Clarify that this is a dynamic programming problem where you compute the maximum reward to reach each cell from the top-left, considering only moves from the top or left. Then present an O(m*n) time and O(n) space solution, explaining how to handle negative values and edge cases.
Pro tip: Emphasize that negative rewards are allowed, so you cannot greedily choose the locally larger neighbor; you must consider all paths. Also, mention that you can optimize space to O(n) by keeping only the previous row, which shows strong DP optimization skills.
Confirm that the grid can contain negative values, that movement is only right or down, and that you start at (0,0) and end at (m-1,n-1). Ask about grid size limits to determine if O(m*n) is acceptable.
Let dp[i][j] be the maximum reward to reach cell (i,j). Then dp[i][j] = grid[i][j] + max(dp[i-1][j], dp[i][j-1]), with base cases for the first row and first column.
Use a 2x2 or 3x3 grid with negative values to demonstrate how the DP table is filled and why greedy fails. Show the final answer at dp[m-1][n-1].
State that time complexity is O(m*n) and space can be reduced from O(m*n) to O(n) by keeping only the previous row (or O(min(m,n)) by choosing the smaller dimension).
Mention handling of 1x1 grid, grids with all negative values, and large grids. Be prepared to discuss variations like allowing all four directions or obstacles.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Treat the maze as an unknown graph and use a systematic exploration algorithm like DFS or BFS with backtracking, leveraging only the provided APIs. To maximize reward, prioritize exploring branches that may contain cheese, and once the goal is found, compute an optimal path that collects as much reward as possible.
Pro tip: Explicitly discuss the trade-off between exploration and exploitation: sometimes it's worth taking a longer path to collect more cheese, but you must ensure you can still reach the goal. Also, mention that you'd cache the maze structure as you explore to avoid redundant moves.
Ask about move() behavior (does it move in a direction and return success?), canMove() (checks if a move is possible without moving?), and isCheese() (checks current cell?). Confirm if moves are reversible and if there's a limit on moves.
Use DFS with backtracking to explore the maze systematically, or BFS if you want to find the shortest path to the goal first. Since there's no global view, you'll need to remember visited cells and paths.
When you detect cheese (via isCheese()), decide whether to collect it immediately or mark it for later. If collecting, ensure you can return to the main path. Consider using a priority system to visit cheese-rich areas first.
Once the goal is found, you may need to backtrack to collect missed cheese. Alternatively, during exploration, keep track of all cheese locations and plan a route that maximizes reward while still reaching the goal.
Discuss time and space complexity in terms of maze size and number of cheese. Address edge cases like unreachable cheese, cycles, and the possibility of infinite loops if not careful.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
DP breaks because you can revisit cells and the subproblem ordering falls apart, especially with negative rewards.
Acknowledge that the DP approach for counting paths with only right/down moves does not directly extend to four-direction movement because cycles are introduced, making the problem about shortest paths or reachability rather than simple path counting. Then pivot to discussing appropriate algorithms like BFS for unweighted grids or Dijkstra for weighted grids, and mention how obstacles or constraints affect the choice.
Pro tip: Show awareness that the original DP relied on a DAG (no cycles), and that adding up/left moves breaks that assumption—demonstrating you understand the underlying reason, not just the algorithm. Also, briefly mention that if the goal is counting simple paths (no revisits), the problem becomes #P-complete, which is a great way to show depth.
Ask whether the goal is still to count all paths, find the shortest path, or just determine reachability. Also confirm if revisiting cells is allowed and if there are obstacles or weights.
State that the original DP works because moves only go right/down, forming a DAG with a topological order. With four directions, cycles exist, so DP over cells without additional state (like visited set) is invalid.
For shortest path in an unweighted grid, use BFS. For weighted grids, use Dijkstra. For counting simple paths, note it's #P-complete and likely infeasible for large grids.
Compare BFS vs. Dijkstra vs. A* based on grid size, weights, and whether we need all shortest paths or just one. Mention that if the grid is small, DFS with backtracking can count simple paths.
Conclude that the DP approach no longer applies directly, but the problem transforms into a classic graph search, and the choice depends on the exact requirement.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.