The core DP part clicked pretty fast for me.
Recognize this as the weighted interval scheduling problem. Sort jobs by end time, use DP where dp[i] = max reward up to job i, and binary search to find the latest non-overlapping job. To reconstruct the selected jobs, store backpointers during DP.
Pro tip: Mention that sorting by end time is crucial for the binary search to work, and that the DP state can be optimized to O(N) space by storing only the previous state and using a parent array for reconstruction.
Confirm the problem: jobs are non-overlapping if one ends before the other starts. The goal is to maximize total reward and return both the max reward and the selected job indices.
Sort jobs by end time. For each job, precompute p(j) = the largest index i < j such that job i is compatible with job j (i.e., end time of i <= start time of j). This can be done with binary search.
Define dp[j] as the maximum reward considering jobs 1..j. Recurrence: dp[j] = max(dp[j-1], reward[j] + dp[p(j)]). Iterate j from 1 to N, filling dp and storing the choice made (include job j or not) for reconstruction.
After computing dp[N], backtrack using the stored choices to collect the indices of selected jobs. Start from j = N and move backwards: if job j was included, add j to the result and jump to p(j); otherwise, move to j-1.
Sorting takes O(N log N). Binary search for each job takes O(log N), so O(N log N) total. DP and reconstruction take O(N). Space is O(N) for dp and parent arrays.
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I knew I needed to store back-pointers but my first attempt was sloppy.
Explain that reconstructing the chosen jobs requires storing parent pointers or decisions during the dynamic programming computation, then backtracking from the optimal state. Emphasize that this adds O(n) space but enables retrieval of the actual sequence without recomputing the DP.
Pro tip: Mention that you can avoid extra space by recomputing the DP values on the fly during backtracking, but clarify the trade-off between time and space. This shows you understand optimization beyond the basic solution.
Clearly state the DP state, e.g., dp[i] = max reward using jobs up to i, and the recurrence: dp[i] = max(dp[i-1], reward[i] + dp[prev[i]]).
During DP computation, store for each i whether job i was chosen or not, or store the index of the previous compatible job to enable backtracking.
Start from the final state (e.g., dp[n]) and follow the stored decisions backwards to collect the chosen jobs.
Ensure the backtracking correctly handles cases where no job is chosen or multiple optimal solutions exist, and return the list of jobs in the correct order.
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State the time and space complexity of your algorithm in Big-O notation, then explain how memory usage scales with N up to 200,000, including constant factors and worst-case scenarios. Emphasize any optimizations or trade-offs you made to handle large inputs efficiently.
Pro tip: Mention concrete numbers: for N=200,000, an O(N) algorithm uses ~200K operations, while O(N log N) is ~3.6M—this shows you think about practical performance, not just theory.
Clearly state the Big-O time complexity of your algorithm, e.g., O(N log N), and briefly explain why (e.g., sorting step dominates).
State the Big-O space complexity, e.g., O(N), and specify what data structures contribute to memory usage (e.g., hash map, arrays).
Describe how memory usage grows with N, e.g., linear growth, and estimate actual memory in MB if possible (e.g., 200K integers ~ 1.6MB).
Mention any trade-offs (e.g., time vs. space) and optimizations you applied to keep memory reasonable, such as in-place operations or streaming.
Note worst-case scenarios (e.g., all elements distinct) and how they affect complexity, and mention any edge cases like N=0 or N=200,000.
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I mentioned coordinate compression as a way to handle the large reward values and dense time ranges, and brought up segment trees as an alternative for range-max queries.
Acknowledge that both segment trees and coordinate compression are valid alternatives, then compare them against binary search on time/space complexity, implementation complexity, and constraints. Emphasize that the best choice depends on the problem's specific requirements, such as query frequency, data size, and update patterns.
Pro tip: Always tie your trade-off analysis back to the actual constraints and expected usage; interviewers value pragmatic reasoning over theoretical purity. Mention that binary search is often preferred for its simplicity unless the problem demands dynamic updates or many queries.
Restate the problem to ensure you understand the input size, query types (static vs dynamic), and performance requirements. This determines which data structures are even applicable.
Describe how binary search, segment tree, and coordinate compression would solve the problem, focusing on their core mechanics and typical use cases.
Analyze the asymptotic complexity of each approach for preprocessing, query, and update operations. Highlight scenarios where one outperforms the others.
Weigh factors like code complexity, ease of debugging, memory overhead, and flexibility to handle updates or additional queries.
Give a clear recommendation for the given problem, justifying why one approach is preferable, and note any edge cases or assumptions.
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This one tripped me up more than I expected.
Start by clarifying the constraints of the streaming scenario (e.g., unbounded input, latency requirements, memory limits) and how they differ from the offline case. Then propose an incremental algorithm that processes each job as it arrives, maintaining only necessary state, and discuss trade-offs between optimality and efficiency. Finally, relate your approach to real-world systems like online scheduling or streaming analytics.
Pro tip: Acknowledge that perfect optimality is often impossible in streaming settings; instead, focus on competitive ratios or approximation guarantees, and mention how you'd monitor and adapt the algorithm in production.
Ask about arrival rate, job size distribution, memory limits, and whether jobs can be buffered or must be processed immediately. This shows you understand the problem space before jumping to solutions.
Briefly state what the original solution assumes (e.g., full knowledge of all jobs) and why that breaks in a streaming context. This highlights the gap you need to bridge.
Describe a strategy that processes jobs one by one, such as maintaining a priority queue, using a greedy heuristic, or applying a sliding window. Explain how it handles new arrivals and updates state efficiently.
Discuss time/space complexity, competitive ratio, and potential suboptimality. Mention if the algorithm is deterministic or randomized, and how it performs under worst-case vs. average-case inputs.
Talk about implementation details like handling out-of-order arrivals, late jobs, or failures. Suggest monitoring metrics (e.g., latency, throughput) and how you might adapt the algorithm dynamically.
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