by Kevin Wayne, Robert Sedgewick
Okay, so the first step is to be able to make some observations about the running time of the programs. And for analysis of algorithms that’s easier than in a lot of scientific disciplines, as we’ll see.
For a running example we’re going to use the so-called 3-sum problem. And it’s an easy to state problem. If you’ve got N distinct integers, how many triple sum to exactly zero? For example in this file 8ints.text. Text which has eight integers in it. There’s four triples that sum to zero. 30 – 40, ten. 30 – twenty – ten and so forth and so our goal is to write a program that can compute this quantity for any input file, any set of N integers. This is actually a, an extremely important computation that’s deeply related to many problems in computational geometry which is a branch of computer science that covers the algorithms and underlying science related to graphics and movies and geometric models of all sort. So this is a actually an important practical problem.
But it’s a simple one to write code for in a view you could write down this program without much effort. It’s a, got a static method count that is going to go ahead and take a integer array as an argument. And, is that, that’s a number of integers, that’s the length of the array. We will start with a variable count equals zero, and then a triple for loop, that checks each triple i j k, we go I from one and j from i+1 to n, and k from j+1 to n, so that we get each triple just once. And then if i+j, ai + aj + ak = zero, we increment the count. Alright. And after that triple four loop, we return the count. And then the main method, in this simple class just reads in, all the integers, and prints out the count. So that’s a brute force algorithm that is a fine method for solving the three sum problem, now what we’re interested in is how much time does this take as a function of’ n?
Well, one to time our program is to is just look at the watch. If you have a stopwatch, or look at the clock or your phone, or whatever you might need you can just go ahead and time it if you want or we have, Java has this part of it’s standard library, a stopwatch class that will go ahead and compute a lapse time. So, in order, anytime you run a program, if it is setup to easily take input of different sizes, a natural thing to do, is just run it for bigger sizes.
So for eight ints this program takes not too much time, for 1000 ints it takes half a second. For 2,000. Takes more time. That’s 3.7 seconds run it again, still takes 3.7 seconds for 4,000, so each time we’re doubling the size of the input and it’s definitely taking more time each time. And actually as we’ll see if programmers who get in the habit of testing or any time on their program in this way can get so that you can actually pretty easily and quickly evaluate when it’s going to finish. In fact. While you’re waiting for it to finish you can often figure it out. So that one took 30 seconds for 4K and definitely we could figure it out how long it’s going to take for 8K before it finishes, and you’ll see how in just a second. I’m not going to wait right now.You can think about what you think.
Okay so [cough] that’s empirical analysis, analysis. Run it for various input sizes and measure their running time. Now if this were some scientific problem where we were counting something that happen in the natural world. The number of ants in an ant hill or whatever then we’d have only a few data points and we would try to understand whats was going on by doing a plot of or running time with quite interested in on the Y axis and problem size with the X axis.
Hit a curve like this and actually whats science usually do because of some many problems fall into out of this class is do the plot as a lg, lg plot. If you do it as a lg, lg plot very often you’ll get a straight line. And the slope of the straight line is the key to what’s going on. In this case, the slope of the straight line is three and so you can run what’s called a regression to fit a late, the straight line through the data points. And then, it’s not difficult to show to do the math to show that if you get a straight line and the slope is B, then your function is proportional to A, N^B. That’s called the power law. And that’s true of many, many scientific problems including most algorithms. So here’s a little bit of the math for that. So the straight line means that since we did a lg, lg plot with powers of two, that lg(T(N) = B lg N + C. And we have our empirical values of B and C and then if you raise both sides of that equation to two to that power then you get T(N) = a constant times N^B.
So right away just from observation we have a pretty good model for the running time for our program, we can figure and do the math and figure out that it seems as though the running time is about ten^-10 N^3 seconds.
We can use that hypothesis to go ahead and make predictions. Just plug in for different values of N and it says it will take us 400 seconds for 16,000. 400 seconds is plenty of time but now we can go ahead and invest and run that experiment and sure enough we’re pretty close to that 408 seconds when we run it. And now we can make a prediction for 32,000 or for or for whatever else we might be interested in. The model helps us do predictions without investing the expense to run the experiments.
In fact, in this situation if there is a power law, and again in a very great majority of computer algorithm running times is going to be a power law. What we can do is just double the size of the input each time the way we were and take the ratio of the running times for N and 2N. And if you do that, that ratio going to converge to a constant. And in fact the log of the ratio is going to converge to that constant, which is the exponent of N and the running time. And you just need a little math to check that one, but that’s a very easy and natural way to go ahead and predict running times.
So that’s what I said before is, so we have this quick way to estimate B in the power law relationship. How do we estimate A? Well we can just run it and solve for A. So once we’ve decided that, that exponent is three let’s run it for some big N and we get pretty close model to the one we had from plotting things. So it’s almost identical hypothesis and we just got it by running the program double N each time.
Okay so there is a lot of effects in trying to understand the running time of a program on, on your machine. [cough] So. Key effects are independent of what computer it is. And that’s the algorithm you’re using and what’s the data. And that’s going to really determine the exponent in the power law. And then there’s a lot of, system dependent effects. What kind of hardware do you have? Do you have a fast computer or a slow one? What kind of software? What’s going on in your computer? All of those things really determine the constant A in the power law. So. In modern systems it is so much going on in the hardware and software, it’s sometimes difficult to get really precise measurements. But on the other hand we don’t have to sacrifice animals, or fly to another planet the way they do in other sciences, we can just run a huge number of experiments and usually take care of understanding these kind of effects.