62. Unique Paths

/*
 * @lc app=leetcode id=62 lang=cpp
 *
 * [62] Unique Paths
 *
 * https://leetcode.com/problems/unique-paths/description/
 *
 * algorithms
 * Medium (65.00%)
 * Likes:    16985
 * Dislikes: 457
 * Total Accepted:    2.1M
 * Total Submissions: 3.2M
 * Testcase Example:  '3\n7'
 *
 * 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 * 10^9.
 *
 *
 * 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
 *
 *
 *
 * Constraints:
 *
 *
 * 1 <= m, n <= 100
 *
 *
 */

// @lc code=start
class Solution {
public:
    int uniquePaths(int m, int n)
    {
        vector<vector<int>> dp(m, vector<int>(n, 1));
        for (int i = 1; i < m; i++)
        {
            for (int j = 1; j < n; j++)
            {
                dp[i][j] = dp[i - 1][j] + dp[i][j - 1];
            }
        }
        return dp[m - 1][n - 1];
    }
};
// @lc code=end

• T: $O(M \cdot N)$
• S: $O(M \cdot N)$