An efficient algorithm for testing the compatibility of phylogenies with nested taxa

Background

The tree compatibility problem is a basic special case of the supertree problem. A supertree method is a way to synthesize a collection of phylogenetic trees with partially overlapping taxon sets into a single supertree that represents the information in the input trees. The supertree approach, proposed in the early 90s [5, 6], has been used successfully to build large-scale phylogenies [7].

The original supertree methods were limited to input trees where only the leaves are labeled. Page [8] was among the first to note the need to handle phylogenies where internal nodes are labeled, and taxa are nested. A major motivation is the desire to incorporate taxonomies as input trees in large-scale supertree analyses, as way to circumvent one of the obstacles to building comprehensive phylogenies: the limited taxonomic overlap among different phylogenetic studies [9]. Taxonomies group organisms according to a system of taxonomic rank (e.g., family, genus, and species); two examples are the NCBI taxonomy [10] and the Angiosperm taxonomy [11]. Taxonomies spanning a broad range of taxa provide structure and completeness that might be hard to obtain otherwise. A recent example of the utility of taxonomies is the Open Tree of Life, a draft phylogeny for over 2.3 million species [12].

Taxonomies are not, strictly speaking, phylogenies. In particular, their internal nodes and some of their leaves are labeled with higher-order taxa. Nevertheless, taxonomies have many of the same mathematical characteristics as phylogenies. Indeed, both phylogenies and taxonomies are semi-labeled trees [13, 14]. We will use this term throughout the rest of the paper to refer to trees with nested taxa.

The fastest previous algorithm for testing ancestral compatibility, based on earlier work by Daniel and Semple [15], is due to Berry and Semple [16]. Their algorithm runs in \(O \left( \log ^2 n \cdot \tau _\mathcal {P}\right)\) time using \(O \left( \tau _\mathcal {P}\right)\) space. Here, n is the number of distinct taxa in \(\mathcal {P}\) and \(\tau _\mathcal {P}= \sum _{i = 1}^k \sum _{v \in I(\mathcal {T}_i)} d(v)^2\), where \(I(\mathcal {T}_i)\) is the set of internal nodes of \(\mathcal {T}_i\), for each \(i \in \{1, \ldots , k\}\), and d(v) is the degree of node v. While the algorithm is polynomial, its dependence on node degrees is problematic: semi-labeled trees can be highly unresolved (i.e., contain nodes of high degree), especially if they are taxonomies.

Our contributions

As stated earlier, our main result is an algorithm to test ancestral compatibility that runs in \(O(M_{\mathcal {P}}\log ^2 M_{\mathcal {P}})\) time, using \(O(M_{\mathcal {P}})\) space. These bounds are independent of the degrees of the nodes of the input trees, a valuable characteristic for large datasets that include taxonomies. To achieve our result, we extend ideas from our recent algorithm for testing the compatibility of ordinary phylogenetic trees [2]. As in that algorithm, a central notion in the current paper is the display graph of profile \(\mathcal {P}\), denoted \(H_{\mathcal {P}}\). This is the graph obtained from the disjoint union of the trees in \(\mathcal {P}\) by identifying nodes that have the same label (see the section titled  "Testing ancestral compatibility"). The term “display graph” was introduced by Bryant and Lagergren [17], but similar ideas have been used elsewhere. In particular, the display graph is closely related to Berry and Semple’s restricted descendancy graph [16], a mixed graph whose directed edges correspond to the (undirected) edges of \(H_{\mathcal {P}}\) and whose undirected edges have no correspondence in \(H_{\mathcal {P}}\). The second kind of edges are the major component of the \(\tau _\mathcal {P}\) term in the time and space complexity of Berry and Semple’s algorithm. The absence of such edges makes \(H_{\mathcal {P}}\) significantly smaller than the restricted descendancy graph. Display graphs also bear some relation to tree alignment graphs [18].

Here, we exploit the display graph more extensively than in our previous work. Although the display graph of a collection of semi-labeled trees is more complex than that of a collection of ordinary phylogenies, we are able to extend several of the key ideas—notably, that of a semi-universal label—to the general setting of semi-labeled trees. As in [2], the implementation relies on a dynamic graph data structure, but it requires a more careful amortized analysis based on a weighing scheme.

