My first instinct was shortest path and I almost reached for Bellman-Ford out of habit.
Transform the problem into a shortest path problem by taking negative logarithms of the probabilities, then apply Dijkstra's algorithm to find the path with minimum sum of negative logs, which corresponds to maximum product. Alternatively, modify Dijkstra to use max-product relaxation directly. Handle unreachable nodes by returning 0.
Pro tip: Mention that this is a classic 'widest path' variant and that using Dijkstra with a max-heap is optimal because the product operation is monotonic and the graph has non-negative weights (after log transform). Also, note that if probabilities can be zero, the log transform fails, so handle zeros separately.
Confirm that edge weights are probabilities (0 to 1), that the path can be any simple path, and that we need the maximum product. Ask about graph size, whether probabilities can be zero, and if negative probabilities exist (they shouldn't).
Recognize that maximizing product is equivalent to minimizing sum of negative logs. Since log is monotonic, the optimal path remains the same. Use Dijkstra's algorithm with a max-heap on the product directly, or a min-heap on negative logs.
Initialize distances to 0 (or -inf for max-product) except source = 1. Use a priority queue to always expand the node with the highest current probability. Relax edges by multiplying the current probability with the edge probability.
If the target is never reached, return 0. If any edge probability is 0, the product becomes 0, so paths through such edges are invalid unless no other path exists. Ensure the algorithm correctly handles disconnected graphs.
Time complexity is O(E log V) with a binary heap, which is optimal for this problem. Discuss space complexity O(V+E). Mention that Bellman-Ford could work but is slower, and that BFS won't work because edge weights are not uniform.
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