62. Unique Paths

Leetcode

https://leetcode.com/problems/unique-paths/description/

題目

There is a robot on an m x n grid. The robot is initially located at the top-left corner (i.e., grid[0][0]). The robot tries to move to the bottom-right corner (i.e., grid[m - 1][n - 1]). The robot can only move either down or right at any point in time.

Given the two integers m and n, return the number of possible unique paths that the robot can take to reach the bottom-right corner.

The test cases are generated so that the answer will be less than or equal to 2 * 109.

Example 1:

Input: m = 3, n = 7
Output: 28

Example 2:

Input: m = 3, n = 2
Output: 3
Explanation: From the top-left corner, there are a total of 3 ways to reach the bottom-right corner:
1. Right -> Down -> Down
2. Down -> Down -> Right
3. Down -> Right -> Down

解答

• 方法一 - DP

範例：m = 3, n = 2

使用 dp[i][j] 代表走到 i + 1 列 j + 1 行時所有可能的步數，從左上角開始走：

• dp[0][0] = 1

  第一列第一行為起始點，只有一種可能 => dp[0][0] = 1

• dp[1][0] = 1

  走到第二列第一行唯一的可能性為起始點往下走，也就是說 dp[1][0] = dp[0][0] = 1

• dp[2][0] = 1

  走到第三列第一行唯一的可能性為 dp[1][0] 往下走，也就是說 dp[2][0] = dp[1][0] = 1

• dp[0][1] = 1

  走到第一列第二行唯一的可能性為起始點往右走，也就是說 dp[0][1] = dp[0][0] = 1

• dp[1][1] = 2

  走到第二列第二行的可能性為 dp[1][0] 往右走以及 dp[0][1] 往下走，也就是說

  dp[1][1] = dp[1][0] + dp[0][1] = 1 + 1 = 2

• dp[2][1] = 3

  走到第三列第二行的可能性為 dp[2][0] 往右走以及 dp[1][1] 往下走，也就是說

  dp[2][1] = dp[2][0] + dp[1][1] = 1 + 2 = 3

var uniquePaths = function (m, n) {
    const dp = Array.from({ length: m }, () => []);

    for (let i = 0; i < m; i++) {
        for (let j = 0; j < n; j++) {
            if (i === 0 && j === 0) {
                dp[i][j] = 1;
            } else {
                dp[i][j] = 0;
                // 如果 i - 1 沒有超過上面邊界的話，加入上面那個點的 DP
                if (i - 1 >= 0) {
                    dp[i][j] += dp[i - 1][j];
                }
                // 如果 j - 1 沒有超過左邊邊界的話，加入左邊那個點的 DP
                if (j - 1 >= 0) {
                    dp[i][j] += dp[i][j - 1];
                }
            }
        }
    }

    return dp[m - 1][n - 1];
};

參考資料

62. Unique Paths · leetcodeleetcode.wang