# Taming the Knight's Tour: Minimizing Turns and Crossings

## TL;DR

This paper introduces new metrics for evaluating knight's tours based on turns and crossings, presents an efficient algorithm with provable approximation bounds, and extends these techniques to various board configurations and leapers.

## Contribution

It proposes a novel algorithm for knight's tours that minimizes turns and crossings with proven approximation ratios and extends these methods to diverse chessboard variants and leapers.

## Key findings

- Algorithm achieves $9.25n+O(1)$ turns and $12n+O(1)$ crossings.
- Lower bounds of $(6-\epsilon)n$ turns and $4n-O(1)$ crossings established.
- Algorithm runs in linear time and is fully parallelizable.

## Abstract

We introduce two new metrics of "simplicity" for knight's tours: the number of turns and the number of crossings. We give a novel algorithm that produces tours with $9.25n+O(1)$ turns and $12n+O(1)$ crossings on an $n\times n$ board, and we show lower bounds of $(6-\epsilon)n$ and $4n-O(1)$ on the respective problems of minimizing these metrics. Hence, our algorithm achieves approximation ratios of $9.25/6+o(1)$ and $3+o(1)$. Our algorithm takes linear time and is fully parallelizable, i.e., the tour can be computed in $O(n^2/p)$ time using $p$ processors in the CREW PRAM model. We generalize our techniques to rectangular boards, high-dimensional boards, symmetric tours, odd boards with a missing corner, and tours for $(1,4)$-leapers. In doing so, we show that these extensions also admit a constant approximation ratio on the minimum number of turns, and on the number of crossings in most cases.