Lecture Notes: Sum of Subarray Ranges

Introduction

• Recap of the previous lesson.
• Topic: Sum of subarray ranges.

Problem Statement

• Given an array of numbers, generate all subarrays.
• Return the sum of the ranges of these subarrays.
• Subarray: A contiguous part of an array.
• Subarray Range: Difference between the largest and smallest element in a subarray.

Examples

Generating Subarrays

• Subarrays for the array [1, 4, 3, 2]:
  • [1]
  • [1, 4]
  • [1, 4, 3]
  • [1, 4, 3, 2]
  • [4]
  • [4, 3]
  • [4, 3, 2]
  • [3]
  • [3, 2]
  • [2]

Calculating Subarray Ranges

• For each subarray, calculate the largest and smallest elements:

  • [1]: largest = 1, smallest = 1, range = 0
  • [1, 4]: largest = 4, smallest = 1, range = 3
  • [1, 4, 3]: largest = 4, smallest = 1, range = 3
  • [1, 4, 3, 2]: largest = 4, smallest = 1, range = 3
  • [4]: largest = 4, smallest = 4, range = 0
  • [4, 3]: largest = 4, smallest = 3, range = 1
  • [4, 3, 2]: largest = 4, smallest = 2, range = 2
  • [3]: largest = 3, smallest = 3, range = 0
  • [3, 2]: largest = 3, smallest = 2, range = 1
  • [2]: largest = 2, smallest = 2, range = 0

• Sum of all ranges: 0 + 3 + 3 + 3 + 0 + 1 + 2 + 0 + 1 + 0 = 13

Brute Force Approach

• Generate all subarrays using nested loops.
• Maintain a sum variable initialized to 0.
• Loop through the array:
  • Track the largest and smallest elements for each subarray.
  • Update the sum with the difference (largest - smallest).

Complexity Analysis

• Time Complexity: O(n²) due to nested loops.
• Space Complexity: O(1).

Optimization Strategies

• Need to optimize from O(n²) to O(n).
• Utilize previously solved problems:
  • Sum of Subarray Minimums
  • Sum of Subarray Maximums
• Combine results:
  • Result = Sum of all largest - Sum of all smallest.

Final Implementation

• Write functions for sum of maximums and minimums.
• Use these functions to get results for the current problem.
• Expected time complexity should be around O(n) for both functions, leading to an overall complexity close to O(n).

Conclusion

• Importance of understanding subarray concepts for interview success.
• Encourage audience to like and subscribe for more content.

Note: Make sure you understand each concept and approach discussed in the lecture.