Contents

This paper has five sections, in addition to this introduction. The section titled "Preliminaries" presents basic definitions regarding graphs, semi-labeled trees, and ancestral compatibility. The section titled "The display graph" introduces the display graph and discusses its properties. The section titled "Testing ancestral compatibility" presents \(\mathrm{BuildNT}\), our algorithm for testing ancestral compatibility. We first present the algorithm recursively, and then show how to transform it into an iterative algorithm, \(\mathrm{BuildNT}_\mathrm{N}\), that is easier to implement. We also give an example of the execution of \(\mathrm{BuildNT_N}\). The "Implementation" section gives the implementation details for \(\mathrm{BuildNT_N}\). The "Discussion" section gives some concluding remarks.

Graph notation

Let G be a graph. V(G) and E(G) denote the node and edge sets of G. The degree of a node \(v \in V(G)\) is the number of edges incident on v. A tree is an acyclic connected graph. In this paper, all trees are assumed to be rooted. For a tree T, r(T) denotes the root of T. Suppose \(u, v \in V(T)\). Then, u is an ancestor of v in T, denoted \(u \le _T v\), if u lies on the path from v to r(T) in T. If \(u \le _T v\), then v is a descendant of u. Node u is a proper descendant of v if u is a descendant of v and \(v \ne u\). If \(\{u,v\} \in E(T)\) and \(u \le _T v\), then u is the parent of v and v is a child of u. If neither \(u \le _T v\) nor \(v \le _T u\) hold, then we write \(u \parallel _T v\) and say that u and v are not comparable in T.

Semi-labeled trees

A semi-labeled tree is a pair \(\mathcal {T}= (T,\phi )\) where T is a tree and \(\phi\) is a mapping from a set \(L(\mathcal {T})\) to V(T) such that, for every node \(v \in V(T)\) of degree at most two, \(v \in \phi (L(\mathcal {T}))\). \(L(\mathcal {T})\) is the label set of \(\mathcal {T}\) and \(\phi\) is the labeling function of \(\mathcal {T}\).

For every node \(v \in V(T)\), \(\phi ^{-1}(v)\) denotes the (possibly empty) subset of \(L(\mathcal {T})\) whose elements map into v; these elements as the labels of v (thus, each label is a taxon). If \(\phi ^{-1}(v) \ne \emptyset\), then v is labeled; otherwise, v is unlabeled. Note that, by definition, every leaf in a semi-labeled tree is labeled. Further, any node, including the root, that has a single child must be labeled. Nodes with two or more children may be labeled or unlabeled. A semi-labeled tree \(\mathcal {T}= (T,\phi )\) is singularly labeled if every node in T has at most one label; \(\mathcal {T}\) is fully labeled if every node in T is labeled.

Semi-labeled trees, also known as X -trees, generalize ordinary phylogenetic trees, also known as phylogenetic X -trees [14]. An ordinary phylogenetic tree is a semi-labeled tree \(\mathcal {T}= (T,\phi )\) where r(T) has degree at least two and \(\phi\) is a bijection from \(L(\mathcal {T})\) into leaf set of T (thus, internal nodes are not labeled).

Let \(\mathcal {T}= (T,\phi )\) be a semi-labeled tree and let \(\ell\) and \(\ell '\) be two labels in \(L(\mathcal {T})\). If \(\phi (\ell ) \le _T \phi (\ell ')\), then we write \(\ell \le _\mathcal {T}\ell '\), and say that \(\ell '\) is a descendant of \(\ell\) in \(\mathcal {T}\) and that \(\ell\) is an ancestor of \(\ell '\). We write \(\ell <_\mathcal {T}\ell '\) if \(\phi (\ell ')\) is a proper descendant of \(\phi (\ell )\). If \(\phi (\ell ) \parallel _T \phi (\ell ')\), then we write \(\ell \parallel _\mathcal {T}\ell '\) and say that \(\ell\) and \(\ell '\) are not comparable in \(\mathcal {T}\). If \(\mathcal {T}\) is fully labeled and \(\phi (\ell )\) is the parent of \(\phi (\ell ')\) in T, then \(\ell\) is the parent of \(\ell '\) in \(\mathcal {T}\) and \(\ell '\) is a child of \(\ell\) in \(\mathcal {T}\); two labels with the same parent are siblings.

Two semi-labelled trees \(\mathcal {T}= (T,\phi )\) and \(\mathcal {T}' = (T', \phi ')\) are isomorphic if there exists a bijection \(\psi : V(T) \rightarrow V(T')\) such that \(\phi ' = \psi \circ \phi\) and, for any two nodes \(u, v \in V(T)\), \((u,v) \in E(T)\) if and only \((\psi (u), \psi (v)) \in E(T')\).

