Having completed our first analysis of an algorithm, namely an upper bound on the
running time of the Merge Short algorithm. What I wanna do next is take a step back,
and be explicit about three assumptions, three biases that we made when we did this
analysis of Merge Short, and interpreted the results. These three assumptions we
will adopt as guiding principles for how to reason about algorithms, and how to
define a so called fast algorithm for the rest of the course. So, the first guiding
principle is that we used what's often called worst case analysis. By worst case.
Analysis, I simply mean that our upper bound of six N log N plus six N. Applies
to the number of lines of executed for every single input array of length end. We
made absolutely no assumptions about the input, where it comes from, what it looks
like beyond what the input length N was. Put differently, if, hypothetically, we
had some adversary whose sole purpose in life was to concoct some malevolent input
designed to make our algorithm run as slow as possible. The worst this adversary
could do, is. Upper bounded by this same number, 6N log N + 6N. Now, this, so,
sort of worst case guarantee popped out so naturally from our analysis of Merge
Short, you might well be wondering, what else could you do? Well, two other methods
of analysis, which do have their place, although we won't really dicuss them in
this course, are quote unquote, average case analysis. And also the use of a set
of prespecified benchmarks. By average case analysis, I mean, you analyze the
average running time of an algorithm under some assumption about the relative
frequencies of different inputs. So, for example, in the sorting problem, one thing
you could do, although it's not what we do here. You could assume that every possible
input array is equally unlikely, and then analyze the average running time of
an algorithm. By benchmarks, I just mean that one agrees up front about some set,
say ten or twenty, Benchmark inputs, which are thought to represent practical or
typical inputs for the algorithm. Now, both average-case analysis and benchmarks
are useful in certain settings, but for them to make sense, you really have to
have domain knowledge about your problem. You need to have some understanding of
what inputs are more common than others, what inputs better represent typical
inputs than others. By contrast, in worst-case analysis, by definition you're
making absolutely no assumptions about where the input comes from. So, as a
result, worst-case analysis is particularly appropriate for
general-purpose sub-routines. Sub-routines that you design. Find without having any
knowledge of how they will be used or what kind of inputs they will be used on. And
happily, another bonus of doing worst case analysis, as we will in this course, it's
usually mathematically much more attractable than trying to analyze the
average performance of an algorithm under some distribution over inputs. Or to
understand the detailed behavior of an algorithm on a particular set of benchmark
inputs. This mathemetical tractabilty was reflected in our Merge Sort analysis,
where we had no a priori goal of analyzing the worst case, per se. But it's
naturally what popped out of our reasoning about the algorithm's running time. The
second and third guiding principles are closely related. The second one is that,
in this course, when we analyze algorithms, we won't worry unduly about
small constant factors or lower order terms. We saw this philosophy at work very
early on in our analysis of merge sort. When we discussed the number of lines of
code that the merge subroutine requires. We first upper-bounded it by 4m plus two,
for an array of length m, and then we said, eh, let's just think about it as 6m
instead. Let's have a simpler, sloppy upper-bound and work with that. So, that
was already an example of not worrying about small changes in the constant
factor. Now, the question you should be wondering about is, why do we do this, and
can we really get away with it? So let me tell you about the justifications for this
guiding principle. So the first motivation is clear, and we used it already in our
merge short analysis. Which is simply way easier mathematically, if we don't have
to, precisely pin down what the [inaudible] constant factors and lower-order terms are.
The second justification is a little less obvious, but is extremely important. So, I
claim that, given the level at which we're describing and analyzing algorithms in
this course, it would be totally inappropriate to obsess unduly about
exactly what the constant factors are. Recall our discussion of the merge
subroutine. So, we wrote that subroutine down in pseudocode, and we gave it
analysis of 4m plus two on the number of lines of code executed, given an input of
length m. We also noted that, it was somewhat ambiguous exactly how many lines
of code we should count it as, depending on how you count loop increments and so on. So
even there it's small constant factors could creep in given the under
specification of the pseudo code. Depending on how that pseudo code gets
translated into an actual program language like C or Java. You'll see the number of
lines of code deviate even further, not but a lot but again by small constant
factors. When such a program is then compiled down into machine code, you'll
see even greater variance depending on the exact processor, the compiler, the
compiler optimizations, the programming implementation, and so on. So to
summarize, because we're going to describe algorithms at a level. That transcends any
particular programming language. It would be inappropriate to specify precise
constants. The precise constants were ultimately determined by more machine
dependent aspects like who the programmer is, what the compiler is, what the
processor is, and so on. And now the third justification is frankly, we're just going
to be able to get away with it. [sound] That is, one might be concerned that
ignoring things like small constant factors leads us astray. That we wind up deriving
results which suggest that an algorithm is fast when it's really slow in practice, or
vice versa. And for the problems we discuss in this course we'll get extremely
accurate predictive power. Even though we won't be keeping track of lower terms and
constant factors. When the mathematical analysis we do suggests that an algorithm
is fast, indeed it will be. When it suggests that it's not fast, indeed that
will be the case. So we lose a little bit of granularity of information. But we
don't lose in what we really care about, which is accurate guidance about what
algorithms are gonna be faster than others. So the first two justifications, I
think, are pretty self evident. This third justification is more of an assertion, but
it's one we'll be baking up over and over again as we proceed through this course.
