Optimal Algorithm to Reconstruct a Tree from a Subtree Distance

TL;DR

This paper presents a linear-time algorithm for reconstructing a tree from a subtree distance, extending the tree metric concept, and proves the uniqueness of the minimal representation.

Contribution

It introduces a novel linear-time algorithm for reconstructing a tree from subtree distances and establishes the uniqueness of the minimal representation.

Findings

The algorithm runs in linear time.

The minimal representation of a subtree distance is unique.

The method extends the tree metric concept.

Abstract

This paper addresses the problem of finding a representation of a subtree distance, which is an extension of the tree metric. We show that a minimal representation is uniquely determined by a given subtree distance, and give a linear time algorithm that finds such a representation. This algorithm achieves the optimal time complexity.

Equations

$$d(x,y)=d_{\mathcal{T}}(\psi (x),\psi (y))(x,y\in X),$$

$$d(x,y)+d(z,w)\le \max \{d(x,z)+d(y,w),d(x,w)+d(y,z)\}$$

$$d(x,y)=d_{\mathcal{T}}(\varphi (x),\varphi (y))(x,y\in X),$$

$$\displaystyle d(x,y)+d(z,w)\leq\max\left\{\begin{array}[]{l}d(x,z)+d(y,w),d(x,w)+d(y,z),\\ d(x,y),d(z,w),\\ \frac{d(x,y)+d(y,z)+d(z,x)}{2},\frac{d(x,y)+d(y,w)+d(w,x)}{2},\\ \frac{d(x,z)+d(z,w)+d(w,x)}{2},\frac{d(y,z)+d(z,w)+d(w,y)}{2}\end{array}\right\}$$

$$I(r,a;x,b)=\{u\in \text{path}(\varphi (r),\varphi (x)):d_{\mathcal{T}}(\varphi (r),u)\ge a,d_{\mathcal{T}}(\varphi (x),u)\ge b\}.$$

$$\overline{\varphi (z)}=\bigcup _{x\in L\setminus \{r\}}I(r,d(r,z);x,d(x,z)).$$

$$\overline{\varphi (z)}$$

$$=\bigcup _{x\in L\setminus \{r\}}\left(\text{path}(\varphi (r),\varphi (x))\cap \overline{\varphi (z)}\right)$$

$$=\bigcup _{x\in L\setminus \{r\}}I(r,d(r,z);x,d(x,z)).$$

Full text

1 Introduction and the results

A phylogenetic tree represents an evolutionary relationship among the species that are investigated. Estimating a phylogenetic tree from experimental data is a fundamental problem in phylogenetics [8]. One of the commonly used approaches to achieve this task is the use of a distance-based method. In this approach, we first compute the dissimilarity (i.e., a nonnegative and symmetric function) between the species by, e.g., the edit distance between the genome sequences. Then, we find a weighted tree having the shortest path distance that best fits the given dissimilarity. The most popular method for this approach is the neighbor-joining method [6].

A weighted tree $\mathcal{T}$ is specified by the set of vertices $\mathcal{V}(\mathcal{T})$, the set of edges $\mathcal{E}(\mathcal{T})$, and the nonnegative edge weight $l:\mathcal{E}(\mathcal{T})\to\mathbb{R}_{+}$. Let us consider the case in which a given dissimilarity $d:X\times X\to\mathbb{R}$ exactly fits some weighted tree; i.e., there exists a weighted tree $\mathcal{T}$ and a mapping $\psi:X\to\mathcal{V}(\mathcal{T})$ such that

[TABLE]

where $d_{\mathcal{T}}(u,v)$ is the distance between $u$ and $v$ in $\mathcal{T}$ for $u,v\in\mathcal{V}(\mathcal{T})$. In this case, the dissimilarity $d:X\times X\to\mathbb{R}$ is called a tree metric, and the pair $(\mathcal{T},\psi)$ is called a representation of $d$. It is known that a dissimilarity $d$ is a tree metric if and only if it satisfies an inequality called the four-point condition [9, 2], which is given by

[TABLE]

for any $x,y,z,w\in X$. For any tree metric $d$, there exists a unique minimal representation $(\mathcal{T},\psi)$ of $d$ [2]. Here, a representation is minimal if there is no representation $(\mathcal{T}^{\prime},\psi^{\prime})$ of $d$, such that $\mathcal{T}^{\prime}$ is obtained by removing some vertices and edges and/or by contracting some edges of $\mathcal{T}$ (i.e., $\mathcal{T}^{\prime}$ is a proper minor of $\mathcal{T}$). Furthermore, such a representation is constructed in $O(n^{2})$ time [3], where $n=|X|$.

