Classic binary search on the answer, which I did recognize pretty quickly.
Recognize this as a binary search on the answer problem: the feasibility of a given rate k is monotonic (if k works, any larger rate also works). Define a helper function that checks if sum(ceil(v/k)) <= h, then binary search the smallest k in the range [1, max(vaults)].
Pro tip: Mention that the upper bound can be max(vaults) because k=1 is the slowest possible rate, and if k >= max(vaults) each vault takes at most 1 hour, so if h >= number of vaults, max(vaults) is always feasible. Also note that using integer arithmetic for ceil division avoids floating-point errors.
Restate the problem: given an array of vault sizes and a time limit h, find the minimum integer rate k (units per hour) such that the total hours needed, sum(ceil(v/k)), is at most h. Confirm edge cases like empty array, h less than number of vaults, and large values.
Write a function feasible(k) that computes the total hours required at rate k: sum over vaults of ceil(v/k). Use integer arithmetic: (v + k - 1) // k. Return true if total <= h.
Observe that if rate k is feasible, any rate > k is also feasible. Set lower bound low = 1 and upper bound high = max(vaults) (or sum(vaults) if h is very small, but max is sufficient because at k = max(vaults), each vault takes at most 1 hour).
While low < high, compute mid = (low + high) // 2. If feasible(mid), set high = mid; else set low = mid + 1. Return low as the minimum feasible rate.
Time complexity: O(n log(max(vaults))), space O(1). Walk through a small example to verify correctness, and discuss potential overflow if sums are large (use 64-bit integers).
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Start by clarifying the problem constraints (e.g., input size, whether negative numbers are allowed, memory limits). Then propose a dynamic programming solution using a boolean array to track achievable sums, and discuss trade-offs with other approaches like recursion with memoization or meet-in-the-middle. Finally, analyze time and space complexity and consider optimizations for the specific constraints.
Pro tip: Mention that the DP can be optimized to use a 1D array and iterate backwards to avoid reusing elements, and discuss how to handle large targets with bitset or meet-in-the-middle. This shows awareness of practical performance concerns.
Ask about input size, range of integers, whether negative numbers are allowed, and if the target can be zero. This determines the best algorithm and prevents incorrect assumptions.
For small n and target, use DP with a boolean array; for large n but small target, DP is still good; for large target and moderate n, consider meet-in-the-middle. Discuss trade-offs.
Write clean code for the chosen approach, handling edge cases like empty set or target 0. For DP, initialize dp[0]=true and iterate through numbers, updating dp from target down to num.
State time and space complexity (e.g., O(n*target) time, O(target) space for DP). Mention optimizations like bitset or pruning for large targets.
Walk through a small example to verify correctness, including cases where the subset exists and where it doesn't. Mention potential pitfalls like integer overflow.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.