Isomorphic unordered labeled trees up to substitution ciphering

TL;DR

This paper introduces a linear-time method to significantly reduce the search space for tree isomorphisms with labeled unordered trees, aiding in solving the marked tree isomorphism problem more efficiently.

Contribution

It presents a novel approach that exponentially decreases the isomorphism search space in linear time, improving the efficiency of solving the marked tree isomorphism problem.

Findings

Reduces the isomorphism search space exponentially on average

Operates in linear time relative to tree size

Enhances the efficiency of solving tree isomorphism problems

Abstract

Given two messages - as linear sequences of letters, it is immediate to determine whether one can be transformed into the other by simple substitution cipher of the letters. On the other hand, if the letters are carried as labels on nodes of topologically isomorphic unordered trees, determining if a substitution exists is referred to as marked tree isomorphism problem in the literature and has been show to be as hard as graph isomorphism. While the left-to-right direction provides the cipher of letters in the case of linear messages, if the messages are carried by unordered trees, the cipher is given by a tree isomorphism. The number of isomorphisms between two trees is roughly exponential in the size of the trees, which makes the problem of finding a cipher difficult by exhaustive search. This paper presents a method that aims to break the combinatorics of the isomorphisms search space. We show that in a linear time (in the size of the trees), we reduce the cardinality of this space by an exponential factor on average.