Finding the Median from Two Databases with Logarithmic Queries

Finding the Median from Two Databases Efficiently

This document describes an algorithm to find the median of 2n values, where the values are distributed across two databases. The algorithm leverages a binary search approach to minimize the number of database queries, resulting in a time complexity of O(log n).

Problem Statement

You are tasked with finding the median of 2n unique numerical values stored across two databases. Each database contains n values. The only way to access these values is by querying each database individually. A single query involves specifying a value k to a database, and the database returns the kth smallest value it contains. Due to the cost of querying, minimizing the number of queries is crucial.

Algorithm

The algorithm employs a binary search strategy to efficiently determine the median.

1. Initialization: Set the minimum value (min_val) to the smallest possible value and the maximum value (max_val) to the largest possible value. This defines the initial search range.
2. Binary Search: While min_val is less than or equal to max_val: a. Calculate the middle value (mid_val) as the average of min_val and max_val. b. Query the first database with mid_val and obtain the kth smallest value (k1). c. Query the second database with mid_val and obtain the kth smallest value (k2). d. Median Found: If k1 is equal to k2, then mid_val is the median, so return k1. e. Adjust Search Range: If k1 is less than k2, update min_val to mid_val + 1. Otherwise, update max_val to mid_val - 1. This narrows the search range.
3. Return Median: When the loop terminates, min_val will represent the median value. Return min_val.

Pseudo-code

function findMedian(n):
    min_val = -infinity
    max_val = +infinity
    while min_val <= max_val:
        mid_val = (min_val + max_val) / 2
        k1 = query(database1, mid_val)
        k2 = query(database2, mid_val)
        if k1 == k2:
            return k1
        elif k1 < k2:
            min_val = mid_val + 1
        else:
            max_val = mid_val - 1
    return min_val

Correctness

The algorithm's correctness stems from its binary search strategy. Each iteration halves the search space, guaranteeing convergence to the median value within a logarithmic number of steps.

Complexity

• Time Complexity: The binary search performs O(log n) iterations. In each iteration, two database queries are executed. Therefore, the total number of queries and the overall time complexity is O(log n).
• Space Complexity: The algorithm utilizes a constant amount of extra space to store variables like min_val, max_val, and mid_val. Thus, the space complexity is O(1).

This algorithm provides an efficient and optimized solution for determining the median from two databases, significantly reducing the number of expensive database queries.