Why does adding an index to what you sort reduce the amount of sorting work?

Question

From here: http://dev.mysql.com/doc/refman/5.0/en/order-by-optimization.html

In some cases, MySQL can use an index to satisfy an ORDER BY clause without doing any extra sorting.

I thought that indexes helped extract certain pieces of data (e.g. indexes in an array), giving you O (1) instead of O (n) when indexing. But when sorting, I assumed that they use any O (nlogn) algorithm or something else based on the sort column, however, apparently, indexing the columns you are sorting can reduce the amount of work.

How it works? (I'm not sure if this is a common SQL thing or a MySQL thing).

Answer 1

: , ( !) , " " ( ORDER BY ), , " ".

: @Will , , .

Answer 2

, , O (log n) .

, O (n). , , , .

SQL, MySQL.