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DSA Visualizations /

Overview of TechniquesOverview of Techniques14
❯

  1. 1Overview of DSA Problem Solving Techniques
  2. 2Brute Force Technique in DSA
  3. 3Greedy Algorithm Technique in DSA
  4. 4Divide and Conquer Technique in DSA
  5. 5Dynamic Programming Technique in DSA
  6. 6Backtracking Technique in DSA
  7. 7Recursion Technique in DSA
  8. 8Sliding Window Technique in DSA
  9. 9Two Pointers Technique
  10. 10Binary Search Technique
  11. 11Tree / Graph Traversal Technique in DSA
  12. 12Bit Manipulation Technique in DSA
  13. 13Hashing Technique
  14. 14Heaps Technique in DSA

SortingSorting10
❯

  1. 1Bubble Sort
  2. 2Selection Sort
  3. 3Insertion Sort
  4. 4Merge Sort
  5. 5Quick Sort
  6. 6Heap Sort
  7. 7Radix Sort
  8. 8Counting Sort
  9. 9Bucket Sort
  10. 10Shell Sort

ArraysArrays32
❯

  1. 1Find Maximum and Minimum in Array using Loop
  2. 2Find Second Largest in Array
  3. 3Find Second Smallest in Array
  4. 4Reverse Array using Two Pointers
  5. 5Check if Array is Sorted
  6. 6Remove Duplicates from Sorted Array
  7. 7Left Rotate an Array by One Place
  8. 8Left Rotate an Array by K Places
  9. 9Move Zeroes in Array to End
  10. 10Linear Search in Array
  11. 11Union of Two Arrays
  12. 12Find Missing Number in Array
  13. 13Max Consecutive Ones in Array
  14. 14Find Kth Smallest Element
  15. 15Longest Subarray with Given Sum (Positives)
  16. 16Longest Subarray with Given Sum (Positives and Negatives)
  17. 17Find Majority Element in Array (more than n/2 times)
  18. 18Find Majority Element in Array (more than n/3 times)
  19. 19Maximum Subarray Sum using Kadane's Algorithm
  20. 20Print Subarray with Maximum Sum
  21. 21Stock Buy and Sell
  22. 22Rearrange Array Alternating Positive and Negative Elements
  23. 23Next Permutation of Array
  24. 24Leaders in an Array
  25. 25Longest Consecutive Sequence in Array
  26. 26Count Subarrays with Given Sum
  27. 27Sort an Array of 0s, 1s, and 2s
  28. 28Two Sum Problem
  29. 29Three Sum Problem
  30. 304 Sum Problem
  31. 31Find Length of Largest Subarray with 0 Sum
  32. 32Find Maximum Product Subarray

Arrays - Binary SearchArrays - Binary Search28
❯

  1. 1Binary Search in Array<br><span class="highlight">Using Iteration</span>
  2. 2Find Lower Bound in Sorted Array
  3. 3Find Upper Bound in Sorted Array
  4. 4Search Insert Position in Sorted Array (Lower Bound Approach)
  5. 5Floor and Ceil in Sorted Array
  6. 6First Occurrence in a Sorted Array
  7. 7Last Occurrence in a Sorted Array
  8. 8Count Occurrences in Sorted Array
  9. 9Search Element in a Rotated Sorted Array
  10. 10Search in Rotated Sorted Array with Duplicates
  11. 11Minimum in Rotated Sorted Array
  12. 12Find Rotation Count in Sorted Array
  13. 13Search Single Element in Sorted Array
  14. 14Find Peak Element in Array
  15. 15Square Root using Binary Search
  16. 16Nth Root of a Number using Binary Search
  17. 17Koko Eating Bananas
  18. 18Minimum Days to Make M Bouquets
  19. 19Find the Smallest Divisor Given a Threshold
  20. 20Capacity to Ship Packages within D Days
  21. 21Kth Missing Positive Number
  22. 22Aggressive Cows Problem
  23. 23Allocate Minimum Number of Pages
  24. 24Split Array - Minimize Largest Sum
  25. 25Painter's Partition Problem
  26. 26Minimize Maximum Distance Between Gas Stations
  27. 27Median of Two Sorted Arrays of Different Sizes
  28. 28K-th Element of Two Sorted Arrays

