Counting Solutions: Difference between revisions
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(Created page with "{{DISPLAYTITLE:Counting Solutions (n-Queens Problem)}} == 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 == <pre>n: number of queens, size of chessboard</pre> == Tab...") |
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== Parameters == | == Parameters == | ||
$n$: number of queens, size of chessboard | |||
== Table of Algorithms == | == Table of Algorithms == | ||
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| [[Naive Algorithm (Counting Solutions; Constructing solutions n-Queens Problem)|Naive Algorithm]] || 1848 || $O(n^n)$ || $O(n)$ || Exact || Deterministic || | |||
|- | |||
| [[Naive + 1 queen per row restriction (Counting Solutions; Constructing solutions n-Queens Problem)|Naive + 1 queen per row restriction]] || 1850 || $O(n!)$ || $O(n)$ || Exact || Deterministic || | |||
|- | |||
| [[Dijkstra (Counting Solutions; Constructing solutions n-Queens Problem)|Dijkstra]] || 1972 || $O(n!)$ || $O(n)$ || Exact || Deterministic || [https://dl-acm-org.ezproxy.canberra.edu.au/citation.cfm?id=1243380 Time] | |||
|- | |||
| [[Nauck (Counting Solutions; Constructing solutions n-Queens Problem)|Nauck]] || 1850 || $O(n!)$ || || Exact || Deterministic || | |||
|- | |||
| [[Gunther Determinants solution (Counting Solutions; Constructing solutions n-Queens Problem)|Gunther Determinants solution]] || 1874 || $O(n!)$ || $O(n!)$ ? || Exact || Deterministic || | |||
|- | |||
| [[Rivin, Zabih (Counting Solutions n-Queens Problem)|Rivin, Zabih]] || 1992 || $O({8}^n*poly(n)$) || $O({8}^n*n^{2})$ || Exact || Deterministic || [http://www.cs.cornell.edu/~rdz/Papers/RZ-IPL92.pdf Time & Space] | | [[Rivin, Zabih (Counting Solutions n-Queens Problem)|Rivin, Zabih]] || 1992 || $O({8}^n*poly(n)$) || $O({8}^n*n^{2})$ || Exact || Deterministic || [http://www.cs.cornell.edu/~rdz/Papers/RZ-IPL92.pdf Time & Space] | ||
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|} | |} | ||
== Time Complexity | == Time Complexity Graph == | ||
[[File:n-Queens Problem - Counting Solutions - Time.png|1000px]] | [[File:n-Queens Problem - Counting Solutions - Time.png|1000px]] | ||
== References/Citation == | == References/Citation == | ||
https://dl-acm-org.ezproxy.canberra.edu.au/citation.cfm?id=1243380 | https://dl-acm-org.ezproxy.canberra.edu.au/citation.cfm?id=1243380 | ||
Latest revision as of 10:12, 28 April 2023
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
References/Citation
https://dl-acm-org.ezproxy.canberra.edu.au/citation.cfm?id=1243380