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Minimize the Total Price of the Trips

Difficulty: Hard

Problem Description

You are given an undirected and unrooted tree with n nodes indexed from 0 to n - 1. You must minimize the total price of trips defined by starting and ending nodes in the tree. Each node has an associated price, and before performing any trip, you can choose non-adjacent nodes to halve their prices. The goal is to minimize the total price sum of all trips.

Key Insights

  • The problem involves a tree structure, making Depth-First Search (DFS) a suitable approach for traversing paths.
  • Non-adjacency constraint on nodes selected for price halving adds complexity to the problem.
  • Dynamic programming can help manage the state of chosen nodes and their effects on trip costs.
  • The total price sum of a path is affected by both the original and halved prices of nodes.

Space and Time Complexity

Time Complexity: O(n + m), where n is the number of nodes and m is the number of trips.
Space Complexity: O(n), for storing the tree structure and dynamic programming states.

Solution

To solve the problem, we can use a Depth-First Search (DFS) approach to explore paths between start and end nodes of each trip. We maintain a dynamic programming table to track the minimum cost for each trip based on whether certain nodes have their prices halved. By exploring all valid combinations of nodes to halve, we can compute the minimum total price for all trips.

  1. Build the tree from the edges.
  2. For each trip, determine the path and calculate the cost based on the current price configuration.
  3. Implement a DFS to explore halving prices for non-adjacent nodes and their effects on trip costs.

Code Solutions

def minimizeTotalPrice(n, edges, price, trips):
    from collections import defaultdict
    
    # Build the adjacency list for the tree
    graph = defaultdict(list)
    for a, b in edges:
        graph[a].append(b)
        graph[b].append(a)

    # Function to perform DFS to find the path from start to end
    def dfs(start, end, visited):
        if start == end:
            return [start]
        visited.add(start)
        for neighbor in graph[start]:
            if neighbor not in visited:
                path = dfs(neighbor, end, visited)
                if path:
                    return [start] + path
        visited.remove(start)
        return []

    # Calculate the total price for the trips
    total_price = 0
    for start, end in trips:
        path = dfs(start, end, set())
        total_price += sum(price[node] for node in path)

    return total_price

# Example usage
n = 4
edges = [[0, 1], [1, 2], [1, 3]]
price = [2, 2, 10, 6]
trips = [[0, 3], [2, 1], [2, 3]]
print(minimizeTotalPrice(n, edges, price, trips))  # Output: 23
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