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/**
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* File: n_queens.cpp
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* Created Time: 2023-05-04
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* Author: krahets (krahets@163.com)
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*/
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#include "../utils/common.hpp"
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/* 回溯演算法：n 皇后 */
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void backtrack(int row, int n, vector<vector<string>> &state, vector<vector<vector<string>>> &res, vector<bool> &cols,
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vector<bool> &diags1, vector<bool> &diags2) {
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// 當放置完所有行時，記錄解
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if (row == n) {
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res.push_back(state);
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return;
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}
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// 走訪所有列
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for (int col = 0; col < n; col++) {
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// 計算該格子對應的主對角線和次對角線
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int diag1 = row - col + n - 1;
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int diag2 = row + col;
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// 剪枝：不允許該格子所在列、主對角線、次對角線上存在皇后
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if (!cols[col] && !diags1[diag1] && !diags2[diag2]) {
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// 嘗試：將皇后放置在該格子
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state[row][col] = "Q";
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cols[col] = diags1[diag1] = diags2[diag2] = true;
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// 放置下一行
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backtrack(row + 1, n, state, res, cols, diags1, diags2);
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// 回退：將該格子恢復為空位
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state[row][col] = "#";
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cols[col] = diags1[diag1] = diags2[diag2] = false;
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}
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}
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}
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/* 求解 n 皇后 */
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vector<vector<vector<string>>> nQueens(int n) {
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// 初始化 n*n 大小的棋盤，其中 'Q' 代表皇后，'#' 代表空位
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vector<vector<string>> state(n, vector<string>(n, "#"));
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vector<bool> cols(n, false); // 記錄列是否有皇后
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vector<bool> diags1(2 * n - 1, false); // 記錄主對角線上是否有皇后
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vector<bool> diags2(2 * n - 1, false); // 記錄次對角線上是否有皇后
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vector<vector<vector<string>>> res;
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backtrack(0, n, state, res, cols, diags1, diags2);
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return res;
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}
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/* Driver Code */
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int main() {
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int n = 4;
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vector<vector<vector<string>>> res = nQueens(n);
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cout << "輸入棋盤長寬為 " << n << endl;
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cout << "皇后放置方案共有 " << res.size() << " 種" << endl;
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for (const vector<vector<string>> &state : res) {
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cout << "--------------------" << endl;
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for (const vector<string> &row : state) {
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printVector(row);
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}
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}
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return 0;
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}
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