Majority Element Moore's Voting Algorithm in DSA C - Time & Space Complexity

Time Complexity: Majority Element Moore's Voting Algorithm

O(n)

Understanding Time Complexity

We want to understand how the time needed to find the majority element grows as the input size increases.

Specifically, how many steps does Moore's Voting Algorithm take when the input array gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

int findMajorityElement(int arr[], int n) {
    int count = 0, candidate = -1;
    for (int i = 0; i < n; i++) {
        if (count == 0) {
            candidate = arr[i];
        }
        if (arr[i] == candidate) {
            count++;
        } else {
            count--;
        }
    }
    return candidate;
}
    

This code finds the majority element by keeping track of a candidate and its count while scanning the array once.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Single for-loop scanning the array once.
• How many times: Exactly n times, where n is the array size.

How Execution Grows With Input

As the array size grows, the number of steps grows directly in proportion.

Input Size (n)	Approx. Operations
10	About 10 steps
100	About 100 steps
1000	About 1000 steps

Pattern observation: Doubling the input size roughly doubles the work done.

Final Time Complexity

Time Complexity: O(n)

This means the time to find the majority element grows linearly with the size of the input array.

Common Mistake

[X] Wrong: "Since there are two loops in some versions, the time complexity is O(n²)."

[OK] Correct: The algorithm uses only one main loop that runs n times; any second pass is separate and also linear, so overall time is still O(n), not squared.

Interview Connect

Understanding this linear time solution shows you can find efficient answers by clever counting, a skill valued in problem solving.

Self-Check

"What if we added a verification step that counts the candidate's occurrences after finding it? How would the time complexity change?"

Practice

1. What is the main purpose of Moore's Voting Algorithm in arrays?

easy

A. To sort the array in ascending order

B. To find the element that appears more than half the time

C. To find the smallest element in the array

D. To reverse the array elements

Solution

1. Step 1: Understand the problem Moore's Voting Algorithm solves

   It is designed to find the majority element, which is the element that appears more than half the time in the array.

2. Step 2: Compare other options

   Sorting, finding smallest, or reversing are unrelated to majority element detection.

3. Final Answer:

   To find the element that appears more than half the time -> Option B

4. Quick Check:

   Majority element = element with > n/2 occurrences [OK]

Hint: Majority element appears more than half times [OK]

Common Mistakes:

• Confusing majority element with most frequent element
• Thinking it sorts the array
• Assuming it finds smallest or largest element

2. Which of the following is the correct initialization step in Moore's Voting Algorithm in C?

easy

A. int count = 0, candidate = -1;

B. int count = 1, candidate = 0;

C. int candidate = 0, count = 0;

D. int candidate = -1, count = 1;

Solution

1. Step 1: Identify correct variable initialization

   Moore's Voting Algorithm starts with count = 0 and candidate = -1 (or any invalid index) to indicate no candidate chosen yet.

2. Step 2: Check other options

   Other options start count or candidate incorrectly, which can cause logic errors.

3. Final Answer:

   int count = 0, candidate = -1; -> Option A

4. Quick Check:

   Start with no candidate and zero count [OK]

Hint: Initialize count=0 and candidate invalid (-1) [OK]

Common Mistakes:

• Starting count at 1 instead of 0
• Assigning candidate to 0 prematurely
• Swapping order of variables

3. What is the output of the following C code snippet using Moore's Voting Algorithm?

int arr[] = {2, 2, 1, 1, 1, 2, 2};
int n = 7;
int count = 0, candidate = -1;
for (int i = 0; i < n; i++) {
  if (count == 0) {
    candidate = arr[i];
  }
  if (arr[i] == candidate) {
    count++;
  } else {
    count--;
  }
}
printf("%d", candidate);

medium

A. 1

B. 0

C. 2

D. No majority element

Solution

1. Step 1: Trace the candidate and count updates

   Start count=0, candidate=-1
   i=0: count=0, candidate=2, count=1
   i=1: arr[i]=2 == candidate=2, count=2
   i=2: arr[i]=1 != candidate=2, count=1
   i=3: arr[i]=1 != candidate=2, count=0
   i=4: count=0, candidate=1, count=1
   i=5: arr[i]=2 != candidate=1, count=0
   i=6: count=0, candidate=2, count=1

2. Step 2: Final candidate after loop

   Candidate is 2 after loop ends.

3. Final Answer:

   2 -> Option C

4. Quick Check:

   Majority element found = 2 [OK]

Hint: Track count and candidate changes stepwise [OK]

Common Mistakes:

• Not resetting candidate when count hits zero
• Confusing candidate with last element
• Ignoring count decrement on mismatch

4. Identify the bug in this Moore's Voting Algorithm implementation snippet:

int count = 0, candidate = -1;
for (int i = 0; i < n; i++) {
  if (count == 0) {
    candidate = arr[i];
  }
  if (arr[i] != candidate) {
    count++;
  } else {
    count--;
  }
}

medium

A. Candidate initialized incorrectly

B. Candidate is not updated when count is zero

C. Loop does not run for all elements

D. Count increment and decrement conditions are reversed

Solution

1. Step 1: Check count update logic

   Correct logic: increment count if element equals candidate, decrement otherwise.
   Given code increments count when arr[i] != candidate, which is wrong.

2. Step 2: Confirm candidate update and loop correctness

   Candidate updates correctly when count == 0.
   Loop runs correctly for all elements.
   Candidate initialization is valid.

3. Final Answer:

   Count increment and decrement conditions are reversed -> Option D

4. Quick Check:

   Increment on match, decrement on mismatch [OK]

Hint: Increment count on match, decrement on mismatch [OK]

Common Mistakes:

• Swapping increment and decrement conditions
• Not updating candidate when count is zero
• Incorrect loop bounds

5. Given an array int arr[] = {3, 3, 4, 2, 4, 4, 2, 4, 4}, how can you verify if the candidate found by Moore's Voting Algorithm is actually a majority element?

hard

A. Count occurrences of candidate and check if it is more than n/2

B. Check if candidate is at the first position of the array

C. Verify if candidate is the maximum element in the array

D. No verification needed; candidate is always majority

Solution

1. Step 1: Understand Moore's Voting Algorithm limitation

   The algorithm finds a candidate but does not guarantee it is majority if none exists.

2. Step 2: Verify candidate by counting occurrences

   Count how many times candidate appears in array.
   If count > n/2, candidate is majority; else no majority element.

3. Final Answer:

   Count occurrences of candidate and check if it is more than n/2 -> Option A

4. Quick Check:

   Verify candidate count > n/2 to confirm majority [OK]

Hint: Always verify candidate count > n/2 after algorithm [OK]

Common Mistakes:

• Assuming candidate is always majority without check
• Checking candidate position instead of count
• Confusing majority with max element