In the realm of algorithmic problems, often, the straightforward approach doesn’t yield the most optimal solution. Leetcode’s “Maximize Sum Of Array After K Negations” problem is an illustrative example. It involves a blend of sorting, greedy algorithms, and mathematical insights. In this article, we’ll unpack the problem, delve into its intricacies, and craft a Pythonic solution.

### Table of Contents

- Problem Statement
- Intuitive Understanding and Observations
- Problem-solving Strategy
- Python Implementation
- Time and Space Complexity Analysis
- Conclusion

### 1. Problem Statement

Given an array `A`

of integers, we must modify it in the following way: we choose an `i`

and replace `A[i]`

with `-A[i]`

, and we repeat this process `K`

times in total. We may choose the same index `i`

multiple times. Return the largest possible sum of the array after modifying it in this way.

**Example:**

```
Input: A = [4,2,3], K = 1
Output: 5
Explanation: Choose indices (1,) and A becomes [4,-2,3].
```

### 2. Intuitive Understanding and Observations

- Flipping a negative number makes it positive, which will increase the sum of the array.
- Flipping a positive number will decrease the sum of the array, so it might be beneficial to flip the smallest positive number if necessary.
- If there’s an even number of negative values and
`K`

is also even, it would be optimal to flip all the negative values. - If
`K`

is odd, after flipping all possible negative numbers, it’s optimal to flip the smallest absolute value in the array, which could be a previously flipped negative number or a positive number.

### 3. Problem-solving Strategy

**Sort the Array:**A sorted array will help us quickly identify the negatives and the smallest numbers.**Flip All Negatives:**As long as`K > 0`

and there are negatives in the array, flip them. This is a greedy choice to increase the total sum.**Optimally Use Remaining Flips:**If`K`

is still greater than 0, and it’s odd (because with an even K, we can flip any number twice to get back the original array), we should flip the smallest absolute number.**Calculate the Sum:**Sum up all the numbers in the array.

### 4. Python Implementation

```
def largestSumAfterKNegations(A, K):
# Step 1: Sort the array
A.sort()
# Step 2: Flip all negatives while K is positive
for i in range(len(A)):
if A[i] < 0 and K > 0:
A[i] = -A[i]
K -= 1
# Step 3: Use remaining flips optimally
if K % 2:
min_index = A.index(min(A))
A[min_index] = -A[min_index]
# Step 4: Calculate the sum
return sum(A)
```

### 5. Time and Space Complexity Analysis

**Time Complexity:**- Sorting the array will take O(nlogn).
- Flipping all negatives and using the remaining flips will take O(n).
- Thus, the overall time complexity is O(nlogn).

**Space Complexity:**- We’re not using any extra space proportional to the input size.
- Thus, the space complexity is O(1).

### 6. Conclusion

The “Maximize Sum Of Array After K Negations” problem, while seemingly simple, requires a blend of insights to solve optimally. It’s a classic example of problems where sorting and greedy algorithms shine, allowing for efficient and elegant solutions.