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How do you find the minimum sum of a path?

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Traversing over arrays and matrix are basic operations for programmers. But have you ever thought to find the minimum path for traversing till the last element? The minimum path sum problem will help you in doing it. 

There are a lot of programmers who are finding it difficult for finding the minimum path for reaching the last element. It is because they are not clear about the approach and the problem itself.

To guide you towards the right path of solving the minimum path problem, the mentioned approach will surely help you!

About the Minimum Sum of a Path Problem?

The minimum sum path of the problem is an interesting problem based on a matrix or 2D arrays. For the given problem, we have a minimum path sum through which we can traverse from the top left to bottom right with minimum sum.

The problem is based on traversing over the matrix by which we can get the minimum path sum. We all know that there are different ways that are available for traversing over the matrix, however, we will have to find the most efficient path which will be known as the minimum path sum.

Let us understand the problem with some examples.

Example 1

For the given matrix, find the minimum path sum.

            1          2          3

arr=     4          5          4

7          5          9

 

Solution: The minimum path sum for the given problem is: 19. 

Explanation: If you see the problem, then we have to find the shortest possible way according to the element sum. However, an important thing that we have to keep in mind is that we can only traverse in the right or down on the matrix. Thus, if we find the all possible way for reaching the bottom right element, then we have different cases which are:

Case 1: Starting from down, we will keep on traversing till we reach the limit, after it, we will traverse to the right.

1 + 4 + 7 + 5 + 9 = 26

Case 2: We will traverse on the right, after it on the down when we reach the last element.

1 + 2 + 3 + 4 + 9 = 19

Case 3: Traversing on the right for one time, after it, traversing to the down till the last element and again traversing on the right till the last element.

1 + 2 + 5 + 5 + 9 = 22

Case 4: Similarly to case 3, we will traverse on the down for one time, after it, we will traverse on the right till the last element. After doing it, we will again traverse down.

1 + 4 + 5 + 4 + 9 = 23

Through the given cases, we hope that you have got why 19 is the answer to the minimum path sum. We have provided you with some of the cases that can be used to reach the last element. However, we have to find the best possible path, thus we will compare it with all the cases.

After doing it, we got our answer. If you have noticed, then we are making different partitions which is similar to the partition equal subset sum.

Through the given example, we hope that now you are able to understand the problem. Now, take a look at the approach which we have used to find the minimum sum of a path problem.

How To Solve Minimum Sum of a Path Problem

For finding the minimum path sum of a matrix, you will have to know about its approach. Without proper knowledge, you will not be able to solve the problem. We will tell you the right approach that you can use for solving the problem. Along with it, we will also do the dry run through which you will be able to code it in your preferred language.

Let’s understand it with the help of the given problem. 

Problem Statement 

For the given matrix, find the minimum path sum.

            1          2          3

arr=     4          5          4

7          5          9

Answer: 19. 

Approach

  • For finding the minimum sum of the given matrix, we will have to traverse over the whole matrix.
  • Once we are done with the traverse, then we will store the element and will sum them with a similar column and rows.
  • When we are adding the elements, then we have to make sure that it will start from the 1st index of our matrix and will end on the last element.
  • Thus, we have to make all the possible cases for reaching the last element.
  • We will add those elements and will display the output of the minimum sum.
  • When we are displaying the minimum sum, then we will have to make sure that it is the least number. For doing it, we will compare the number with each of the cases that we have found.
  • After it, we will display the output.

Dry Run

  • We will start our iteration for the given matrix. After that, we will look for every possible case.
  • The possible cases for the matrix are 1, 2, 3, 4, and 9. After that, we will look for all similar cases.

Case 1:

1 + 4 + 7 + 5 + 9 = 26

Case 2:

1 + 2 + 3 + 4 + 9 = 19

Case 3:

1 + 2 + 5 + 5 + 9 = 22

Case 4:

1 + 4 + 5 + 4 + 9 = 23

  • After it, we will add the whole element. The sum that we will get is 26, 19, 22, and 23.
  • Then we will display the output. For displaying the output, we will compare each of the elements. On comparing each of them, we will look for the least number.
  • Once we have compared the all elements, then we will print it.

Conclusion

The minimum path sum is an interesting problem that you should attempt to get your hands on different coding challenges. You can easily do this by taking help from our mentioned approach. Do the code in your preferred programming language to check whether you are able to code the problem or still facing any issue with it.

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