Solution Explanation for Shortest Distance in a Line

The problem asks to find the shortest distance between any two points in a table named Point which contains a single column x representing the x-coordinate of a point on a line.

Approach 1: Self-Join

This approach uses a self-join to compare every point with every other point. It leverages the fact that the shortest distance will always exist between two adjacent points when the points are sorted.

SQL (MySQL):

SELECT MIN(p2.x - p1.x) AS shortest
FROM
    Point AS p1
    JOIN Point AS p2 ON p1.x < p2.x;

Explanation:

1. SELECT MIN(p2.x - p1.x) AS shortest: This selects the minimum difference between two x-coordinates and aliases it as shortest. The difference p2.x - p1.x represents the distance between two points. MIN() function finds the smallest distance among all pairs.

2. FROM Point AS p1 JOIN Point AS p2 ON p1.x < p2.x: This performs a self-join of the Point table with itself.

   • Point AS p1: The table is aliased as p1.
   • Point AS p2: The table is aliased as p2.
   • ON p1.x < p2.x: This join condition ensures that we only compare pairs where p1.x is less than p2.x. This avoids redundant comparisons and comparisons of a point with itself.

Time Complexity: O(n^2), where n is the number of points. This is because a self-join creates a comparison of every point with every other point resulting in a quadratic number of comparisons.

Space Complexity: O(1) – Constant space is used, regardless of the input size.

Approach 2: Window Function (LAG())

This approach is more efficient if the Point table is already sorted in ascending order by the x-coordinate. It uses the LAG() window function to access the previous row's value.

SQL (MySQL):

SELECT MIN(x - LAG(x) OVER (ORDER BY x)) AS shortest
FROM Point;

Explanation:

1. LAG(x) OVER (ORDER BY x): The LAG() function retrieves the value of x from the previous row based on the ordering of x. OVER (ORDER BY x) specifies that the ordering should be done according to the x values.

2. x - LAG(x) OVER (ORDER BY x): This subtracts the previous row's x value from the current row's x value which computes the distance between consecutive points.

3. MIN(...) AS shortest: This finds the minimum distance among all consecutive distances.

Time Complexity: O(n), where n is the number of points. This is because the window function processes each row only once.

Space Complexity: O(1) – Constant space is used, regardless of the input size.

Note: The original solution uses LIMIT 1, 1 which is not necessary for finding the minimum distance. The MIN() function already handles this. The revised solution is more concise and efficient. The LIMIT 1,1 clause is only necessary if you are guaranteed that the shortest distance is between adjacent points. The MIN() function ensures the shortest distance is found in all cases.