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ALBERT R. MEYER: Let's take
a quick look at some blunders

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that people regularly
make in dealing

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with asymptotic notation,
in particular with big O

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notation, which tends
to confuse people.

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So the most important
thing to remember

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is that this notation,
something equals O of something

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else-- 1/x equals O of
1, say-- is actually

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to be understood as just a
not such attractive notation,

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misleading notation for a binary
relation between two functions.

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This is supposed to
be a function there,

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and this is supposed
to be a function there.

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And this is saying that there's
a relation between the growth

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rates of these two functions.

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O of f is not quantity.

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And you mustn't
treat it as such.

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So, for example-- and
the equality, of course,

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is not an inequality.

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Once upon a time, we tried
to get the equality replaced

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by an epsilon, which
makes much better sense--

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that is, a membership symbol.

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But there was a sense that this
notation was so deeply embedded

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in the mathematical culture--
multiple mathematical

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communities-- that there was no
way we were going to change it.

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In particular, a confusion where
you think that that equality

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sign means some
kind of an equality

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is to write instead
of f equals O of g,

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well, if f equals O of
g by symmetry, O of g

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ought to equal f.

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Don't write that.

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The reason is that it's
a recipe for confusion,

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because look at this.

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I know that x is
O of x trivially,

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which would suggest
that O of x is

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equal to x, if you
believe in symmetry

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and you think of O of
x as being quantity.

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Well, remember,
though, that 2x is also

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equal to O of x by
definition of O.

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So what we wind up with
is combining 2x equals

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O of x with O of x equals x is
I get 2x is equal to this thing,

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is equal to x.

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I conclude that 2x is equal
to x, which is absurd.

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So that's nonsense.

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It's the kind of
trouble that you

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can get into if you start
thinking of this equality

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as meaning equality
between two quantities,

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as opposed to just being a
part of the name of a relation.

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Another mistake that
people make is less serious

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but it's sloppy, is
to think that big O

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corresponds to a lower bound,
so that people will say things

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like f is at least
O of n squared.

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Well, again, at
least O of n squared

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is starting to treat O of
n squared like a quantity.

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You could say that f is
equal to O of n square,

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but that means that n
squared is an upper bound

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on f to within a constant
factor after a certain point.

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If you want to say intuitively
that n squared is a lower

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bound on f, then
all you have to do

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is say that n squared is O of f.

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And that is a proper use of
O of f of getting a lower

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bound on a function, by
saying that the lower bound is

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O of the function.

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Another example of
the kind of nonsense

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that you see-- this is
a stretch, but let's

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look at it as a reminder
of things not to do.

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I'm going to prove to you that
the sum from i equals 1 to n

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of i-- that is that 1 plus 2
plus 3 up to n-- is O of n.

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Now, of course, it's not.

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We know that the
sum of the first n

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integers n times
n plus 1 over 2,

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which is O of n squared--
theta of n squared actually.

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So I'm going to prove
something false.

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Watch carefully how I do it.

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Here's the false proof.

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Let's, first of all, notice
that any constant is O of 1.

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So 0 is O of 1, 1 is O of
1, 2 is O of 1, and so on.

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Any constant function is O
of the constant function 1.

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OK, that's true.

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So that means that each i
in this sum, i is a number,

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so that means it might be 1,
it might be 2, it might be 3,

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it might be 50.

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Whatever it is, it's O of 1.

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And that means that I could
think of this sum from i

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equals 1 to n as O of 1
plus O of 1 plus O of 1.

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And that's, of course, n
times O of 1, which is O of n.

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Now, there's all kinds
of weird arithmetic

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rules of things being used here,
none of which are justified.

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But it's just a heads up.

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You do see stuff like this
from inexperienced students.

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And I hope that you
won't fall into this kind

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of a sloppy trap.

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O of something is
not a quantity.

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It's part of the
name of a relation.