Highest Trees of Random Mappings

TL;DR

This paper determines the precise asymptotic probability that a random mapping's underlying graph has a unique highest tree, a property key to solving the Road Coloring Problem and automaton synchronization.

Contribution

It provides the exact asymptotic probability for the unique highest tree property in random mappings, linking it to important problems in automata theory.

Findings

Exact asymptotic probability derived

Connected to the Road Coloring Problem

Implications for automaton synchronization

Abstract

We prove the exact asymptotic $1-\left({\frac{2\pi}{3}-\frac{827}{288\pi}}+o(1)\right)/{\sqrt{n}}$ for the probability that the underlying graph of a random mapping of $n$ elements possesses a unique highest tree. The property of having a unique highest tree turned out to be crucial in the solution of the famous Road Coloring Problem as well as the generalization of this property in the proof of the author's result about the probability of being synchronizable for a random automaton.