Counting Solutions

Description

How many ways can one put $n$ queens on an $n \times n$ chessboard so that no two queens attack each other? In other words, how many points can be placed on an $n \times n$ grid so that no two are on the same row, column, or diagonal?

Related Problems

Related: Constructing Solutions, n-Queens Completion

Parameters

$n$: number of queens, size of chessboard

Table of Algorithms

Name	Year	Time	Space	Approximation Factor	Model	Reference
Naive Algorithm	1848	$O(n^n)$	$O(n)$	Exact	Deterministic
Naive + 1 queen per row restriction	1850	$O(n!)$	$O(n)$	Exact	Deterministic
Dijkstra	1972	$O(n!)$	$O(n)$	Exact	Deterministic	Time
Nauck	1850	$O(n!)$		Exact	Deterministic
Gunther Determinants solution	1874	$O(n!)$	$O(n!)$ ?	Exact	Deterministic
Rivin, Zabih	1992	$O({8}^n*poly(n)$)	$O({8}^n*n^{2})$	Exact	Deterministic	Time & Space

Time Complexity Graph

Space Complexity Graph

Time-Space Tradeoff

References/Citation

https://dl-acm-org.ezproxy.canberra.edu.au/citation.cfm?id=1243380