Solution Explanation: Maximum Path Quality of a Graph

This problem asks to find the maximum quality of a path in a graph, starting and ending at node 0, with a maximum allowed time. The quality is the sum of unique node values visited.

Approach: Depth-First Search (DFS)

The most efficient approach is to use Depth-First Search (DFS) to explore all possible paths within the time constraint. Since the maximum time is relatively small and each node has at most four edges, a brute-force DFS is feasible.

Algorithm:

1. Graph Representation: Represent the graph using an adjacency list. Each node stores a list of its neighbors and the corresponding edge weights (times).

2. DFS Function: The core of the solution is a recursive DFS function: dfs(u, cost, value, visited):

   • u: The current node.
   • cost: The accumulated time cost of the path so far.
   • value: The accumulated quality (sum of unique node values) of the path so far.
   • visited: A set to track visited nodes (to ensure unique node value summation).

3. Base Case: If the current node u is 0 (the starting node) and we've returned, update the maximum quality (ans) if the current path's quality is higher.

4. Recursive Step: For each neighbor v of the current node u, check if adding the edge time to the current cost exceeds maxTime. If not:

   • If v is already visited, recursively call dfs(v, cost + time, value, visited).
   • If v is not visited, add it to visited, update value with values[v], recursively call dfs(v, cost + time, value, visited), and then remove v from visited (backtracking).

5. Initialization: Start the DFS from node 0 with initial cost 0, value values[0], and an empty visited set.

Time and Space Complexity:

• Time Complexity: O(N * 4T/min_time), where N is the number of nodes and T is maxTime. The exponential factor comes from the branching in the DFS; in the worst case, we explore all possible paths, which can grow exponentially with the maximum allowed time. The min_time is the shortest edge weight which limits the maximum depth of the DFS tree.
• Space Complexity: O(N + E + T/min_time), where E is the number of edges. The space is dominated by the recursion stack (depth proportional to T/min_time) and the adjacency list.

Code (Python):

def maximalPathQuality(values, edges, maxTime):
    n = len(values)
    graph = [[] for _ in range(n)]
    for u, v, t in edges:
        graph[u].append((v, t))
        graph[v].append((u, t))
 
    ans = 0
 
    def dfs(u, cost, value, visited):
        nonlocal ans
        if u == 0:
            ans = max(ans, value)
            return
 
        for v, t in graph[u]:
            if cost + t <= maxTime:
                new_visited = set(visited)
                if v not in new_visited:
                    new_visited.add(v)
                    dfs(v, cost + t, value + values[v], new_visited)
                else:
                    dfs(v, cost + t, value, new_visited)
 
    visited = {0}
    dfs(0, 0, values[0], visited)
    return ans

The code in other languages (Java, C++, Go, TypeScript) follows a very similar structure, using their respective data structures and syntax. The key algorithmic idea remains the same – a depth-first search that explores paths within the time constraint and tracks visited nodes for quality calculation.