let a denote the node s, T (u), y denote the sibling of v in T, and T' = TBRT(v, x, y) for x e V (Tv). Depending on a and fT (u) there are five possible cases: (i) a 4 V (Tv), (ii) a = rt(T,) andfT (u) = 1, (iii) a= rt(Tv) andfT (u) = 0, (iv) a eV (T,)\rt(T,) andfT (u) = 1, and (v) a e V (T,)\rt(T,) and fT (u) = 0. Lemmas 4 through 8 characterize the value f7r (u) for each of these five cases respectively. Lemma 4. If a 4 V (Tv), then fT' (u) = fT (u) for any x V (Tv). Proof Follows directly from Lemma 3. E Lemma 5. If a = rt(Tv) and fT (u) = 1, then f (u) = 1 for all x V (Tv). Proof. Since we have a = rt(Tv) and f (u) = 1, by Lemma 2 we must have (S,) = (Tv). Thus, for any x e V (Tv), &s, T' (u) must be the root of the subtree RR (Tv, x). The lemma follows. E Lemma 6. Let L denote the set (T,)\(Su), and let b = Ica (L). If a = rt(Tv) andfT (u) = 0, then, 1. for x XT b, fT' (u) = 0, and, 2. for x T, b,fTr (u) = 1 if and only if ILl = I(Tb). Proof Since a = rt(Ty) and f (u) = 0, by Lemma 2 we must have L(S,) L(Tv). We analyze each part of the lemma separately. 1. x XT, b: For this case to be valid, we must have b 0. Therefore, we must have b Lemma 2 implies thatfrT (u) = 0. 2. x rt(T,). Therefore, let b' denote the parent of b in tree Tv. Now consider the tree T'. The set ( Ty,) must be identical to L(Su). Hence, fT' (u) = 1 in this case. (b) IL IL(Tb)l: We claim that there does not exist any edge (pa(w), w) e E(Tv) such that L(T,) is either L(S,) or L. Let us suppose, for the sake of contradiction, that such an edge exists. If _i (T,) = L(Su) then we must have a = w, which is a contradiction since a = rt(T,). If L(T,) = g L then we must have b = w, and, consequently, ILI # L_ (Tb) which is, again, a contradiction. Thus, such an edge (pa(w), w) e E(T,) cannot exist. Hence, we must have fT (u) = 0 for every x e V (Tv) in this case. The lemma follows. E Lemma 7. If a e V (Tv)\rt(Tv) and f(u) (u) = 0 if and only ifx < Tv a. 1, then fT, Page 6 of 12 Bansal et al. Algorithms for Molecular Biology 2010, 5:18 http://www.almob.org/content/5/1/18 Proof By Lemma 2 we must have (S,) = (T,). We have two cases: 1. x XT, a: In this case we must have &s, T' (u) = a, and L(T,) = L(T). Thus, L(S,) = (T') and hence, fr (u) = 1. 2. x and L(S,) L(Tv), Lemma 2 implies thatfr (u) = 0. The lemma follows. El Lemma 8. If a e V (T,)\rt(T,) and fT (u) = 0, then fT' (u) = Ofor all x e V (T). Proof By Lemma 2 we must have (S,) (Ta). We have two possible cases: 1. x XT, a: In this case we must have s&, T' (u) = a, and L(T,) = L(T'). Thus, L(S,) # L(T') and hence, frT (u) = 0 2. x and L(S,) L(Tv), Lemma 2 implies thatfrT (u) = 0. The lemma follows. The Algorithm. For any x e V (T,) let A(x) denote the cardinality of the set {u e I(S):fT (u) = 0, butfT' (u) = 1}, and B(x) the car dinality of the set {u e I(S):fT (u) = 1, but fT (u) = 0}, where T' = TBRT (v x, y). By definition, to solve the Rooting problem we must find a node x e V (Tv) for which IA(x) IB(x)l is maxi mized. Our algorithm computes, at each node x e V (Tv), the values A(x) and B(x). In a preprocessing step, our algorithm computes the mapping &s, T as well as the size of each cluster in S and T, and creates and initializes (to 0) two counters a (x) and P(x) at each node x e V (Tv). This takes O(n) time. When the algorithm terminates, the values a(x) and P(x) at any x e V (Tv) will be the values a(x) and /3 (x). Recall that, given u e I(S), a denotes the node &s, T (u). Thus, any given u e I(S) must satisfy the precondi tion (given in terms of a) of exactly one of the the Lem mas 4 through 8. Moreover, the precondition of each of these lemmas can be checked in 0(1) time. The algorithm then traverses through S and considers each node u e I(S). There are three cases: 1. If u satisfies the preconditions of Lemmas 4, 5, or 8 then we must have fTr (u) = fT (u). Consequently, we do nothing in this case. 2. If u satisfies the precondition of Lemma 7, then we increment the value of ((x) at each node x e V (Ta)\{a} (where a is as in the statement of Lemma 7). To do this efficiently we can simply increment a counter at node a such that, after all u e I(S) have been considered, a single preorder traversal of Tv can be used to compute the correct values of P(x) at each x e V (Tv). 