# HARD 0004. Median of Two Sorted Arrays¶

## Problem Statement¶

Leetcode Link

Given two sorted arrays nums1 and nums2 of size m and n respectively, return the median of the two sorted arrays.

### Constraints¶

nums1.length == m
nums2.length == n
0 <= m <= 1000
0 <= n <= 1000
1 <= m + n <= 2000
-106 <= nums1[i], nums2[i] <= 106


Follow up: The overall run time complexity should be $$O(log (m+n))$$.

Tip

Remember that the median of a list is that item that divides the list into two equal halves, by population.

Tip

This problem uses principles I’ve come across when learning the Merge Sort algorithm. It should help to re-read that section before trying to solve this problem.

## Solution¶

This is a good explanation targeting this problem: Binary Search - Median of Two Sorted Arrays.

1. Partition the two arrays into two other arrays which are of equal halves.

2. Ensure that every element on the left side is less than or equal to every element on the right side.

Solution
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 """Solution for Leetcode #0004 - Median of Two Sorted Arrays https://leetcode.com/problems/median-of-two-sorted-arrays/ """ import statistics def median_two_sorted_arrays(nums1: list[int], nums2: list[int]) -> int: """Return a median of two sorted arrays in O(log(m+n)) """ # Note: Please read the explanation / watch the linked video # *before* trying to understand this. if 0 < len(nums1) < len(nums2): smaller_array = nums1 larger_array = nums2 elif 0 < len(nums2) < len(nums1): smaller_array = nums2 larger_array = nums1 elif 0 < len(nums1) == len(nums2): smaller_array = nums1 larger_array = nums2 elif len(nums2) == 0 and len(nums1) > 0: return statistics.median(nums1) elif len(nums1) == 0 and len(nums2) > 0: return statistics.median(nums2) else: return None # perform binary search on the smaller array. start = 0 end = len(smaller_array) - 1 found_partition = False counter = 0 while not found_partition: counter += 1 partition_smaller_array = (end + start) // 2 partition_larger_array = ( len(smaller_array) + len(larger_array) + 1 ) // 2 - partition_smaller_array left_smaller = smaller_array[:partition_smaller_array] left_larger = larger_array[:partition_larger_array] right_smaller = smaller_array[partition_smaller_array:] right_larger = larger_array[partition_larger_array:] max_left_smaller = left_smaller[-1] if len( left_smaller) > 0 else -float('inf') min_right_larger = right_larger if len( right_larger) > 0 else float('inf') max_left_larger = left_larger[-1] if len( left_larger) > 0 else -float("inf") min_right_smaller = right_smaller if len( right_smaller) > 0 else float("inf") if max_left_smaller <= min_right_larger and max_left_larger <= min_right_smaller: found_partition = True elif max_left_smaller > min_right_larger: end -= 1 else: start += 1 # now the median will be in the last four digits: total_length = len(smaller_array) + len(larger_array) if total_length % 2 == 0: # this is even median = (max(max_left_smaller, max_left_larger ) + min(min_right_larger, min_right_smaller))/2 else: median = max(max_left_smaller, max_left_larger) return median def test_base(): import random import statistics list_1 = [random.randint(0, 100) for _ in range(random.randint(10**5, 10**6))] list_2 = [random.randint(0, 100) for _ in range(random.randint(10**5, 10**6))] list_1.sort() list_2.sort() calculated_median = median_two_sorted_arrays(list_1, list_2) merged_list = sorted(list_1 + list_2) median = statistics.median(merged_list) assert calculated_median == median, f"Returned value {calculated_median=} is wrong! {median=}" def test_edgecase(): list_1 = [1, 3] list_2 =  calculated_median = median_two_sorted_arrays(list_1, list_2) median = 2 assert calculated_median == median, f"Returned value {calculated_median=} is wrong! {median=}" def test_equal_case(): list_1 = list_2 = [1, 2, 3, 4] calculated_median = median_two_sorted_arrays(list_1, list_2) median = statistics.median(list_1) assert calculated_median == median, f"Returned value {calculated_median=} is wrong! {median=}"