The prime constraint is what makes this annoying.
Model the problem as a dynamic programming problem where dp[i] represents the maximum score to reach index i. Precompute all prime gaps up to k, then for each index i, consider all previous indices j such that i-j is a prime gap and update dp[i] = max(dp[i], dp[j] + score[i]). Return dp[n-1] if reachable, else null.
Pro tip: Clarify edge cases upfront: what if k < 2 (no prime gaps), or if the array has only one element? Also, mention that negative scores mean you might need to skip certain jumps, so DP is necessary rather than greedy.
Restate the rules: jumps only forward, gap ≤ k, gap must be prime. Note that scores can be negative, so maximizing total score may involve avoiding certain indices.
Generate all prime numbers up to k using a sieve or simple primality check. These are the only allowed jump distances.
Let dp[i] be the max score to reach index i. Initialize dp[0] = score[0], others as -infinity. For each i, for each prime p ≤ k, if i-p ≥ 0 and dp[i-p] is reachable, update dp[i] = max(dp[i], dp[i-p] + score[i]).
After filling dp, if dp[n-1] is still -infinity, return null; otherwise return dp[n-1]. Discuss time complexity O(n * number of primes ≤ k) and space O(n).
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Binary search on the answer felt right and I went with it.
Model the problem as maximizing the minimum throughput after allocating expansions under a budget. Use binary search on the target throughput T, and for each T compute the minimum cost to make every service's throughput ≥ T. If the total cost ≤ budget, T is feasible; otherwise, it's not.
Pro tip: Clarify whether expansions are discrete (integer number) or continuous, and whether throughputs and costs are integers. This affects the feasibility check and binary search bounds. Also, mention that if expansions are discrete, the cost function is a step function, but binary search still works if you compute the exact number of expansions needed.
Restate the problem: we have n services in series, each with initial throughput t_i and expansion multiplier m_i (or additive increase) and cost c_i per expansion. We need to allocate expansions to maximize the minimum throughput under a budget B. Clarify if expansions are integer or continuous, and if throughputs are integers.
For a given target T, compute the minimum cost to raise each service's throughput to at least T. If expansions multiply throughput by m_i, the required expansions for service i is the smallest integer k such that t_i * (m_i)^k ≥ T. The cost is k * c_i. Sum over all services.
Binary search on T between the initial minimum throughput and an upper bound (e.g., max initial throughput * (max multiplier)^(B/min_cost)). For each mid, check if total cost ≤ B. Adjust bounds accordingly.
The binary search takes O(log(range)) iterations, each costing O(n) to compute the total cost. Discuss edge cases: budget insufficient for any expansion, very large multipliers, and the possibility of not expanding some services.
Mention that a greedy approach (repeatedly expand the current bottleneck) may not be optimal because expanding a non-bottleneck could become beneficial later. Binary search is efficient and optimal for this problem. Also, consider if the problem can be solved with dynamic programming for small n and B.
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