How do you solve Path with Maximum Probability on LeetCode?

Path with Maximum Probability (LeetCode #1514) can be solved using Brute Force or Optimal Solution. The optimal approach is Optimal Solution with O(E log V) time complexity and O(V) space complexity. Key insight: Understanding how to represent graphs is crucial.

What is the brute force solution for Path with Maximum Probability?

The brute force approach for Path with Maximum Probability is Brute Force, with O(n²) time complexity and O(1) space complexity. The brute force approach involves exploring all possible paths from the start node to the end node and calculating the probability for each path. We then select the maximum probability found. This is straightforward but inefficient for larger graphs. The optimal Optimal Solution approach improves this to O(E log V).

What algorithm or data structure is used to solve Path with Maximum Probability?

Path with Maximum Probability uses the following concepts: Array, Graph Theory, Heap (Priority Queue), Shortest Path. The recommended patterns to study are: Graph Traversal, Priority Queue, Dijkstra's Algorithm.

What are the interview tips for Path with Maximum Probability?

Always clarify the problem statement and constraints. Discuss your thought process while coding. Test your code with edge cases after implementation.

What are common mistakes when solving Path with Maximum Probability?

Confusing probability multiplication with addition. Not considering edge cases where no path exists.

#1514

Path with Maximum Probability

Medium

ArrayGraph TheoryHeap (Priority Queue)Shortest PathGraph TraversalPriority QueueDijkstra's Algorithm

LeetCode ↗

Approaches

Brute ForceOptimal

Complexity Comparison

	Brute Force	Optimal Solution★
Time	O(n²)	O(E log V)
Space	O(1)	O(V)

💡

Intuition

Time O(E log V)Space O(V)

The optimal solution uses a priority queue (max-heap) to explore the graph, similar to Dijkstra's algorithm, but instead of minimizing distances, we maximize probabilities. This approach is efficient and handles larger graphs well.

⚙️

Algorithm

3 steps

1Step 1: Create a graph representation using an adjacency list.
2Step 2: Use a max-heap to explore nodes, starting from the start node with a probability of 1.
3Step 3: For each node, update the maximum probability for its neighbors and push them into the heap if a higher probability path is found.

solution.py26 lines

1# Full working Python code
2import heapq
3from collections import defaultdict
4
5def maxProbability(n, edges, succProb, start, end):
6    graph = defaultdict(list)
7    for (a, b), prob in zip(edges, succProb):
8        graph[a].append((b, prob))
9        graph[b].append((a, prob))
10
11    max_prob = [0] * n
12    max_prob[start] = 1.0
13    heap = [(-1.0, start)]  # max-heap (using negative for max)
14
15    while heap:
16        prob, node = heapq.heappop(heap)
17        prob = -prob  # revert back to positive
18        if node == end:
19            return prob
20        for neighbor, edge_prob in graph[node]:
21            new_prob = prob * edge_prob
22            if new_prob > max_prob[neighbor]:
23                max_prob[neighbor] = new_prob
24                heapq.heappush(heap, (-new_prob, neighbor))
25
26    return 0.0

ℹ

Complexity note: The time complexity is O(E log V) because we process each edge and use a priority queue to manage the nodes, where E is the number of edges and V is the number of vertices. The space complexity is O(V) due to the storage of the graph and the max probabilities.

1Understanding how to represent graphs is crucial.
2Using priority queues can significantly optimize pathfinding problems.

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