The brute force explanation part tripped me up more than the actual DP.
Start by clarifying the problem constraints (array size, value ranges, threshold) and explaining why brute force enumeration of all 2^n subsequences is infeasible. Then present a dynamic programming approach that counts subsequences by sum, using a DP table or map, and analyze its time and space complexity. Finally, discuss potential optimizations or trade-offs based on constraints.
Pro tip: Mention that if the threshold is small, a DP over sums is efficient; if the array is large but values are small, a meet-in-the-middle approach can be used. Also, note that counting subsequences (not subsets) means order doesn't matter, but each element can be included or not, so it's equivalent to subsets.
Ask about array size, value ranges (including negatives), and threshold magnitude. Confirm that a subsequence is any subset of indices in order, but since order doesn't affect sum, it's equivalent to subsets.
Brute force would enumerate all 2^n non-empty subsequences, which is exponential and infeasible for n > ~20. Even with pruning, worst-case remains exponential.
Use DP where dp[s] = number of subsequences with sum s. Initialize dp[0]=1 (empty subsequence). For each element x, update dp[s] += dp[s-x] for s from T down to x (if x positive) or adjust bounds for negatives. Finally, sum dp[s] for s=1..T.
Time O(n*T), space O(T). If T is large, use a hash map to store only reachable sums, but worst-case still O(n*T). Mention meet-in-the-middle for large n and small T, or if values are small, use generating functions.
Address negative numbers (shift sums or use map), empty array, and threshold <=0. Summarize that DP is optimal for moderate T, and mention alternative approaches if constraints differ.
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Start by clarifying the requirements: INSERT adds a job with a priority, POP removes and returns the job with the smallest priority, and DECREASE_KEY lowers a job's priority. Use a binary min-heap with an auxiliary hash map from job ID to heap index to achieve O(log n) for all operations, and handle edge cases like duplicate IDs, invalid IDs, and empty heap.
Pro tip: Emphasize stable tie-breaking by storing an insertion sequence number with each job and comparing it when priorities are equal; this shows attention to detail and avoids subtle bugs.
Ask about expected input sizes, whether job IDs are unique, and what should happen on invalid operations (e.g., DECREASE_KEY on non-existent ID, POP on empty heap).
Propose a binary min-heap where each element stores (priority, insertion_order, job_id) and a hash map mapping job_id to its current index in the heap.
For INSERT: add to heap end, update map, bubble up. For POP: swap root with last, remove last, update map, bubble down. For DECREASE_KEY: update priority, bubble up.
Check if job_id exists in map for DECREASE_KEY; if not, return error. For INSERT, if job_id already exists, either reject or update (clarify). For POP on empty heap, return null or throw exception.
Confirm all operations are O(log n) due to heap height. Walk through examples including ties 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.
Start by clarifying the problem and defining the objective function. Then propose a greedy heuristic for a quick baseline, followed by simulated annealing for improvement. Discuss initialization, stopping criteria, and how to measure approximation quality against the greedy baseline and possibly an exact solver for small instances.
Pro tip: Emphasize the trade-off between solution quality and runtime, and suggest using the greedy solution as the initial state for simulated annealing to speed up convergence. Also, mention the importance of tuning parameters like temperature and cooling schedule based on problem size.
Restate the problem: select k items from n to minimize total score = sum of weights + sum of pairwise penalties. Confirm that penalties are symmetric and only apply to selected pairs.
Design a greedy algorithm: start with an empty set, iteratively add the item that minimizes the increase in total score (weight + penalties with already selected items) until k items are selected. This provides a fast baseline.
Define state representation (set of k items), neighborhood moves (swap one selected item with an unselected one), initial temperature, cooling schedule, and stopping criteria (e.g., max iterations, no improvement for X iterations, or temperature below threshold).
Compare simulated annealing result to greedy baseline and, for small n, to an exact solution (e.g., brute force or integer programming). Report relative improvement and runtime trade-offs.
Outline how to implement efficiently (e.g., incremental score updates, data structures) and discuss trade-offs between solution quality, runtime, and parameter tuning.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.