Majority Element

An element holding more than half the total votes can survive being paired off and canceled against every other competing element combined. Discarding any two differing elements leaves the true majority element invariant, which means one candidate and one tally variable are sufficient to isolate the winner without saving past values.

Full solution

Boyer-Moore voting: pairing off differing elements cannot eliminate a value that holds more than half the votes, so one candidate and one counter find it in O(n) time and O(1) space. No verification pass is needed because the statement guarantees the majority element exists.

def majority_element(nums: list[int]) -> int:
    candidate = 0
    count = 0
    for num in nums:
        if count == 0:
            candidate = num  # every prior vote has been cancelled; restart here
        count += 1 if num == candidate else -1
    return candidate

function majorityElement(nums) {
  let candidate = 0;
  let count = 0;
  for (const num of nums) {
    if (count === 0) candidate = num; // every prior vote has been cancelled; restart here
    count += num === candidate ? 1 : -1;
  }
  return candidate;
}

The theory behind it

Arrays — the ground this problem stands on. All Arrays problems

What Arrays is

An array is a row of fixed boxes laid side by side in computer memory, like numbered lockers in a hallway. Because each box occupies identical space and sits directly next to its neighbors, jumping to locker zero or locker ten thousand takes the exact same tiny fraction of time. Every box holds an item of the same type, addressed by an offset number called an index.

When to reach for it

Reach for an array when items arrive in a known sequence and need immediate retrieval by position number. Problems asking for running totals, prefix accumulations, cyclic rotations, or in-place rearrangements signal array mechanics. Whenever constraints require constant-time random lookups or contiguous cache scans across fixed collections, a flat sequence is the default container.

How the pattern works

Visualize a tape with zero-indexed slots stretching from start to end. Keep track of write and read cursors when modifying contents without allocating helper buffers. For running computations, maintain an invariant such as having processed all elements left of the current index while pending elements wait to the right. When modifying entries in place, consider scanning backwards from the end so unread data is not overwritten.