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Minimize Increments to Equalize Path Costs

ByteDanceSoftware Engineer · Technical Screen
MediumSoftware EngineerTechnical ScreenCoding & Algorithms

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This problem tests your ability to manipulate tree node values to satisfy a global constraint using a bottom-up dynamic programming or greedy approach.

Problem Statement

You are given a binary tree where each node has an integer value. A 'path' is defined as the sequence of nodes from the root to any leaf node. The 'cost' of a path is the sum of the values of the nodes along that path.

You are allowed to increase the value of any node by any non-negative integer amount. Your goal is to make the cost of every root-to-leaf path in the tree equal, while minimizing the total amount by which you increment the node values.

Implement the function: def min_increments_to_equalize(root: TreeNode) -> int:

Constraints

  • The tree contains between 1 and 10^4 nodes.
  • Node values are integers between 0 and 10^6.
  • You must return the minimum total increment required.

Example 1

Input: [1, 2, 3, 4, 5, 6, 7] Path costs: 1 -> 2 -> 4 = 7 1 -> 2 -> 5 = 8 1 -> 3 -> 6 = 10 1 -> 3 -> 7 = 11 Output: 10 Explanation: We must equalize all paths to the maximum current path cost (11). Incrementing path 1-2-4 by 4, 1-2-5 by 3, and 1-3-6 by 1.

Edge Cases

  • Single node tree: Return 0.
  • Already equal paths: Return 0.
  • Skewed trees (linked list structure).

Hints

Hint 1

Identify the maximum path sum in the current tree; all paths must eventually sum to at least this value.

Hint 2

Process the tree using a post-order traversal to calculate the maximum path sum in the subtree rooted at the current node.

Hint 3

For any node, the difference between the maximum path sum of its left and right children determines how much the smaller path must be increased to match the larger one.

Follow-up questions

  • How would your algorithm change if you were allowed to decrease node values instead of only increasing them?
  • What is the time and space complexity of your solution, and can you optimize the space complexity for a very deep tree?
  • If the tree were an N-ary tree instead of a binary tree, how would the logic for balancing sub-paths change?