# How do you visualize difference between O(log n) and O(n log n)?

### Tags:

#### big-o

Binary search has a average case performance as `O(log n)` and Quick Sort with `O(n log n)` is `O(n log n)` is same as O(n) + O(log n)

### 4 Answers

Imagine a database with with every person in the world. That's 6.7 billion entries. O(log n) is a lookup on an indexed column (e.g. primary key). O(n log n) is returning the entire population in sorted order on an unindexed column.

• O(log n) was finished before you finished reading the first word of that sentence.
• O(n log n) is still calculating...

Another way to imagine it:

`log n` is proportional to the number of digits in n.

`n log n` is n times greater.

Try writing the number `1000` once versus writing it one thousand times. The first takes O(log n) time, the second takes O(n log n) time.

Now try that again with `6700000000`. Writing it once is still trivial. Now try writing it 6.7 billion times. Even if you could write it once per second you'd be dead before you finished.

138
answered Nov 16 '22 16:11

#### Mark Byers

You could visualize it in a plot, see here for example:

20
answered Nov 16 '22 17:11

#### cjg

No, `O(n log n)` = `O(n) * O(log n)`

In mathematics, when you have an expression (i.e. e=mc^2), if there is no operator, then you multiply.

Normally the way to visualize O(n log n) is "do something which takes `log n` computations `n` times."

If you had an algorithm which first iterated over a list, then did a binary search of that list (which would be `N + log N`) you can express that simply as `O(n)` because the `n` dwarfs the `log n` for large values of `n`

27
answered Nov 16 '22 16:11

#### tster

A `(log n)` plot increases, but is concave downward, which means:

• It increases when n gets larger
• It's rate of increasing decreases when n gets larger

A `(n log n)` plot increases, and is (slightly) concave upward, which means:

• It increases when n gets larger
• It's rate of increasing (slightly) increases when n gets larger
36
answered Nov 16 '22 17:11

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