Classic BFS setup once you see it, but I spent an embarrassing amount of time trying to think about it as a graph problem before realizing it basically IS a graph problem and BFS just falls out naturally.
Model the problem as a shortest path search on a graph where each state is a 4-digit combination and edges represent single-digit rotations. Use BFS to find the minimum moves from '0000' to the target, skipping forbidden states. This ensures optimality because each move has uniform cost.
Pro tip: Mention that BFS is optimal for unweighted graphs and that you can optimize by using a bidirectional BFS or A* with a heuristic like the sum of circular distances to the target, but only if needed. Also, clarify edge cases like the start or target being forbidden.
Confirm that each move rotates one digit by ±1 with wrap-around, and that forbidden states cannot be visited. Ask if the start or target can be forbidden and if the target is guaranteed reachable.
Treat each 4-digit combination as a node. Connect nodes that differ by one rotation on a single digit. This forms an unweighted graph with up to 10,000 nodes.
Use BFS from '0000' to find the minimum moves to the target, skipping forbidden nodes. BFS explores level by level, guaranteeing the first time we reach the target is via the shortest path.
Use a queue for BFS and a set for forbidden states. For each state, generate up to 8 neighbors by rotating each digit up and down. Track visited states to avoid cycles.
Time complexity is O(10^4) since each state is visited once. Space is O(10^4). Handle cases where start or target is forbidden, and if BFS exhausts without reaching target, return -1.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.