The non-standard move set tripped me up a bit.
Clarify the movement rules and grid dimensions, then model the problem as counting paths in a directed graph or using dynamic programming. Recognize that the ability to move left, up, and down introduces cycles, so a naive DP fails; instead, consider that the problem may be equivalent to counting simple paths, which is #P-complete in general, and discuss constraints or alternative interpretations.
Pro tip: Interviewers often expect you to identify that allowing left, up, and down makes the problem significantly harder (likely #P-complete) and to discuss how the answer changes if moves are restricted to right and up. Demonstrating this awareness shows depth and prevents you from diving into an incorrect solution.
Ask about grid size, starting and ending corners, and whether paths can revisit cells. Confirm that moves are left, up, and down (no right), which creates cycles.
Explain that with cycles, counting simple paths is #P-complete, so no polynomial-time algorithm exists unless P=NP. Mention that if moves were only right and up, the answer would be a simple binomial coefficient.
For small grids, suggest backtracking with memoization on visited cells (bitmask DP) or DFS with pruning. Discuss time complexity O(2^(m*n)) and space O(m*n).
If the interviewer intended only right and up moves, present the combinatorial solution: C(m+n-2, m-1). If only up and down, the answer is 1 (straight line).
Restate that the problem as stated is computationally intractable for large grids, and offer to implement a solution for small grids or under different movement constraints.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
For each problem, first clarify the input/output and constraints, then identify the need for a sweep line by recognizing events (e.g., start/end points) that can be sorted and processed in order. Explain how to maintain state (e.g., active intervals, current overlap count) as you sweep, and analyze time/space complexity. Practice communicating the algorithm clearly and handling edge cases.
Pro tip: Demonstrate deep understanding by discussing how to handle duplicate coordinates and whether to process starts before ends (or vice versa) based on the problem's definition of overlap. Also, mention that sweep line is often combined with a data structure like a heap or balanced BST for dynamic updates.
Ask questions to confirm input format, output requirements, constraints, and edge cases (e.g., empty input, single interval, duplicate points).
Determine if the problem involves intervals, points, or events that can be sorted along an axis, and if a running state can be maintained efficiently.
Specify what constitutes an event (e.g., interval start/end) and how to sort them (e.g., by coordinate, with tie-breaking rules).
Choose an appropriate data structure (e.g., counter, heap, BST) to track active intervals or relevant information as the sweep progresses.
Derive time and space complexity, then walk through examples and edge cases to verify correctness.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.