【Leecode】Leecode刷题之路第37天之解数独

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题目出处

37-解数独-题目出处

题目描述

37-解数独-题目描述1
37-解数独-题目描述2
37-解数独-题目描述3

个人解法

思路:

todo

代码示例:(Java)

todo

复杂度分析

todo

官方解法

37-解数独-官方解法

37-解数独-官方解法-前言

方法1:回溯

思路:

37-解数独-回溯法-思路

代码示例:(Java)

public class Solution1 {
    private boolean[][] line = new boolean[9][9];
    private boolean[][] column = new boolean[9][9];
    private boolean[][][] block = new boolean[3][3][9];
    private boolean valid = false;
    private List<int[]> spaces = new ArrayList<int[]>();

    public void solveSudoku(char[][] board) {
        for (int i = 0; i < 9; ++i) {
            for (int j = 0; j < 9; ++j) {
                if (board[i][j] == '.') {
                    spaces.add(new int[]{i, j});
                } else {
                    int digit = board[i][j] - '0' - 1;
                    line[i][digit] = column[j][digit] = block[i / 3][j / 3][digit] = true;
                }
            }
        }

        dfs(board, 0);
    }

    public void dfs(char[][] board, int pos) {
        if (pos == spaces.size()) {
            valid = true;
            return;
        }

        int[] space = spaces.get(pos);
        int i = space[0], j = space[1];
        for (int digit = 0; digit < 9 && !valid; ++digit) {
            if (!line[i][digit] && !column[j][digit] && !block[i / 3][j / 3][digit]) {
                line[i][digit] = column[j][digit] = block[i / 3][j / 3][digit] = true;
                board[i][j] = (char) (digit + '0' + 1);
                dfs(board, pos + 1);
                line[i][digit] = column[j][digit] = block[i / 3][j / 3][digit] = false;
            }
        }
    }


}

复杂度分析

  • 时间复杂度:O(1)。数独共有 81 个单元格,只需要对每个单元格遍历一次即可。
  • 空间复杂度:O(1)。由于数独的大小固定,因此哈希表的空间也是固定的。

方法2:位运算优化

思路:

37-解数独-位运算优化-思路

代码示例:(Java)

public class Solution2 {
    private int[] line = new int[9];
    private int[] column = new int[9];
    private int[][] block = new int[3][3];
    private boolean valid = false;
    private List<int[]> spaces = new ArrayList<int[]>();

    public void solveSudoku(char[][] board) {
        for (int i = 0; i < 9; ++i) {
            for (int j = 0; j < 9; ++j) {
                if (board[i][j] == '.') {
                    spaces.add(new int[]{i, j});
                } else {
                    int digit = board[i][j] - '0' - 1;
                    flip(i, j, digit);
                }
            }
        }

        dfs(board, 0);
    }

    public void dfs(char[][] board, int pos) {
        if (pos == spaces.size()) {
            valid = true;
            return;
        }

        int[] space = spaces.get(pos);
        int i = space[0], j = space[1];
        int mask = ~(line[i] | column[j] | block[i / 3][j / 3]) & 0x1ff;
        for (; mask != 0 && !valid; mask &= (mask - 1)) {
            int digitMask = mask & (-mask);
            int digit = Integer.bitCount(digitMask - 1);
            flip(i, j, digit);
            board[i][j] = (char) (digit + '0' + 1);
            dfs(board, pos + 1);
            flip(i, j, digit);
        }
    }

    public void flip(int i, int j, int digit) {
        line[i] ^= (1 << digit);
        column[j] ^= (1 << digit);
        block[i / 3][j / 3] ^= (1 << digit);
    }


}

复杂度分析

  • 时间复杂度:O(1)。数独共有 81 个单元格,只需要对每个单元格遍历一次即可。
  • 空间复杂度:O(1)。由于数独的大小固定,因此哈希表的空间也是固定的。

方法3:枚举优化

思路:

37-解数独-枚举优化-思路

代码示例:(Java)

public class Solution3 {
    private int[] line = new int[9];
    private int[] column = new int[9];
    private int[][] block = new int[3][3];
    private boolean valid = false;
    private List<int[]> spaces = new ArrayList<int[]>();

    public void solveSudoku(char[][] board) {
        for (int i = 0; i < 9; ++i) {
            for (int j = 0; j < 9; ++j) {
                if (board[i][j] != '.') {
                    int digit = board[i][j] - '0' - 1;
                    flip(i, j, digit);
                }
            }
        }

        while (true) {
            boolean modified = false;
            for (int i = 0; i < 9; ++i) {
                for (int j = 0; j < 9; ++j) {
                    if (board[i][j] == '.') {
                        int mask = ~(line[i] | column[j] | block[i / 3][j / 3]) & 0x1ff;
                        if ((mask & (mask - 1)) == 0) {
                            int digit = Integer.bitCount(mask - 1);
                            flip(i, j, digit);
                            board[i][j] = (char) (digit + '0' + 1);
                            modified = true;
                        }
                    }
                }
            }
            if (!modified) {
                break;
            }
        }

        for (int i = 0; i < 9; ++i) {
            for (int j = 0; j < 9; ++j) {
                if (board[i][j] == '.') {
                    spaces.add(new int[]{i, j});
                }
            }
        }

        dfs(board, 0);
    }

    public void dfs(char[][] board, int pos) {
        if (pos == spaces.size()) {
            valid = true;
            return;
        }

        int[] space = spaces.get(pos);
        int i = space[0], j = space[1];
        int mask = ~(line[i] | column[j] | block[i / 3][j / 3]) & 0x1ff;
        for (; mask != 0 && !valid; mask &= (mask - 1)) {
            int digitMask = mask & (-mask);
            int digit = Integer.bitCount(digitMask - 1);
            flip(i, j, digit);
            board[i][j] = (char) (digit + '0' + 1);
            dfs(board, pos + 1);
            flip(i, j, digit);
        }
    }

    public void flip(int i, int j, int digit) {
        line[i] ^= (1 << digit);
        column[j] ^= (1 << digit);
        block[i / 3][j / 3] ^= (1 << digit);
    }


}

复杂度分析

  • 时间复杂度:O(1)。数独共有 81 个单元格,只需要对每个单元格遍历一次即可。
  • 空间复杂度:O(1)。由于数独的大小固定,因此哈希表的空间也是固定的。

考察知识点

收获

1.位运算

Gitee源码位置

37-解数独-源码

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