Nonexistence of minimal-time solutions for some variations of the firing squad synchronization problem having simple geometric configurations

TL;DR

This paper proves that for certain simple geometric configurations in the firing squad synchronization problem, no minimal-time solutions can exist, extending previous nonexistence results to new configurations.

Contribution

It establishes the nonexistence of minimal-time solutions for specific geometric configurations in the FSSP, including L-shaped paths and rectangular walls with fixed ratios.

Findings

No minimal-time solutions for L-shaped configurations with fixed ratio.

No minimal-time solutions for rectangular walls with fixed ratio.

Extends previous nonexistence results to new geometric configurations.

Abstract

We prove nonexistence of minimal-time solutions for three variations of the firing squad synchronization problem (FSSP, for short). Configurations of these variations are paths in the two-dimensional grid space having simple geometric shapes. In the first variation a configuration is an L-shaped path such that the ratio of the length of horizontal line to that of the vertical line is fixed. The general may be at any position. In the second and the third variations a configuration is a rectangular wall such that the ratio of the length of the two horizontal walls to that of the two vertical walls is fixed. The general is at the left down corner in the second variation and may be at any position in the third variation. We use the idea used in the proof of Yamashita et al's recent similar result for variations of FSSP with sub-generals.