【LeetCode】【动态规划】表格移动问题

前言

这里总结了两道表格移动的问题，分别是：Unique Paths 和 

题一：Unique Paths

描述

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.

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

 

思路：动态规划

因为只能向右或者向下移动，所以

 dp[0][j] = 1

 dp[i][0] = 1

 dp[i][j] = dp[i -1][j] + dp[i][j-1]

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];
    }
};

 

题二：Minimum Path Sum

描述

Given a m x n grid filled with non-negative numbers, find a path from top left to bottom right which minimizes the sum of all numbers along its path.

Note: You can only move either down or right at any point in time.

Example:

Input:
[
  [1,3,1],
  [1,5,1],
  [4,2,1]
]
Output: 7
Explanation: Because the path 1→3→1→1→1 minimizes the sum.

思路

这个是表格的变形，这是将线路上的权值相加，求最小值，所以dp数组更新的是线路上最小的权值和：递推公式如下：

 

 

dp[0][0] = grid[0][0]

dp[i][0] = dp[i - 1][0] + grid[i][0]

dp[0][j] = dp[0][j-1] + grid[0][j]

dp[i][j] = min(dp[i-1][j], dp[i][j-1]) + grid[i][j]

class Solution {
public:
    int minPathSum(vector<vector<int>>& grid) {
        if(grid.size() == 0) return 0;
        int n = grid.size(), m = grid[0].size();
        cout<<m<<" "<<n<<endl;
        vector<vector<int> > dp(n, vector<int>(m, 0));
        for(int i = 0;i<n;++i){
            for(int j=0;j<m;++j){
                if(i == 0 && j == 0)
                    dp[i][j] = grid[i][j];
                else if(i == 0 && j != 0)
                    dp[i][j] = dp[i][j-1] + grid[i][j];
                else if(j == 0 && i != 0)
                    dp[i][j] = dp[i-1][j] + grid[i][j];
                else
                    dp[i][j] = min(dp[i-1][j], dp[i][j-1]) + grid[i][j];
            }
        }
        return dp[n-1][m-1];
    }
};

或者将for循环里的 if  else提取出来分别处理：

class Solution {
public:
    int minPathSum(vector<vector<int>>& grid) {
        if(grid.size() == 0) return 0;
        int n = grid.size(), m = grid[0].size();
        cout<<m<<" "<<n<<endl;
        vector<vector<int> > dp(n, vector<int>(m, 0));
        dp[0][0] = grid[0][0];
        for(int i = 1;i<n;++i)
            dp[i][0] = dp[i - 1][0] + grid[i][0];
        for(int j = 1;j<m;++j)
            dp[0][j] = dp[0][j-1] + grid[0][j];
        for(int i = 1;i<n;++i){
            for(int j= 1;j<m;++j){
                dp[i][j] = min(dp[i-1][j], dp[i][j-1]) + grid[i][j];
            }
        }
        return dp[n-1][m-1];
    }
};