Minimum Score After Removals On A Tree - Leetcode Solution

Problem Statement:

Given a tree in the form of an undirected graph consisting of n nodes numbered from 0 to n-1, and an integer x, you need to delete the edges of the tree in such a way that after deleting the edges, the maximum element in each connected component of the tree is less than or equal to x. Your task is to determine the minimum possible sum of the remaining edge weights after deleting these edges. If it is impossible to obtain connected components with maximum element less than or equal to x, return -1.

Solution Approach:

The minimum score can be obtained by using a binary search on the maximum elements that could be present in each connected component of the tree. For each maximum element guess, we perform a DFS starting from each node in the tree and collecting the maximum elements in each connected component. If all the maximum elements in the connected components are less than or equal to x, then we can keep guessing for lower values. Otherwise, we should guess for higher values.

The DFS is implemented using a stack and the maximum element encountered during the DFS is kept track of using a variable. The minimum score for a given maximum element guess is obtained using a counting sort. We count the number of elements that are greater than x for each connected component of the tree and add the products of these counts with the corresponding edge weights.

Time Complexity:

The overall time complexity of the solution is O(nlogn+mlogm), where m is the number of edges in the tree. The DFS is performed n times, once for each node, and each DFS has O(m) time complexity. The binary search has O(logn) time complexity and for each maximum element guess, counting sort is performed which has O(mlogm) time complexity.

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