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Difference Between Maximum and Minimum Price Sum

Difficulty: Hard

Problem Description

Given an undirected tree with n nodes and an associated price for each node, determine the maximum difference between the maximum and minimum price sums of paths starting from any root node.

Key Insights

  • The problem involves finding paths in a tree structure, which is a connected acyclic graph.
  • Paths can vary greatly in price sums based on the choice of the root node.
  • The maximum price sum path can be found by traversing the tree using Depth-First Search (DFS) or Breadth-First Search (BFS).
  • The minimum price sum path is typically the price of the root node itself.
  • The goal is to maximize the difference between the maximum and minimum price sums across all possible root nodes.

Space and Time Complexity

Time Complexity: O(n) - Each node and edge is processed once. Space Complexity: O(n) - Space used by the adjacency list and recursive call stack in DFS.

Solution

To solve the problem, we can use a Depth-First Search (DFS) approach to explore all paths starting from each node in the tree. We will compute the maximum and minimum price sum paths for each root choice and keep track of the highest difference encountered.

  1. Data Structure: Use an adjacency list to represent the tree.
  2. Algorithm:
    • Traverse the tree using DFS.
    • For each node, calculate the maximum price sum path and the minimum price sum path.
    • Update the maximum difference found when considering each node as a root.

Code Solutions

def maxCost(n, edges, price):
    from collections import defaultdict

    # Create adjacency list
    tree = defaultdict(list)
    for a, b in edges:
        tree[a].append(b)
        tree[b].append(a)

    def dfs(node, parent):
        max_sum = price[node]
        min_sum = price[node]
        
        for neighbor in tree[node]:
            if neighbor == parent:
                continue
            path_sum = dfs(neighbor, node)
            max_sum = max(max_sum, price[node] + path_sum)
            min_sum = min(min_sum, price[node] + path_sum)

        return max_sum, min_sum

    max_difference = 0
    for root in range(n):
        max_sum, min_sum = dfs(root, -1)
        max_difference = max(max_difference, max_sum - min_sum)

    return max_difference
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