MCS-177 Lab: Orders of Growth

Introduction

In this lab, you will measure the time the computer takes to do computations of varying size using different methods. You will use theoretical analysis to produce predictions as to how the time will vary with problem size for each method, and then you will compare your actual observations to those predictions. The computations you will measure are all based on the sum-of-divisors procedure introduced in section 3.3.

Prelab

Do exercise 3.6 on page 60, if you haven't already. Be sure to compare low² to n rather than comparing low to the square root of n. The square root method will not work reliably due to roundoff errors; the computer might determine that the square root of 9 is 2.99999 instead of 3.00000.

In lab

To load into Scheme a procedure called run-time for timing how long computations take, copy the following expression:
```
  (load "~mc27/public/time.scm")
```
into your definition window and execute it. To get the text's definitions of sum-of-divisors and divides?, copy the procedures in the following file into your definitions window and execute them:
```
  ~mc27/labs/sum-divisors/sum-of-divisors.scm
```
Now you should time how long it takes to find the sum of divisors of 100, 1000, 10000, 100000, and 1000000. You should do the same timing repeatedly, at least for the smaller numbers, to get some idea how precisely repeatable the timings are. To illustrate how you would do a timing, here is a sample timing of the sum-of-divisors of 100:
```
(run-time sum-of-divisors 100)  ;Time: .04695454545454616   ;Value: 217
```
The time is given in seconds; e.g., in the above example, the evaluation took a little less than five hundredths of a second. The time reported will not agree well with your own impression of the time taken, because it doesn't include time the computer spends doing other things, and also because for fast evaluations (such as the above) the evaluation is repeated until at least a second has elapsed and an average time for the repetitions is reported.
What is the ratio of times from each measurement to the next? (That is: What is the ratio of times going from 100 to 1000? From 1000 to 10000? And so forth.) How does this compare with your expectation? (Note: When answering questions like these in your lab write-up, be sure to back up your answers with adequate arguments. In other words, give reasons, probably in the form of big Theta arguments, which explain your expectations.)
Now evaluate the improved definition from exercise 3.6 and repeat the measurements. Again, what are the time ratios, and how do they compare with your prediction?
One use for the sum-of-divisors procedure other than finding perfect numbers is to measure the relative frequency of abundant and deficient numbers. As mentioned in section 3.3, very few numbers are perfect. The remaining numbers either have a sum of divisors that is more than twice the number, in which case they are called abundant, or less than twice the number, in which case they are called deficient. The empirical question we'd like to ask is, are these two kinds of numbers equally common, or does one predominate? If so, by how much? The usual mathematical way to formulate this is to count how many numbers less than or equal to n are abundant and how many are deficient, and take the ratio. Then repeat this for larger and larger values of n. If the ratio approaches closer and closer to some limiting value as n increases, then this limiting ratio tells us the relative frequency of abundant and deficient numbers. For example, if the ratio approaches a limit of 1, then abundant and deficient numbers are equally frequent. On the other hand, a limiting value of 1/2 would mean that there are roughly twice as many deficient numbers as abundant numbers.
Write a procedure that generates an iterative process for computing this ratio, given n. Most of the work can be done by a subsidiary procedure that keeps count of how many abundant numbers it has found and how many deficient numbers it has found as it scans the range from 1 to n. Once the entire range has been scanned, one count can be divided by the other. Test your procedure to be sure that it works.
Suppose that you are using the text's definition of sum-of-divisors with your procedure for computing the ratio, and double the size of the range being tested. By roughly what factor do you expect the number of divides? tests (and hence time) to increase? How about if you use the sum-of-divisors procedure from exercise 3.6? (You can use the same general style of reasoning as on page 81 to produce a big Theta asymptotic order of growth.)
Now measure how long your procedure takes to find the ratio for n equal to 200, 400, 800, 1600 and 3200, and calculate the ratio of each consecutive pair of times. Do this with each of the two definitions for sum-of-divisors. How well do your measurements agree with your predictions? Do the abundant/deficient ratios seem to approach some simple limiting value? If so, what?

Postlab

The main purpose of this lab is to measure the time the computer takes to do computations of varying size and to compare the results with theoretical predictions. Thus, your lab report should concentrate on your theoretical analyses (based on the procedures) and your empirical results (based on the timing data). In particular, you will want to include not only the procedures you used and the measurements you made, but asymptotic orders of growth (in big Theta notation) for each of the two methods of doing each of the two computations, explanations of these orders of growth, predictions based on them, and explicit comparisons of the predictions with the measurements.

You ought to use tables and/or graphs to more clearly show your results. Keep in mind that tables and graphs are most clear if axes are clearly labeled. Also, if your goal is to compare the results of two experiments, it's best if the data from both can be placed in the same table or graph. It's confusing for the reader if the data appear on separate pages.

You might also want to make some additional predictions that go beyond your measurements. Suppose you were interested in pursuing the abundant/deficient ratio study well beyond n=3200, perhaps to n=102400. How long would you expect each method to take? Would you want to experimentally verify this? A harder (optional) question is, how large an n would the computer be able to handle in a day? a week? a year?

When you write the lab report, remember that your audience is generally knowledgeable about computer science and Scheme, but not about what you did.

As with lab 1, you should also consult the document entitled Suggestions for clear lab reports in computer science courses.