3. If u satisfies the precondition of Lemma 6, then we proceed as follows: Let a and L be as in the statement of Lemma 6. According to the Lemma, if we can find a node b e V(Tv) such that b = IcaT (L) and L(Tb) = ILI, then we increment the value of a(x) at each node x e V (Tb); otherwise, if such a b does not exist, we do nothing. As before, to do this efficiently, we only increment a single counter at node b such that, after all u e I(S) have been considered, a preorder traversal of Tv suffices to compute the correct values of a(x) at each x e V (Tv). In order to prove the O(n) runtime for this algorithm we will now explain how to precompute such a corresponding node b (if it exists), for each u e I(S) satisfying the precondition of Lemma 6, within O(n) time. Note that any edge in a tree bipartitions its leaf set. Construct the tree S' = S [L(Tv)]. Observe that, given any candidate u, the corresponding node b exists if and only if the parti tion of L(S) induced by the edge (u, pa(u)) E(S), is also induced by some edge, e, in the tree Tv If such an e exists, then b must be that node on e which is farther away from the root, i.e. the edge e must be the edge (b, pa(b)) in Tv This edge e (or its absence) can be precomputed, for all candidate u, as follows: Com pute the strict consensus of the unrooted variants of the trees S' and Tv. Every edge in this strict consensus corresponds to an edge in S' and an edge in Tv that induce the same bipartitions in the two trees. Thus, for all candidate u that lie on such an edge, the corresponding node b can be inferred in 0(1) time (by using the association between the edges of the strict consensus and the edges of S' and Tv), and for all candi date u that do not lie on such an edge, we know that the corresponding node b does not exist. This strict consensus of the unrooted variants of S' and Tv can be precomputed within O(n) time by using the algorithm of Day [30]. Hence, the Rooting problem for a profile consisting of a single tree can be solved in O(n) time; yielding the fol lowing theorem. Theorem 3. The Rooting problem can be solved in 0 (kn) time. Solving the SPRRS Problem We will show how to solve the SPRRS problem on instance ((S), T, v) in O(n) time. Consider the tree R = SPRT (v, rt(T)) Observe that, since SPRR(v) = SPRs(v), Page 7 of 12 Bansal et al. Algorithms for Molecular Biology 2010, 5:18 http://www.almob.org/content/5/1/18 solving the SPRRS problem on instance ((S), T, v) is equivalent to solving it on the instance ((S), R, v). Thus, in the remainder of this section, we will work with tree R instead of tree S. The following four lemmas let us efficiently infer, for any u e I(S), whether fT (u) = 1 or fT' (u) = 0, for any given T' e SPRR(v). For brevity, let a denote the node s, R(u), and let Q denote the set V (R)\(V (R,) U {rt(R)}). Let T' = SPRR(V, x), for any x e Q. Depending on a and fR(u) there are four possible cases: (i) a e V (Rv), (ii) a e Q and f(u) = 1, (iii) a e Q andfR(u) = 0, and (iv) a = rt(R). Lemmas 9 through 12 characterize the value fr (u) for each of these four cases respectively. l Lemma 9. If a e V (Rv), then fT' (u) = fR(u) for any x Q. Proof Observe that TBRR(v, v) = SPRR(v). Lemma 3 now immediately completes the proof. E Lemma 10. If a e Q andfR(u)= 1, then, 1. fr (u) = O, for x Proof Since fR(u) = 1, Lemma 2 implies that I Cs (u) = SCR (a)l. Let T' = SPRR(V, x); we now have two cases. 1. x (by Lemma 2). 2. x 'R a: In this case &/s, T' (u) = a, and since  Cs (u) I = I CR (a) = C'(a) 1, we must have f, (u) = 1 (by Lemma 2). The lemma follows. El Lemma 11. If a e Q and f(u) = 0, then f, (u) = Ofor any x e Q. Proof Since fR(u) = 0, Lemma 2 implies that I Cs (u) I SCR (a) Thus, by the definition of LCA mapping, I Cs (u) < I CR (a). Let T' = SPRR(V, x); we now have two cases. 1. x (by Lemma 2). 2. x (by Lemma 2). The lemma follows. El For the next lemma, let S' be the tree obtained from S by suppressing all nodes s for which &s, R(s) e R,. Lemma 12. If a = rt(R) and b = &gs, R(u), then, f, (u)= 1 if and only ifx Proof First, observe that, since a = rt(R), the mapping %s' R(u) is well defined. Second, since b = .s' R(u), we must have L( Su) L(Rb), which implies that L(S,) c (Rv) L(Rb). We now have the following three cases: 1. x 'R b: In this case we must have s T' (u) = IcaT' (x, b). By Lemma 2 we know that f, (u) = 1 only if Cs (u) = CT'(.s, T' (u)) However, since we have (S,) L(Rv) L(Rb), and x 'R b, we must have Cs (u) < I Cr,(s, T' (u)) ; and hence, fT' (u) = 0. 