The sequential (distributional) topological complexity of the ordered configuration space of disks in a strip

2024/12/27 by Nicholas Wawrykow, Wawrykow, Nicholas · 1 citation
Mathematics · #55M30 #Algebraic Topology (math.AT) #FOS: Mathematics #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2412.19943

openalex publication_date 2024/12/27 · openalex created_date 2025/01/01 · openalex updated_date 2026/07/28

Abstract

How hard is it to program n robots to move about a long narrow aisle while making a series of r-2 intermediate stops, provided only w of the robots can fit across the width of the aisle? In this paper, we answer this question by calculating the r^th-sequential topological complexity of conf(n,w), the ordered configuration space of n open unit-diameter disks in the infinite strip of width w, as well as its r^th-sequential distributional topological complexity. We prove that as long as n is greater than w, the r^th-sequential (distributional) topological complexity of conf(n,w) is r(n-\lceil(n)/(w)\rceil). This shows that any non-looping program moving the n robots between arbitrary initial and final configurations, with r-2 intermediate stops, must consider at least r(n-\lceil(n)/(w)\rceil) cases.

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