Can Walker Localize The Middle Point of A Line-segment?

TL;DR

This paper investigates whether an oblivious walker on a line can localize the midpoint in finite steps using limited observations, addressing a fundamental problem in distributed autonomous robot localization.

Contribution

It introduces algorithms for three variants of the problem with minimal relaxation and proves an impossibility result for symmetric algorithms.

Findings

Algorithms successfully localize the midpoint in the variants studied.

An impossibility theorem is established for bilaterally symmetric algorithms.

The problem's solvability depends on the specific variant and symmetry considerations.

Abstract

This paper poses a question about a simple localization problem. The question is if an {\em oblivious} walker on a line-segment can localize the middle point of the line-segment in {\em finite} steps observing the direction (i.e., Left or Right) and the distance to the nearest end point. This problem is arisen from {\em self-stabilizing} location problems by {\em autonomous mobile robots} with {\em limited visibility}, that is a widely interested abstract model in distributed computing. Contrary to appearances, it is far from trivial if this simple problem is solvable or not, and unsettled yet. This paper is concerned with three variants of the problem with a minimal relaxation, and presents self-stabilizing algorithms for them. We also show an easy impossibility theorem for bilaterally symmetric algorithms.