Linear algorithm for solution n-Queens Completion problem

TL;DR

This paper introduces a linear algorithm for solving the n-Queens Completion problem that efficiently determines solutions or impossibility for any queen configuration, with high accuracy and reduced average placement time as n increases.

Contribution

The paper presents a novel linear algorithm for the n-Queens Completion problem, including probabilistic decision rules, random models, and an improved backtracking organization.

Findings

Solution probability increases with n

Over 35% of solutions avoid backtracking for large n

Error probability in decision-making is below 0.0001

Abstract

A linear algorithm is described for solving the n-Queens Completion problem for an arbitrary composition of k queens, consistently distributed on a chessboard of size n x n. Two important rules are used in the algorithm: a) the rule of sequential risk elimination for the entire system as a whole; b) the rule of formation of minimal damage in the given selection conditions. For any composition of k queens (1<= k<n), a solution is provided, or a decision is made that this composition can't be completed. The probability of an error in making such a decision does not exceed 0.0001, and its value decreases, with increasing n. It is established that the average time, required for the queen to be placed on one row, decreases with increasing value of n. A description is given of two random selection models and the results of their comparative analysis. A model for organizing the Back Tracking procedure is proposed based on the separation of the solution matrix into two basic levels. Regression formulas are given for the dependence of basic levels on the value of n. It was found that for n=(7-100000) the number of solutions in which the Back Tracking procedure has never been used exceeds 35%. Moreover, for n=(320-22500), the number of such cases exceeds 50 %. A quick algorithm for verifying the correctness of n-Queens problem solution or arbitrary composition of k queens is given.