Finding the Median of Two Sorted Arrays

You are given two arrays, each sorted in ascending order. Your task is to find the median of the two merged arrays as if they were combined into a single sorted array. This is a common problem in algorithm design and is crucial for understanding efficient merging and searching techniques.

Problem Description

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)).

Key Requirements:

The input arrays nums1 and nums2 are guaranteed to be sorted in ascending order.
You need to calculate the median of the combined elements from both arrays.
The median is the middle element in a sorted list of numbers. If the list has an even number of elements, the median is the average of the two middle elements.
The solution must achieve a time complexity of O(log(m+n)).

Expected Behavior:

If the total number of elements (m + n) is odd, the median is the single middle element.
If the total number of elements (m + n) is even, the median is the average of the two middle elements.

Edge Cases to Consider:

One or both arrays can be empty.
The arrays can contain duplicate elements.
The arrays can have vastly different lengths.

Examples

Example 1:

Input: nums1 = [1, 3], nums2 = [2]
Output: 2.00000
Explanation: merged array = [1, 2, 3] and median is 2.

Example 2:

Input: nums1 = [1, 2], nums2 = [3, 4]
Output: 2.50000
Explanation: merged array = [1, 2, 3, 4] and median is (2 + 3) / 2 = 2.5.

Example 3:

Input: nums1 = [0, 0], nums2 = [0, 0]
Output: 0.00000
Explanation: merged array = [0, 0, 0, 0] and median is (0 + 0) / 2 = 0.0.

Example 4:

Input: nums1 = [], nums2 = [1]
Output: 1.00000
Explanation: merged array = [1] and median is 1.

Example 5:

Input: nums1 = [2], nums2 = []
Output: 2.00000
Explanation: merged array = [2] and median is 2.

Constraints

m == nums1.length
n == nums2.length
0 <= m <= 1000
0 <= n <= 1000
1 <= m + n <= 2000
-10^6 <= nums1[i], nums2[i] <= 10^6
nums1 and nums2 are sorted in non-decreasing order.
The solution must run in O(log(m+n)) time complexity.

Notes

A naive solution would be to merge the two arrays and then find the median. However, this would typically take O(m+n) time. You are expected to find a more efficient solution.
Consider how you can find the k-th element in two sorted arrays efficiently, as the median problem can be reduced to this.
Binary search is likely to be a key component of an efficient solution.
Pay close attention to how you handle the partitioning of the arrays to correctly identify the elements that form the median.