[Paper Review] A rearrangement distance for fully-labelled trees

Giulia Bernardini, Paola Bonizzoni+2 more|arXiv (Cornell University)|Apr 2, 2019

Data Management and Algorithms33 references4 citations

TL;DR

This paper introduces a novel rearrangement distance for fully-labelled trees—where every vertex bears a unique label—by combining topology-changing operations (link-and-cut) and label-permuting operations. It proves the problem is NP-hard but fixed-parameter tractable, and provides a 4-approximation algorithm when one tree is binary, enabling efficient comparison of tumor phylogenies in cancer evolution studies.

ABSTRACT

The problem of comparing trees representing the evolutionary histories of cancerous tumors has turned out to be crucial, since there is a variety of different methods which typically infer multiple possible trees. A departure from the widely studied setting of classical phylogenetics, where trees are leaf-labelled, tumoral trees are fully labelled, i.e., \emph{every} vertex has a label. In this paper we provide a rearrangement distance measure between two fully-labelled trees. This notion originates from two operations: one which modifies the topology of the tree, the other which permutes the labels of the vertices, hence leaving the topology unaffected. While we show that the distance between two trees in terms of each such operation alone can be decided in polynomial time, the more general notion of distance when both operations are allowed is NP-hard to decide. Despite this result, we show that it is fixed-parameter tractable, and we give a 4-approximation algorithm when one of the trees is binary.

Motivation & Objective

• Address the challenge of comparing multiple fully-labelled evolutionary trees inferred from cancer sequencing data, where all nodes (not just leaves) are labelled with mutations.
• Define a new rearrangement distance measure that accounts for both topological changes (link-and-cut) and label permutations in fully-labelled trees.
• Establish computational complexity bounds for the new distance measure, showing NP-hardness in general but fixed-parameter tractability.
• Provide a polynomial-time 4-approximation algorithm when one of the input trees is binary, enabling practical application to tumor progression analysis.

Proposed method

• Define two atomic operations: (1) link-and-cut to modify tree topology, and (2) permutation of vertex labels without altering topology.
• Prove that the distance under each operation alone is computable in polynomial time, but the combined distance is NP-hard.
• Leverage Lemma 2.1 to reorder operations: all permutations precede all link-and-cut operations in an optimal transformation sequence.
• Use bounded search tree technique with parameter $k$ to explore all possible label permutation sequences of size at most $k$, bounded by $O((4k)^{2k^2})$ sequences.
• For each resulting intermediate tree $T^*$, compute the link-and-cut distance to the target tree $T_2$ in $O(n)$ time using Lemma 3.1.
• Combine $d_\pi(T_1, T^*)$ and $d_\ell(T^*, T_2)$ to compute the total distance and track the minimum over all $T^*$.

Experimental results

Research questions

• RQ1 Can a meaningful rearrangement distance be defined for fully-labelled trees, where every vertex has a unique label, in contrast to classical phylogenetics where only leaves are labelled?
• RQ2 Is the problem of computing the minimum number of combined topology and label operations to transform one fully-labelled tree into another computationally tractable?
• RQ3 What is the computational complexity of the rearrangement distance for fully-labelled trees, particularly when vertex degrees are bounded or when one tree is binary?
• RQ4 Can a constant-factor approximation algorithm be designed for the rearrangement distance when one input tree is binary?

Key findings

• The rearrangement distance between two fully-labelled trees is NP-hard to compute when both link-and-cut and label permutation operations are allowed.
• The problem is fixed-parameter tractable with respect to the distance parameter $k$, solvable in $O((4k)^{2k^2}n)$ time.
• A 4-approximation algorithm exists for the case where one of the two input trees is binary, based on the link-and-cut distance approximation and the bound $d(T_1,T_2) \geq \frac{d_\ell(T_1,T_2)}{4}$.
• The family partition $\mathcal{P}$ of vertex pairs across trees provides a lower bound: $d(T_1,T_2) \geq |\mathcal{P}|/2$, which is used to prune the search space in the fixed-parameter algorithm.
• The link-and-cut distance $d_\ell(T^*, T_2)$ can be computed in linear time $O(n)$ for any intermediate tree $T^*$, enabling efficient evaluation in the search process.
• The optimal transformation sequence can be reordered so that all label permutations occur before any topology changes, simplifying the algorithmic design.