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22. Median of Two Sorted Arrays

hardAsked at ByteDance

Find the median of two sorted arrays in logarithmic time — ByteDance uses it to test binary-search-on-answer reasoning that mirrors quantile estimation in their ranking pipeline.

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

Problem

Given two sorted arrays nums1 and nums2 of size m and n, return the median of the combined sorted array. The expected overall run time is O(log(min(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 then index

Merge the two arrays into one sorted array; return the middle element(s).

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

Tradeoff:

2. Binary partition on smaller array

Pick a cut in the smaller array so left halves combined hold the lower half of all elements. Adjust until max(left) <= min(right).

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;
    const 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:

ByteDance-specific tips

ByteDance interviewers reward the explicit invariant that the left partition holds exactly half the elements — the same precondition their quantile-estimation jobs assert before publishing ranking thresholds.

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Output

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