Meet Alex, a bright student who is dealing with an array challenge today. He has been given an array a1, a2, ..., aN of length N. The intriguing aspect of this challenge is that Alex can perform various operations on the array elements.In a single operation, Alex is allowed to choose two indices l and r, where 1 ≤ l ≤ r ≤ n. He can then multiply all the elements in the subarray [al, al+1, ..., ar] by -1, effectively reversing their signs.Being pressed for time as he is running late for school, Alex is seeking your assistance in determining the maximum possible sum of numbers in the array. Additionally, he is eager to find out the minimum number of operations required to achieve this maximum sum.
Question
Meet Alex, a bright student who is dealing with an array challenge today. He has been given an array a1, a2, ..., aN of length N. The intriguing aspect of this challenge is that Alex can perform various operations on the array elements.In a single operation, Alex is allowed to choose two indices l and r, where 1 ≤ l ≤ r ≤ n. He can then multiply all the elements in the subarray [al, al+1, ..., ar] by -1, effectively reversing their signs.Being pressed for time as he is running late for school, Alex is seeking your assistance in determining the maximum possible sum of numbers in the array. Additionally, he is eager to find out the minimum number of operations required to achieve this maximum sum.
Solution
To solve this problem, we need to follow these steps:
-
First, we need to find the minimum absolute value in the array and count the number of negative numbers. Let's call the minimum absolute value
min_absand the count of negative numbersneg_count. -
If
neg_countis even, then the maximum sum of the array can be obtained without any operations because we can pair the negative numbers to make them positive. So, the maximum sum is the sum of absolute values of all numbers in the array. -
If
neg_countis odd, then we need to perform one operation to make the sum maximum. We should choose the subarray which contains themin_abs. By doing this, we can make themin_abspositive and the sum will be maximum. So, the maximum sum is the sum of absolute values of all numbers in the array minus2*min_abs. -
The minimum number of operations is either 0 (when
neg_countis even) or 1 (whenneg_countis odd).
This solution works because multiplying a negative number by -1 makes it positive, and the sum of positive numbers is always greater than or equal to the sum of negative numbers. Also, we should perform the minimum number of operations to save time, so we should choose the subarray wisely.
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