How do you solve Maximum Path Quality of a Graph on LeetCode?

Maximum Path Quality of a Graph (LeetCode #2065) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(n^2) time complexity and O(n) space complexity. Key insight: Exploring all paths can lead to exponential time complexity.

What is the brute force solution for Maximum Path Quality of a Graph?

The brute force approach for Maximum Path Quality of a Graph is Brute Force, with O(n!) time complexity and O(n) space complexity. The brute force approach explores all possible paths starting and ending at node 0, checking each path's total time and the unique nodes visited. This method guarantees finding the maximum quality but is inefficient due to the exponential number of paths. The optimal Optimal Solution approach improves this to O(n^2).

What algorithm or data structure is used to solve Maximum Path Quality of a Graph?

Maximum Path Quality of a Graph uses the following concepts: Array, Backtracking, Graph Theory. The recommended patterns to study are: Backtracking, Graph Traversal.

What are the interview tips for Maximum Path Quality of a Graph?

Always clarify the constraints and edge cases before diving into the solution. Discuss your thought process and approach before coding. Test your solution with different scenarios to ensure correctness.

What are common mistakes when solving Maximum Path Quality of a Graph?

Not considering the time constraint during path exploration. Failing to track unique nodes correctly.

#2065

Maximum Path Quality of a Graph

Hard

ArrayBacktrackingGraph TheoryBacktrackingGraph Traversal

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n!)	O(n^2)
Space	O(n)	O(n)

💡

Intuition

Time O(n^2)Space O(n)

The optimal solution uses a backtracking approach with pruning to explore paths efficiently. By keeping track of the total time and unique nodes visited, we can avoid unnecessary paths and focus on those that maximize quality within the time limit.

⚙️

Algorithm

3 steps

1Step 1: Build the graph representation from the edges.
2Step 2: Use a backtracking function that explores paths recursively, keeping track of the current time and unique nodes visited.
3Step 3: If the current path returns to node 0 and the time is within limits, calculate the quality and update the maximum quality.

solution.py20 lines

1def maxPathQuality(values, edges, maxTime):
2    from collections import defaultdict
3    graph = defaultdict(list)
4    for u, v, t in edges:
5        graph[u].append((v, t))
6        graph[v].append((u, t))
7
8    def dfs(node, timeSpent, visited):
9        if timeSpent > maxTime:
10            return 0
11        if node == 0:
12            return sum(values[i] for i in visited)
13        maxQuality = 0
14        for neighbor, travelTime in graph[node]:
15            visited.add(neighbor)
16            maxQuality = max(maxQuality, dfs(neighbor, timeSpent + travelTime, visited))
17            visited.remove(neighbor)
18        return maxQuality
19
20    return dfs(0, 0, {0})

ℹ

Complexity note: The time complexity is quadratic due to the backtracking approach that explores paths while keeping track of visited nodes. The space complexity is linear due to the storage of the graph and visited nodes.

1Exploring all paths can lead to exponential time complexity.
2Using backtracking with pruning can significantly reduce unnecessary computations.

Solutions and explanations are original Tejav content. Problem titles © LeetCode — use the LeetCode button above for the full problem statement.