Problem Statement:
Given an array arr[] consisting of N integers, the task is to find the sum of the differences between the maximum and minimum elements of all possible subarrays from the given array.
Approach:
The key insight in solving this problem is to efficiently calculate how many times each element in the array acts as a minimum or maximum in the subarrays that include it. By doing so, we can compute the sum of the differences between the maximum and minimum elements across all subarrays.
Step-by-Step Explanation:
Step 1: Understand Contribution of Each Element
For each element in the array:
- Determine how many subarrays it is the maximum element.
- Determine how many subarrays it is the minimum element.
The contribution of an element as a maximum is positive, while its contribution as a minimum is negative. Therefore, for each element, the net contribution to the sum of differences is:
Net Contribution = Number of subarrays where it is max - Number of subarrays where it is min x * element value
Step 2: Utilize Monotonic Stacks
To efficiently calculate the number of subarrays in which each element is the maximum or minimum:
- Use a monotonic increasing stack to find how many subarrays an element is the minimum.
- Use a monotonic decreasing stack to find how many subarrays an element is the maximum.
Monotonic Increasing Stack:
- Helps in finding how many subarrays have the current element as the minimum by counting subarrays extending to the left and right.
Monotonic Decreasing Stack:
- Helps in finding how many subarrays have the current element as the maximum by similarly counting subarrays extending to the left and right.
Step 3: Calculate the Sum Using Contributions
Once the counts are obtained for each element:
- Calculate the net contribution for each element using the derived formula.
- Accumulate these contributions to get the final result.
Time Complexity:
- Time Complexity: The algorithm runs in
O(N)time since each element is processed using stacks. - Space Complexity: The space complexity is
O(N)for storing the stacks and additional arrays for left and right boundaries.