In some applications, we are interested in the distance between groups of species (e.g., genus, tribe, or family). In such a case, we aim to identify a group as a connected subgraph in a phylogenetic tree. The subtree distance, which was introduced by Hirai [5], is an extension of the tree metric that can be adopted for use in such situations. A function $d:X\times X\to\mathbb{R}$ is called a subtree distance if there exists a weighted tree $\mathcal{T}$ and a mapping $\phi:X\to 2^{\mathcal{V}(\mathcal{T})}$ such that $\phi(x)$ induces a subtree of $\mathcal{T}$ (i.e., a connected subgraph of $\mathcal{T}$) for $x\in X$ and equations

[TABLE]

hold, where $d_{\mathcal{T}}(U,W)=\min\{d_{\mathcal{T}}(u,w)\mid u\in U,w\in W\}$ for $U,W\subseteq\mathcal{V}(\mathcal{T})$. We say that a pair $(\mathcal{T},\phi)$ is a representation of $d$. Note that a subtree distance is not necessarily a metric because it may not satisfy the non-degeneracy ($d(x,y)>0$ for $x\neq y$) and the triangle inequality ($d(x,z)\leq d(x,y)+d(y,z)$). Hirai proposed a characterization of subtree distances in which a dissimilarity $d:X\times X\to\mathbb{R}$ is a subtree distance if and only if it satisfies an inequality called the extended four-point condition, which is given by

[TABLE]

for any $x,y,z,w\in X$. The extended four-point condition yields an $O(n^{4})$ time algorithm to recognize a subtree distance.

This paper addresses the following problem, which we call the subtree distance reconstruction problem.

Problem 1**.**

Given a subtree distance $d:X\times X\to\mathbb{R}$ on a finite set $X$, find a representation $(\mathcal{T},\phi)$ of $d$.

Ando and Sato [1] proposed an $O(n^{3})$ time algorithm for this problem. Their algorithm consists of three steps: (1) identify a subset $V_{0}=\{x\in X\mid d(y,z)\leq d(x,y)+d(x,z)\ (y,z\in X)\}$, (2) find a representation $(\mathcal{T},\phi)$ for the restriction of $d$ onto $V_{0}$, and (3) for $x\in X\setminus V_{0}$, locate $\phi(x)$ in $\mathcal{T}$ by examining a connected components of $\mathcal{T}\setminus\phi(x)$.

In this study, we propose the following theorems. We define a minimal representation for a subtree distance in the same manner as a minimal representation for a tree metric.

Theorem 2**.**

For a subtree distance $d:X\times X\to\mathbb{R}$, a minimal representation $(\mathcal{T},\phi)$ is uniquely determined by $d$.

Theorem 3**.**

There exists an $O(n^{2})$ time algorithm that finds, for any subtree distance $d:X\times X\to\mathbb{R}$, its unique minimal representation, where $n=|X|$.

The proof of Theorem 3 is constructive. Similar to [1], our algorithm consists of three parts: (1) identify the set of objects $L$ that are mapped to the leaves, (2) find the minimal representation $(\mathcal{T},\phi)$ for the restriction of $d$ onto $L$, and (3) for $x\in X\setminus L$, locate $\phi(x)$ in ${\mathcal{T}}$ by measuring the distances from the leaves. Since Steps 1 and 3 can be implemented with a time complexity of $O(n^{2})$, and there is an $O(n^{2})$ time algorithm for Step 2 [3], the total time complexity of the algorithm is $O(n^{2})$. Note that even if we know $d\colon X\times X\to\mathbb{R}$ is a tree metric, $\Omega(n^{2})$ time is required to reconstruct a tree [4]. Therefore, our algorithm achieves the optimal time complexity.

