Efficient Median Finding from Two Databases: Binary Search Approach

Finding the Median from Two Databases: A Binary Search Approach

This problem involves finding the median value of two separate databases, each containing 'n' unique numerical values, totaling 2n values. The challenge is that we can only access these values through queries, which are expensive. Our goal is to find the median (the nth smallest value) using the fewest queries possible.

Algorithm:

1. Initialization:
   • Define two variables, left and right, representing the smallest and largest possible values, respectively. Initialize left to negative infinity and right to positive infinity.
2. Iterative Search:
   • While left is less than or equal to right: a. Calculate the mid value as the average of left and right. b. Query both databases for the mid-th smallest value. Let value1 be the value from the first database and value2 be the value from the second. c. Comparison: If value1 is less than value2, update left to mid + 1. This indicates that the median lies within the values larger than mid. d. Otherwise, update right to mid - 1, implying the median lies within values smaller than or equal to mid.
3. Return: After the loop, return left as the median value.

Pseudo-code:

def findMedian(database1, database2, n):
    left = float('-inf')
    right = float('inf')
    
    while left <= right:
        mid = (left + right) / 2
        
        value1 = query(database1, mid)
        value2 = query(database2, mid)
        
        if value1 < value2:
            left = mid + 1
        else:
            right = mid - 1
    
    return left

Correctness:

The algorithm utilizes a binary search approach, effectively narrowing down the search space for the median. It starts by considering the entire range of possible values. In each iteration, the algorithm intelligently adjusts the search range based on the comparison of the mid-th smallest values from both databases. By repeatedly halving the search space, the algorithm converges towards the nth smallest value, which is the median.

Complexity Analysis:

The time complexity of the binary search algorithm is O(log n) due to the halving of the search range in each iteration. Since two queries are performed in every iteration, the total number of queries required is 2 * log n. Therefore, the overall time complexity of the algorithm remains O(log n), signifying efficient query usage despite the large dataset size.