O(log N) == O(1) - Why not?

Question

Whenever I consider algorithms/data structures I tend to replace the log(N) parts by constants. Oh, I know log(N) diverges - but does it matter in real world applications?

Answer 1

Yes, log(N) < 100 for most practical purposes, and No, you can not always replace it by constant.

For example, this may lead to serious errors in estimating performance of your program. If O(N) program processed array of 1000 elements in 1 ms, then you are sure it will process 10⁶ elements in 1 second (or so). If, though, the program is O(N*logN), then it will take it ~2 secs to process 10⁶ elements. This difference may be crucial - for example, you may think you've got enough server power because you get 3000 requests per hour and you think your server can handle up to 3600.

Another example. Imagine you have function f() working in O(logN), and on each iteration calling function g(), which works in O(logN) as well. Then, if you replace both logs by constants, you think that your program works in constant time. Reality will be cruel though - two logs may give you up to 100*100 multiplicator.