I went with a matching approach where you count mismatches per floor and figure out the minimum swaps needed globally rather than floor by floor.
Model the problem as a transportation problem where each floor has a supply (initial count) and demand (target count). The minimum number of moves equals the total surplus (or deficit) across all floors, which can be computed by summing the positive differences between initial and target counts. Argue optimality by showing each move can reduce the total surplus by at most 1, and a greedy algorithm achieves this bound.
Pro tip: Emphasize that the problem reduces to a simple counting argument, not a complex assignment, and that the minimum moves is independent of which specific apartments move—only the per-floor counts matter. This shows you can abstract away unnecessary details.
Represent the initial and target arrangements as counts per floor. A move takes one apartment from a floor with surplus to a floor with deficit.
For each floor, calculate the difference between initial and target counts. Positive differences are surpluses, negative are deficits.
The minimum number of moves equals the sum of all positive differences (or equivalently, the sum of all deficits). This is the total number of apartments that must be relocated.
Show that each move can reduce the total surplus by at most 1, so at least S moves are needed. Then describe a greedy algorithm that always moves from a surplus floor to a deficit floor, achieving exactly S moves.
Mention that the algorithm runs in O(F) time where F is the number of floors, and handles cases where initial and target counts are already equal (zero moves).
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Straightforward enough since the number of floors is fixed at 4.
Start by clearly stating the time and space complexity using Big O notation, then explain how you derived them by analyzing the algorithm's loops, recursion, and data structures. Relate the complexity to the problem constraints and discuss any trade-offs you made between time and space.
Pro tip: At Optiver, interviewers value practical optimization: mention how your complexity scales with input size and whether it meets the performance requirements for real-time trading systems. If applicable, note that you considered constant factors and low-level optimizations, as these matter in latency-sensitive environments.
Clearly state the time and space complexity in Big O notation, e.g., O(n log n) time and O(n) space.
Walk through the algorithm step-by-step, identifying loops, recursive calls, and data structures that contribute to the complexity.
Discuss how the complexity varies with input and mention the worst-case scenario, as it's often the most critical.
Explain any trade-offs between time and space and whether you could improve one at the expense of the other.
Connect the complexity to the problem's input size and performance requirements, showing awareness of practical implications.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
First, clarify the problem setup: what exactly is being optimized (e.g., total time, cost) and what constraints exist. Then, analyze whether the greedy choice of fixing Floor 1 first is always part of an optimal solution by testing small cases or constructing a counterexample. If a counterexample exists, present it clearly; otherwise, outline a proof using exchange arguments or dynamic programming.
Pro tip: In interviews, it's often more impressive to find a counterexample than to attempt a complex proof. Start by testing simple cases (e.g., 2 floors) to see if the greedy choice fails, and if it does, generalize the counterexample.
Restate the problem to ensure you understand the objective and constraints. Ask clarifying questions about what 'fixing' entails, the costs involved, and whether floors can be fixed in any order.
Define what 'optimal' means in this context (e.g., minimizing total time, cost, or number of operations). This will guide your analysis.
Consider the smallest non-trivial cases (e.g., 2 or 3 floors) and compute the optimal solution by brute force. Compare with the greedy strategy to see if it matches.
If the greedy strategy seems optimal, try to prove it using an exchange argument or induction. If it fails, construct a counterexample and explain why it invalidates the greedy approach.
State your conclusion clearly. If a counterexample exists, discuss what it implies about the greedy strategy and suggest alternative approaches (e.g., dynamic programming).
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.
Felt like the easier part after the earlier stuff.
Explain that you would use BFS to find the shortest path and store parent pointers for each visited state. Then, to output one valid sequence, backtrack from the goal state to the start using any stored parent, ensuring the sequence is valid and minimal.
Pro tip: Mention that you can break ties arbitrarily (e.g., by move order) but must ensure consistency; also note that storing all parents is unnecessary if only one sequence is needed, which saves memory.
Define states as nodes and moves as edges. This abstraction helps apply standard shortest-path algorithms.
Use BFS from the start state to compute the minimum number of moves to each state, including the goal.
When visiting a new state, record the predecessor state and the move that led to it. For multiple optimal paths, any valid predecessor works.
Starting at the goal, follow parent pointers until the start state, collecting moves in reverse order.
Reverse the collected moves to get the sequence from start to goal, which is one valid minimal sequence.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.