Notes on Majority Element Problem

Introduction

• Video from Strivers A to Z DSA Course.
• World's most in-depth course in Data Structures and Algorithms (DSA).
• Covering problem of finding the Majority Element in an array.

Definition of Majority Element

• Majority Element: An element that appears more than n/2 times in an array of integers.
  • Example: For n = 8, an element must appear more than 4 times.
  • For n = 9, the floor value is used, so it must appear more than 4 times.

Problem Explanation

• Given an array, identify the element that appears more than n/2 times.
• For example, in an array of length 7, if 2 appears 4 times, 2 is the Majority Element.

Brute Force Solution

• Approach: For each element in the array:
  • Count occurrences by scanning through the entire array.
  • If any element's count exceeds n/2, return that element.
• Time Complexity: O(n²) due to nested loops.
• Example Code Structure:
  • Loop through each element.
  • Inner loop to count occurrences.
  • Return element if count > n/2, otherwise return -1 if no majority element exists.

Optimized Solution

• Observation: The brute force solution is inefficient; we need a better approach.
• Better Approach: Use hashing to count occurrences.
  • Create a hashmap to store each element and its count.
  • Iterate through the array to populate the hashmap.
  • After populating, iterate through the hashmap to find any element that appears more than n/2 times.
• Time Complexity: O(n log n) or O(n) depending on the map implementation.
• Space Complexity: O(n) for storing counts in the hashmap.

Most Optimal Algorithm: Moore's Voting Algorithm

• Key Concepts:
  • Use two variables: element and count.
  • Initialize count to 0 and element to None.
  • Iterate through the array:
    • If count is 0, choose the current element as the new element and set count to 1.
    • If the current element equals element, increment count.
    • If it doesn't match, decrement count.
  • At the end of the iteration, check if the chosen element is indeed the majority by counting its occurrences again.
• Intuition:
  • If an element appears more than n/2 times, it cannot be canceled out by other elements in pairs.
• Verification: After determining a candidate, verify by counting occurrences in the original array.
• Time Complexity: O(n) for the first pass and O(n) for verification (if needed).
• Space Complexity: O(1) since no additional space is used for storage.

Conclusion

• Understanding the intuition and the process behind the algorithms is crucial for interviews.
• Consider practicing coding and discussing the thought process behind the algorithms.
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