Majority Element Moore's Voting Algorithm in DSA Python - 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.

def majority_element(nums):
    count = 0
    candidate = None
    for num in nums:
        if count == 0:
            candidate = num
        count += (1 if num == candidate else -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 loop that goes through each number 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 algorithm checks each number once, so the work grows directly with the list size.

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

Pattern observation: The number of operations grows in a straight line with input size.

Final Time Complexity

Time Complexity: O(n)

This means the time taken grows directly in proportion to the number of elements in the list.

Common Mistake

[X] Wrong: "The algorithm checks pairs or does nested loops, so it must be slower than O(n)."

[OK] Correct: The algorithm only scans the list once, updating counts without extra loops inside, so it stays fast and linear.

Interview Connect

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

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 numbers?

easy

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

B. To sort the array in ascending order

C. To find the smallest element in the array

D. To count the total number of unique elements

Solution

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

   The algorithm 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 with other options

   Sorting, finding smallest element, or counting unique elements are not the goals of this algorithm.

3. Final Answer:

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

4. Quick Check:

   Majority element = element with >50% frequency [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 counts unique elements

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

easy

A. count = 0; candidate = 0

B. count = 1; candidate = 0

C. count = None; candidate = 0

D. count = 0; candidate = None

Solution

1. Step 1: Recall initialization in Moore's Voting Algorithm

   We start with count = 0 and candidate = None to indicate no candidate chosen yet.

2. Step 2: Check other options for correctness

   Starting count at 1 or candidate at 0 without context is incorrect; None is used to indicate no candidate.

3. Final Answer:

   count = 0; candidate = None -> Option D

4. Quick Check:

   Initialize count=0 and candidate=None [OK]

Hint: Start count at 0 and candidate as None [OK]

Common Mistakes:

• Starting count at 1 instead of 0
• Assigning candidate to 0 prematurely
• Using None for count instead of candidate

3. What will be the output of the following code snippet?

def majority_element(nums):
    count = 0
    candidate = None
    for num in nums:
        if count == 0:
            candidate = num
        count += (1 if num == candidate else -1)
    return candidate

print(majority_element([2,2,1,1,1,2,2]))

medium

A. 2

B. 1

C. None

D. Error

Solution

1. Step 1: Trace the candidate and count updates

   Start with count=0, candidate=None. Iterate: 2 (count=0->1, candidate=2), 2 (count=2), 1 (count=1), 1 (count=0), 1 (count=1, candidate=1), 2 (count=0), 2 (count=1, candidate=2).

2. Step 2: Final candidate after iteration

   The candidate ends as 2 after the loop finishes.

3. Final Answer:

   2 -> Option A

4. Quick Check:

   Majority element in [2,2,1,1,1,2,2] is 2 [OK]

Hint: Track candidate changes when count hits zero [OK]

Common Mistakes:

• Not updating candidate when count is zero
• Confusing candidate with last element
• Assuming candidate is None if count resets

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

def majority_element(nums):
    count = 0
    candidate = None
    for num in nums:
        if count == 0:
            candidate = num
        if num == candidate:
            count += 1
        else:
            count -= 1
    return candidate

print(majority_element([3,3,4]))

medium

A. Count increment/decrement logic is reversed

B. No bug, code works correctly

C. Candidate not updated when count is zero

D. Function does not handle empty list

Solution

1. Step 1: Analyze candidate update and count logic

   Candidate updates when count is zero. Count increments if current num equals candidate, else decrements.

2. Step 2: Test with input [3,3,4]

   Iteration: 3 (count=1, candidate=3), 3 (count=2), 4 (count=1). Returns candidate 3 correctly.

3. Final Answer:

   No bug, code works correctly -> Option B

4. Quick Check:

   Code correctly finds majority element [OK]

Hint: Check candidate update and count changes carefully [OK]

Common Mistakes:

• Thinking candidate is not updated when count is zero
• Confusing increment and decrement logic
• Ignoring empty list edge case

5. Given an array where no element appears more than half the time, what will Moore's Voting Algorithm return?
Example: [1,2,3,4]

hard

A. Error or exception

B. The element with highest frequency

C. None or incorrect candidate

D. The first element of the array

Solution

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

   The algorithm assumes a majority element exists (more than half the array). If none exists, it still returns a candidate but it may be incorrect.

2. Step 2: Apply to example [1,2,3,4]

   No element appears more than 2 times (half of 4). The algorithm returns a candidate but it is not a true majority.

3. Final Answer:

   None or incorrect candidate -> Option C

4. Quick Check:

   No majority means candidate may be wrong [OK]

Hint: Algorithm assumes majority exists; else candidate may be wrong [OK]

Common Mistakes:

• Assuming algorithm returns error if no majority
• Thinking it returns highest frequency element always
• Believing it returns first element by default