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22. Median of Two Sorted Arrays

hardAsked at Postman

Find the median of two sorted arrays in logarithmic time.

By Alex Chen, Founder, InterviewChamp.AI · Last verified

Problem

Given two sorted arrays nums1 and nums2 of size m and n respectively, return the median of the two sorted arrays. The overall run time complexity should be O(log(m+n)).

Constraints

  • 0 <= m, n <= 1000
  • 1 <= m + n <= 2000
  • -10^6 <= nums[i] <= 10^6

Examples

Example 1

Input
nums1 = [1,3], nums2 = [2]
Output
2.0

Example 2

Input
nums1 = [1,2], nums2 = [3,4]
Output
2.5

Approaches

1. Merge and pick

Merge both arrays in O(m+n) then pick the middle element(s).

Time
O(m+n)
Space
O(m+n)
const a = [...nums1, ...nums2].sort((x,y)=>x-y);
return a.length % 2 ? a[a.length>>1] : (a[a.length/2-1] + a[a.length/2]) / 2;

Tradeoff:

2. Binary search on partitions

Binary search the smaller array for a partition i such that left-half of A + left-half of B has correct size and maxLeft <= minRight on both sides.

Time
O(log(min(m,n)))
Space
O(1)
function findMedian(A, B) {
  if (A.length > B.length) [A, B] = [B, A];
  const m = A.length, n = B.length, half = (m + n + 1) >> 1;
  let lo = 0, hi = m;
  while (lo <= hi) {
    const i = (lo + hi) >> 1, j = half - i;
    const aL = i === 0 ? -Infinity : A[i-1];
    const aR = i === m ? Infinity : A[i];
    const bL = j === 0 ? -Infinity : B[j-1];
    const bR = j === n ? Infinity : B[j];
    if (aL <= bR && bL <= aR) {
      if ((m+n) % 2) return Math.max(aL, bL);
      return (Math.max(aL, bL) + Math.min(aR, bR)) / 2;
    } else if (aL > bR) hi = i - 1;
    else lo = i + 1;
  }
}

Tradeoff:

Postman-specific tips

Postman likes the partition-search insight because it generalizes to merging sorted response pages from paginated APIs without buffering both pages.

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