What is the median of two sorted arrays?

Problem Statement

Given two sorted arrays nums1 and nums2 of size m and n respectively, find the median of the two sorted arrays. The overall runtime complexity should be O(log(m+n)).

Examples

• Input: nums1 = [1, 3], nums2 = [2]
  Output: 2.0
  Explanation: The arrays merge into [1, 2, 3] and the middle value is 2.0.

• Input: nums1 = [1, 2], nums2 = [3, 4]
  Output: 2.5
  Explanation: The arrays merge into [1, 2, 3, 4] and the median is the average of the two middle numbers (2 and 3) which is 2.5._

Constraints

• nums1 and nums2 cannot be both empty.
• The total number of elements in both arrays will not exceed 2,000.

Approaches to Finding the Median of Two Sorted Arrays

Direct Merge and Find Method

Method: Merge the two sorted arrays into one and then calculate the median from the combined array.

Explanation: This method involves combining both arrays into a single sorted array from which the median can be directly calculated. It’s a straightforward approach but might not be the most efficient for large arrays due to its reliance on sorting.

Time Complexity: O((m + n) log (m + n)), due to the sorting step involved in merging the arrays.

Space Complexity: O(m + n), as additional space is required to store the merged array.

Binary Search Approach

Method: Use binary search to find the median without fully merging the arrays.

Explanation: This approach leverages binary search techniques to locate the median by considering elements from both arrays without merging them. It's particularly effective because it significantly reduces the time complexity by avoiding the merge step, making it scalable for larger datasets.

Time Complexity: O(log(min(m, n))), since the search is conducted on the shorter of the two arrays.

Space Complexity: O(1), as it does not require additional space beyond the input arrays. n text here.

Application

Finding the median of two sorted arrays is essential for statistical analysis in data science, real-time data monitoring, and other applications where it is crucial to quickly determine the central tendency of combined data sets.

Conclusion

Mastering the techniques to efficiently find the median of two sorted arrays is vital for coding interviews and practical software development, particularly for applications that involve large data sets and require high-performance solutions.