From 67d647ab59a52824c3d5af9213f255f41d73b22e Mon Sep 17 00:00:00 2001 From: Yudong Jin Date: Thu, 4 May 2023 05:27:17 +0800 Subject: [PATCH] feat: Add the section of n queens problem (#483) * Add the section of n queens problem * Update n_queens.py * Update n_queens.java * Update n_queens.cpp * Update n_queens.java --- codes/cpp/chapter_backtracking/n_queens.cpp | 65 ++++++++++ codes/java/chapter_backtracking/n_queens.java | 77 ++++++++++++ codes/python/chapter_backtracking/n_queens.py | 62 ++++++++++ .../n_queens_cols_diagonals.png | Bin 0 -> 113653 bytes .../n_queens_constraints.png | Bin 0 -> 82018 bytes .../n_queens_placing.png | Bin 0 -> 71900 bytes .../solution_4_queens.png | Bin 0 -> 29251 bytes docs/chapter_backtracking/n_queens_problem.md | 115 ++++++++++++++++++ .../permutations_problem.md | 6 + mkdocs.yml | 1 + 10 files changed, 326 insertions(+) create mode 100644 codes/cpp/chapter_backtracking/n_queens.cpp create mode 100644 codes/java/chapter_backtracking/n_queens.java create mode 100644 codes/python/chapter_backtracking/n_queens.py create mode 100644 docs/chapter_backtracking/n_queens_problem.assets/n_queens_cols_diagonals.png create mode 100644 docs/chapter_backtracking/n_queens_problem.assets/n_queens_constraints.png create mode 100644 docs/chapter_backtracking/n_queens_problem.assets/n_queens_placing.png create mode 100644 docs/chapter_backtracking/n_queens_problem.assets/solution_4_queens.png create mode 100644 docs/chapter_backtracking/n_queens_problem.md diff --git a/codes/cpp/chapter_backtracking/n_queens.cpp b/codes/cpp/chapter_backtracking/n_queens.cpp new file mode 100644 index 000000000..f89fac96e --- /dev/null +++ b/codes/cpp/chapter_backtracking/n_queens.cpp @@ -0,0 +1,65 @@ +/** + * File: n_queens.cpp + * Created Time: 2023-05-04 + * Author: Krahets (krahets@163.com) + */ + +#include "../utils/common.hpp" + +/* 回溯算法:N 皇后 */ +void backtrack(int row, int n, vector> &state, vector>> &res, vector &cols, + vector &diags1, vector &diags2) { + // 当放置完所有行时,记录解 + if (row == n) { + res.push_back(state); + return; + } + // 遍历所有列 + for (int col = 0; col < n; col++) { + // 计算该格子对应的主对角线和副对角线 + int diag1 = row - col + n - 1; + int diag2 = row + col; + // 剪枝:不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后 + if (!(cols[col] || diags1[diag1] || diags2[diag2])) { + // 尝试:将皇后放置在该格子 + state[row][col] = "Q"; + cols[col] = diags1[diag1] = diags2[diag2] = true; + // 放置下一行 + backtrack(row + 1, n, state, res, cols, diags1, diags2); + // 回退:将该格子恢复为空位 + state[row][col] = "#"; + cols[col] = diags1[diag1] = diags2[diag2] = false; + } + } +} + +/* 求解 N 皇后 */ +vector>> nQueens(int n) { + // 初始化 n*n 大小的棋盘,其中 'Q' 代表皇后,'#' 代表空位 + vector> state(n, vector(n, "#")); + vector cols(n, false); // 记录列是否有皇后 + vector diags1(2 * n - 1, false); // 记录主对角线是否有皇后 + vector diags2(2 * n - 1, false); // 记录副对角线是否有皇后 + vector>> res; + + backtrack(0, n, state, res, cols, diags1, diags2); + + return res; +} + +/* Driver Code */ +int main() { + int n = 4; + vector>> res = nQueens(n); + + cout << "输入棋盘长宽为 " << n << endl; + cout << "皇后放置方案共有 " << res.size() << " 种" << endl; + for (const vector> &state : res) { + cout << "--------------------" << endl; + for (const vector &row : state) { + printVector(row); + } + } + + return 0; +} diff --git a/codes/java/chapter_backtracking/n_queens.java b/codes/java/chapter_backtracking/n_queens.java new file mode 100644 index 000000000..b9a7e7702 --- /dev/null +++ b/codes/java/chapter_backtracking/n_queens.java @@ -0,0 +1,77 @@ +/** + * File: n_queens.java + * Created Time: 2023-05-04 + * Author: Krahets (krahets@163.com) + */ + +package chapter_backtracking; + +import java.util.