Big O Notation

Hi, I’m Gayle Laakmann McDowell, author of Cracking the Coding Interview. Today we’re going to cover one of my favorite topics,
which is big O, or algorithmic efficiency. I’m going to start off with a
story. Back in 2009 there was this company in South Africa and they had
really slow internet they were really really frustrated by, and they actually had
two offices about 50 miles away, so they set up this little test to see if it
would be faster to transfer big chunks of data over the internet with their
really slow internet service provider or via carrier pigeon. Literally they took a bunch of USB
sticks or probably actually one giant one strapped it to a pigeon and flew it
from one office to the next, and they got a ton of press over this test and media
kind of love this stuff right. And of course the pigeon won right. It wouldn’t
be a funny story otherwise. And so they got a whole bunch of press over,
and people are like oh my god I can’t believe that a pigeon beat the internet.
But the problem was it was really actually kind of a ridiculous test because
here’s the thing. Transferring data over the Internet
takes longer and longer and longer with how much data there is. With certain
simplifications and assumptions, that’s not really the case with pigeons. Pigeons
might be really slow or fast but it always basically takes the same amount
of time for a pigeon to transfer one, transfer some chunk of data from one
office to the next. It just has to fly 50 miles. It’s not going to take longer
because that USB stick contains more data, so of course for a sufficiently large
file the pigeon’s going to win. So in big O, the way we talk about this is we
describe the pigeon transfer speed as O of 1, its constant time with
respect to the size of the input. It doesn’t take longer with more input.
Again with certain simplifications and assumptions about the pigeons ability to
carry a USB stick. But the internet time is internet transfer time is O of n. It
scales linearly with respect to the amount of input. Twice amount of data is
going to take roughly twice as much time. So that’s what Big O is. Big O is
essentially an equation that describes how the runtime scales with respect to some input
variables. So I’m going to talk about four specific roles, but first let me show you
what this might look like in code. So let’s consider this simple function that
just walks through an array and checks if it contains a particular value. So you
would describe this is O of n, where n is the size of the array. What about this method that walks
through an array and prints all pairs of values in the array? Note that this will
print if it has the elements 5 and 2 in the array it’ll print both 2 comma 5 and
5 comma 2. So you describe this probably as of O of n squared and where n is the size of the array. You can see that because it has n
elements in the array, so it has n squared pairs, so the amount of time
it’s going to take you to run that function is going to increase with
respect to n squared. Ok so that’s kind of basics of Big O. Let’s take another real-world example. How would you describe the run time to
mow a square plot of land? So a square plot of grass. So you have this plot of
grass and you need to mow all of it. What’s the runtime of this? Well it’s
kind of an interesting question, but you could describe it one of two ways. One of
which, one way you can describe it is O of a where a is the amount of area in
that plot of land. Remember that this is just an equation that expresses how the
time increases so you don’t have to use n in your equation, you can use other
variables and often it make sense to do that. So you just described this as O of
a where a is that the amount of area there. You could also describe this as
O of s squared where s is the amount of, is this length of one side,
because it is a square plot of land so the amount of area is s squared. So it’s
important that you realize that there’s O of a, and O of s squared are obviously saying essentially the same thing, just like when you describe the area of a square. Well you could describe it as 25 or you
describe it as 5 squared, just because it has a square in it doesn’t make it bigger.
So both are valid ways of describing the runtime. It
just depends on what might be clearer for that problem. So with that said, let me go on to 4 important rules to know with the big O. The first one is if you have two different
steps in your algorithm, you add up those steps, so if you have a first step that
takes O of a time and a second step that takes O of b you would add up
those run times and you get O of a plus b. So for example you have this runtime
that, you have this algorithm, that first walks through one array, and then it walks
the other array. It’s going to be the amount of time it takes to walk through each of
them. So you’ll add up their runtimes. The second way, the second rule is that
you drop constants. So let’s look at these two different ways that you can print out the min and max element in an array. One finds the min element
and then finds the max element, the other one finds the min and the max simultaneously.
Now these are fundamentally doing the same thing, they’re doing essentially the
exact same operations, just doing them in slightly different orders. Both of these
get described as O of n, where n is the size of the array. Now it’s really
tempting for people to sometimes see two different loops and describe it as O of 2n
and so it’s important that you remember that you drop constants in big O. You
do not describe things as O of 2n or O of 3n because you’re just looking for how
things scale roughly. Is it a linear relationship, is it a quadratic
relationship, so you always drop constants. The third rule is that if you
have different inputs you’re usually going to use different variables to
represent them. So let’s take this example where we have two arrays and
we’re walking through them to figure out the number of elements in common
between the two arrays. Some people mistakenly call this O of n squared but that’s not correct. When you just talk about runtime if you
describe things as O of n or O of n squared or O of n log n, n must have a meaning,
it’s not like things are inherently one or other. So when you described this as
O of n squared it really doesn’t make sense because it doesn’t,
what does n mean? n is not the size of the array, if there’s two
different arrays. What you want to describe instead is O of a times b. Again
fundamentally, big O is an equation that expresses how the runtime changes, how
its scales, so you describe this as O of a times b. The fourth rule is that you drop
non-dominant terms. So let’s suppose you have this code here that walks through
an array and prints out the biggest element and then for some reason it goes and
prints all pairs of values in the array. So that first step takes O of n time,
the second step takes O of n squared time, so you could first get as a
less simplified form, you could describe this as O of n + n squared, but
note that, compare this to O of n and the runtime, or compare this to the
runtime of O of n squared, and the runtime of O of n squared plus n squared. Both of
those two runtimes reduce down to n squared and what’s interesting here is
that O of n plus n squared, should logically be between them. So this n squared thing
kind of dominates this O of n thing, and so you drop that non dominant term. It’s
the n squared that’s really going to drive how the runtime changes. And if you
want you can look up a more academic explanation for when exactly you drop
things and when exactly you can’t, but this is kinda layman’s explanation for
it. We’ve now covered the very basic pieces
of big O. This is an incredibly important concept to master for your
interviews so don’t be lazy here, make sure you really really understand this
and try a whole bunch of exercises to solidify your understanding. Good luck.

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