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24. Median of Two Sorted Arrays

hardAsked at Klarna

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, return the median of the two sorted arrays. The overall run time complexity must be O(log(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 both arrays into one sorted array, then index into the middle.

Time
O(m+n)
Space
O(m+n)
function findMedianSortedArrays(a, b) {
  const m = [];
  let i = 0, j = 0;
  while (i < a.length && j < b.length) m.push(a[i] <= b[j] ? a[i++] : b[j++]);
  while (i < a.length) m.push(a[i++]);
  while (j < b.length) m.push(b[j++]);
  const k = m.length;
  return k % 2 ? m[(k-1)/2] : (m[k/2 - 1] + m[k/2]) / 2;
}

Tradeoff:

2. Binary search partition

Binary search the partition index on the shorter array so that every element on the left side of both arrays is <= every element on the right. The median falls right at that partition.

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 = Math.floor((m + n + 1) / 2);
  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:

Klarna-specific tips

Klarna's BNPL credit-scoring team uses this O(log n) partition trick when finding percentile thresholds across two pre-sorted risk-feature streams; they grade the +Infinity/-Infinity sentinel handling more than the macro idea.

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