StringsStrings2
❯

  1. 1Remove Outermost Parentheses in String
  2. 2Edit Distance Problem

MatrixMatrix3
❯

  1. 1Set Matrix Zeroes
  2. 2Rotate Matrix by 90 Degrees Clockwise
  3. 3Print Matrix in Spiral Manner

Matrix - Binary SearchMatrix - Binary Search5
❯

  1. 1Row with Maximum Number of 1's
  2. 2Search in a Sorted 2D Matrix
  3. 3Search in Row and Column-wise Sorted Matrix
  4. 4Find Target in 2D Matrix
  5. 5Matrix Median

Binary TreesBinary Trees36
❯

  1. 1Preorder Traversal of a Binary Tree using Recursion
  2. 2Preorder Traversal of a Binary Tree using Iteration
  3. 3Postorder Traversal of a Binary Tree Using Recursion
  4. 4Postorder Traversal of a Binary Tree using Iteration
  5. 5Level Order Traversal of a Binary Tree using Recursion
  6. 6Level Order Traversal of a Binary Tree using Iteration
  7. 7Reverse Level Order Traversal of a Binary Tree using Iteration
  8. 8Reverse Level Order Traversal of a Binary Tree using Recursion
  9. 9Find Height of a Binary Tree
  10. 10Find Diameter of a Binary Tree
  11. 11Find Mirror of a Binary Tree - Todo
  12. 12Inorder Traversal of a Binary Tree using Recursion
  13. 13Inorder Traversal of a Binary Tree using Iteration
  14. 14Left View of a Binary Tree
  15. 15Right View of a Binary Tree
  16. 16Top View of a Binary Tree
  17. 17Bottom View of a Binary Tree
  18. 18Zigzag Traversal of a Binary Tree
  19. 19Check if a Binary Tree is Balanced
  20. 20Diagonal Traversal of a Binary Tree
  21. 21Boundary Traversal of a Binary Tree
  22. 22Construct a Binary Tree from a String with Bracket Representation
  23. 23Convert a Binary Tree into a Doubly Linked List
  24. 24Convert a Binary Tree into a Sum Tree
  25. 25Find Minimum Swaps Required to Convert a Binary Tree into a BST
  26. 26Check if a Binary Tree is a Sum Tree
  27. 27Check if All Leaf Nodes are at the Same Level in a Binary Tree
  28. 28Lowest Common Ancestor (LCA) in a Binary Tree
  29. 29Solve the Tree Isomorphism Problem
  30. 30Check if a Binary Tree Contains Duplicate Subtrees of Size 2 or More
  31. 31Check if Two Binary Trees are Mirror Images
  32. 32Calculate the Sum of Nodes on the Longest Path from Root to Leaf in a Binary Tree
  33. 33Print All Paths in a Binary Tree with a Given Sum
  34. 34Find the Distance Between Two Nodes in a Binary Tree
  35. 35Find the kth Ancestor of a Node in a Binary Tree
  36. 36Find All Duplicate Subtrees in a Binary Tree

Binary Search TreesBinary Search Trees14
❯

  1. 1Find a Value in a Binary Search Tree
  2. 2Delete a Node in a Binary Search Tree
  3. 3Find the Minimum Value in a Binary Search Tree
  4. 4Find the Maximum Value in a Binary Search Tree
  5. 5Find the Inorder Successor in a Binary Search Tree
  6. 6Find the Inorder Predecessor in a Binary Search Tree
  7. 7Check if a Binary Tree is a Binary Search Tree
  8. 8Find the Lowest Common Ancestor of Two Nodes in a Binary Search Tree
  9. 9Convert a Binary Tree into a Binary Search Tree
  10. 10Balance a Binary Search Tree
  11. 11Merge Two Binary Search Trees
  12. 12Find the kth Largest Element in a Binary Search Tree
  13. 13Find the kth Smallest Element in a Binary Search Tree
  14. 14Flatten a Binary Search Tree into a Sorted List

Find Majority Element in Array Using Extended Boyer-Moore Voting Algorithm - Optimal Solution

Topic Contents

ProblemExamplesSolutionVisualizationCodeTime ComplexitySpace Complexity

Problem Statement

Given an integer array arr, your task is to find all elements that appear more than ⌊n/3⌋ times in the array, where n is the size of the array.

If no such element exists, return an empty list.

