4. median of two sorted arrays

Guri Bee

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17-10-2024

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Finding the Median of Two Sorted Arrays: A Comprehensive Guide

Finding the median of two sorted arrays is a classic problem in computer science, often encountered in coding interviews. It challenges us to efficiently determine the middle value when merging two ordered lists. This guide will break down the problem, explore different solutions, and provide practical insights for optimal implementation.

Understanding the Problem

Given two sorted arrays, nums1 and nums2, we need to find the median of the combined array, as if they were merged into a single sorted list.

Example:

nums1 = [1, 3]
nums2 = [2]

Median of the combined array = 2 

Why is this important?

This problem tests your understanding of algorithms, particularly in the areas of:

• Binary Search: Efficiently finding elements in sorted data.
• Divide and Conquer: Breaking down a complex problem into smaller, easier-to-solve subproblems.
• Edge Cases: Handling scenarios like empty arrays or arrays of different lengths.

Solutions

There are several approaches to finding the median of two sorted arrays. We'll explore two prominent methods:

1. Merging and Sorting:

This straightforward approach involves merging the two arrays into a single sorted list. Then, we can easily find the median by calculating the middle element (or the average of the two middle elements if the combined length is even).

Code Example (Python):

def findMedianSortedArrays(nums1, nums2):
  merged = sorted(nums1 + nums2)
  n = len(merged)
  if n % 2 == 0:
    return (merged[n//2 - 1] + merged[n//2]) / 2
  else:
    return merged[n//2]

# Example usage
nums1 = [1, 3]
nums2 = [2]
median = findMedianSortedArrays(nums1, nums2)
print(f"Median: {median}") # Output: Median: 2

Analysis:

• Time Complexity: O(m + n), where m and n are the lengths of the two arrays. This is due to the merging and sorting operations.
• Space Complexity: O(m + n) for storing the merged array.

2. Binary Search Approach:

A more efficient solution involves using binary search to find the partition point in the two arrays that would result in the median. The key idea is to maintain a balanced partition, meaning both sides have an equal number of elements.

Code Example (Python):

def findMedianSortedArrays(nums1, nums2):
  n1 = len(nums1)
  n2 = len(nums2)

  if n1 > n2:
    nums1, nums2, n1, n2 = nums2, nums1, n2, n1  # Ensure nums1 is the shorter array

  low = 0
  high = n1

  while low <= high:
    partitionX = (low + high) // 2
    partitionY = ((n1 + n2 + 1) // 2) - partitionX

    maxLeftX = nums1[partitionX - 1] if partitionX > 0 else float('-inf')
    minRightX = nums1[partitionX] if partitionX < n1 else float('inf')

    maxLeftY = nums2[partitionY - 1] if partitionY > 0 else float('-inf')
    minRightY = nums2[partitionY] if partitionY < n2 else float('inf')

    if maxLeftX <= minRightY and maxLeftY <= minRightX:
      if (n1 + n2) % 2 == 0:
        return (max(maxLeftX, maxLeftY) + min(minRightX, minRightY)) / 2
      else:
        return max(maxLeftX, maxLeftY)

    elif maxLeftX > minRightY:
      high = partitionX - 1
    else:
      low = partitionX + 1

# Example usage
nums1 = [1, 3]
nums2 = [2]
median = findMedianSortedArrays(nums1, nums2)
print(f"Median: {median}") # Output: Median: 2

Analysis:

• Time Complexity: O(log(min(m, n))), as we are repeatedly dividing the search space in half.
• Space Complexity: O(1), as we are only using a few variables to store the partition information.

Important Notes:

• The code examples provided are in Python, but the algorithms can be adapted to other programming languages.
• The binary search approach offers a significant performance improvement over the merging and sorting method, especially for large arrays.
• The provided solutions handle edge cases like empty arrays and arrays of different lengths.

Additional Resources:

For a more in-depth understanding of the algorithm, consider exploring the following resources:

• LeetCode - Median of Two Sorted Arrays - This website offers practice problems and solutions, including detailed explanations and code examples.
• GeeksforGeeks - Median of Two Sorted Arrays - This website provides comprehensive articles and tutorials on various data structures and algorithms, including the median of two sorted arrays problem.

Conclusion

Finding the median of two sorted arrays presents a fascinating algorithmic challenge. Understanding different approaches, their time and space complexities, and handling edge cases is crucial for mastering this problem. By leveraging techniques like binary search and divide and conquer, we can achieve efficient and optimal solutions.