Efficient Algorithm for Finding the Median from Two Databases

Finding the Median from Two Databases with Minimal Queries

This article presents an efficient algorithm for finding the median of 2n values distributed across two separate databases. The algorithm utilizes a binary search approach to minimize the number of expensive database queries.

Problem Statement

We have two databases, each containing 'n' unique numerical values. Our goal is to determine the median of the combined 2n values, which is defined as the nth smallest value. The constraint is that we can only interact with the databases through queries. A single query involves specifying a value 'k' to a chosen database, which returns the kth smallest value within that database. The objective is to find the median using the fewest queries possible.

Algorithm

1. Initialization: Determine the smallest (L) and largest (U) values across both databases. These values serve as initial lower and upper bounds for our search.

2. Iterative Search: Repeat the following steps while L is less than U: a. Calculate Midpoint: Find the middle value M = (L + U) / 2. b. Query Databases: Send a query to each database for the count of values less than or equal to M. c. Adjust Bounds: * If the total count from both databases is less than n, the median must be larger than M. Update the lower bound: L = M + 1. * If the total count exceeds n, the median must be smaller than M. Update the upper bound: U = M. * If the total count equals n, M is the median. Return M.

3. Final Result: If the loop completes (L is no longer less than U), L represents the median. Return L.

Pseudo-codepythonfunction findMedian(n, database1, database2): L = minimum value in database1 and database2 U = maximum value in database1 and database2 while L < U: M = (L + U) / 2 count1 = number of values <= M in database1 count2 = number of values <= M in database2 if count1 + count2 < n: L = M + 1 else if count1 + count2 > n: U = M else: return M return L

Correctness

The algorithm employs a binary search strategy. It initializes a search range defined by lower and upper bounds (L and U). In each iteration, it halves this range by querying for the number of values less than or equal to the midpoint (M).

The logic of adjusting the bounds based on the query results ensures that the search space progressively narrows down, converging on the true median. When the sum of counts equals n, we have found the nth smallest value (the median).

Complexity Analysis

• Time Complexity: The binary search nature of the algorithm yields a time complexity of O(log n), where 'n' is the number of values per database. Each iteration involves two queries, each with a time complexity of O(log n) (assuming efficient database indexing). Thus, the overall time complexity remains O(log n).

• Space Complexity: The algorithm uses a constant amount of extra space to store variables like L, U, M, and the query counts. Therefore, the space complexity is O(1).

Conclusion

This algorithm provides an efficient way to find the median of values spread across two databases by minimizing the number of expensive database queries. Its logarithmic time complexity makes it scalable for large datasets.