Problem Description
Given an undirected tree with n nodes numbered from 0 to n - 1, and a 2D integer array edges representing the edges and their weights, your task is to remove zero or more edges such that each node has at most k edges with other nodes, maximizing the sum of the weights of the remaining edges.
Key Insights
- The problem revolves around managing the degree of each node in the tree.
- A priority-based approach can be used to keep the highest-weight edges while ensuring each node maintains a degree limit of k.
- Depth-first search (DFS) can be used to traverse the tree and calculate the best configuration of edges to keep.
- Dynamic programming can help in keeping track of the maximum sum of weights while adhering to the degree constraints.
Space and Time Complexity
Time Complexity: O(n log n) - due to sorting edges based on weights. Space Complexity: O(n) - for storing the tree and the maximum weights.
Solution
The approach involves representing the tree using an adjacency list, sorting the edges by their weights in descending order, and then using a union-find data structure to keep track of the connected components. We iterate over the edges and, for each edge, check if adding it would exceed the degree limit of either node. If not, we add it to the result.