Let \(\mathcal {T}= (T,\phi )\) be a semi-labeled tree. For each \(u \in V(T)\), X(u) denotes the set of all labels in the subtree of T rooted at u; that is, \(X(u) = \bigcup _{v: u \le _T v} \phi ^{-1}(v)\). X(u) is called a cluster of T. \({\mathrm{Cl}}(\mathcal {T})\) denotes the set of all clusters of \(\mathcal {T}\). It is well known [14, Theorem 3.5.2] that a semi-labeled tree \(\mathcal {T}\) is completely determined by \({\mathrm{Cl}}(\mathcal {T})\). That is, if \({\mathrm{Cl}}(\mathcal {T}) = {\mathrm{Cl}}(\mathcal {T}')\) for some other semi-labeled tree \(\mathcal {T}'\), then \(\mathcal {T}\) is isomorphic to \(\mathcal {T}'\).

Suppose \(A \subseteq L(\mathcal {T})\) for a semi-labeled tree \(\mathcal {T}= (T,\phi )\). The restriction of \(\mathcal {T}\) to A, denoted \(\mathcal {T}|A\), is the semi-labeled tree whose cluster set is \({\mathrm{Cl}}(\mathcal {T}| A) = \{X \cap A : X \in {\mathrm{Cl}}(\mathcal {T}) \text { and } X \cap A \ne \emptyset \}.\) Intuitively, \(\mathcal {T}| A\) is obtained from the minimal rooted subtree of T that connects the nodes in \(\phi (A)\) by suppressing all vertices of degree two that are not in \(\phi (A)\).

Profiles and ancestral compatibility

Throughout the rest of this paper \(\mathcal {P}= \{\mathcal {T}_1, \mathcal {T}_2, \ldots , \mathcal {T}_k\}\) denotes a set where, for each \(i \in [k]\), \(\mathcal {T}_i = (T_i, \phi _i)\) is a semi-labeled tree. We refer to \(\mathcal {P}\) as a profile, and write \(L(\mathcal {P})\) to denote \(\bigcup _{i\in [k]} L(\mathcal {T}_i)\), the label set of \(\mathcal {P}\). Figure 1 shows a profile where \(L(\mathcal {P}) = \{a,b,c,d,e,f,g,h,i\}\). We write \(V(\mathcal {P})\) for \(\bigcup _{i\in [k]} V(T_i)\) and \(E(\mathcal {P})\) for \(\bigcup _{i\in [k]} E(T_i)\), The size of \(\mathcal {P}\) is \(M_{\mathcal {P}}= |V(\mathcal {P})| + |E(\mathcal {P})|\).

\(\mathcal {P}\) is ancestrally compatible if there is a rooted semi-labeled tree \(\mathcal {T}\) that ancestrally displays each of the trees in \(\mathcal {P}\). If \(\mathcal {T}\) exists, we say that \(\mathcal {T}\) ancestrally displays \(\mathcal {P}\) (see Fig. 2).

For technical reasons, fully labeled trees are easier to handle than those that are not. Suppose \(\mathcal {P}\) contains trees that are not fully labeled. We can convert \(\mathcal {P}\) into an equivalent profile \(\mathcal {P}'\) of fully-labeled trees as follows. For each \(i \in [k]\), let \(l_i\) be the number of unlabeled nodes in \(T_i\). Create a set \(L'\) of \(n' = \sum _{i \in [k]} l_i\) labels such that \(L' \cap L(\mathcal {P}) = \emptyset\). For each \(i \in [k]\) and each \(v \in V(T_i)\) such that \(\phi _i^{-1}(v) = \emptyset\), make \(\phi _i^{-1}(v) = \{\ell \}\), where \(\ell\) is a distinct element from \(L'\). We refer to \(\mathcal {P}'\) as the profile obtained by adding distinct new labels to \(\mathcal {P}\) (see Fig. 1).

From this point forward, we make the following assumption.

By Lemma 3, no generality is lost in assuming that all trees in \(\mathcal {P}\) are fully labeled. The assumption that the trees are singularly labeled is inessential; it is only for clarity. Note that, even with the latter assumption, a tree that ancestrally displays \(\mathcal {P}\) is not necessarily singularly labeled. Figure 2 illustrates this fact.