2. x C(Rv) I I L(S,)I, we must have (S,) c (R,) U IL (Rb), which implies that I Cs (u) I < I C" ( Ms, T (u)) . Thus, by Lemma 2, we must have fr (u) = 0. 3. x (Rb)I + IL(Rv) = I(Su), we must have  Cs (u) = I CT' (.s, T' (u)) . Thus, by Lemma 2, we must have fT' (u) = 1. The lemma follows. E The Algorithm. Note that SPRT (v) = SPRR(V) = Ux Q SPRR(V, x). For any x e Q, let A(x) = {u e I(S):fR(u) = 0, butfT' (u) = 1}1, and B(x) = {u e I(S):fR(u) = 1, butfT' (u) = 0}, where T' = SPRn(v, x). By definition, to solve the SPRRS problem on instance ((S), T, v) we must find a node x e Q for which A(x) B(x) is maximized. Our algorithm computes, at each node x e Q, the values A(x) and B(x). In a preprocessing step, our algorithm first constructs the tree R computes the mapping &s, R as well as the size of each cluster in S and R, and creates and initia lizes (to 0) two counters a(x) and P(x) at each node x e Q. This takes a total of O(n) time. When the algorithm terminates, the values a(x) and P(x), at any x e Q will be the values A(x) and B(x). Recall that, given u e I(S), a denotes the node 4Ms, R(u). Thus, any given u e I(S) must satisfy the pre condition (given in terms of a) of exactly one of the the Lemmas 9 through 12. Moreover, the precondi tion of each of these lemmas can be checked in 0(1) time. The algorithm then traverses through S and considers each node u e I(S). There are three cases: 1. If u satisfies the preconditions of Lemmas 9 or 11 then we must have f, (u) = f(u) Consequently, we do nothing in this case. 2. If u satisfies the precondition of Lemma 10, then we increment the value of P(x) at each node x e V (Ta)\{a} (where a is as in the statement of Lemma Page 8 of 12 Bansal et al. Algorithms for Molecular Biology 2010, 5:18 http://www.almob.org/content/5/1/18 10). This can be done efficiently as shown in part (2) of the algorithm for the Rooting problem. 3. If u satisfies the precondition of Lemma 6, and if (Rb) + I(Rv) = IL(Su), then we increment the value of a(x) at each node x e V (Tb)\{b} (where a and b are as in the statement of Lemma 6). Again, to do this efficiently, we increment a counter at node b, and perform a subsequent preorder traversal. Note also that the mapping .&s, R can be computed in O(n) time in the preprocessing step, and hence the node b can be inferred in 0(1) time. The condition L(Rb)\ + I(Rv) = I(S.u) is also verifiable in 0(1) time. Hence, the SPRRS problem for a profile consisting of a single tree can be solved in O(n) time; yielding the fol lowing theorem. Theorem 4. The SPRRS problem can be solved in 0 (kn) time. Remark. To improve the performance of local search heuristics in phylogeny construction, the starting tree for the first local search step is often constructed using a greedy 'stepwise addition' procedure. This greedy pro cedure builds a starting species tree stepbystep by add ing one taxon at a time at its locally optimal position. In the context of RF supertrees, our algorithm for the SPRRS problem also yields a 0(n) speedup over naive algorithms for this greedy procedure. Experimental Evaluation In order to evaluate the performance of the RF super tree method, we implemented an RF heuristic based on the SPR local search algorithm. We focused on the SPR local search because it is faster and simpler to imple ment than TBR, and in analyses of MRF and triplet supertrees, the performance of SPR and TBR was very similar [22,38]. We compared the performance of the RF supertree heuristic to MRP and the triplet supertree method (which seeks a supertree with the most shared triplets with the collection of input trees) using pub lished supertree data sets from sea birds [40], marsupials [41], placental mammals [42], and legumes [43]. The published data sets contain between 7 and 726 input trees and between 112 and 571 total taxa (Table 1). There are a number of ways to implement any local search algorithm. Preliminary analyses of the RF heuris tic