Path with Maximum Probability
Problem Description
You are given an undirected weighted graph of n nodes (0-indexed), represented by an edge list where edges[i] = [a, b] is an undirected edge connecting the nodes a and b with a probability of success of traversing that edge succProb[i].
Given two nodes start and end, find the path with the maximum probability of success to go from start to end and return its success probability.
If there is no path from start to end, return 0. Your answer will be accepted if it differs from the correct answer by at most 1e-5.
Example 1:
Input: n = 3, edges = [[0,1],[1,2],[0,2]], succProb = [0.5,0.5,0.2], start = 0, end = 2
Output: 0.25000
Explanation: There are two paths from start to end, one having a probability of success = 0.2 and the other has 0.5 * 0.5 = 0.25.
Example 2:
Input: n = 3, edges = [[0,1],[1,2],[0,2]], succProb = [0.5,0.5,0.3], start = 0, end = 2
Output: 0.30000
Example 3:
Input: n = 3, edges = [[0,1]], succProb = [0.5], start = 0, end = 2
Output: 0.00000
Explanation: There is no path between 0 and 2.
Constraints:
2 <= n <= 10^40 <= start, end < nstart != end0 <= a, b < na != b0 <= succProb.length == edges.length <= 2*10^40 <= succProb[i] <= 1- There is at most one edge between every two nodes.
Solve it on LeetCode
Approaches
1. Breadth-First Search (BFS)
In this approach, we will utilize a queue to explore all possible paths from the start node, keeping track of the maximum probability of reaching each node. This is similar to BFS, but we focus on maximizing probability (a multiplicative metric) rather than minimizing distance or edges.
Intuition:
We start from the given starting node and try to explore its neighbors, iteratively updating the maximum probability to reach each neighbor node. We continue this process until we either visit all nodes or exhaust all possibilities, aiming to find the path to the target node with the highest probability.
Code:
- Time Complexity: O(V + E), where V is the number of vertices and E is the number of edges. The algorithm explores all edges once.
- Space Complexity: O(V + E), mainly for storing the graph and probability tracking.
2. Dijkstra's Algorithm
Dijkstra's Algorithm can be adapted to find the path with maximum probability by using a priority queue (max-heap) to ensure that we always expand the node with the highest accumulated probability.
Intuition:
Similar to finding the shortest path, where we keep track of minimum distances, here we keep track of maximum probabilities. We utilize a priority queue to explore nodes in the order of their maximum current path probabilities.
Code:
- Time Complexity: O((E + V) log V), using a priority queue gives a log-factor overhead for each of the operations.
- Space Complexity: O(V + E), primarily for the storage of the graph and data structures needed for Dijkstra's algorithm.