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Create Components With Same Value

Difficulty: Hard

Problem Description

You are given an undirected tree with n nodes and a value associated with each node. Your goal is to determine the maximum number of edges that can be deleted such that every connected component in the tree has the same total value.

Key Insights

  • The sum of values in each connected component must be equal after deleting edges.
  • The total sum of the values in the tree must be divisible by the number of components to ensure equal values in each.
  • The problem can be approached using Depth-First Search (DFS) to explore the tree and calculate subtree sums.

Space and Time Complexity

Time Complexity: O(n) - Each node and edge is visited once. Space Complexity: O(n) - For storing the tree and recursion stack.

Solution

To solve this problem, we can use a Depth-First Search (DFS) approach. First, we calculate the total sum of the node values. Then, we look for divisors of this total sum to find possible component values. For each divisor, we perform a DFS to check if we can partition the tree into connected components that each sum to this divisor. Every time we successfully create a new component, we can increase the count of deleted edges.

  1. Calculate the total sum of the node values.
  2. Find all divisors of the total sum.
  3. For each divisor, use DFS to check if we can achieve that component value.
  4. Count the maximum edges that can be deleted based on successful partitions.

Code Solutions

def max_edges_to_delete(nums, edges):
    from collections import defaultdict

    def dfs(node, parent):
        current_sum = nums[node]
        for neighbor in graph[node]:
            if neighbor != parent:
                child_sum = dfs(neighbor, node)
                if child_sum > 0:
                    current_sum += child_sum
                    if child_sum == target:
                        nonlocal edges_deleted
                        edges_deleted += 1
        return current_sum

    total_sum = sum(nums)
    graph = defaultdict(list)
    for a, b in edges:
        graph[a].append(b)
        graph[b].append(a)

    edges_deleted = 0
    possible_values = []

    for i in range(1, total_sum + 1):
        if total_sum % i == 0:
            possible_values.append(i)

    for target in possible_values:
        if dfs(0, -1) == target:
            return edges_deleted

    return 0
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