Maximum Path Sum in an Indexed Binary Tree

Question

This question asks the candidate to find the maximum path sum in an indexed binary tree, where the path can start and end at any node in the tree.

Codemia · Accepted Answer

To solve the maximum path sum problem in a binary tree, we can use a depth-first search (DFS) approach. The idea is to traverse each node and calculate the maximum path sum that can be obtained from that node to any of its descendants. We will keep track of the maximum path sum encountered during the traversal.

We can implement the algorithm using a recursive function. The function will return the maximum path sum that can be obtained from the current node to its children. At each node, we will calculate the maximum path sum that includes the current node and either of its left or right child. The overall maximum path sum will be updated at each node.

```python
class TreeNode:
    def __init__(self, value=0, left=None, right=None):
        self.value = value
        self.left = left
        self.right = right

class Solution:
    def maxPathSum(self, root: TreeNode) -> int:
        self.max_sum = float('-inf')
        def dfs(node):
            if not node:
                return 0
            left = max(dfs(node.left), 0)
            right = max(dfs(node.right), 0)
            self.max_sum = max(self.max_sum, node.value + left + right)
            return node.value + max(left, right)
        dfs(root)
        return self.max_sum
```

The time complexity of this algorithm is O(N), where N is the number of nodes in the binary tree. This is because we visit each node exactly once. The space complexity is O(H), where H is the height of the tree, due to the recursion stack. In the worst case, for a skewed tree, H can be equal to N.

We need to consider edge cases such as an empty tree, where the maximum path sum should be defined (e.g., return 0 or negative infinity). Additionally, we should handle trees where all node values are negative, ensuring our solution correctly identifies the maximum path sum as the least negative value.

Interviewers are looking for clarity in the candidate's explanation of the approach and the ability to articulate the thought process behind the algorithm. They also evaluate the candidate's coding style, efficiency, and how well they handle edge cases. Demonstrating a solid understanding of time and space complexity is crucial for a strong impression.

Maximum Path Sum in an Indexed Binary Tree

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