Positions

Our compatibility algorithm processes the trees in \(\mathcal {P}\) from the top down, starting at the roots. We refer to the set of nodes in \(\mathcal {P}\) currently being considered as a “position”. The algorithm advances from the current position to the next by replacing certain nodes in the current position by their children. Formally, a position (for \(\mathcal {P}\)) is a vector \(U = (U(1), U(2), \ldots , U(k))\), where \(U(i) \subseteq L(\mathcal {T}_i)\), for each \(i \in [k]\). Since labels may be shared among trees, we may have \(U(i) \cap U(j) \ne \emptyset\), for \(i, j \in [k]\) with \(i \ne j\). For each \(i \in [k]\), let \(\mathrm{Desc}_i(U) = \{\ell : \ell ' \le _{\mathcal {T}_i} \ell , \text { for some } \ell ' \in U(i)\}\), and let \(\mathrm{Desc}_\mathcal {P}(U) = \bigcup _{i \in [k]} \mathrm{Desc}_i(U)\).

For any valid position U, \(H_{\mathcal {P}}(U)\) denotes the subgraph of \(H_{\mathcal {P}}\) induced by \(\mathrm{Desc}_\mathcal {P}(U)\).

Semi-universal labels

Let U be a valid position, and let \(\ell\) be a label in U. Then, \(\ell\) is semi-universal in U if \(U(i) = \{\ell \}\), for every \(i \in [k]\) such that \(\ell \in L(\mathcal {T}_i)\). In Fig. 3, labels 1 and 2 are semi-universal in \(U_\mathrm{root}\), but g is not, since g is in both \(L(\mathcal {T}_2)\) and \(L(\mathcal {T}_3)\), but \(U_\mathrm{root}(2) \ne \{g\}\).

The term “semi-universal”, borrowed from Pe’er et al. [19], derives from the following fact. Suppose that \(\mathcal {P}\) is ancestrally compatible, that \(\mathcal {T}\) is a tree that ancestrally displays \(\mathcal {P}\), and that \(\ell\) is a semi-universal label for some valid position U. Then, as we shall see, \(\ell\) must label the root \(u_\ell\) of a subtree of \(\mathcal {T}\) that contains all the descendants of \(\ell\) in \(\mathcal {T}_i\), for every i such that \(\ell \in L(\mathcal {T}_i)\). The qualifier “semi” is because this subtree may also contain labels that do not descend from \(\ell\) in any input tree, but descend instead from some other semi-universal label \(\ell '\) in U. In this case, \(\ell '\) also labels \(u_\ell\). We exploit this property of semi-universal labels in our ancestral compatibility algorithm and its proof of correctness (see "Testing ancestral compatibility").

For each label \(\ell \in L(\mathcal {P})\), let \(k_\ell\) denote the number of input trees that contain label \(\ell\). We can obtain \(k_\ell\) for every \(\ell \in L(\mathcal {P})\) in \(O(M_{\mathcal {P}})\) time during the construction of \(H_{\mathcal {P}}\).

Successor positions

Let U be a valid position, and W be a subset of \(\mathrm{Desc}_\mathcal {P}(U)\). Then, U|W denotes the position \((U(1) \cap W, U(2) \cap W, \ldots , U(k) \cap W)\). In Fig. 3, the components of \(H_{\mathcal {P}}(U')\), where \(U'\) is the successor of \(U_\mathrm{root}\), are \(W_1 = \{3,4,a,b,c,d,e,g\}\) and \(W_2 = \{f,h,i\}\). Thus, \(U' | W_1 = (\{3\}, \{e,g\}, \{g\})\) and \(U' | W_2 = (\{f\}, \{f\}, \emptyset )\). We have the following result.

Overview of the algorithm

\(\mathrm{BuildNT}\) (Algorithm 1) is our algorithm for testing compatibility of semi-labeled trees. Its argument, U, is a valid position in \(\mathcal {P}\) such that \(H_{\mathcal {P}}(U)\) is connected. \(\mathrm{BuildNT}\) relies on the fact—proved later, in Theorem 1—that if \(\mathcal {P}|\mathrm{Desc}_\mathcal {P}(U)\) is compatible, then U must contain a nonempty set S of semi-universal labels. If such a set S exists, the algorithm replaces U by its successor \(U'\) with respect to S. It then processes each connected component of \(H_{\mathcal {P}}(U')\) recursively, to determine if the associated sub-profile is compatible. If all the recursive calls are successful, then their results are combined into a supertree for \(\mathcal {P}|\mathrm{Desc}_\mathcal {P}(U)\).