This algorithm can also be used to recognize a subtree distance by checking the failure or inconsistency during the process and by verifying equations (1.3) after the reconstruction.

Corollary 4**.**

There exists an $O(n^{2})$ time algorithm that determines whether a given input $d:X\times X\to\mathbb{R}$ is a subtree distance or not, where $n=|X|$.

2 Proofs

We assume that there are no objects $x,y\in X$ such that $d(x,z)=d(y,z)$ for all $z\in X$. This assumption is satisfied by removing such elements after lexicographic sorting, which requires $O(n^{2})$ time [7]. Clearly, this preprocessing does not change the minimal representation. We also assume that $|X|\geq 3$. Otherwise, the theorems trivially hold.

First, we prove Theorem 2. We identify the properties of a minimal representation.

Lemma 5**.**

Let $(\mathcal{T},\phi)$ be a minimal representation of a subtree distance $d:X\times X\to\mathbb{R}$. Then, the following properties hold.

For each edge $e\in\mathcal{E}(\mathcal{T})$, the length of $e$ is positive. 2. 2.

For each leaf vertex $u\in\mathcal{V}(\mathcal{T})$, there exists $x\in X$ such that $\phi(x)=\{u\}$.

Proof.

1. If there is an edge $e$ of zero length, we can contract $e$ from the representation.

2. Let $u$ be a leaf vertex of $\mathcal{T}$. If there is no $x\in X$ with $u\in\phi(x)$, we can remove $u$ from the representation to obtain a smaller representation. Suppose that, for all $x\in X$ with $u\in\phi(x)$, $\phi(x)$ contains at least two elements. Then, these $\phi(x)$s contain the unique adjacent vertex $v$ of $u$. Since $d(v,w)\leq d(u,w)$ for all $w\in\mathcal{V}(\mathcal{T})\setminus\{u\}$, $u$ does not contribute any shortest paths in the tree. Therefore, we can remove $u$ from the representation. ∎

Motivated by Property 2 in Lemma 5, we introduce the following definition. For a minimal representation $(\mathcal{T},\phi)$, an object $x\in X$ is a leaf object if $\phi(x)=\{u\}$ for some leaf $u\in\mathcal{V}(\mathcal{T})$.

We defined a leaf object by specifying a minimal representation. However, as shown below, the set of leaf objects is uniquely determined by $d$. First, we show that there exists an object that is a leaf object for any minimal representation.

Lemma 6**.**

Let $(r,r^{\prime})\in\operatornamewithlimits{argmax}_{x,y\in X}d(x,y)$. Then, for any minimal representation, $r$ and $r^{\prime}$ are leaf objects.

Proof.

For any tree, the farthest pair is attained by a pair of leaves. ∎

Next, we show that the leaf objects are characterized by $d$ and a leaf object $r$.

Lemma 7**.**

Let $(\mathcal{T},\phi)$ be a minimal representation of subtree distance $d:X\times X\to\mathbb{R}$, and let $r\in X$ be a leaf object. An object $x\in X\setminus\{r\}$ is a leaf object if and only if $d(y,r)<d(x,y)+d(x,r)$ for all $y\in X\setminus\{x,r\}$.

Proof.

(The “if” part). Suppose $x$ is not a leaf object. Then, there is another leaf object $y\in X\setminus\{x,r\}$ such that the path from $\phi(r)$ to $\phi(y)$ intersects $\phi(x)$. By considering distances among $\phi(r)$, $\phi(y)$ and $\phi(x)$ on this path, we have $d(r,y)\geq d(x,r)+d(x,y)$.

(The “only if” part). Suppose $x$ is a leaf object. Then, for any $y\in X\setminus\{x,r\}$ the shortest path from $\phi(y)$ to $\phi(r)$ never intersects $\phi(x)$. Thus, we have $d(y,r)<d(x,y)+d(x,r)$. Here, we used the assumption that there is no $x,y\in X$ such that $d(x,z)=d(y,z)$ for all $z\in X$. ∎

Since we can take a leaf object $r$ universally by Lemma 6, and the condition in Lemma 7 is described without specifying the underlying representation, we can conclude that the set of leaf objects is uniquely determined by $d$. Thus, we obtain the following corollary.

