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You have completed Introduction to Algorithms!
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Space complexity is a measure of how much working storage, or extra storage, is needed as a particular algorithm grows. In this video let's take a look at the space complexity of our algorithms
Glossary
- Space Complexity - a measure of how much working storage, or extra storage, is needed as a particular algorithm grows
- Tail call - a call, inside a function, to the same function itself as the last operation. Also called tail recursion
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At the beginning of this series,
0:00
I mentioned that there were two ways of
measuring the efficiency of an algorithm.
0:01
The first was time complexity, or
0:05
how the runtime of an algorithm
grows as N grows larger.
0:08
The second is space complexity.
0:11
We took a pretty long route
to build up this example, but
0:14
now we're in a good place to
discuss space complexity.
0:17
Space complexity is a measure
of how much working storage or
0:20
extra storage is needed as
a particular algorithm grows.
0:25
We don't think about it much these days,
but
0:29
every single thing we do on
a computer takes up space in memory.
0:32
In the early days of computing,
considering memory usage was of paramount
0:36
importance because memory was limited and
really expensive.
0:41
These days we're spoiled,
our devices are rich with memory.
0:45
This is okay when we write everyday
code because most of us aren't dealing
0:48
with enormously large data sets.
0:53
When we write algorithms however, we
need to think about this because we want
0:55
to design our algorithms to
perform as efficiently as it can,
1:00
as the size of the data set,
N, grows really large.
1:04
Like time complexity,
1:07
space complexity is measured in the
worst-case scenario using big O notation.
1:09
Since you are familiar with
the different kinds of complexities,
1:14
let's dive right into an example.
1:19
In our iterative implementation of binary
search, the first one we wrote that uses
1:21
a while loop, let's look at what happens
to our memory usage as N gets large.
1:26
Let's bring up that function.
1:31
Let's say we start off with
a list of ten elements.
1:34
Now inspecting the codes we see that our
solution relies heavily on these two
1:38
variables first and last.
1:43
First points to the start of the list and
last, to the end.
1:44
When we eliminate a set of values,
we don't actually create a sublist.
1:48
Instead we just redefine first and
last as you see here,
1:52
to point to a different
section of the list.
1:57
Since the algorithm only considers
the values between first and
2:01
last when determining the midpoint,
by redefining first and
2:05
last as the algorithm proceeds we can find
the solution using just the original list.
2:09
This means that for
any value of n the space complexity of
2:14
the interative version of
binary search is constant.
2:18
Or that the iterative version of
binary search takes constant space.
2:22
Remember that we would
write this as big O of one.
2:27
This might seem confusing,
because as n grows,
2:31
we need more storage to account for
that larger list size.
2:34
Now this is true, but
2:37
that storage is not what space
complexity cares about measuring.
2:39
We care about what additional storage
is needed as the algorithm runs and
2:43
tries to find a solution.
2:48
If we assume something simple, say that
for a given size of a list, represented
2:50
by a value N, it takes N amount of
space to store it, whatever that means.
2:55
Then for the iterative version of
binary search, regardless of how large
3:00
the list is, at the start, middle,
and end of the algorithm process,
3:05
the amount of storage required
does not get larger than N.
3:10
And this is why we consider
it to run in constant space.
3:15
Now, this is an entirely different story
with the recursive version, however.
3:18
In the recursive version of binary search
we don't make use of variables to keep
3:23
track of which portion of
the list we're working with.
3:27
Instead, we create new lists every
time with a subset of values, or
3:30
sublists, with every
recursive function call.
3:35
Let's assume we have a list of size n and
in the worst case scenario,
3:39
the target element is
the last in the list.
3:43
Calling the Recursive Implementation
of binary search on this list and
3:46
target would lead to a scenario like this.
3:50
The function would call itself and
3:53
create a new list that goes from
the midpoint to the end of the list.
3:55
Since we're discarding half the values,
the size of the sublist is N by 2.
3:59
This function will keep calling itself,
4:05
creating a new sublist that's
half the size of the current one,
4:07
until it arrives at a single element list,
and a stopping condition.
4:11
This pattern that you see here,
where the size of the sublist is reduced
4:15
by a factor on each execution
of the algorithmic logic,
4:19
well we've seen that pattern before.
4:23
Do you remember where?
4:25
This is exactly how binary search works.
4:26
[SOUND] It discards half the values
every time until it finds a solution.
4:29
Now we know that because of this pattern,
4:33
the running time of binary
search is logarithmic.
4:36
In fact,
4:38
this space complexity of the recursive
version of binary search is the same.
4:39
If we start out with the memory allocation
of size N that matches the list,
4:44
on each function call of
recursive binary search,
4:49
we need to allocate additional
memory of size N/2, N/4, and so
4:53
on until we have a sublist that is
either empty or contains a single value.
4:57
Because of this, we say that
the recursive version of the binary
5:02
search algorithm runs in logarithmic
time with a big O of log N.
5:07
Now, there's an important caveat here.
5:12
This totally depends on the language.
5:14
Remember how I said that a programming
language like Swift can do some tricks to
5:16
where recursion depth doesn't matter?
5:21
The same concept applies here.
5:23
If you care to read more about this
concept, it's called tail optimization.
5:25
It's called tail optimization because
if you think of a function as
5:30
having a head and a tail, if the recursive
function call is the last line
5:35
of code in the function, as it is in
our case, we call this tail recursion,
5:40
since it's the last part of
the function that calls itself.
5:45
Now the trick that Swift does to
reduce the amount of space, and
5:49
therefore computing overhead,
to keep track of this recursive calls,
5:53
is called tail call optimization,
or tail call elimination.
5:58
It's one of those things that
you'll see thrown around a lot
6:02
in algorithm discussions, but
may not always be relevant to you.
6:06
Now, what of any this is relevant to us?
6:09
Well, Python does not implement
tail call optimization.
6:12
So the recursive version of binary
search takes logarithmic space.
6:16
If we had to choose between
the two implementations,
6:20
given that time complexity or run-time
of both versions, the iterative and
6:24
the recursive version are the same,
we should definitely go with the iterative
6:28
implementation in Python,
since it runs in constant space.
6:33
Okay, that was a lot.
6:37
But all of this, with all of this we've
now established two important ways
6:38
to distinguish between algorithms
that handle the same task,
6:43
and determine which one we should use.
6:47
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