Anti Tai Mapping for Unordered Labeled Trees
Abstract
The wellstudied Tai mapping between two rooted labeled trees $T_1(V_1, E_1)$ and $T_2(V_2, E_2)$ defines a onetoone mapping between nodes in $T_1$ and $T_2$ that preserves ancestor relationship. For unordered trees the problem of finding a maximumweight Tai mapping is known to be NPcomplete. In this work, we define an anti Tai mapping $M\subseteq V_1\times V_2$ as a binary relation between two unordered labeled trees such that any two $(x,y), (x', y')\in M$ violate ancestor relationship and thus cannot be part of the same Tai mapping, i.e. $(x\le x' \iff y\not \le y') \vee (x'\le x \iff y'\not \le y)$, given an ancestor order $x<x'$ meaning that $x$ is an ancestor of $x'$. Finding a maximumweight anti Tai mapping arises in the cutting plane method for solving the maximumweight Tai mapping problem via integer programming. We give an efficient polynomialtime algorithm for finding a maximumweight anti Tai mapping for the case when one of the two trees is a path and further show how to extend this result in order to provide a polynomially computable lower bound on the optimal anti Tai mapping for two unordered labeled trees. The latter result stems from the special class of anti Tai mapping defined by the more restricted condition $x\sim x' \iff y\not\sim y'$, where $\sim$ denotes that two nodes belong to the same roottoleaf path. For this class, we give an efficient algorithm that solves the problem directly on two unordered trees in $O(V_1^2V_2^2)$.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.08292
 Bibcode:
 2021arXiv210708292B
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Discrete Mathematics