2656. Maximum Sum With Exactly K Elements
Description
You are given a 0-indexed integer array nums
and an integer k
. Your task is to perform the following operation exactly k
times in order to maximize your score:
- Select an element
m
fromnums
. - Remove the selected element
m
from the array. - Add a new element with a value of
m + 1
to the array. - Increase your score by
m
.
Return the maximum score you can achieve after performing the operation exactly k
times.
Example 1:
Input: nums = [1,2,3,4,5], k = 3 Output: 18 Explanation: We need to choose exactly 3 elements from nums to maximize the sum. For the first iteration, we choose 5. Then sum is 5 and nums = [1,2,3,4,6] For the second iteration, we choose 6. Then sum is 5 + 6 and nums = [1,2,3,4,7] For the third iteration, we choose 7. Then sum is 5 + 6 + 7 = 18 and nums = [1,2,3,4,8] So, we will return 18. It can be proven, that 18 is the maximum answer that we can achieve.
Example 2:
Input: nums = [5,5,5], k = 2 Output: 11 Explanation: We need to choose exactly 2 elements from nums to maximize the sum. For the first iteration, we choose 5. Then sum is 5 and nums = [5,5,6] For the second iteration, we choose 6. Then sum is 5 + 6 = 11 and nums = [5,5,7] So, we will return 11. It can be proven, that 11 is the maximum answer that we can achieve.
Constraints:
1 <= nums.length <= 100
1 <= nums[i] <= 100
1 <= k <= 100
Solutions
Solution 1: Greedy + Mathematics
We notice that to make the final score maximum, we should make each choice as large as possible. Therefore, we select the largest element $x$ in the array for the first time, $x+1$ for the second time, $x+2$ for the third time, and so on, until the $k$th time we select $x+k-1$. This way of selection ensures that the element selected each time is the largest in the current array, so the final score is also the largest. The answer is $k$ $x$ sum plus $0+1+2+\cdots+(k-1)$, that is, $k \times x + (k - 1) \times k / 2$.
Time complexity is $O(n)$, where $n$ is the length of the array. Space complexity is $O(1)$.
1 2 3 4 |
|
1 2 3 4 5 6 7 8 9 |
|
1 2 3 4 5 6 7 |
|
1 2 3 4 |
|
1 2 3 4 |
|
1 2 3 4 5 6 7 8 9 10 11 12 13 |
|
Solution 2
1 2 3 4 5 6 7 |
|