Now, don't get me wrong. I'm not saying constant factors aren't important in
practice. Obviously, for crucial programs the constant factors are hugely important.
If you're running the sorta crucial loop, you know, your start up's survival depends
on, by all means optimize the constant like crazy. The point is just that
understanding tiny constant factors in the analysis is an inappropriate level of
granularity for the kind of algorithm analysis we're going to be doing in this
course. Okay, lets move on the, the third and final guiding principle. So the third
principle is that we're going to use what's called asymptotic analysis, by
which I mean we will focus on the case of a large input sizes. The performance of an
algorithm as the size N of the input grows large, that is, tends to infinity. Now
this focus on large input size is it was already evident when we interpreted our
bound on Merge Sort. So, how did we describe the bound on Merge Sort? We said,
oh, well, it needs a number of operations proportional, a constant fact or times in
login. And we very cavalierly declared that this was better than any algorithm
which has quadratic dependence of it's running time on the number of operations.
So for example, we argued that merge sort is a better, faster algorithm than
something like insertion sort, without actually discussing the constant factors
at all. So mathematically. We were saying the running time of merge short, which we
know, which we can represent as the function. Six N log base two of N + 6N
is better than any function which has a quadratic dependence on n. Even one with a
small constant like lets say 1/2 N squared which might roughly be the running
time of insertion sort. And this is a mathematical statement that is true if and
only if N is sufficiently large once N grows large it is certainly true that
the expression on the left is smaller than the expression on the right but for small
N the expression on the right is actually going to be smaller because of the smaller
leading term so in saying that merge sort is superior to insertion sort the bias
is that we're focusing on problems with a large N so the question you should have is
that reasonable is that a justified assumption to focus on large input sizes
and the answer is certainly yes. So the reason we focus on large input sizes is
because, frankly, those are the only problems which are even, which are at all
interesting. If all you need to do is sort 100 numbers, use whatever method you want,
and it's gonna happen instantaneously on modern computers. You don't need to know
say, the divide and conquer paradigm, if all you need to do is sort 100 numbers. So
one thing you might be wondering is if, with computers getting faster all the time
according to Moore's Law, if really, it doesn't even matter to think about
algorithmic analysis, if eventually all problem sizes will just be trivially
solvable on super fast computers. But, in fact, the opposite is true. Moore's Law,
with computers getting faster, actually says that our computational ambitions will
naturally grow. We naturally focus on ever larger problem sizes. And the gulf between
an N squared algorithm and an m log n algorithm will become ever wider. A
different way to think about it is in terms of how much bigger a problem size
you can solve. As computers get faster. If you are using an algorithm with a running
time which is proportional to the input size then the computers get faster by a
factor of four then you can solve problems that are factor of four or larger. Whereas
if you are using an algorithm whose running time is proportional to the square
of the input size then a computer gets faster by a factor of four, you can only
solve double the problem size and we'll see even starker examples of this gulf
between different algorithm approaches as the time goes on. So to drive this point
home. Let me show you a couple of graphs. So what we're looking at here, is we're
looking at a graph, of two functions. So the solid function. Is the upper bound
that we proved on merge sort. So this is gonna be 6nlog(base2)n plus 6n. And the
dotted line is an estimate. A rather generous estimate about the running time
of, [inaudible] sort. Namely one-half times N. Squared. And we see here in the
graph exactly the behavior that we discussed earlier, which is that the small
N. Down here. In fact because one-half N. Squared has a smaller leading constant
it's actually a smaller function. And this is true up to this crossing point of maybe
90 or so. Again, beyond n=90. The quadratic growth in the N squared term.
