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Abhishek Chaudhary

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# Maximum Path Quality of a Graph

There is an undirected graph with `n` nodes numbered from `0` to `n - 1` (inclusive). You are given a 0-indexed integer array `values` where `values[i]` is the value of the `ith` node. You are also given a 0-indexed 2D integer array `edges`, where each `edges[j] = [uj, vj, timej]` indicates that there is an undirected edge between the nodes `uj` and `vj`, and it takes `timej` seconds to travel between the two nodes. Finally, you are given an integer `maxTime`.

A valid path in the graph is any path that starts at node `0`, ends at node `0`, and takes at most `maxTime` seconds to complete. You may visit the same node multiple times. The quality of a valid path is the sum of the values of the unique nodes visited in the path (each node's value is added at most once to the sum).

Return the maximum quality of a valid path.

Note: There are at most four edges connected to each node.

Example 1:

Input: values = [0,32,10,43], edges = [[0,1,10],[1,2,15],[0,3,10]], maxTime = 49
Output: 75
Explanation:
One possible path is 0 -> 1 -> 0 -> 3 -> 0. The total time taken is 10 + 10 + 10 + 10 = 40 <= 49.
The nodes visited are 0, 1, and 3, giving a maximal path quality of 0 + 32 + 43 = 75.

Example 2:

Input: values = [5,10,15,20], edges = [[0,1,10],[1,2,10],[0,3,10]], maxTime = 30
Output: 25
Explanation:
One possible path is 0 -> 3 -> 0. The total time taken is 10 + 10 = 20 <= 30.
The nodes visited are 0 and 3, giving a maximal path quality of 5 + 20 = 25.

Example 3:

Input: values = [1,2,3,4], edges = [[0,1,10],[1,2,11],[2,3,12],[1,3,13]], maxTime = 50
Output: 7
Explanation:
One possible path is 0 -> 1 -> 3 -> 1 -> 0. The total time taken is 10 + 13 + 13 + 10 = 46 <= 50.
The nodes visited are 0, 1, and 3, giving a maximal path quality of 1 + 2 + 4 = 7.

Constraints:

• `n == values.length`
• `1 <= n <= 1000`
• `0 <= values[i] <= 108`
• `0 <= edges.length <= 2000`
• `edges[j].length == 3`
• `0 <= uj < vj <= n - 1`
• `10 <= timej, maxTime <= 100`
• All the pairs `[uj, vj]` are unique.
• There are at most four edges connected to each node.
• The graph may not be connected.

SOLUTION:

``````from collections import defaultdict

class Solution:
def getPath(self, node, graph, visited, time, values, maxTime):
if node == 0:
currQuality = sum([values[i] for i in visited])
if currQuality >= self.maxQuality:
self.maxQuality = currQuality
for j, t in graph[node]:
if time + t <= maxTime:
self.getPath(j, graph, visited.union({j}), time + t, values, maxTime)

def maximalPathQuality(self, values: List[int], edges: List[List[int]], maxTime: int) -> int:
self.maxQuality = float('-inf')
graph = defaultdict(list)
for a, b, t in edges:
graph[a].append((b, t))
graph[b].append((a, t))
self.getPath(0, graph, {0}, 0, values, maxTime)
return self.maxQuality
``````