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23. Median of Two Sorted Arrays

hardAsked at Squarespace

Compute the median of two sorted arrays without merging them; Squarespace uses it to test binary-search-on-partition fluency.

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

Problem

Given two sorted arrays nums1 and nums2 of sizes m and n, return the median of the two sorted arrays combined. The overall runtime should be O(log(min(m, n))).

Constraints

  • 0 <= m, n <= 1000
  • 1 <= m + n <= 2000
  • -10^6 <= nums1[i], nums2[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 then index

Merge into one sorted array and pick the middle element(s). Simple but O(m + n).

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

Tradeoff:

2. Binary search the partition

Pick a partition of the shorter array so left halves sum to half the total. Verify left max <= right min on both sides; binary search until valid.

Time
O(log min(m,n))
Space
O(1)
function findMedianSortedArrays(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:

Squarespace-specific tips

Squarespace likes when you sketch the partition diagram first and tie the log-time win to their analytics-rollup p50 percentile probe over two pre-sorted time buckets.

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Output

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