[MUSIC] So in the last video,
we got all tied up in questions of what an operation is.
How do we count every single last operation
when we execute a piece of code on a given input?
And now, what we'd like to do is just clean away all that clutter,
clean away all that mess, and talk about asymptotic analysis.
So by the end of this video, you'll be able to explain why asymptotic analysis is
so useful, and then start calculating the big O class,
which is that indicator of asymptotics of particular code snippets.
So we've already got some of that motivation based on the pain that we went
through in the previous video.
What we'd like to do is not have to worry about things that we can't control.
So we'd like to let go some of the details, and
think about the big picture of how our algorithm or our problem solving strategy
behaves, in particular, relative to big data and big inputs.
So we're going to think about putting away the details of the initialization time of
the execution.
So if there are certain things that we need to do at the beginning,
some variables we need to declare.
Or some tidying up stuff that we need to do at the beginning of our code.
We don't want to have to worry about that.
We also want to not have to worry about implementing specific operations.
How do those happen?
Especially if they don't have anything to do with how our input size
determines the runtime.
We don't need to think about them at all.
And so what we'd like to do is come up with a principled theoretical way of
analyzing our algorithm.
But something that's a little bit more sophisticated than counting each
instruction and that has this wider purpose of not having to
worry about the nitty picky details of the specific operations and
things that have to do with initialization time or small inputs for our algorithm.
Okay, so our focus is going to be on how the performance scales,
how our algorithm runs as we give it bigger and bigger inputs.
So, that motivating question that we talked about earlier is,
if our input is twice as big, how many more operations do we need in general?
Okay, and so, that's going to be motivating us throughout.
So let's start to think about some specific code snippets.
And these are, you'll recognize them,
taken from that hasLetter method that we were thinking about before.
So let's think about this, just the if branch that we have right over here.
And think about how it behaves, with respect to word.
So our input here is that string word.
And when we're thinking about performance,
we're thinking about it relative to the input size.
So if we have a string, then the size of that string is the length,
the number of characters in that string.
And you'll notice that whenever we talk about asymptotic analysis,
when we talk about inputs, we usually use the letter n
to represent the size of the inputs that we're given.
So as we work through these examples, whenever you see a little n pop up,
that's our representation of the size of the input, okay.
So for this particular piece of code, when the word is our input, and
so its size is n, the number of characters in the string.
What we'd like to think about is how long does it take to execute this if branch.
And so, in our previous model, we counted operations.
But right now what we'd like to do is maybe not have to count all
the operations, but think about how does the number of operations change
as the word gets bigger or smaller, as we have different inputs?
And if we think about what we're doing in this if branch, we're doing
one comparison, comparing a particular character with another character, and
then either doing zero operations, or one operation.
Okay, so this if branch is either going to have one operation associated with it,
just the comparison, or two operations associated with it, the comparison and
the return statement.
And notice that all of that calculation,
what we just did, didn't depend on the size of the word at all.
We had to think about two cases based on the particular character, but
nothing to do with the size of the input.
And so the number of operations is constant if all we're thinking about is
the size of the input.
Okay, so this is an example of a constant time code snippet.
So let's look at another code snippet, and in this code snippet,
what we're doing really is just counting the number of characters in our string.
And so let's think about how the runtime of this code snippet changes
as our input string grows and grows and grows.
And so what we're doing over is here is we're taking the input, which is still
the string word, and notice that here we're still going to call the size n.
And again it's the number of characters in the word.
But now the size is starting to play a role in the execution of the code.
Because what we're doing here is we're stepping through, in this for
loop, through each character in the string, and so
the number of operations is going to grow as the string grows.
And how is it going to grow?
For every new character in the string,
we're going to have to go through one more iteration of the for loop.
Because the length is going to be just one bigger.
So every time we add one character to our input, one additional
character to our string, we're adding one more iteration to the for loop.
And in that for loop, well, we're just doing one instruction.
So, every new character in our string adds one more instruction overall.
And this relationship, well, if we go through that calculation,
we have one instruction for the initial variable, declaration and assignment.
And then, n times through this loop, we're going to check
our loop counter, increment the count, and then add one to our loop counter.
And that's gonna happen n times.
So overall, the run time of this piece of code is going to be one for
that initial piece, and then within that body of the for loop,
we're going to have one for declaring the loop counter i.
And then for each iteration of the for loop,
we're going to have three instructions.
And then we're going to have one additional instruction at the very end,
when we've incremented the i too many times and it's bigger than
the length of the string, and that's gonna kick us out of the for loop.
Okay, so overall the number of instructions that we execute is 3n + 3,
putting all of those pieces together.
Notice that that's a linear function of n.
Okay so thinking back to calculus and definitions of linear functions,
when we have an input being mapped to some constant times that input,
maybe plus another constant, that's a linear function.
Okay, so the run time in terms of the number of operations of this piece of code
is linear, so it's linear time.
So we've seen a code snippet that was constant time,
relative to the size of its input.
And now we've seen another code snippet that's linear time,
relative to the size of its input.
And now what we'd like to do, is phrase this more broadly,
in the sense of big O classes.
And so, when we say that one function is big O of another function, what we
mean by that is that those two functions grow in the same way as their input grows.
Because what we'd like to do in this big O notation is say I don't wanna
care about the constants I don't wanna have to keep track of the plus one here,
and the plus three there.
What I like to think about is the rate of growth of the functions.
And so when we looked at that second function over there,
the one that had the linear time, we don't want to focus on the fact that it was
three n plus three, but rather that notion of it being a linear function of n.
That it grew just a little bit with every time that we increased the input
by a little bit.
Okay, and so this big O notation is going to be our way of capturing
the rate of growth of two functions, and so we say that
two functions are in the same big O class if they have the same rate of growth.
Okay, now disclaimer, in industry and
in practice, this is the way that most people use big O.
The way I just said it.
The big O class means that the two functions grow in the same way.
Now, if you look at the text books, if you Google asymptotics,
you'll notice that there are a lot of different ways of measuring asymptotics.
So, there's big O, which we'll be talking about.
There's also something called big theta, big omega,
little o, all of these different kinds of classes.
And they're going to represent a finer-grained analysis of the asymptotics.
Sometimes we just care about lower bounds or
upper bounds, and that's going to be useful as well.
But what we're going to do in this course and in our analysis,
is focus on the big O, and we're going to use that as shorthand for the tight bound.
Now, that's not exactly how it's used In the formal definition, but
for us big O is gonna be the tightest description
of the asymptotics of the function that we can come up with.
