Finding the Median of Two Databases Using Efficient Queries

Finding the Median of Two Databases Using Efficient Queries

Imagine you're tasked with analyzing hard-to-obtain data spread across two separate databases. Each database holds 'n' unique numerical values, totaling 2n values. Your goal is to find the median of these 2n values, defined as the nth smallest value. However, accessing the data comes with a constraint: you can only query each database individually, asking for the kth smallest value it contains. Since queries are expensive, you want to minimize their use. This article presents an algorithm that achieves this goal using a clever adaptation of the binary search strategy.

Algorithm Overview

The core idea is to progressively narrow down the range where the median could lie. By repeatedly querying for the middle value within a shrinking search space, we efficiently converge on the true median. Let's break down the steps:

1. Initialization: We start by defining the search range, initially from 1 to 'n'. This represents all possible positions for the median.

2. Iterative Search: As long as our search range hasn't converged to a single point, we repeat the following: * Calculate Midpoint: Find the middle value of the current search range. * Query Databases: Query both databases for the value at this midpoint. * Compare and Update: Compare the results from the two databases. If the first database's value is smaller or equal to the second's, the median must lie within the upper half of our search range. Conversely, if the first database's value is larger, the median must lie within the lower half. We update the search range accordingly.

3. Median Found: Once the search range shrinks to a single point, this point represents the position of the median in the combined dataset.

Pseudo-code

## Correctness Proof

The algorithm's correctness stems from its ability to effectively eliminate half of the search space in each iteration. By comparing the middle values from each database, we gain crucial information about the relative position of the median. If the first database's middle value is smaller than or equal to the second's, it implies that at least 'mid' values from the first database are smaller than or equal to the median. Consequently, the median cannot lie within the lower half of the search range. A symmetrical argument applies if the first database's middle value is larger. This iterative elimination guarantees that we eventually converge on the true median.


## Complexity Analysis

The algorithm utilizes a binary search approach, leading to a logarithmic time complexity. In each step, we halve the search range. Starting with a range of 'n', we need at most log₂(n) iterations to reduce it to a single point. As each iteration involves two database queries, the total number of queries is 2 * log₂(n). This falls under the O(logn) complexity class, fulfilling the requirement for query efficiency.


## Conclusion

This algorithm provides an efficient solution for finding the median of two databases with limited access. Its logarithmic query complexity makes it particularly useful when dealing with large datasets and expensive queries. By leveraging the power of binary search, we achieve both accuracy and efficiency in determining the median value.