What makes the middle element in a sorted array the majority element?

While working on this particular problem on Leetcode, I came across a solution that left me puzzled.

The solution suggested to "sort the array and the middle element will be the majority."

My question is:

"Why is the middle element of a sorted array considered the majority element?"

Can someone please provide an explanation for this?

This particular problem requires:

Given an array of size n, find the majority element. The majority element is the element that appears more than ⌊ n/2 ⌋ times.

You may assume that the array is non-empty and that the majority element always exists in the array.

Below is the code snippet for the solution provided:

var majorityElement = function(nums) {
    // sort the array and the middle is the majority
    nums.sort((a,b) => a - b);
    return nums[Math.floor(nums.length/2)];
}; 
console.log(majorityElement([3,2,3]))
console.log(majorityElement([2,2,1,1,1,2,2]))

Answer №1

Supposing that the provided array contains a dominant element, it will appear at least (n / 2) + 1 times within the array. If the predominant element is the smallest value in the array, then the arranged array will resemble:

MMMMXX
  ^^ mid (even)

or

MMMMMXXX
    ^ mid (odd)

where M denotes the majority element, and X represents any other element. It can be observed that M consistently occupies the middle of the array. In case the majority element is the largest value in the array, then it would appear as follows:

XXMMMM
  ^^ mid (even)

or

XXXMMMM
    ^ mid (odd)

The M still resides centrally.

If M does not rank as the highest or lowest element in the array, the sorted array will continue to position M in the middle, despite attempts to alter the range:

XXMMMM
XMMMMX
MMMMXX
  ^^

or

XXXMMMM
XXMMMMX
XMMMMXX
MMMMXXX
   ^

These instances are specifically tailored for arrays of size 6 and 7, but the concept extends likewise to arrays of all dimensions.

Answer №2

Here is my interpretation.

Let's assume that the middle element is not the majority element. This means the majority element is located either to the right or left of the middle element. There are ⌊ n/2 ⌋ elements to the left and n - ⌊ n/2 ⌋ elements to the right.

Considering the majority element appears more than ⌊ n/2 ⌋ times, it cannot solely be on one side of the middle element since both sides have <= ⌊ n/2 ⌋ elements. Therefore, if a majority element exists, it must involve the middle element as well.

Answer №3

One explanation is that when an array contains an element that appears more than half of the time, after the array is sorted, that element will always be located at the midpoint. This is why the value at index n/2 is being returned.

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