Leetcode
Array
Dynamic Programming
A robot is located at the top-left corner of a m times n grid (marked 'Start' in the diagram below).
The robot can only move either down or right at any point in time. The robot is trying to reach the bottom-right corner of the grid (marked 'Finish' in the diagram below).
How many possible unique paths are there?
Above is a 7 x 3 grid. How many possible unique paths are there?
Note: m and n will be at most 100.
Example 1:
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 -> Right -> Down 2. Right -> Down -> Right 3. Down -> Right -> Right
Example 2:
Input: m = 7, n = 3 Output: 28
分析¶
这道题目要求路径的数量,又是二维棋盘,一看就是典型的动态规划题目。设想机器人站在终点坐标(m-1, n-1)上,那么它的上一步来自哪里呢?有且仅有两种可能,来自(m-2, n-1)和来自(m-1, n-2)。那么答案非常明显了。这种自顶向下的方法形成的代码如下:
public int uniquePaths(int m, int n) { if (m == 0 || n == 0) return 0; int[][] numPaths = new int[m + 1][n + 1]; numPaths[1][1] = 1; for (int i = 1; i < m + 1; i++) for (int j = 1; j < n + 1; j++) { if (i == 1 && j == 1) continue; numPaths[i][j] = numPaths[i][j - 1] + numPaths[i - 1][j]; } return numPaths[m][n]; }
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