The sequential (distributional) topological complexity of the ordered configuration space of disks in a strip
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915292965240832 |
|---|---|
| author | Wawrykow, Nicholas |
| author_facet | Wawrykow, Nicholas |
| contents | 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 $r^{\text{th}}$-sequential topological complexity of $\text{conf}(n,w)$, the ordered configuration space of $n$ open unit-diameter disks in the infinite strip of width $w$, as well as its $r^{\text{th}}$-sequential distributional topological complexity. We prove that as long as $n$ is greater than $w$, the $r^{\text{th}}$-sequential (distributional) topological complexity of $\text{conf}(n,w)$ is $r\big(n-\big\lceil\frac{n}{w}\big\rceil\big)$. 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 $r\big(n-\big\lceil\frac{n}{w}\big\rceil\big)$ cases. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19943 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The sequential (distributional) topological complexity of the ordered configuration space of disks in a strip Wawrykow, Nicholas Algebraic Topology 55M30 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 $r^{\text{th}}$-sequential topological complexity of $\text{conf}(n,w)$, the ordered configuration space of $n$ open unit-diameter disks in the infinite strip of width $w$, as well as its $r^{\text{th}}$-sequential distributional topological complexity. We prove that as long as $n$ is greater than $w$, the $r^{\text{th}}$-sequential (distributional) topological complexity of $\text{conf}(n,w)$ is $r\big(n-\big\lceil\frac{n}{w}\big\rceil\big)$. 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 $r\big(n-\big\lceil\frac{n}{w}\big\rceil\big)$ cases. |
| title | The sequential (distributional) topological complexity of the ordered configuration space of disks in a strip |
| topic | Algebraic Topology 55M30 |
| url | https://arxiv.org/abs/2412.19943 |