Performance difference: select top 1 order by vs. select min(val)

Question

Question is simple. Which query will be faster:

SELECT TOP 1 value FROM table ORDER BY value

or

SELECT TOP 1  MIN(value) FROM table

We can assume that we have two cases, Case 1. No index and Case 2. With index on value.
Any insights are appreciated. Thanks!

Answer 1

In the case where no index exists:

• MIN(value) should be implemented in O(N) time with a single scan;
• TOP 1 ... ORDER BY will require O(N Log N) time because of the specified sort (unless the DB engine is smart enough to read intent, which I would hate to rely on in production code).

When an index does exist:

• Both should require only O(1) time, using the index.