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

TL;DR

This work determines the sequential (and distributional) topological complexity of conf(n,w), the ordered configuration space of $n$ disks in a strip of width $w$, with a sharp formula in terms of $n$ and $w$. Using the cell model cell(n,w) and a wheel-based homology framework, the authors prove that for $n>w$ both $\textbf{TC}_{\mathbf{r}}(\text{conf}(n,w))$ and $\textbf{dTC}_{\mathbf{r}}(\text{conf}(n,w))$ equal $r\left(n-\left\lceil\frac{n}{w}\right\rceil\right)$, while for $n\le w$ the spaces are homotopy equivalent to $F_{n}(\mathbb{R}^{2})$ and the sequential topological complexities reduce to the known point-configuration values. The nontrivial lower bounds come from two homologically decomposable disjoint tori built from concatenations of wheels, extending Knudsen’s method to the sequential and distributional setting, and the upper bounds come from the cell-model equivalence. In the $n\le w$ regime, the results recover $\textbf{dTC}_{\mathbf{r}}(\text{conf}(n,w))=\textbf{TC}_{\mathbf{r}}(\text{conf}(n,w))=r(n-1)-1$, matching the $F_{n}(\mathbb{R}^{2})$ case. Overall, the paper provides exact complexity counts for constrained multi-robot motion planning in a corridor and links these counts to explicit homological constructions, yielding a concrete bridge between disk configurations and sequential motion-planning complexity.

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 $r^{\text{th}}$-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.

Figures (6)

• Figure 1: A point in $\text{conf}(5,2)$. Note that two disks can be aligned vertically, but three cannot.
• Figure 2: The cellular complex $\text{cell}(3,2)$. The loop corresponding the $1$-cells $1\,2|3$ and $2\,1|3$ corresponds to disks $1$ and $2$ orbiting each other while disk $3$ sits to the right of them. See the right side of Figure \ref{['disjointtoriconf32']}.
• Figure 3: The wheel $W(1,3,2)$ in $\text{conf}(3,3)$. Disks $1$ and $3$ orbit each other, while disk $2$ independently orbits them.
• Figure 4: The tori $A=W(3,1)W(2)$ and $B=W(2,1)W(3)$ in $\text{conf}(3,2)$. This is a disjoint pair of homologically decomposable tori, see Figure \ref{['disjointtoricell32']}.
• Figure 5: The tori $A=W(3,1)W(2)$ and $B=W(2,1)W(3)$ in $\text{cell}(3,2)$. In the $\text{cell}(3,2)$ model of $\text{conf}(3,2)$ it is obvious that these tori are homologically disjoint. Compare with Figure \ref{['disjointtoriconf32']}.

Theorems & Definitions (28)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Example 2.1
• Example 2.2
• Example 2.3
• Example 2.4
• Proposition 2.5
• Proposition 2.6
• proof