Princeton Univerisy Lectures: Algorithms Part I – Analysis of Algorithms(3) – Mathematical Models

by Kevin Wayne, Robert Sedgewick

Observing what’s happening as we did in the last section it gives us a, a way to predict performance but it really doesn’t help us understand what the algorithm’s doing. So next, we’re going to look at mathematical model. A way to get a better concept of what’s really happening.


Again, this concept was really developed and popularized by Don Knuth starting in the late 60s. At that time, computer systems were really becoming complicated for the first time. And computer scientists were concerned about whether we really were going to be able to understand what’s going on. And Knuth was very direct in saying that this is something that we certainly can do. We can calculate the total running time of a program by identifying all the basic operations, figuring out the cost, figuring out the frequency of execution and summing up the cost times frequency for all the operations.

You have to analyze the program to determine what set of operations and the cost depends on the machine and the computer in the system is what we talked about before. The frequency leads us to mathematics because it depends on the algorithm and input data. Knuth has written a series of books that give very detailed and all exact analyses within a particular computer model for a wide range of algorithms. So, from Knuth, we know that in principle, we can get accurate mathematical models for the performance of algorithms or programs and operation.


All right. So what, what does this process look like? Well you can, if you want run experiments. In, in ancient times, we would actually look at the computer manual and every computer came with a manual that said precisely how long each instruction would take. But nowadays, it’s a little more complicated. So, we run experiments and, and you can go ahead and do a billion ads and figure out that maybe on your computer, an ad takes 2.1 nano seconds. Or you can do more complicated function s like computer sign or an arc tangent although that’s already getting close to the analysis of algorithms.


So, there’s some way to determine the costs of the basic operations. And so, we’ll just in most, most of the cases we’ll just postulate that it’s some constant and you can figure out what the constant is. Although when we’re working with a collection of objects, of anobjects there are some things that takes time proportional to N like if you’re going to allocate a array of size N it takes time proportional to N because in Java the default is that all the elements in the array initialize to zero.

In other operations it depends on the system implementation and an important one is string concatenation. If you concatenate two strings the running time is proportional to the length of the string. In many novices programming in Java, make a mistake of assuming that’s a constant time operation when its not.


Alright, so that’s the cost of each operation. More interesting is the frequency of operation, of execution of the operations. So this is a, a, it’s a very simple variant of the three sum problem. That’s the one sum problem. That’s how many numbers are actually equal to zero? How many single numbers add up to zero? So, that one, it’s just one four loop, and we go through, and we tested the number zero and increment or count. And by analyzing that code you can see that I and count have to be declared and then they have to be assigned to zero. There’s compares of i against N and there’s N + one of them. There’s compares of A(i) against zero, there’s N of those, N array axises and the number incremented is number of times there’s an increment is variable. I has incremented N times, but count could be incremented any number from zero to N times. And so that frequency is dependent on the input data. Or we might need a model for describing that or maybe there’s other operations that are more e xpensive and we won’t need to worry about that.


So, let’s look at the next more complicated problem is what about the frequency of execution of instructions in this program which is the two sum problem, how many pairs of integers sum to zero? Well, in this case, you have to do a little bit of math to see that when we when i goes from zero to N, and j goes from i + a to N the number of compares that we do work, plus array axises that we do is two for each time the if statement is executed for Ai and Aj and that time is, thing is executed N – one times the first time through the loop and N -two^2 and so forth. It’s the sum of the integers from zero up to N – one which is a simple discrete sum one-half N, (N – one) and since, and since we’re doing it twice the number of array axises is N, N – one. So, we can go ahead and get these actual exact counts. But already, it’s getting a little bit tedious to do that.


And as far back as Turing who also knew that and as well as Babbage did, that we want to have a measure of the amount of work involved in the process. He recognized that you didn’t want to necessarily go through and do it in full detail. It’s still helpful to have a crude estimate. So, you could count up the number of times that every operation is applied, give it weights and, and count the [inaudible] and so forth. But maybe we should just count the ones that are most expensive that’s what Turing said in 1947, and realistically that’s what we do nowadays.


So rather than going in and counting every little detail, we take some basic operation that’s maybe the most expensive and or and or the one that’s executed the most often. The one that cost and frequency is the highest and use that as a proxy for running time. Essentially, making the hypothesis that the running time is, is going to grow like a constant times [inaudible], So, in this case, were going to pick array axises.


So, that’s the first simplification. And the second simplification is that we’re going to ignore low order terms in the formulas that we derive. And there’s an easy way to do that. It’s called the tilde notation and, and the idea is when N is large in a formula like this the N^3 term is much, much higher than the N term or sixteen. In fact, so much so that we wouldn’t even hardly notice these low order terms. So, all of these formulas are tilde one-sixth N^3 and that’s a fine representative or approximate, approximation to these quantities. And it greatly simplifies their calculations to for a, through a way to lower, lower to terms like this. So, by focusing on one operation and , throwing away the tildes, the lower the terms and this is the technical definition of tilde.


It’s just, F(N) tilde G (N) means the limit as FN or GN equals one, and you can check that that’s going to hold in these kinds of situations. So, that greatly simplifies the frequency counts.


And if we’re only picking one thing we’re just talking about tilde N^2 and maybe another tilde N^2 for the increment for the two sum problems, okay. So again, when N is large, the terms are negligible and when N is really small, they’re not negligible but we don’t really care because we’re trying to estimate running times for large N and running times for small N are going to be small no matter what.


All right, so now, we’re using both the cost model and the tilde notation and then we can simply say, that this program uses tilde N^2 squared array axises and have implicit the hypothesis that we think the running time is going to be tilde, a constant, times N squared. Okay, we now what about three sums, let’s do a, a real problem. So now, we have the triple loop.


And then, we have to do a more complicated combinatorial problem in is not that big a deal really we are looking at the distinct number of ways you can chose three things out of N and that ‘s binomial coefficient. And again, doing the math and using the tilde, it’s just tilde one-sixth N^3 three ray axises for each triple so we can say one-half N^3. So we’re not computing and summing the costs of all operations that’s too much work. We’re picking the most expensive in terms of cost times frequency and approximating that and trying to get a good model for the running time.


So now most, we’re not going to do of a full discrete mathematics in this course but there’s some basic things that we’ll want to use and are, are not that difficult to understand. So, a lot of times we find out that we need to come up with an estimate of a discrete sum. Like we did for one + two up to N. Or some of the squares or other things like the three sum triple loop. And so actually if you’ve had basic calculus, one way to think of it as to just replace the sum with an interval, integral. That usually works or we can do the math and use the so-called Euler–Maclaurin summation formula to get a true approximation. But if you think of it this way you’ll believe us when we say that, that thing is tilde one-half N^2 or sum of one+ one-half + one-third up to one / N. That’s like integral from x = one to N1 / x and that’s natural log of N. Now even the three sum triple loop kind of if you’re used to multiple integrals, I will quickly give you the one-sixth N^3.

There’s many more and other techniques that we could use for this. And we’re not going to teach all that, but we’ll sometimes refer to results of this type.


Alright, so in principle, Knuth tells us that accurate mathematical models are available in practice, we can get really complicated formulas. We also might need some advance mathematics that the theoretician will revel in. But that maybe people learning algorithms for the first time might not be expected to know.

So in the end careful exact models are best, best left for exit, experts. There’s really a lot of things that can go on. On the other hand approximate models are definitely worthwhile. And for all the algorithms that we consider we’ll try to communicate a reasonable approximate model that can be used to describe the running time. Sometimes we’ll give the mathematical proofs and other times we’ll have to just cite the work of some expert.

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