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 fast Moore's Voting Algorithm finds the majority element in a list.

How does the time needed grow as the list gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

int findMajorityElement(const std::vector<int>& nums) {
    int count = 0, candidate = 0;
    for (int num : nums) {
        if (count == 0) {
            candidate = num;
        }
        count += (num == candidate) ? 1 : -1;
    }
    return candidate;
}
    

This code finds the majority element by counting and updating a candidate as it scans the list once.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: One single loop that goes through each element in the list once.
• How many times: Exactly once for each element, so n times if the list has n elements.

How Execution Grows With Input

As the list gets bigger, the number of steps grows directly with the number of elements.

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

Pattern observation: The steps increase in a straight line as the list size grows.

Final Time Complexity

Time Complexity: O(n)

This means the time to find the majority element grows directly with the size of the list.

Common Mistake

[X] Wrong: "The algorithm checks every pair of elements, so it must be O(n²)."

[OK] Correct: The algorithm only scans the list once, updating counts without nested loops, so it is much faster than checking pairs.

Interview Connect

Knowing this algorithm's time complexity shows you can find efficient solutions that scan data just once, a skill valued in real-world coding.

Self-Check

"What if we added a second pass to verify the candidate's count? How would the time complexity change?"

Practice

1. What is the main purpose of Moore's Voting Algorithm in an array of integers?

easy

A. To count the total number of unique elements

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

C. To find the smallest element in the array

D. To sort the array in ascending order

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 times in the array.

2. Step 2: Compare other options

   Sorting, finding smallest element, or counting unique elements are unrelated to Moore's Voting Algorithm's goal.

3. Final Answer:

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

4. Quick Check:

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

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

Common Mistakes:

• Thinking it sorts the array
• Confusing majority element with smallest element
• Assuming it counts unique elements

2. Which of the following is the correct syntax to initialize the candidate and count variables for Moore's Voting Algorithm in C++?

easy

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

B. int candidate, count = 1;

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

D. int candidate = 0; int count;

Solution

1. Step 1: Check initialization for candidate and count

   Candidate is usually initialized to an invalid value like -1 or first element, count starts at 0.

2. Step 2: Validate options

   int candidate = -1, count = 0; initializes candidate to -1 and count to 0, which is standard and safe before iteration.

3. Final Answer:

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

4. Quick Check:

   Candidate = -1, Count = 0 initialization [OK]

Hint: Initialize candidate to -1 and count to 0 before loop [OK]

Common Mistakes:

• Not initializing count to 0
• Setting count to negative value
• Leaving candidate uninitialized

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 candidate = -1, count = 0;
for (int i = 0; i < 7; i++) {
  if (count == 0) candidate = arr[i];
  if (arr[i] == candidate) count++;
  else count--;
}
std::cout << candidate << std::endl;

medium

A. 1

B. No output

C. 0

D. 2

Solution

1. Step 1: Trace the loop and update candidate and count

   Start with candidate=-1, count=0.
   i=0: count=0, candidate=2, arr[0]=2 matches candidate, count=1
   i=1: arr[1]=2 matches candidate, count=2
   i=2: arr[2]=1 not match, count=1
   i=3: arr[3]=1 not match, count=0
   i=4: count=0, candidate=1, arr[4]=1 matches candidate, count=1
   i=5: arr[5]=2 not match, count=0
   i=6: count=0, candidate=2, arr[6]=2 matches candidate, count=1

2. Step 2: Final candidate after loop

   The candidate is 2 after the loop ends.

3. Final Answer:

   2 -> Option D

4. Quick Check:

   Final candidate = 2 [OK]

Hint: Update candidate when count is zero, increment/decrement count [OK]

Common Mistakes:

• Not updating candidate when count is zero
• Confusing increment and decrement conditions
• Assuming candidate is last element always

4. Identify the error in the following Moore's Voting Algorithm implementation snippet:

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

medium

A. Candidate update should happen before count increment/decrement

B. Count should never be decremented

C. Candidate should be initialized to -1

D. Loop should run from i = 1 to n

Solution

1. Step 1: Analyze order of candidate update and count changes

   The candidate must be updated when count is zero before comparing arr[i] with candidate to decide increment or decrement.

2. Step 2: Identify the error in code order

   Here, count is changed before candidate update, which causes wrong counting because candidate may be outdated.

3. Final Answer:

   Candidate update should happen before count increment/decrement -> Option A

4. Quick Check:

   Update candidate first when count=0 [OK]

Hint: Update candidate before count changes when count is zero [OK]

Common Mistakes:

• Updating candidate after changing count
• Initializing candidate incorrectly
• Wrong loop range

5. Given an array arr = {3, 3, 4, 2, 4, 4, 2, 4, 4}, apply Moore's Voting Algorithm to find the majority element. What is the correct majority element and why?

hard

A. 2, because it appears twice

B. 3, because it appears first in the array

C. 4, because it appears 5 times which is more than half of 9

D. No majority element exists

Solution

1. Step 1: Apply Moore's Voting Algorithm to find candidate

   Start candidate=-1, count=0.
   i=0: count=0, candidate=3, arr[0]=3 matches candidate, count=1
   i=1: arr[1]=3 matches candidate, count=2
   i=2: arr[2]=4 not match, count=1
   i=3: arr[3]=2 not match, count=0
   i=4: count=0, candidate=4, arr[4]=4 matches candidate, count=1
   i=5: arr[5]=4 matches candidate, count=2
   i=6: arr[6]=2 not match, count=1
   i=7: arr[7]=4 matches candidate, count=2
   i=8: arr[8]=4 matches candidate, count=3

2. Step 2: Verify candidate frequency

   Candidate is 4. Count occurrences of 4 in array: 5 times.
   Since 5 > 9/2 (4.5), 4 is majority element.

3. Final Answer:

   4, because it appears 5 times which is more than half of 9 -> Option C

4. Quick Check:

   Count(4) = 5 > 9/2 [OK]

Hint: Count candidate frequency after algorithm to confirm majority [OK]

Common Mistakes:

• Assuming first element is majority
• Not verifying candidate frequency
• Ignoring that majority must be > n/2