based on the SPR local search indicated that, as with other phylogenetic methods, the starting tree can affect the estimate of the final supertree. Occasionally the SPR searches got caught in local optima with relatively high RFdistance scores. To ameliorate this potential pro blem, we implemented a ratchet search heuristic for RF supertrees based on the parsimony ratchet [44]. In gen eral, a ratchet search performs a number of iterations  in our case 25 that consist of two local SPR searches: Table 1 Experimental Results Data Set Supertree Method RFDistance Parsimony Score Marsupial (272 RFRatchet 1514 2528 taxa; 158 trees) RFMRP 1502 2513 MRPTBR 1514 2509 MRPRatchet 1514 2509 Triplet 1604 2569 Sea Birds (121 RFRatchet 61 223 taxa; 7 trees) RFMRP 61 223 MRPTBR 63 221 MRPRatchet 63 221 Triplet 61 223 Placental RFRatchet 5686 8926 Mammals (116 RFMRP 5690 8890 taxa; 726 trees) MRPTBR 5694 8878 MRPRatchet 5694 8878 Triplet 6032 9064 Legumes (571 RFRatchet 1556 965 taxa; 22 trees) RFMRP 1534 882 MRPTBR 1554 856 MRPRatchet 1552 854 Triplet N/A N/A Experimental results comparing the performance of the RF supertree method to MRP and triplet supertree methods. We used five different supertree analyses: RF supertrees using our SPR local search algorithm with a ratchet starting from either random addition sequence trees (RFratchet) or MRP trees (RFMRP), MRP with TBR branch swapping with (MRPratchet) and without (MRPTBR) a ratchet search, and triplet supertrees with a TBR local search (Triplet). We measured the RF distance to the collection of input trees (RF distance) and the parsimony score of a best found supertree based on the matrix representation of the input trees. The best RF distance and parsimony scores are in bold. one in which the characters (input trees) are equally weighted, and another in which the set of the characters are reweighted. We reweighted the characters by ran domly removing approximately twothirds of the input trees. The goal of reweighting the characters is to alter the tree space to avoid getting caught in a globally sub optimal part of the tree space. At the end of each itera tion, the best tree is taken as the starting point of the next iteration. For each data set, we started RF ratchet searches from 20 random sequence addition starting trees, and we also ran three replicates starting from an optimal MRP supertree. All RF supertree analyses were performed on an 3 GHz Intel Pentium 4 based desktop computer with 1 GB of main memory. The RFratchet runs took between 5 seconds (for the Sea Birds data set) and 90 minutes (for the legume dataset) when starting from a random sequence addition tree. RFratchet runs starting from optimal MRP trees were at least twice as fast because they required fewer search steps. For our MRP analyses, we also tried two heuristic search methods, both implemented using PAUP* [18]. First, we performed 20 replicates of TBR branch swap ping from trees built with random addition sequence Page 9 of 12 Bansal et al. Algorithms for Molecular Biology 2010, 5:18 http://www.almob.org/content/5/1/18 starting trees. Next, we performed 20 replicates of a par simony ratchet search with TBR branch swapping. Based on the results of trial analyses, each ratchet search con sisted of 25 iterations, each reweighting 15% of the char acters. The PAUP* command block for the parsimony ratchet searches was generated using PAUPRat [45]. For each data set, we performed 20 replicates of a TBR local search heuristic starting with random addition sequence trees. Triplet supertrees were constructed using the pro gram from Lin et al. [38]. We were unable to perform ratchet searches with the existing triplet supertree soft ware, and also, due to memory limitations, we were unable to perform triplet supertree analyses on the legume data set. Our analyses demonstrate the effectiveness of our local search heuristics for the RF supertree problem. In all four data sets, RFratchet searches found the super trees with the lowest total RF distance to the input trees (Table 1). MRP also generally performs well, finding supertrees with RF distances between 0.14% (placental mammals) and 3.3% (sea birds) higher than the best score found by the RF supertree heuristics (Table 