Distributing Labels on Infinite Trees

TL;DR

This paper extends the concept of Sturmian words, which are balanced and minimally complex infinite sequences, to infinite trees, showing that balanced trees with similar properties can be constructed and may have applications in scheduling problems.

Contribution

It introduces the concept of balanced infinite trees analogous to Sturmian words and demonstrates their construction using mechanical processes for irrational cases.

Findings

Balanced infinite trees exist and can be constructed mechanically.

Such trees have minimal factor complexity.

Potential for extending extremal scheduling properties to trees.

Abstract

Sturmian words are infinite binary words with many equivalent definitions: They have a minimal factor complexity among all aperiodic sequences; they are balanced sequences (the labels 0 and 1 are as evenly distributed as possible) and they can be constructed using a mechanical definition. All this properties make them good candidates for being extremal points in scheduling problems over two processors. In this paper, we consider the problem of generalizing Sturmian words to trees. The problem is to evenly distribute labels 0 and 1 over infinite trees. We show that (strongly) balanced trees exist and can also be constructed using a mechanical process as long as the tree is irrational. Such trees also have a minimal factor complexity. Therefore they bring the hope that extremal scheduling properties of Sturmian words can be extended to such trees, as least partially. Such possible extensions are illustrated by one such example.