Today, we’ll learn to solve the problem of finding the minimum cost path using dynamic programming in C++. It’s a popular dynamic programming problem on LeetCode, CodeForces, and CodeChef. Dynamic programming is a great choice to solve the minimum cost path and maximum cost path problems. So without wasting time, let’s get started.

## Problem Statement for Minimum Cost Path

*You are standing in a maze of size M x N units. Some cells of the maze might be blocked. You are standing at the cell (1, 1) in the maze. The cost of traveling through a cell in the maze is the value stored in that cell. Your objective is to reach the cell (x, y) in the maze minimizing the total cost of traversal. Note: You can only travel along the right and the down directions. *

### Input Format

The input will be a single integer *N*. And the following line will contain N integers representing the value of coins.

For example,

```
M: 3
N: 3
Maze:
1 5 2
7 1 1
8 1 3
X: 2
Y: 2
Solution:
Minimum cost of travel: 7
Path: 1 --> 5 --> 1
```

## Approach for Minimum Cost Path

The recursive approach is not enough for large values of *N*. The time complexity of the recursive solution is exponential with the number of turns(i.e. time complexity = *O(2 ^{N})*). The recursive solution is functional only up to

*N = 24*.

### How To Deal With Dynamic Programming Problems

Dynamic programming problems are recursive problems but by storing the intermediate states, we reduce a large number of recomputations. Similar to recursive solutions, the dynamic programming solutions compute all the subproblems of a problem but we store our solutions using some data structure.

*Dynamic Programming = Recursion + Memoisation = Never solve the same problem twice*

*Recursion: Compute all the subproblems of a problem.*

*Memoisation: Store the results of expensive computations.*

### Pseudocode

Below is the step by step process that we’ll follow to code the solution.

- Base cases:
*(i == 0 && j == 0): dp[i][j] = cost[0][0];**i == 0: dp[i][j] = dp[0][j – 1] + cost[i][j];**j == 0: dp[i][j] = dp[i][0] + cost[i][j];*

- Recursive case:
- We can reach the cell
*(i, j)*using two routes- Through
*(i – 1, j)* - Or via,
*(i, j – 1)*

- Through
- Our answer will be the minimum of both the paths.

*dp[i][j]=min(dp[i – 1][j], dp[i][j – 1]) + cost[i][j];*

- We can reach the cell
- Store this value in the dp[i][j] cell of the dynamic programming array. And use this variable later to avoid the computation of pre-computed subproblems.

## Minimum Cost Path Using Dynamic Programming In C++ Program

```
#include <iostream>
using namespace std;
int minCost(int R, int C, int cost[][100])
{
int dp[R][C] = {0};
for(int i = 0; i < R; i++)
{
for(int j = 0; j < C; j++)
{
//base case
if(i == 0 and j == 0)
{
dp[i][j] = cost[0][0];
}
else if(i == 0)
{
dp[i][j] = dp[0][j - 1] + cost[i][j];
}
else if(j == 0)
{
dp[i][j] = dp[i][0] + cost[i][j];
}
else
{
dp[i][j]=min(dp[i - 1][j], dp[i][j - 1]) + cost[i][j];
}
}
}
// let's have a look at the dynamic programming array
/*cout<<"The dp array is:"<<endl;
for(int i = 0; i < R; i++)
{
for(int j = 0; j < C; j++)
{
cout << dp[i][j] << " \t";
}
cout << endl;
}*/
//tracing our path
cout << "The path followed is: " << endl;
cout << cost[0][0] << " --> ";
int i = 0,j = 0;
while(i < R - 1 && j < C - 1)
{
if(dp[i + 1][j] < dp[i][j + 1])
{
cout << cost[i + 1][j]<< " --> ";
i++;
}
else
{
cout << cost[i][j + 1] << " --> ";
j++;
}
}
cout << cost[R - 1][C - 1] << endl;
cout << endl << "The minimum cost needed to reach the destination is: " << dp[R - 1][C - 1];
cout << endl;
return dp[R - 1][C - 1];
}
int main()
{
cout << "Enter the values of N and M" << endl;
int m, n;
cin >> m >> n;
int dp[m][n] = {0};
cout << "Enter the cost of each cell" << endl;
int cost[m][100];
// initialize the array values
for(int i = 0; i < m; i++)
{
for(int j = 0; j < n; j++)
{
cin >> cost[i][j];
}
}
cout << "Enter the co-ordinates of the destination" << endl;
int x, y;
cin >> x >> y;
minCost(x, y, cost);
return 0;
}
```

## Output

## Conclusion

Today, we solved the problem of a minimum-cost path using dynamic programming in C++. We used the concept of recursion + memoisation(Dynamic Programming) to solve this problem. Dynamic programming solutions are of two types. The first is the bottom-up approach and the second one is the top-down approach. In this article, we used the bottom-up approach to develop this solution In the end, we implemented a C++ program to demonstrate the working of our solution. The time and the space complexity of this approach is *O(N ^{2})*. That’s all for today, thanks for reading.

## Further Readings

To learn more about C++ programming, data structures, and algorithms, you can go through the following articles.

https://www.journaldev.com/56663/breadth-first-search-algorithm-graphs-cpp

https://www.journaldev.com/56532/queue-class-in-cpp