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title: Majority Element
slug: majority-element
difficulty: easy
leetcode_id: 169
leetcode_url: https://leetcode.com/problems/majority-element/
categories:
- arrays
- hash-tables
patterns:
- greedy
function_signature: "def majority_element(nums: list[int]) -> int:"
test_cases:
visible:
- input: { nums: [3, 2, 3] }
expected: 3
- input: { nums: [2, 2, 1, 1, 1, 2, 2] }
expected: 2
- input: { nums: [1] }
expected: 1
hidden:
- input: { nums: [6, 5, 5] }
expected: 5
- input: { nums: [1, 1, 1, 2, 2, 2, 2] }
expected: 2
- input: { nums: [3, 3, 4] }
expected: 3
description: |
Given an array `nums` of size `n`, return *the majority element*.
The majority element is the element that appears **more than** `⌊n / 2⌋` times. You may assume that the majority element always exists in the array.
constraints: |
- `n == nums.length`
- `1 <= n <= 5 * 10^4`
- `-10^9 <= nums[i] <= 10^9`
- The input is generated such that a majority element will exist in the array
examples:
- input: "nums = [3,2,3]"
output: "3"
explanation: "The element 3 appears twice out of 3 elements, which is more than ⌊3/2⌋ = 1 time."
- input: "nums = [2,2,1,1,1,2,2]"
output: "2"
explanation: "The element 2 appears 4 times out of 7 elements, which is more than ⌊7/2⌋ = 3 times."
explanation:
intuition: |
Imagine a rowdy crowd where two groups are shouting different slogans. If one group has **more than half** the people, their voice will always dominate — no matter how the other groups combine.
This is the core insight behind the **Boyer-Moore Voting Algorithm**. Think of it as a "battle royale" where each element fights against others:
- When you encounter the same element, it gains strength (count increases)
- When you encounter a different element, they cancel each other out (count decreases)
Since the majority element appears more than `n/2` times, it's guaranteed to have "survivors" at the end. Even if every other element teams up against it, they can't outnumber it — the majority element will always be the last one standing.
The key insight is that if we pair up different elements to "eliminate" each other, the majority element will always have at least one unpaired instance remaining.
approach: |
We solve this using the **Boyer-Moore Voting Algorithm**:
**Step 1: Initialise candidate tracking**
- `candidate`: Will store our current guess for the majority element
- `count`: Set to `0`, tracks the "strength" of our current candidate
&nbsp;
**Step 2: First pass — find the candidate**
- For each element in the array:
- If `count == 0`, adopt the current element as our new `candidate`
- If the current element equals `candidate`, increment `count` (gains strength)
- Otherwise, decrement `count` (different elements cancel out)
&nbsp;
**Step 3: Return the candidate**
- Since the problem guarantees a majority element exists, our candidate is the answer
- No verification pass is needed (but would be required if existence wasn't guaranteed)
&nbsp;
This works because the majority element, appearing more than half the time, cannot be fully cancelled out by all other elements combined.
common_pitfalls:
- title: Using Extra Space with Hash Map
description: |
A common first approach is to count occurrences using a hash map:
- Iterate through the array, counting each element
- Return the element with count > `n/2`
While this works and runs in O(n) time, it uses **O(n) space** for the hash map. The follow-up specifically asks for O(1) space, which the Boyer-Moore algorithm achieves.
wrong_approach: "Hash map counting with O(n) space"
correct_approach: "Boyer-Moore Voting Algorithm with O(1) space"
- title: Sorting and Taking the Middle
description: |
Another approach is to sort the array and return the middle element. Since the majority element appears more than `n/2` times, it must occupy the middle position after sorting.
This works but has **O(n log n)** time complexity due to sorting. The Boyer-Moore algorithm achieves O(n) time.
wrong_approach: "Sorting with O(n log n) time"
correct_approach: "Single pass with O(n) time"
- title: Forgetting to Reset the Candidate
description: |
A critical part of Boyer-Moore is resetting the candidate when `count` reaches zero. If you only decrement without adopting a new candidate, you'll miss the majority element.
When `count == 0`, it means all previously seen elements have cancelled out, so we start fresh with the current element as our new candidate.
wrong_approach: "Only incrementing/decrementing without resetting candidate"
correct_approach: "Reset candidate when count becomes zero"
key_takeaways:
- "**Boyer-Moore Voting Algorithm**: A brilliant technique for finding majority elements in O(n) time and O(1) space"
- "**Cancellation principle**: Different elements cancel each other out; the majority survives because it can't be fully cancelled"
- "**Space-time optimisation**: When a hash map solution exists, ask if there's a pattern-based approach using constant space"
- "**Foundation for variations**: This extends to finding elements appearing more than `n/3` times (Boyer-Moore generalisation)"
time_complexity: "O(n). We traverse the array exactly once, performing constant-time operations at each step."
space_complexity: "O(1). We only use two variables (`candidate` and `count`), regardless of input size."
solutions:
- approach_name: Boyer-Moore Voting Algorithm
is_optimal: true
code: |
def majority_element(nums: list[int]) -> int:
# Current candidate for majority element
candidate = None
# Count tracks the "strength" of our candidate
count = 0
for num in nums:
# If count is zero, adopt current element as new candidate
if count == 0:
candidate = num
# Same as candidate? Gains strength. Different? Cancel out.
if num == candidate:
count += 1
else:
count -= 1
# Candidate is guaranteed to be the majority element
return candidate
explanation: |
**Time Complexity:** O(n) — Single pass through the array.
**Space Complexity:** O(1) — Only two variables used regardless of input size.
The algorithm works by maintaining a candidate and a count. When elements match, we increase confidence. When they differ, they cancel out. Since the majority element appears more than half the time, it will always be the survivor.
- approach_name: Hash Map Counting
is_optimal: false
code: |
from collections import Counter
def majority_element(nums: list[int]) -> int:
# Count occurrences of each element
counts = Counter(nums)
n = len(nums)
# Find the element appearing more than n/2 times
for num, count in counts.items():
if count > n // 2:
return num
# Problem guarantees majority exists, so we'll always return above
return -1
explanation: |
**Time Complexity:** O(n) — Single pass to build the counter.
**Space Complexity:** O(n) — Hash map stores up to n/2 distinct elements in worst case.
This approach is intuitive and easy to implement. It counts all elements and returns the one exceeding the threshold. While correct, it uses more space than necessary.
- approach_name: Sorting
is_optimal: false
code: |
def majority_element(nums: list[int]) -> int:
# Sort the array
nums.sort()
# The majority element must be at the middle index
# Since it appears > n/2 times, it spans the middle
return nums[len(nums) // 2]
explanation: |
**Time Complexity:** O(n log n) — Dominated by the sorting step.
**Space Complexity:** O(1) or O(n) — Depends on sorting algorithm (in-place vs. not).
After sorting, the majority element must occupy the middle position because it appears more than half the time. Simple but slower than optimal.