Overwhelms the fact that it had a smaller constant and it starts being bigger than
this other function six of N + six N so in the regime below 90 it's predicting that
the insertion store will be better and in the regime above 90 it's predicting that
merge sort will be faster. Now here's what's interesting let's scale the X axis
let's look well beyond this crossing point of 90 let's just increase it in order of
magnitude up to a raise in size 1500. And I want to emphasize these are still very
small problem sizes. If all you need to do is sort arrays of size 1500 you really don't
need to know Divide-and-conquer or anything else we'll talk about -- that's a
pretty trivial problem on modern computers. [sound]. So what we're seeing
is, that even for very modest problem sizes here, array of, of, size, say 1500.
The quadratic dependence in the insertion sort bound is more than dwarfing the
fact, that it had a lower constant factor. So in this large regime, the gulf between
the two algorithms is growing. And of course, if I increased it another 10X or
100x or 1000x to get to genuinely interesting problem sizes, the gap between
these two algorithms would be even bigger, it would be huge. That said, I'm not
saying you should be completely ignorant of constant factors when you implement
algorithms. It's still good to have a general sense of what these constant
factors are so for example in highly tuned versions of Merge Sort which you'll find
in mny programming libraries. In fact, because of the difference in
constant factors, the algorithm will actually switch from Merge Sort over to
insertion sort, once the problem size drops below some particular threshold, say
seven elements, or something like that. So for small problem sizes, you use the
algorithm with smaller constant factors, and the insertion sort for larger problem
sizes, you use the algorithm and better rate of growth, mainly merge short. So, to
review our first guiding principal is that we're going to pursue worse case analysis.
We're going to look to bounds on the performance, on the running time of an
algorithm which make no domain assumptions, which make no assumptions
about which input of a given length the algorithm is provided. The second guiding
principal is we're not going to focus on constant factors or lower returns, that
would be inappropriate, given the level of granularity at which we're describing
algorithms and third is where going to focus on the rate of growth of algorithms
for large problem sizes. Putting these three principles together, we get a
mathematical definition of a fast algorithm. Namely, we're gonna pursue
algorithms whose worst case running time grows slowly as a function of the input
size. So let me tell you how you should interpret what I just wrote down in this
box. So on the left hand side is clearly what we want. Okay, we want algorithms
which run quickly if we implement them. And on the right hand side is a proposed
mathematical surrogate of a fast algorithm. Right, the left hand side is
not a mathematical definition. The right hand side is, as well become clear in the
next set of lectures. So we're identifying fast algorithms, which, those that have
good asymptotic running time which grows slowly with the input size. Now, what
would we want from a mathematical definition? We'd want a sweet spot. On one
hand we want something we can actually reason about. This is why we zoom out and
squint and ignore things like constant factors and lower terms. We can't keep
track of everything. Otherwise we'd never be able to analyze stuff. On the other hand
we don't want to throw out the baby with the bath water, we want to retain
predictive power and this turns out this definition turns out for the problems
we're going to talk about in this course, to be the sweet spot for reasoning about
algorithms okay worst case analysis using the asymptotic running time. We'll be able to
prove lots of theorems. We'll be able to establish a lot of performance guarantees
for fundamental algorithms but at the same time we'll have good predictive power what
the theory advocates will in fact be algorithms that are known to be best in
practice. So, the final explanation I owe you, is, what do I mean by, the running
time grows slowly, with respect to the input size? Well, the answer depends a
little bit on the context, but, for almost all of the problems we're going to
discuss, the holy grail will be to have what's called a linear time algorithm, an
algorithm whose number of instructions grows proportional to the input size. So,
we won't always be able to achieve linear time, but that's, in some sense, the best
case scenario. Notice, linear time is even better than what we achieved with our
merge short algorithm for sorting. Merge short runs a little bit superlinear, it's
n times log n, running as the input size. If possible, we. To be linear time. It's
not always gonna be possible, but that is what we will aspire toward. For most of
the problems we'll discuss in this course. Looking ahead, the next series of videos
is going to have two goals. First of all, on the analysis side, I'll describe
formally what I mean by asymptotic running time. I'll introduce "Big Oh"
notation and its variants, explain its mathematical definitions, and give a
number of examples. On the design side, we'll get more experience applying the
divide and conquer paradigm to further problems. See you then.