In detail, \(\mathrm{BuildNT}\) proceeds as follows. Line 1 computes the set S of semi-universal labels in U. If S is empty, then, \(\mathcal {P}|\mathrm{Desc}_\mathcal {P}(U)\) is incompatible, and, thus, so is \(\mathcal {P}\). This fact is reported in Line 3. Line 4 creates a tentative root \(r_U\), labeled by S, for the tree \(\mathcal {T}_U\) for L(U). Line 5 checks if S contains exactly one label \(\ell\), with no proper descendants. If so, by the connectivity assumption, \(\ell\) must be the sole member of \(\mathrm{Desc}_\mathcal {P}(U)\); that is, \(L(U) = \ell\). Therefore, Line 6 simply returns the tree with a single node, labeled by \(S = \{\ell \}\). Line 7 updates U, replacing it by its successor with respect to S. Let \(W_1, W_2, \dots , W_p\) be the connected components of \(H_{\mathcal {P}}(U)\) after updating U. By Lemma 6, \(U | W_j\) is a valid position, for each \(j \in [p]\). Lines 8–12 recursively invoke \(\mathrm{BuildNT}\) on \(U | W_j\) for each \(j \in [p]\), to determine if there is a tree \(t_j\) that ancestrally displays \(\mathcal {P}| \mathrm{Desc}_\mathcal {P}(U \cap W_j)\). If any subproblem is incompatible, Line 12 reports that \(\mathcal {P}\) is incompatible. Otherwise, Line 13 returns the tree obtained by making the \(t_j\)s the subtrees of root \(r_U\).

Next, we argue the correctness of \(\mathrm{BuildNT}\).

Correctness

An iterative version

We now present \(\mathrm {BuildNT}_\mathrm {N}\) (Algorithm 2), an iterative version of \(\mathrm {BuildNT}\), which lends itself naturally to an efficient implementation. \(\mathrm{BuildNT_N}\) performs a breadth-first traversal of \(\mathrm{BuildNT}\)’s recursion tree, using a first-in first-out queue Q that stores pairs of the form \((U, \mathrm{pred})\), where U is a valid position in \(\mathcal {P}\) and \(\mathrm{pred}\) is a reference to the parent of the node corresponding to U in the supertree built so far. \(\mathrm{BuildNT_N}\) simulates recursive calls in \(\mathrm{BuildNT}\) by enqueuing pairs corresponding to subproblems. We explain this in more detail next.

\(\mathrm {BuildNT_N}\) initializes its queue to contain the starting position, \(U_\mathrm{root}\), with a null parent. It then proceeds to the while loop of Lines 3–14. Each iteration of the loop starts by dequeuing a valid position U, along with a reference \(\mathrm{pred}\) to the potential parent for the subtree for L(U) in the supertree. The body of the loop closely follows the steps performed by a call to \(\mathrm{BuildNT}(U)\). Line 5 computes the set S of semi-universal labels in U. If S is empty, the algorithm reports that \(\mathcal {P}\) is incompatible and terminates (Lines 6–7). The algorithm then creates a tentative root \(r_U\) labeled by S for the tree \(\mathcal {T}_U\) for L(U), and links \(r_U\) to its parent (Line 8). If S consists of exactly one element that has no proper descendants, we skip the rest of the current iteration of the while loop, and continue to the next iteration (Lines 9–10). Line 11 replaces U by its successor with respect to S. Lines 13–14 enqueue each of \(U|W_1, U|W_2, \ldots , U|W_p\), along with \(r_U\), for processing in a subsequent iteration. If the while loop terminates without any incompatibility being detected, the algorithm returns the tree with root \(r_{U_\mathrm{root}}\).

Although the order in which \(\mathrm{BuildNT_N}\) processes connected components differs from that of \(\mathrm {BuildNT}\) —breadth-first instead of depth-first—, it is straightforward to see that the effect is equivalent, and the proof of correctness of \(\mathrm{BuildNT}\) (Theorem 1) applies to \(\mathrm{BuildNT_N}\) as well. We thus state the following without proof.

Let Q be \(\mathrm{BuildNT_N}\)’s first-in first-out queue. In the rest of the paper, we will say that a valid position U is in Q if \((U,\mathrm{pred}) \in Q, \text { for some } \mathrm{pred}.\) Let \(H_Q\) be the subgraph of \(H_{\mathcal {P}}\) induced by \(\bigcup \{\mathrm{Desc}(U): U \text { is in } Q\}.\) By Observation 1, \(H_Q\) is obtained from \(H_{\mathcal {P}}\) through edge and node deletions.

