Question

Formatted question description: https://leetcode.ca/all/62.html

62	Unique Paths

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 3 x 7 grid. How many possible unique paths are there?

Note: m and n will be at most 100.

@tag-array

Algorithm

Using Dynamic Programming to solve, you can maintain a two-dimensional array dp, where dp[i][j] represents the number of different moves to the current position, and then the state transition equation can be obtained as: dp[i][j ] = dp[i-1][j] + dp[i][j-1], here in order to save space, use a one-dimensional array dp, refresh line by line.

Code

Java

public class Unique_Paths {

    class Solution {
        public int uniquePaths(int m, int n) {

            if (m <= 0 || n <= 0) {
                return 0;
            }

            int[][] dp = new int[m][n];

            for (int i = 0; i < m; i++) {
                for (int j = 0; j < n; j++) {

                    if (i == 0 || j == 0) {
                        dp[i][j] = 1;
                    } else { // i,j both not 1
                        dp[i][j] = dp[i-1][j] + dp[i][j-1];
                    }
                }
            }

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


}

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