*; + +public class n_queens { + /* 求解 N 皇后 */ + public static List>> nQueens(int n) { + // 初始化 n*n 大小的棋盘,其中 'Q' 代表皇后,'#' 代表空位 + List> state = new ArrayList<>(); + for (int i = 0; i < n; i++) { + List row = new ArrayList<>(); + for (int j = 0; j < n; j++) { + row.add("#"); + } + state.add(row); + } + boolean[] cols = new boolean[n]; // 记录列是否有皇后 + boolean[] diags1 = new boolean[2 * n - 1]; // 记录主对角线是否有皇后 + boolean[] diags2 = new boolean[2 * n - 1]; // 记录副对角线是否有皇后 + List>> res = new ArrayList<>(); + + backtrack(0, n, state, res, cols, diags1, diags2); + + return res; + } + + /* 回溯算法:N 皇后 */ + public static void backtrack(int row, int n, List> state, List>> res, + boolean[] cols, boolean[] diags1, boolean[] diags2) { + // 当放置完所有行时,记录解 + if (row == n) { + List> copyState = new ArrayList<>(); + for (List sRow : state) { + copyState.add(new ArrayList<>(sRow)); + } + res.add(copyState); + return; + } + // 遍历所有列 + for (int col = 0; col < n; col++) { + // 计算该格子对应的主对角线和副对角线 + int diag1 = row - col + n - 1; + int diag2 = row + col; + // 剪枝:不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后 + if (!(cols[col] || diags1[diag1] || diags2[diag2])) { + // 尝试:将皇后放置在该格子 + state.get(row).set(col, "Q"); + cols[col] = diags1[diag1] = diags2[diag2] = true; + // 放置下一行 + backtrack(row + 1, n, state, res, cols, diags1, diags2); + // 回退:将该格子恢复为空位 + state.get(row).set(col, "#"); + cols[col] = diags1[diag1] = diags2[diag2] = false; + } + } + } + + public static void main(String[] args) { + int n = 4; + List>> res = nQueens(n); + + System.out.println("输入棋盘长宽为 " + n); + System.out.println("皇后放置方案共有 " + res.size() + " 种"); + for (List> state : res) { + System.out.println("--------------------"); + for (List row : state) { + System.out.println(row); + } + } + } +} diff --git a/codes/python/chapter_backtracking/n_queens.py b/codes/python/chapter_backtracking/n_queens.py new file mode 100644 index 000000000..14a607915 --- /dev/null +++ b/codes/python/chapter_backtracking/n_queens.py @@ -0,0 +1,62 @@ +""" +File: n_queens.py +Created Time: 2023-04-26 +Author: Krahets (krahets@163.com) +""" + + +def backtrack( + row: int, + n: int, + state: list[list[str]], + res: list[list[list[str]]], + cols: list[bool], + diags1: list[bool], + diags2: list[bool], +): + """回溯算法:N 皇后""" + # 当放置完所有行时,记录解 + if row == n: + res.append([list(row) for row in state]) + return + # 遍历所有列 + for col in range(n): + # 计算该格子对应的主对角线和副对角线 + diag1 = row - col + n - 1 + diag2 = row + col + # 剪枝:不允许该格子所在 (列 或 主对角线 或 副对角线) 包含皇后 + if not (cols[col] or diags1[diag1] or diags2[diag2]): + # 尝试:将皇后放置在该格子 + state[row][col] = "Q" + cols[col] = diags1[diag1] = diags2[diag2] = True + # 放置下一行 + backtrack(row + 1, n, state, res, cols, diags1, diags2) + # 回退:将该格子恢复为空位 + state[row][col] = "#" + cols[col] = diags1[diag1] = diags2[diag2] = False + + +def n_queens(n: int) -> list[list[list[str]]]: + """求解 N 皇后""" + # 初始化 n*n 大小的棋盘,其中 'Q' 代表皇后,'#' 代表空位 + state = [["#" for _ in range(n)] for _ in range(n)] + cols = [False] * n # 记录列是否有皇后 + diags1 = [False] * (2 * n - 1) # 记录主对角线是否有皇后 + diags2 = [False] * (2 * n - 1) # 记录副对角线是否有皇后 + res = [] + backtrack(0, n, state, res, cols, diags1, diags2) + + return res + + +"""Driver Code""" +if __name__ == "__main__": + n = 4 + res = n_queens(n) + + print(f"输入棋盘长宽为 {n}") + print(f"皇后放置方案共有 {len(res)} 种") + for state in res: + print("--------------------") + for row 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