Maximum Path Sum is an algorithm to find out a path such that sum of element(node) of that path is greater than any other path.

For example, let's we have a we a triangle as shown below.

```
3
7 4
2 4 6
8 5 9 3
```

In above triangle, find the maximum path which has maximum sum.
Answer is, `3 + 7 + 4 + 9 = 23`

To find out the solution, as always we get an idea of Brute-Force method. Brute-Force method is good for this 4 rows triangle but think about a triangle with 100 or more than 100 rows. So, We can not use Brute-Force method to solve this problem.

Let's move to dynamic programming approach.

**Algorithm:**

For each and every node in a triangle or in a binary tree there can be four ways that the max path goes through the node.

- Node only
- Max path through Left Child + Node
- Max path through Right Child + Node
- Max path through Left Child + Node + Max path through Right Child.

**Example of Maximum Path Sum Algorithm:**

**Space Auxiliary:** `O(n)`

**Time Complexity:** `O(n)`