1). The triplet supertree method performs as well as the RF supertree method on the small sea bird data set; how ever, the triplet supertrees for the marsupial and placen tal mammal data sets have a much higher RF distance to the input trees than either the RF or MRP supertrees (Table 1). For all the data sets, the MRP supertrees had the lowest (best) parsimony score based on a binary matrix representation of the input trees (Table 1). Thus, not surprisingly, it appears that optimizing based on the parsimony score or the triplet distance to the input trees does not optimize the similarity of the supertrees to the input trees based on the RF distance metric (see also [11,13]). All of the data sets used in this analysis are from pub lished studies that used MRP. Therefore, it is not surpris ing that MRP performed well (but see [46]). Still, our results demonstrate that MRP leaves some room for improvement. If the goal is to find the supertrees that are most similar to the collection of input trees, the RF searches ultimately provide better estimates than MRP (Table 1). Interestingly, while the MRP trees tend to have rela tively low RFdistance scores, in some cases, such as the legume data set, trees with low RFdistance scores have high parsimony scores (Table 1). Thus, parsimony scores are not necessarily indicative of RF score, and MRP and RF supertree optimality criteria are certainly not equivalent. Still, MRP trees appear to be useful as starting points for RF supertree heuristics. Indeed, in three of the four data sets, the best RF trees were found in ratchet searches beginning from MRP trees (Table 1). Our program for computing RF supertrees is freely available (for Windows, Linux, and Mac OS X) at http://genome.cs.iastate.edu/CBL/RFsupertrees Discussion and Conclusion There is a growing interest in using supertrees for large scale evolutionary and ecological analyses. Yet there are many concerns about the performance of existing super tree methods, and the great majority of published super tree analyses have relied on only MRP [47]. Since the goal of a supertree analysis is to synthesize the phyloge netic data from a collection of input trees, it makes sense that an effective supertree method should directly seek the supertree that is most similar to the input trees. Our new algorithms make it possible, for the first time, to estimate large supertrees by directly optimizing the RF distance from the supertree to the input trees. There are numerous alternate metrics to compare phylogenetic trees besides the RF distance, and any of these can be used for supertree methods (see, for exam ple, [11]). Triplet distance supertrees [11,48], quartetfit and quartet joining supertrees [11,24], maximum splits fit supertrees [11], and most similar supertrees [49] are all, like RF supertrees, estimated by comparing input trees to the supertree using tree distance measures. All of these methods may provide different, and perhaps equally valid, perspectives on supertree accuracy. Based on our experimental analyses using the RF and triplet supertree method, optimizing the supertree based on different distance measures can result in very different supertrees (Table 1). In the future, it will be important to characterize and compare the performance of these methods in more detail (see, for example, [11,50]). The results also suggest several future directions for research. Although heuristics guided by local search pro blems, especially SPR and TBR, have been very effective for many intrinsically difficult phylogenetic inference problems, our experiments indicate that the tree space for RF supertrees is complex. The ratchet approach and also starting from MRP trees appears to improve the per formance in the four examples we tested (Table 1). How ever, more work is needed to identify the most efficient ways to implement our fast local search heuristics. Also, the use of alternative supertrees methods (other than MRP) to generate starting trees might result in a better global strategy to compute RF supertrees and this should be investigated further. We note that the ideas presented in [51] can be directly used to perform efficient NNI based local searches for the RF supertree problem. In particular, we