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Part A felt manageable once I framed it as a constrained path search.
Clarify the problem constraints and edge cases, then propose a DFS with pruning that tracks the maximum remaining energy at each visited node to avoid redundant exploration. Implement the solution, and finally justify correctness and analyze time and space complexity.
Pro tip: Mention that pruning by maximum energy per node is crucial for efficiency, and that the problem is NP-hard in general, so the DFS with pruning is a practical heuristic that works well for many cases but may still be exponential in the worst case.
Ask clarifying questions about graph size, energy bounds, and whether the path must be simple. Define the problem formally: find a path from s to t such that the cumulative sum of hp along the path never exceeds E.
Propose a DFS that explores paths, but prune when the current energy is negative or when reaching a node with energy less than or equal to the maximum energy previously seen at that node. Use a hash map to store the best energy per node.
Write clean code for the DFS with pruning, including path reconstruction. Walk through a small example to verify correctness, and discuss edge cases like s == t or no path.
Explain that the DFS explores all viable paths, and pruning is safe because if a node is reached with less energy than before, any continuation from that state is dominated by the previous visit. Thus, if a path exists, it will be found.
State that the worst-case time complexity is O(n * E) or O(n^2) depending on pruning effectiveness, but in the worst case it can be exponential. Space complexity is O(n) for recursion stack and visited map.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Clarify the problem constraints and define the state as (node, energy, potions collected), but note that energy can be large and potions are one-time. Propose a solution that combines graph traversal with dynamic programming or greedy strategies, leveraging the fact that potions only help and revisiting nodes may be necessary. Discuss how to handle large graphs by using efficient data structures and pruning.
Pro tip: Emphasize that the problem is NP-hard in general (e.g., reduces to longest path), so you'd discuss approximations or special cases, and mention that in practice, you'd use BFS with state compression and priority queues to explore promising paths first.
Restate the problem: find a walk from s to t maximizing distinct rooms visited, with energy never negative, where potions give one-time boosts. Ask about constraints: can nodes be revisited? Are potions consumed on first visit only? What are energy bounds?
Recognize that maximizing rooms visited is akin to longest path, which is NP-hard. The energy constraint and one-time potions add state complexity. Discuss that exact solution may be exponential, so consider heuristics or special cases.
Suggest a BFS/DFS with state (node, energy, potions collected) but note state explosion. Alternatively, model as a resource-constrained shortest path where we maximize rooms, using dynamic programming over subsets of potions if m is small.
For large graphs, use A* with a heuristic (e.g., number of unvisited rooms) or greedy exploration: always take potions when possible, and use Dijkstra-like search on (node, energy) with pruning. Mention that energy can be capped at max needed to reach t.
Explain that the greedy approach may not be optimal, but can be proven optimal under certain conditions (e.g., if potions are abundant). Discuss time/space complexity and potential approximations.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.