算法实现/杂项/N 皇后

八皇后问题是指在一个 8×8 的棋盘上放置八个皇后，使得任何两个皇后都不能互相攻击。皇后必须放置在这样的位置，任何两个皇后都不能攻击对方。因此，解决方案要求任何两个皇后不能在同一行、同一列或同一对角线上。八皇后问题是更一般的 n 皇后问题的示例，它将 n 个皇后放置在一个 n×n 的棋盘上，其中只有 n = 1 或 n ≥ 4 才有解。

C++

一个带有完整注释的回溯实现，用 C++ 编写

#include <iostream>

using namespace std;

const int N = 5;
int position[N];

// Check if a position is safe
bool isSafe(int queen_number, int row_position)
{
	// Check each queen before this one
	for (int i = 0; i < queen_number; i++)
	{
		// Get another queen's row_position
		int other_row_pos = position[i];

		// Now check if they're in the same row or diagonals
		if (other_row_pos == row_position || // Same row
			other_row_pos == row_position - (queen_number - i) || // Same diagonal
			other_row_pos == row_position + (queen_number - i))   // Same diagonal
			return false;
	}
	return true;
}

// Recursively generate a tuple like [0 0 0 0], then [0 0 0 1] then etc...
void solve(int k)
{
	if (k == N) // We placed N-1 queens (0 included), problem solved!
	{
		// Solution found!
		cout << "Solution: ";
		for (int i = 0; i < N; i++)
			cout << position[i] << " ";
		cout << endl;
	}
	else
	{
		for (int i = 0; i < N; i++) // Generate ALL combinations
		{
			// Before putting a queen (the k-th queen) into a row, test it for safeness
			if (isSafe(k, i))
			{
				position[k] = i;
				// Place another queen
				solve(k + 1);
			}
		}
	}
}

int main()
{
	solve(0);

	return 0;
}

C

一个用 C 语言编写的回溯深度优先搜索 (DFS) 解决方案

#include <stdio.h>

int
is_safe(int rows[8], int x, int y)  
{
    int i;
    
    if (y == 0)
            return 1;
    for (i=0; i < y; ++i) {
       if (rows[i] == x || rows[i] == x + y - i || rows[i] == x - y +i)
            return 0;
    }

    return 1;
}
 
void
putboard(int rows[8])  
{
    static int s = 0;
    int x, y;

    printf("\nSolution #%d:\n---------------------------------\n", ++s);
    for (y=0; y < 8; ++y) {
        for (x=0; x < 8; ++x)
            printf(x == rows[y] ? "| Q " : "|   ");
        printf("|\n---------------------------------\n");
    }
}
 
void
eight_queens_helper(int rows[8], int y)
{
    int x;

    for (x=0; x < 8; ++x) {
        if (is_safe(rows, x, y)) {
            rows[y] = x;
            if (y == 7)
              putboard(rows);
            else
              eight_queens_helper(rows, y+1);
        }
    }
}

int
main()
{
    int rows[8];

    eight_queens_helper(rows, 0);

    return 0;
}

Haskell

import Control.Monad

queens n = foldM (\y _ -> [ x : y | x <- [1..n], safe x y 1]) [] [1..n]
safe x [] n = True
safe x (c:y) n = and [ x /= c, x /= c + n, x /= c - n, safe x y (n+1)]

main = mapM_ print $ queens 8

Python

def queensproblem(rows, columns):
    solutions = [[]]
    for row in range(rows):
        solutions = add_one_queen(row, columns, solutions)
    return solutions

def add_one_queen(new_row, columns, prev_solutions):
    return [solution + [new_column]
            for solution in prev_solutions
            for new_column in range(columns)
            if no_conflict(new_row, new_column, solution)]

def no_conflict(new_row, new_column, solution):
    return all(solution[row]       != new_column           and
               solution[row] + row != new_column + new_row and
               solution[row] - row != new_column - new_row
               for row in range(new_row))

for solution in queensproblem(8, 8):
    print(solution)

Mathematica

dispSol[sol_] := sol /. {1 -> "Q", 0 -> "-"} // Grid

solveNqueens[n_] := 
 Module[{c, m, b, vars}, c = cqueens[n]; m = mqueens[n]; 
  vars = mqueens2[n]; b = bqueens[Length[m]]; 
  Partition[LinearProgramming[c, m, b, vars, Integers], n]]

cqueens[n_] := Table[-1, {i, n^2}]

bqueens[l_] := Table[{1, -1}, {i, l}]

mqueens2[n_] := Table[{0, 1}, {i, n^2}]

mqueens[n_] := 
 Module[{t, t2, t3, t4}, t = mqueensh[n]; t2 = Append[t, mqueensv[n]];
   t3 = Append[t2, mqueensd[n]]; t4 = Append[t3, mqueensdm[n]]; 
  Partition[Flatten[t4], n^2]]

mqueensh[n_] := 
 Module[{t}, t = Table[0, {i, n}, {j, n^2}]; 
  For[i = 1, i <= n, i++, 
   For[j = 1, j <= n, j++, t[[i, ((i - 1)*n) + j]] = 1]]; t]

mqueensv[n_] := 
 Module[{t}, t = Table[0, {i, n}, {j, n^2}]; 
  For[i = 1, i <= n, i++, 
   For[j = 1, j <= n, j++, t[[j, ((i - 1)*n) + j]] = 1]]; t]

mqueensd[n_] := 
 Module[{t}, t = Table[0, {i, (2*n) - 1}, {j, n^2}]; 
  For[k = 2, k <= 2 n, k++, 
   For[i = 1, i <= n, i++, 
    For[j = 1, j <= n, j++, 
     If[i + j == k, t[[k - 1, ((i - 1)*n) + j]] = 1]]]]; t]

mqueensdm[n_] := 
 Module[{t}, t = Table[0, {i, Sum[1, {i, 1 - n, n - 1}]}, {j, n^2}]; 
  For[k = 1 - n, k <= n - 1, k++, 
   For[i = 1, i <= n, i++, 
    For[j = 1, j <= n, j++, 
     If[i == j - k, t[[k + n, ((i - 1)*n) + j]] = 1]]]]; t]