DSA 1 Course - Lecture Notes

Introduction

• Lecturer: Anuj
• Topic: Arrays, focusing on Maximum Sum Subarray problem.
• Important algorithm: Kadane's Algorithm
• Companies that have asked this question: Paypal, Amazon, Microsoft.

Problem Definition

• Question: Find a subarray in a given array that has the maximum sum.
• Example Array: [-5, 4, 6, -3, 4, -1]
  • Example Subarray for maximum sum: [4, 6, -3, 4] which sums to 11.

Approach to Solve the Problem

1. Brute Force Solution

• The brute force method involves:
  • Finding all subarrays and calculating their sums.
  • Using two nested loops to explore all possible subarrays:
    • Outer Loop: Iterate through each element
    • Inner Loop: From the current element to the end of the array
• Time Complexity: O(n²) which is not efficient.

2. Optimizing from Brute Force

• Considerations: You can't use sorting, as it will change the order of elements.
• Using space doesn’t help either since it still results in O(n²) complexity.

3. Kadane's Algorithm - Efficient Approach

• Core Idea: Start from the beginning and iteratively track the maximum sum:
  • If current sum becomes negative, reset it to 0.
  • Keep updating the maximum sum found so far.
• Handle positive and negative elements intelligently:
  • If the total of a subarray is negative, discard it to maximize the overall sum.
  • Begin fresh from the next element if the current subarray sum goes below zero.

Example Walkthrough of Kadane's Algorithm:

1. Start at the first element (initially 0): cursum = 0
2. Add current element and check:
   • If cursum > maxsum, update maxsum.
   • If cursum < 0, reset cursum to 0.
3. Continue for the entire array.
4. Final output: maximum subarray sum.

Complexity of Kadane's Algorithm

• Time Complexity: O(n) since it traverses the array only once.
• Space Complexity: O(1) as it does not utilize any extra space.

Additional Tasks

• If all elements are negative, adjustments need to be made to find the correct maximum.
  • Suggestion: Analyze and modify the existing solution accordingly.

Conclusion

• Kadane's Algorithm is an effective method for solving the Maximum Sum Subarray problem, requiring only O(n) time complexity.
• Homework: Consider how to modify Kadane’s Algorithm to handle arrays where all elements are negative.

Closing

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• Thank you for attending!