A Dynamic Programming Solution to the n-Queens Problem

The College Mathematics Journal

The N-queens problem is a popular classic puzzle where numbers of queen were to be placed on an n x n matrix such that no queen can attack any other queen. The Branching Factor grows in a roughly linear way, which is an important consideration for the researchers. However, many researchers have cited the issues with help of artificial intelligence search patterns say DFS, BFS and backtracking algorithms. The proposed algorithm is able to compute one unique solution in polynomial time when chess board size is greater than 7. This algorithm is based on 8 different series. For each series a different approach is taken to place the queen on a given chess board.

Nowadays, permutation problems with large state spaces and the path to solution is irrelevant such as N-Queens problem has the same general property for many important applications such as integrated-circuit design, factory-floor layout, job-shop scheduling, automatic programming, telecommunications network optimization, vehicle routing, and portfolio management. Therefore, methods which are able to find a solution are very important. Genetic algorithm (GA) is one the most well-known methods for solving N-Queens problem and applicable to a wide range of permutation problems. In the absence of specialized solution for a particular problem, genetic algorithm would be efficient. But holism and random choices cause problem for genetic algorithm in searching large state spaces. So, the efficiency of this algorithm would be demoted when the size of state space of the problem grows exponentially. In this paper, the subproblems used based on genetic algorithm to cover this weakness. This proposed method is trying to provide partial view for genetic algorithm by locally searching the state space. This method works to take shorter steps toward the solution. To find the first solution and other solutions in N-Queens problem using proposed method: dividing N-Queens problem into subproblems, which configuring initial population of genetic algorithm. The proposed method is evaluated and compares it with two similar methods that indicate the amount of performance improvement.

International Journal on Advanced Science, Engineering and Information Technology, 2021

This paper presents a study on the N-Queens Problem. Different approaches to its solution discussed in the scientific literature are analyzed. The implementation of an algorithm based on the backtracking method is also presented. The algorithm is optimized to find solutions in a specific subset of configurations among all possible ones. With this approach, the computational complexity of the algorithm is reduced from exponential to quadratic. In this way, the algorithm finds all possible solutions in a shorter time: fundamental and their symmetrical equivalents. The methodology for conducting the experiments is presented. The purpose of the study, the tasks to be performed, and the conditions for conducting the experiments are presented as well. In connection with the research, an application that implements the presented algorithm has been developed. This application generated all the results obtained in this study. The experimental results show that with a linear increase in the number of queens (equivalent to a quadratic increase in the number of fields on the board, the number of recursive calls made by the algorithm increases exponentially. Similarly, the number of possible solutions, as well as the execution time of the algorithm (in the different modes of the application - internal, interactive, and combined), also increases exponentially. However, the algorithm's execution time in the internal mode is significantly shorter than in the other two modes - interactive and combined. The future guidelines for the study are presented.

arXiv: Combinatorics, 2017

Let $D$ be a digraph, possibly with loops. A queen labeling of $D$ is a bijective function $l:V(G)\longrightarrow \{1,2,\ldots,|V(G)|\}$ such that, for every pair of arcs in $E(D)$, namely $(u,v)$ and $(u',v')$ we have (i) $l(u)+l(v)\neq l(u')+l(v')$ and (ii) $l(v)-l(u)\neq l(v')-l(u')$. Similarly, if the two conditions are satisfied modulo $n=|V(G)|$, we define a modular queen labeling. There is a bijection between (modular) queen labelings of $1$-regular digraphs and the solutions of the (modular) $n$-queens problem. The $\otimes_h$-product was introduced in 2008 as a generalization of the Kronecker product and since then, many relations among labelings have been established using the $\otimes_h$-product and some particular families of graphs. In this paper, we study some families of $1$-regular digraphs that admit (modular) queen labelings and present a new construction concerning to the (modular) $n$-queens problem in terms of the $\otimes_h$-product, whi...

2013

Q: Can you place 8 nonattacking queens on an 8 × 8 chessboard? Q: In ho w ma ny wa ys A: Yes! 92 The n-Queens Problem: Find a formula for the number of nonattacking configurations of n queens on an n × n chessboard.

Abstract: Most of the optimization methods do not inherit convergence proofs. One of the measures to rank them is their potential to solve challenging problems specially formulated for this purpose. In this paper the problem involves two main issues. Firstly we present new formulations of the N queens’ configuration problem as optimization problems and secondly we modify a derivative free method so that it may be able to find the optimal configuration of the chessboard.

International Journal of Advance Research and Innovative Ideas in Education, 2017

Genetic algorithm (GAs) is powerful method which uses heuristic approach. It is capable of searching large spaces of possible solutions in an efficient manner. This paper approaches an implementation of Genetic Algorithm for solving n-Queens problem. It provides an efficient way to solve the problem than traditional backtracking method. In proposed genetic algorithm, members of the population are represented by number of chromosomes. Each chromosome is a possible board position in the n queens game. Instead of traditional approach the proposed algorithm gives efficient solution in terms of reduced time and efforts.

1984

a contemporary 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 backtracking search methods. A related problem, placing the N queens on a toroidal board, has been discussed in detail by Polya and Chandra, Their work focused on characterizing the solvable cases and finding regular solutions, board setups that solve the problem while arranging the queens z'n a regular pattern, . This paper describes a new linear time divide-and-conquer algorithm that solves both problems. W'hen applied to the toroidal problem, the algorithm yields a family of non-regular solutions based on number factorization. These 80lutions, in turn, can be modified to 80lve the standard N-queens problem for board 8izes which are unsolvable on a torus.

2015 Science and Information Conference (SAI), 2015

This research proposes the swapping algorithm a new algorithm for solving the n-queens problem, and provides data from experimental performance results of this new algorithm. A summary is also provided of various meta-heuristic approaches which have been used to solve the n-queens problem including neural networks, evolutionary algorithms, genetic programming, and recently Imperialist Competitive Algorithm (ICA). Currently the Cooperative PSO algorithm is the best algorithm in the literature for finding the first valid solution. Also the research looks into the effect of the number of hidden nodes and layers within neural networks and the effect on the time taken to find a solution. This paper proposes a new swapping algorithm which swaps the position of queens.