Model the stops and routes as an unweighted graph, then run BFS from the source to find the shortest path to the target while skipping dangerous stops. First check if source or target is dangerous and return -1 immediately; otherwise, BFS guarantees the minimum number of steps.
Pro tip: Explicitly state that BFS is optimal for unweighted graphs and mention that you would treat dangerous stops as blocked nodes, never enqueuing them. This shows you understand both the algorithm and the problem constraints.
Confirm the graph representation (adjacency list or matrix), whether stops are 0-indexed or 1-indexed, and that dangerous stops are given as a set. Check if source or target is dangerous and return -1 immediately.
Construct an adjacency list from the given routes, ensuring it's undirected if routes are bidirectional. Exclude any edges that involve dangerous stops to simplify traversal.
Initialize a queue with the source stop and a visited set. While the queue is not empty, dequeue a stop, and if it's the target, return the current distance. Otherwise, enqueue all unvisited, non-dangerous neighbors with distance+1.
If BFS completes without finding the target, return -1. Also ensure that the source and target are not dangerous before starting BFS.
State that time complexity is O(V + E) and space complexity is O(V) for the visited set and queue, where V is the number of stops and E is the number of routes.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Model the problem as a graph where each node tracks both the number of dangerous stops visited and the number of steps taken. Use a modified Dijkstra's algorithm with a priority queue ordered lexicographically by (dangerous count, steps) to find the optimal path. Alternatively, use 0-1 BFS if edge weights are binary, but Dijkstra is more general.
Pro tip: Emphasize that the two-level priority can be handled by encoding the cost as a pair and ensuring the priority queue compares them lexicographically. Also, mention that if the graph is large, we can optimize by only considering Pareto-optimal states (dangerous count, steps) for each node.
Define the state as (node, dangerous_count) and the cost as steps. The primary objective is to minimize dangerous_count, and the secondary is to minimize steps. This allows us to treat the problem as a shortest path on an expanded graph.
Use Dijkstra's algorithm with a priority queue that orders by (dangerous_count, steps) lexicographically. Alternatively, if edge weights are 0 or 1, use 0-1 BFS with a deque, but Dijkstra is more straightforward for this two-level priority.
Initialize distances for all (node, dangerous_count) pairs to infinity, except the source with dangerous_count = 0 (or 1 if source is dangerous) and steps = 0. During relaxation, update the state if the new dangerous_count is smaller, or if equal and steps are fewer.
Stop when the target node is popped from the priority queue with the minimal dangerous_count and steps. Reconstruct the path by keeping track of predecessors for each state.
The time complexity is O((V + E) log V) for Dijkstra on the expanded graph, where V is the number of nodes times the maximum possible dangerous count. Space complexity is O(V) for distances and predecessors.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Follow-up came right after and I think I gave a decent answer but it was a bit rambly.
Start by restating the problem: we need to find a path that minimizes dangerous stops first, then total steps. Explain that a simple visited set is insufficient because a node might be reached with more dangerous stops but fewer total steps, and that state must be tracked as (node, dangerous_stops). Then describe how to adapt the visited structure to store the best (dangerous_stops, steps) pair for each node and only revisit when both criteria improve.
Pro tip: Emphasize that the visited structure must support dominance checks: a state is only worth exploring if it is not dominated by a previously seen state with both fewer or equal dangerous stops and fewer or equal steps. This shows you understand multi-criteria optimization beyond simple BFS.
Confirm that the primary objective is to minimize dangerous stops, and the secondary is to minimize total steps. This lexicographic order determines how we compare states.
The state should be (node, dangerous_stops_so_far) because the same node can be reached with different numbers of dangerous stops, affecting future choices. Total steps is the value we optimize.
Use a map from node to a list of (dangerous_stops, steps) pairs that are Pareto-optimal. Alternatively, store the best steps for each (node, dangerous_stops) combination, but prune dominated entries.
When reaching a node with a new (dangerous_stops, steps), check if it is dominated by any existing entry. If not, add it and remove any entries it dominates. Only enqueue the state if it is not dominated.
Since we have two criteria, use a priority queue ordered by (dangerous_stops, steps) to ensure we explore states in lexicographic order, similar to Dijkstra but with a custom comparator.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.