golang_n_queens

The n-queens puzzle is the problem of placing n queens on an n x n chessboard such that no two queens attack each other.

Given an integer n, return all distinct solutions to the n-queens puzzle. You may return the answer in any order.

Each solution contains a distinct board configuration of the n-queens' placement, where 'Q' and '.' both indicate a queen and an empty space, respectively.

Examples

Example 1:

Input: n = 4
Output: [[".Q..","...Q","Q...","..Q."],["..Q.","Q...","...Q",".Q.."]]
Explanation: There exist two distinct solutions to the 4-queens puzzle as shown above

Example 2:

Input: n = 1
Output: [["Q"]]

Constraints:

• 1 <= n <= 9

解析

給定一個正整數 n , 要把 n 個西洋棋的皇后放到 n by n 的棋盤使得 n 個皇后不會相互攻擊

根據西洋棋規則，西洋棋的皇后可以走正反對角線還有上下左右直線。或者說是米字形範圍

這個限制就是用來檢查是否能夠放皇后的規則

要簡化的這個問題， 可以利用上面那個規則

因為每個皇后水平方向會互斥，代表每個列只能放一個皇后

可以簡化成對 n 列， 選擇一個 col 位置給 皇后放

檢查以下3個條件：

1. 該 col 是否已經有皇后選過
2. 該對角線是否已經有皇候選過
3. 該反對角線是否已經有皇候選過

如果都沒有則繼續往下選直到最後一個

否則換下一個位置繼續選

如下圖:

程式碼

package sol

import "strings"

func solveNQueens(n int) [][]string {
	columns := make(map[int]struct{})
	pos_diagonal := make(map[int]struct{})
	neg_diagonal := make(map[int]struct{})

	board := make([][]string, n)
	for row := range board {
		board[row] = make([]string, n)
		for col := 0; col < n; col++ {
			board[row][col] = "."
		}
	}
	result := [][]string{}
	var dfs func(row int)
	dfs = func(row int) {
		if row == n {
			temp := make([]string, n)
			for idx, rowValues := range board {
				temp[idx] = strings.Join(rowValues, "")
			}
			result = append(result, temp)
			return
		}
		for col := 0; col < n; col++ {
			if _, exists := columns[col]; exists {
				continue
			}
			if _, exists := pos_diagonal[row-col]; exists {
				continue
			}
			if _, exists := neg_diagonal[row+col]; exists {
				continue
			}
			columns[col] = struct{}{}
			pos_diagonal[row-col] = struct{}{}
			neg_diagonal[row+col] = struct{}{}
			board[row][col] = "Q"
			dfs(row + 1)
			delete(columns, col)
			delete(pos_diagonal, row-col)
			delete(neg_diagonal, row+col)
			board[row][col] = "."

		}
	}
	dfs(0)
	return result
}

困難點

1. 理解窮舉的規則
2. 理解西洋棋皇后的規則

Solve Point

• Understand what problem to solve
• Analysis Complexity