動態規劃練習1--Unique Paths

題目內容

[引自] https://leetcode.com/problems/unique-paths/description/

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.

僞代碼

核心: uniquePaths(i,j)=uniquePaths(i-1,j)+uniquePaths(i,j-1)
states數組保存狀態值 states[i][j]表示到達i,j位置的不一樣路徑數
邊界值:初始邊界爲(0,x)和 (x,0)
初始化: states[0][0]=0,states[0][x]=0,states[x][0]=0 
(1,1)表示初始位置,states[1][1]=0,states[1][2]=1,states[2][1]=1

算法

int uniquePaths(int m, int n)
    int[][] states =new int[101][101]
    states[1][1]=1
    for  i=1 to m+1
        for j=1 to n+1
            if(i==1 and j==1)
                continue;
            states[i][j]=states[i-1][j]+states[i][j-1]
    states[1][1]=0
    return states[m][n]

java 代碼實現

public int uniquePaths(int m, int n) {
        int[][] states =new int[101][101];
        states[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;
                }
                states[i][j]=states[i-1][j]+states[i][j-1];
            }
        }
        states[1][1]=0;
        return states[m][n];
    }