Corollary 8**.**

Any minimal representation of a subtree distance has the same leaf objects. ∎

Now, we observe that the dissimilarity $d|_{L}:L\times L\to\mathbb{R}$ obtained by restricting $d:X\times X\to\mathbb{R}$ on the set of leaf objects $L\subseteq X$ forms a tree metric because the leaf objects are mapped to singletons. Since the minimal representation of a tree metric is unique [2], any minimal representation of $d$ has the same topology as the minimal representation of $d|_{L}$.

The remaining issue is to show that each non-leaf object $x\in X\setminus L$ is uniquely located in the minimal representation $(\mathcal{T},\phi)$ of $d|_{L}$. This is clear because any connected subgraph in a tree is uniquely identified by the distances from the leaves. More precisely, we obtain the following explicit representation.

We first consider $\mathcal{T}$ as a continuous object. We fix a leaf object $r\in L$. For each leaf object $x\in L\setminus\{r\}$, there exists a unique path $\text{path}(\phi(r),\phi(x))$ from $\phi(r)$ to $\phi(x)$ in $\mathcal{T}$. Let $I(r,a;x,b)$ be the interval on the path having a distance of at least $a$ from $\phi(r)$ and at least $b$ from $\phi(x)$, i.e.,

[TABLE]

For $U\subseteq\mathcal{V}(\mathcal{T})$, we denote by $\overline{U}$ the subgraph of $\mathcal{T}$ induced by $U$. Note that both $I(r,a;x,b)$ and $\overline{U}$ are continuous objects. By using these notations, we obtain the following.

Lemma 9**.**

Let $d:X\times X\to\mathbb{R}$ be a subtree distance and $L$ be the set of leaf objects. Let $(\mathcal{T},\phi)$ be the minimal representation of $d|_{L}$. Fix a leaf object $r\in L$. Then, we have for each non-leaf object $z\in X\setminus L$

[TABLE]

Proof.

Since $\overline{\phi(z)}$ is a connected subgraph, the intersection of $\overline{\phi(z)}$ and the path from $\phi(r)$ to $\phi(x)$ is the interval $I(r,d(r,z);x,d(x,z))$. Since any tree is covered by the paths from a fixed leaf $\phi(r)$ and the other leaves $\phi(x)$ for $x\in L\setminus\{r\}$, we have

[TABLE]

∎

By placing the vertices on the boundaries of $\overline{\phi(z)}$ for $z\in X\setminus L$ and then letting $\phi(z)$ be the vertices intersecting $\overline{\phi(z)}$ for $z\in X\setminus L$, we obtain the minimal representation of $d$. This completes the proof of Theorem 2. Note that the number of the vertices of the minimal representation is $O(n^{2})$ since each $\overline{\phi(z)}$ has at most $|L|=O(n)$ boundaries.

Now, we prove Theorem 3. The above proof of Theorem 2 is constructive, and it provides the following algorithm:

Identify the set $L$ of leaf objects by Lemmas 6 and 7. 2. 2.

Find the minimal representation of $d|_{L}$ by the existing algorithm. 3. 3.

Locate the non-leaf objects by Lemma 9.

We evaluate the time complexity of this algorithm. Step 1 is conducted in $O(n^{2})$ time for finding a leaf object $r\in X$ and $O(n^{2})$ time for finding other leaf objects. Step 2 is performed in $O(n^{2})$ time by using Culberson and Rudnicki’s algorithm [3]. Also, Step 3 is performed in $O(n^{2})$ time by equation (2.2) since, for each $z$, it processes each edge at most once. Hence, Theorem 3 is proved.

Acknowledgments

The authors thank Hiroshi Hirai and anonymous referees for helpful comments. This work was supported by the Japan Society for the Promotion of Science, KAKENHI Grant Number 15K00033.