can show that heuristic searches for the RF supertree problem, which perform a total of p local search steps based on 1, 2, or 3NNI neighborhoods (see [51]), can all be executed in O(kn(n + p)) time; yielding speedups of 0(min{n, p}), 0(n.min{n, p}) and 0(n2.min Page 10 of 12 Bansal et al. Algorithms for Molecular Biology 2010, 5:18 http://www.almob.org/content/5/1/18 {n, p}) for heuristic searches that are based on naive algo rithms for 1, 2 and 3NNI local searches respectively. It would also be interesting to see if heuristics based on TBR perform significantly better than those based on SPR in inferring RF supertrees. In some cases it might be desirable to remove the restriction that the supertree be binary. In the consensus setting, such a median tree can be obtained within poly nomial time [27]; however, finding a median RF tree in the supertree setting is NPhard [52]. One simple way to estimate a nonbinary median tree could be to first com pute an RF supertree and then to iteratively (and perhaps greedily) contract those edges in the supertree that result in a reduction in the total RF distance. Thus, our algo rithms may even be useful for roughly estimating major ityrule() supertrees [28], which are essentially the strict consensus of all optimal, not necessarily binary, median RF trees, and have several desirable properties [29]. These majorityrule() supertrees are also the strict con sensus of all maximumlikelihood supertrees [53]. Also, there are several alternate forms of the RF distance metric that could be incorporated into our local search algorithms. For example, in order to account for biases associated with the different sizes of input trees, we could normalize the RF distance for each input tree, dividing the observed RF distance by the maximum pos sible RF distance based on the tree size. Similarly, we could incorporate either branch length data or phyloge netic support scores (bootstrap values or posterior prob abilities) from the input trees into the RF distance in order to give more weight to partitions that are strongly supported or separated by long branches (e.g., [25,54]). Our current implementation essentially treats all branch lengths as one and all partitions as equal. The addition of branch length or support data may further improve the accuracy of the RF supertree method. Acknowledgements We thank Harris Lin for providing software for the triplet supertree analyses This work was supported in part by NESCent and by NSF grants DEB 0334832 and DEB0829674 MSB was supported in part by a postdoctoral fellowship from the Edmond J Safra Bioinformatics program at TelAviv university Author details ,The Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel Department of Computer Science, Iowa State University, Ames, A 50011, USA Department of Biology, University of Florida, Gainesville, FL 32611, USA Authors' contributions MSB was responsible for algorithm design and program implementation, contributed to the experimental evaluation, and wrote major parts of the manuscript JGB performed the experimental evaluation and the analysis of the results, and contributed to the writing of the manuscript OE and DFB supervised the project and contributed to the writing of the manuscript Al authors read and approved the final manuscript Competing interests The authors declare that they have no competing interests Received: 27 June 2009 Accepted: 24 February 2010 Published: 24 February 2010 References S Davies TJ, Barraclough TG, Chase MW, Soltis PS, Soltis DE, Savolainen V Darwin's abominable mystery: Insights from a supertree of the angiosperms. 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Mol Biol Evol 1994, 11 (3) 459468 doi:10.1186/17487188518 Cite this article as: Bansal et alo RobinsonFoulds Supertrees. Algorithms for Molecular Biology 2010 5'18 O BloMed Central Page 12 of 12 Submit your next manuscript to BioMed Central and take full advantage of: * Convenient online submission * Thorough peer review * No space constraints or color figure charges * Immediate publication on acceptance * Inclusion in PubMed, CAS, Scopus and Google Scholar * Research which is freely available for redistribution Submit your manuscript at www.biomedcentral.com/submit 