An Algorithm of Generalized Knight’s Tour Problem Based on Path Joint

There are a lot of algorithms of knight's tour problem on two-dimensional chessboard, but most of them are hard to extend on N dimensional chessboard. In this paper, knight's tour problem has been considered as looking for Hamiltonian path, a path joint algorithm (PJA) has been given on 2 dimensional chessboard, a few conceptions such as path adjacency matrix and related theorems have been presented. The path joint algorithm is also applied to N dimensional chessboard. For an N dimensional chessboard L= l₁ × l₂ × ⋯l_N, the algorithm's time complexity is O(L⁴) if find all knight's tours while the algorithm's time complexity is O(L³) if find one knight's tour for 2 fixed vertexes, this matched the experimental result.

I. Introduction

A knight's tour is a sequence of moves of a knight on a chessboard such that the knight visits every square only once. The knight's tour problem (KTP) is a mathematical problem of finding a knight's tour. A generalized knight's tour is a sequence of moves of a knight on an N dimension chessboard such that the knight visits every generalized square only once, which the knight visiting rule can be defined.