Examples❯

Input ArrayOutputDescription
[3, 2, 3][3]n = 3 → ⌊3/3⌋ = 1, 3 appears 2 times
[1][1]Single element always satisfies > n/3 condition
[1, 2][1, 2]n = 2 → ⌊2/3⌋ = 0, both 1 and 2 appear > 0 times
[1, 2, 3, 4][]No element appears more than n/3 = 1 times
[1, 1, 1, 3, 3, 2, 2, 2][1, 2]n = 8 → ⌊8/3⌋ = 2, 1 and 2 both appear 3 times
[][]Empty array has no elements, returns empty list
[2, 2][2]n = 2 → ⌊2/3⌋ = 0, 2 appears 2 times
[4, 4, 4, 5, 5, 5, 6][4, 5]n = 7 → ⌊7/3⌋ = 2, 4 and 5 appear 3 times

Solution❯

To solve this problem, we need to find elements that appear more than ⌊n/3⌋ times in a given array. The main challenge is to do it efficiently—ideally in linear time and using constant space.

Understanding the Constraint

When we say we want elements that occur more than n/3 times, it means we're looking for items that appear a significant number of times—but not necessarily the most frequent. Because of this threshold, there can be at most two elements that satisfy this condition. Why two? Because if three distinct numbers all appeared more than n/3 times, their total count would exceed n, which is impossible.

How Do We Approach This?

The trick is to use an extension of the Boyer-Moore Voting Algorithm. The original Boyer-Moore algorithm finds a single majority element (> n/2), but we can extend it to handle two potential candidates:

  • We go through the array once, keeping track of two candidates and their counts.
  • If we see a number matching a candidate, we increment that candidate’s count.
  • If one of the counts is zero, we replace that candidate with the current number and reset the count to 1.
  • If the current number matches neither candidate, we decrement both counts.

This gives us up to two potential candidates. But since they are just candidates, we need a second pass through the array to count how many times they actually appear.

Final Validation

Once we've completed the first pass and identified the two candidates, we run a second loop to count how many times each one appears. Only if their count is greater than n/3, do we include them in the final result.

Case-by-Case Insight

  • Single Element: If the array has only one number, that number is the majority by default.
  • All Unique Elements: If no number repeats more than ⌊n/3⌋ times, we return an empty list.
  • Multiple Valid Candidates: Sometimes two numbers will satisfy the condition—both should be returned.
  • Empty Array: There are no elements to count, so the result is an empty list.

This solution is optimal: it runs in O(n) time and uses O(1) space, as we only track two numbers and their counts during traversal.

Visualization

Algorithm Steps

  1. Given an array arr of size n.
  2. Initialize two candidates candidate1 and candidate2 and their counts to 0.
  3. Traverse the array:
  4. → If the current element equals candidate1, increment count1.
  5. → Else if it equals candidate2, increment count2.
  6. → Else if count1 is 0, set candidate1 to current element and count1 to 1.
  7. → Else if count2 is 0, set candidate2 to current element and count2 to 1.
  8. → Else decrement both count1 and count2.
  9. After the first pass, verify the candidates by counting their actual occurrences in the array.
  10. Return the candidates that appear more than n/3 times.

Code

Python
JavaScript
Java
C++
C
def majority_elements(arr):
    if not arr:
        return []
    
    candidate1 = candidate2 = None
    count1 = count2 = 0

    for num in arr:
        if num == candidate1:
            count1 += 1
        elif num == candidate2:
            count2 += 1
        elif count1 == 0:
            candidate1 = num
            count1 = 1
        elif count2 == 0:
            candidate2 = num
            count2 = 1
        else:
            count1 -= 1
            count2 -= 1

    result = []
    for candidate in [candidate1, candidate2]:
        if arr.count(candidate) > len(arr) // 3:
            result.append(candidate)
    return result

# Sample Input
arr = [3,2,3]
print("Majority Elements:", majority_elements(arr))

Time Complexity

CaseTime ComplexityExplanation
Best CaseO(n)The array is traversed twice: once for candidate selection and once for verification, both in linear time.
Average CaseO(n)Regardless of the input distribution, both candidate identification and validation phases take linear time.
Average CaseO(n)Even in the worst-case (e.g., all elements are different), both passes still involve scanning the entire array.

Space Complexity

O(1)

Explanation: Only a fixed number of variables are used (two candidates and their counts), independent of the input size.

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