MCS-177 Project: Orders of Growth

Problem

Does making a small change in a procedure make any difference in how the time the procedure takes varies with the size of the problem? In this project you will make some changes to the sum-of-divisors procedure introduced in section 3.3 and explore what effect these changes have on the procedure itself and on procedures which call it.

In particular, you will write a second version of sum-of-divisors.Using the techniques we learned in class, you should do a theoretical analysis of each version to produce predictions about how the time the computer takes to do a computation changes with the size of the computation. Next, you will measure the time the computer takes to do computations of varying sizes. (This is known as empirical testing.) Then you can compare your theoretical predictions with your empirical results. You will also write a procedure that uses sum-of-divisors, and do a similar analysis.

Extra deadline and groups

You should submit the procedure for Exercise 3.6,


sum-of-divisors

in class on Monday, October 9. We also highly recommend you write a procedure to compute the relative frequency of abundant and deficient numbers (problem 5 below) for more practice programming. Note that this is after only one day of working on this project in lab. On Tuesday, October 10, you will choose a partners and work on the remaining portions of the lab in groups of two (with perhaps one group of three, should that prove necessary due to an odd number of students).

We warn you that working in a group of two does not mean that your work load will be halved; in fact, some groups may find that the amount of hassle will effectively increase the work. Our intention is that you gain experience in working in teams, since teamwork is very common in today's workplace. You should have a clear understanding of each other's responsibilities, and be dependable and punctual in carrying them out. Also, be sure that neither partner monopolizes the work, but rather that you share it as equally as possible. Finally, we strongly encourage you to set up a group meeting with a lab instructor after your first or second day of working together in lab (October 12), so that we can assess how the project is going and, if need be, suggest strategies for more successfully working together.

Project Report

As in the first project, your lab report should concentrate on answering the central questions for this project. In this case, you have two sets of procedures. In each set, there are two procedures which produce the same result when given the same data. Your job is to compare the amount of time the computer takes to run the procedures, using both theoretical analyses and empirical results.

In writing this report, you will want to include not only the procedures you used and the measurements you made, but a clear analysis of the asymptotic orders of growth (in O-notation) for each of the two methods of doing each of the two computations. In particular, for each method, you will need to

(Theoretical results) Explain the order of growth for each method by looking at the procedures themselves.
(Experimental results) Clearly present time measurements for how long each procedure takes as a function of n.
(Compare the two) Explain how well the experimental measurements match up with your theoretical results. If they differ, do they differ by much? What would explain the differences?

Use tables and/or graphs to more clearly show your results. Keep in mind that tables (or graphs) are most clear if headings (or axes) are clearly labeled with both the variable being measured and the units. 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 project report, remember that your audience is generally knowledgeable about computer science and Scheme, but not about what you did.

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

Computer Exercises

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 round off errors; the computer might determine that the square root of 9 is 2.99999 instead of 3.00000.
What is the running time of each procedure you wrote for Exercise 3.6 expressed in O-notation?
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 project 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. What is the running time of the procedure expressed in O-notation? 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 with your ratio finding procedure? (You can use the same general style of reasoning as on page 81 to produce determine the running time expressed in big Theta notation.)
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?