A robot is located at the top-left corner of a m x 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.
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
题目
给定MxN棋盘,只允许从左上往右下走,每次走一格。共有多少种走法?
思路
Matrix DP(二维DP) 问题
1. 初始化
预处理第一个row: dp[0][j] = 1 因为从左上起点出发,往右走的每一个unique path都是1
预处理第一个col: dp[i][0]= 1 因为从左上起点出发,往下走的每一个unique path 都是1
2. 转移方程
因为要求所有possible unique paths之和
dp[i][j] 要么来自dp[i-1][j] 要么来自dp[i][j-1]
代码
class Solution {
public int uniquePaths(int m, int n) {
int[][]dp = new int[n][m]; // [row][col] // 预处理第一个col
for(int i= 0; i < n; i++){
dp[i][0] = 1;
}
//预处理第一个row
for(int j=0; i < m; i++){
dp[0][j] = 1;
}
for(int i= 1; i < n; i++){
for(int j= 1; j < m; j++){
dp[i][j] = dp[i-1][j] + dp[i][j-1];
}
}
return dp[n-1][m-1];
}
}