Counting Solutions (n-Queens Problem)
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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
