{x}
blog image

Shortest Distance in a Plane

Solution Explanation for Shortest Distance in a Plane

This problem requires finding the shortest distance between any two points in a given table Point2D. The solution involves a self-join of the table to compare all pairs of points and calculating the Euclidean distance between them.

Approach

  1. Self-Join: We perform a self-join of the Point2D table with itself, aliasing it as p1 and p2. This allows us to compare every point with every other point. The ON clause p1.x != p2.x OR p1.y != p2.y ensures we don't calculate the distance of a point to itself.

  2. Distance Calculation: The Euclidean distance between two points (x1, y1) and (x2, y2) is calculated using the formula: sqrt((x2 - x1)^2 + (y2 - y1)^2). In SQL, we use POW() for exponentiation and SQRT() for the square root.

  3. Rounding: The calculated distance is rounded to two decimal places using ROUND(..., 2).

  4. Ordering and Limiting: The results are ordered by the calculated distance in ascending order (ORDER BY 1), and we select only the first row using LIMIT 1 to get the shortest distance.

MySQL Code

SELECT ROUND(SQRT(POW(p1.x - p2.x, 2) + POW(p1.y - p2.y, 2)), 2) AS shortest
FROM
    Point2D AS p1
    JOIN Point2D AS p2 ON p1.x != p2.x OR p1.y != p2.y
ORDER BY 1
LIMIT 1;

Time Complexity Analysis

The time complexity of this solution is O(n^2), where n is the number of points in the Point2D table. This is because the self-join creates a pairwise comparison of all points, resulting in a quadratic number of distance calculations.

Space Complexity Analysis

The space complexity is dominated by the intermediate result set generated by the self-join. In the worst case, this could be O(n^2), although the final result only requires constant space to store the shortest distance. However, the space used during the calculation is the dominant factor, making the overall space complexity O(n^2).

Note: While other approaches might be conceivable (e.g., using a more sophisticated algorithm to reduce the number of pairwise comparisons), they would likely be more complex to implement in SQL and might not offer a significant performance advantage for reasonably sized datasets. The provided solution is straightforward, efficient within the constraints of SQL, and easy to understand.