Divide and conquer under global constraints: A solution to the N-queens problem

Configuring N mutually nonattacking queens on an N -by- N chessboard is a classical problem that was first posed over a century ago. Over the past few decades, this problem has become important to computer scientists by serving as the standard example of a globally constrained problem which is solvable using backtracking search methods. A related problem, placing the N queens on a toroidal board, has been discussed in detail by Poyla and Chandra. Their work focused on characterizing the solvable cases and finding solutions which arrange the queens in a regular pattern. This paper describes a new divide-and-conquer algorithm that solves both problems and investigates the relationship between them. The connection between the solutions of the two problems illustrates an important, but frequently overlooked, method of algorithm design: detailed combinatorial analysis of an overconstrained variation can reveal solutions to the corresponding original problem. The solution is an example of solving a globally constrained problem using the divide-and-conquer technique, rather than the usual backtracking algorithm. The former is much faster in both sequential and parallel environments.