You are given an integer array nums and three integers k, op1, and op2.
You can perform the following operations on nums:
i and divide nums[i] by 2, rounding up to the nearest whole number. You can perform this operation at most op1 times, and not more than once per index.i and subtract k from nums[i], but only if nums[i] is greater than or equal to k. You can perform this operation at most op2 times, and not more than once per index.Note: Both operations can be applied to the same index, but at most once each.
Return the minimum possible sum of all elements in nums after performing any number of operations.
Example 1:
Input: nums = [2,8,3,19,3], k = 3, op1 = 1, op2 = 1
Output: 23
Explanation:
nums[1] = 8, making nums[1] = 5.nums[3] = 19, making nums[3] = 10.[2, 5, 3, 10, 3], which has the minimum possible sum of 23 after applying the operations.Example 2:
Input: nums = [2,4,3], k = 3, op1 = 2, op2 = 1
Output: 3
Explanation:
nums[0] = 2, making nums[0] = 1.nums[1] = 4, making nums[1] = 2.nums[2] = 3, making nums[2] = 0.[1, 2, 0], which has the minimum possible sum of 3 after applying the operations.Constraints:
1 <= nums.length <= 1000 <= nums[i] <= 1050 <= k <= 1050 <= op1, op2 <= nums.lengthThe key idea in #3366 Minimum Array Sum is to strategically apply the allowed operations so that the overall array sum becomes as small as possible. Instead of modifying elements arbitrarily, an efficient approach focuses on prioritizing elements where an operation produces the largest reduction in value.
A common strategy is to use greedy techniques combined with data structures such as a priority queue or sorting. By always selecting the element that benefits most from the operation, we can ensure each step contributes maximally to reducing the total sum. In some variants, dynamic programming can also be used when the number of operations or choices per element is limited and decisions interact with each other.
The goal is to track the current best candidates for modification while efficiently updating the array state. With proper use of heaps or ordered structures, each operation can be processed quickly, leading to an overall time complexity typically around O(n log n), with additional space for auxiliary structures.
| Approach | Time Complexity | Space Complexity |
|---|---|---|
| Greedy with Priority Queue | O(n log n) | O(n) |
| Dynamic Programming Variant | O(n * k) | O(n * k) |
NeetCode
Use these hints if you're stuck. Try solving on your own first.
Think of dynamic programming with states to track progress and remaining operations.
Use <code>dp[index][op1][op2]</code> where each state tracks progress at <code>index</code> with <code>op1</code> and <code>op2</code> operations left.
At each state, try applying only operation 1, only operation 2, both in sequence, or skip both to find optimal results.
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Problems similar to Minimum Array Sum frequently appear in coding interviews at large tech companies. They typically test greedy reasoning, efficient data structure usage, and the ability to minimize or maximize a metric under constraints.
A common optimal strategy uses a greedy approach where operations are applied to elements that produce the maximum reduction in the array's total sum. Priority queues or sorting are often used to efficiently select the best candidate at each step.
Yes, dynamic programming can be used when the number of operations or choices per element affects future decisions. DP helps track optimal results across different states, though greedy solutions are often more efficient when applicable.
A priority queue (max-heap or min-heap depending on the operation) is often the most useful structure. It allows you to repeatedly access and update the element that yields the largest improvement to the array sum.