Lost in Self-stabilization

TL;DR

This paper introduces a minimal-information, local-rule-based model for self-stabilization of a twisted discrete thread, demonstrating how it can reorganize into a straight line, with applications in various domains like robotics and crystallography.

Contribution

It presents a novel stochastic process enabling a lost, local-agent chain to self-organize into a line, bridging multiple fields such as distributed algorithms and language theory.

Findings

Efficient self-organization into a line from a twisted configuration.

Equivalence between reordering a word and reorienting a discrete thread.

Applicable to models of crystallography, cellular automata, and robot chains.

Abstract

One of the questions addressed here is How can a twisted thread correct itself?. We consider a theoretical model where the studied mathematical object represents a 2D twisted discrete thread linking two points. This thread is made of a chain of agents which are lost, i.e. they have no knowledge of the global setting and no sense of direction. Thus, the modifications made by the agents are local and all the decisions use only minimal information about the local neighborhood. We introduce a random process such that the thread reorganizes itself efficiently to become a discrete line between these two points. The second question addressed here is to reorder a word by local flips in order to scatter the letters to avoid long successions of the same letter. These two questions are equivalent. The work presented here is at the crossroad of many different domains such as modeling cooling process in crystallography [2, 3, 8], stochastic cellular automata [6, 7], organizing a line of robots in distributed algorithms (the robot chain problem [5, 11]), and Christoffel words in language theory [1].