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Longest Special Path

Difficulty: Hard

Problem Description

You are given an undirected tree rooted at node 0 with n nodes numbered from 0 to n - 1, represented by a 2D array edges of length n - 1, where edges[i] = [u_i, v_i, length_i] indicates an edge between nodes u_i and v_i with length length_i. You are also given an integer array nums, where nums[i] represents the value at node i.

A special path is defined as a downward path from an ancestor node to a descendant node such that all the values of the nodes in that path are unique. Note that a path may start and end at the same node.

Return an array result of size 2, where result[0] is the length of the longest special path, and result[1] is the minimum number of nodes in all possible longest special paths.

Key Insights

  • The problem involves traversing a tree structure to find paths with unique values.
  • Depth-First Search (DFS) is an effective method to explore all possible paths from each node.
  • A hash table (set) can be utilized to keep track of unique values encountered along a path.
  • The output requires not only the maximum path length but also the minimum number of nodes across all longest paths.

Space and Time Complexity

Time Complexity: O(n) - Each node and edge is visited once during the DFS traversal. Space Complexity: O(n) - The space is used by the recursion stack and the set to track unique values.

Solution

To solve this problem, we will use Depth-First Search (DFS) to traverse the tree. We will maintain a set to track the unique values encountered along the path from each ancestor to its descendants. If we encounter a duplicate value during our traversal, we will backtrack. We will keep track of the longest path length and the minimum number of nodes that constitute this path.

  1. Build the tree from the edges list.
  2. Perform DFS starting from the root node (node 0).
  3. At each node, check if the current value is unique in the context of the current path.
  4. If it is, recursively explore its children, adding the edge lengths to the current path length.
  5. Update the longest path length and the count of nodes when we find a valid path.
  6. Return the results based on the longest path found.

Code Solutions

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

    # Build the adjacency list for the tree
    graph = defaultdict(list)
    for u, v, length in edges:
        graph[u].append((v, length))
        graph[v].append((u, length))

    def dfs(node, parent, current_length, current_nums):
        nonlocal max_length, min_nodes
        # If the current node's value already exists in the path, return
        if nums[node] in current_nums:
            return
        
        # Add the current node's value
        current_nums.add(nums[node])
        
        # Check if this is a new longest path
        if current_length > max_length:
            max_length = current_length
            min_nodes = 1  # Start counting from the current node
        elif current_length == max_length:
            min_nodes += 1  # Found another path with the same length

        # Explore neighbors
        for neighbor, length in graph[node]:
            if neighbor != parent:  # Avoid going back to the parent
                dfs(neighbor, node, current_length + length, current_nums)

        # Backtrack
        current_nums.remove(nums[node])

    max_length = 0
    min_nodes = 0
    dfs(0, -1, 0, set())
    
    return [max_length, min_nodes]
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