In other words, Lemma 8 states that each iteration of \(\mathrm{BuildNT_N}(\mathcal {P})\) deals with a subgraph of \(H_{\mathcal {P}}\), whose connected components are in one-to-one correspondence with the valid positions stored in Q. This is illustrated by the next example.

An example

Figures 4, 5, 6, 7 and 8 illustrate the execution of \(\mathrm{BuildNT_N}\) on the profile \(\mathcal {P}= (\mathcal {T}_1, \mathcal {T}_2, \mathcal {T}_3)\) of Fig. 1. The figures show how the graph \(H_Q\) —initially equal to \(H_{\mathcal {P}}= H_{\mathcal {P}}(U_\mathrm{root})\) (Fig. 3)—evolves as its edges and nodes are deleted.

In each figure, \(H_Q\) is shown on the left and the current supertree is shown on the right. For brevity, the figures only exhibit the state of \(H_Q\) and the supertree after all the nodes at each level are generated. The various valid positions processed by \(\mathrm{BuildNT_N}(\mathcal {P})\) are denoted by \(U_\alpha\), for different subscripts \(\alpha\); \(S_\alpha\) denotes the semi-universal labels in \(U_\alpha\), and \(U_\alpha '\) denotes the successor of \(U_\alpha\) with respect to \(S_\alpha\). We write \(L_\alpha\) as an abbreviation for \(L(U_\alpha )\) The root of the tree for \(L_\alpha\) is \(r_{U_\alpha }\) and is labeled by \(S_\alpha\).

Initially, \(Q = ((U_\mathrm{root},\mathtt {null}))\). In what follows, the elements of Q are listed from front to rear.

Level 1. Refer to Fig. 5. We have \(S_1 = \{3\}\), so \(H_{\mathcal {P}}(U_1')\) has two components \(W_{11}\) and \(W_{12}\). Let \(U_{11} = U_1'|W_{11}\) and \(U_{12} = U_1'|W_{12}\). Then,

$$\begin{aligned} U_{11} = (\{a,d\}, \{g\},\{g\}) \quad \text {and} \quad U_{12} = (\{e\}, \{e\},\emptyset ). \end{aligned}$$

Level 2. Refer to Fig. 6. We have \(S_{11} = \{g\}\), so \(H_{\mathcal {P}}(U_{11}')\) has two components \(W_{111}\) and \(W_{112}\). Let \(U_{111} = U_{11}'|W_{111}\) and \(U_{112} = U_{11}'|W_{112}\). Then,

$$\begin{aligned} U_{111} = (\{a\}, \{a\},\{4\}) \quad \text {and} \quad U_{112} = (\emptyset , \{d\},\{d\}). \end{aligned}$$

The only semi-universal labels in \(U_{12}\), \(U_{21}\), and \(U_{22}\) are, respectively, e, h, and i. Since none of these labels have proper descendants, each of them is a leaf in the supertree.

After level 2 is processed, \(Q = ((U_{111},r_{11}), (U_{112},r_{11}))\).

Level 3. Refer to Fig. 7. We have \(S_{111} = \{4, a\}\), so \(H_{\mathcal {P}}(U_{111}')\) has two components \(W_{1111}\) and \(W_{1112}\). Let \(U_{1111} = U_{111}'|W_{1111}\) and \(U_{1112} = U_{111}'|W_{1112}\). Then,

$$\begin{aligned} U_{1111} = (\{b\}, \emptyset ,\{b\}) \quad \text {and} \quad U_{1112} = (\{c\}, \emptyset ,\{c\}). \end{aligned}$$

The only semi-universal label in \(U_{112}\) is d. Since d has no proper descendants, it becomes a leaf in the supertree.

After level 3 is processed, \(Q = ((U_{1111},r_{111}), (U_{1112},r_{111}))\).

Level 4. Refer to Fig. 8. The only semi-universal labels in \(U_{1111}\) and \(U_{1112}\) are, respectively, b and c. Since neither of these labels have proper descendants, each of them is a leaf in the supertree.

After level 4 is processed, Q is empty, and \(\mathrm{BuildNT_N}(\mathcal {P})\) terminates.

Initializing the data fields

Maintaining the data fields

Suppose that all data fields fields are correctly computed for every connected component that is in Q at the beginning of an iteration of the while loop in 3–14 of \(\mathrm{BuildNT_N}\). We now show how to generate the same fields efficiently for the new connected components created by Line 11.

Summary