CSCI 1312 (Introduction to Programming for Engineering), Fall 2017:
Homework 7

Credit:

40 points.

Reading

Be sure you have read (or at least skimmed) the assigned readings from chapter 8 (not including the material on searching and sorting).

Honor Code Statement

Please include with each part of the assignment the Honor Code pledge or just the word ``pledged'', plus one or more of the following about collaboration and help (as many as apply).¹Text in italics is explanatory or something for you to fill in. For written assignments, it should go right after your name and the assignment number; for programming assignments, it should go in comments at the start of your program(s).

• This assignment is entirely my own work. (Here, ``entirely my own work'' means that it's your own work except for anything you got from the assignment itself -- some programming assignments include ``starter code'', for example -- or from the course Web site. In particular, for programming assignments you can copy freely from anything on the ``sample programs page''.)
• I worked with names of other students on this assignment.
• I got help with this assignment from source of help -- ACM tutoring, another student in the course, the instructor, etc. (Here, ``help'' means significant help, beyond a little assistance with tools or compiler errors.)
• I got help from outside source -- a book other than the textbook (give title and author), a Web site (give its URL), etc.. (Here too, you only need to mention significant help -- you don't need to tell me that you looked up an error message on the Web, but if you found an algorithm or a code sketch, tell me about that.)
• I provided help to names of students on this assignment. (And here too, you only need to tell me about significant help.)

Programming Problems

Do the following programming problems. You will end up with at least one code file per problem. Submit your program source (and any other needed files) by sending mail to bmassing@cs.trinity.edu with each file as an attachment. Please use a subject line that mentions the course and the assignment (e.g., ``csci 1312 hw 7'' or ``CS1 hw 7''). You can develop your programs on any system that provides the needed functionality, but I will test them on one of the department's Linux machines, so you should probably make sure they work in that environment before turning them in.

1. (20 points) C, like many programming languages, has a library function (rand()) that can be used to generate a ``random'' sequence of numbers (quotes because it's not truly random, as mentioned in class). This function can be used to generate a value in a specified range (e.g., between 0 and 5 inclusive if you're trying to simulate rolling a 6-sided die), with a bit of extra work. rand() itself generates a number between 0 and the library-defined constant value RAND_MAX, so to get a value in a smaller range you have to somehow map the larger range to the smaller one. The somewhat obvious way to do this is by computing a remainder (see starter program), but with some implementations of rand() this gives results that aren't very good. The conventional wisdom is therefore to instead try to do a more-direct map (e.g., to map to just two possible values, assign values from 0 through RAND_MAX/2 to 0 and the remaining values to 1).

   Your mission for this problem is to complete a C program that, given a ``seed'' $S$ , a number of samples $N$ , and a number of ``bins'' $B$ , generates a sequence of $N$ ``random'' numbers (using $S$ as described below), uses both methods to map each generated number to a number between 0 and $B-1$ inclusive, and shows the distribution of results as in the sample output below. To help you (I hope!) I'm providing a starter program, link below, which you should use as your starting point.

   Your program should use $S$ to ``seed'' the random sequence: By default rand() always starts with the same value (and produces the same sequence). You can make it produce different results by first calling srand() with an integer ``seed'' ($S$ here).

   Sample execution:
   

   [bmassing@diasw04]$ ./a.out
   seed?
   5
   how many samples?
   1000
   how many bins?
   6
   counts using remainder method:
   (0) 154
   (1) 188
   (2) 171
   (3) 161
   (4) 155
   (5) 171
   counts using quotient method:
   (0) 172
   (1) 175
   (2) 183
   (3) 150
   (4) 168
   (5) 152
   

   If you feel a bit more ambitious you could also have the program print maximum and minimum counts and the difference between them, as a crude measure of how uniform the distribution is:
   

   [bmassing@diasw04]$ ./a.out 
   seed?
   5
   how many samples?
   1000
   how many bins?
   6
   counts using remainder method:
   (0) 154
   (1) 188
   (2) 171
   (3) 161
   (4) 155
   (5) 171
   min = 154, max = 188, difference 34
   counts using quotient method:
   (0) 172
   (1) 175
   (2) 183
   (3) 150
   (4) 168
   (5) 152
   min = 150, max = 183, difference 33
   

   (You will get an extra-credit point for doing this.)

   Here is a starter program that prompts for the seed, generates a few ``random'' numbers, and illustrates the two methods of mapping to a specified range: rands.c.

   Of course, your program should check to make sure all the inputs are positive integers. (Yes, error checking is a pain, but it's an incentive to get better at copy-and-paste?)

2. (20 points) In class we have written two programs to compute an element of the Fibonacci sequence, one using a loop (iteration) and one using recursion. If you compile and run both programs, you will likely notice that the one using recursion can be quite slow with all but small inputs. The reason for that is fairly obvious if you think about how it works -- the recursive function does quite a lot of duplicate computation. One way to improve the performance of such a function is with a technique referred to as memoization², in which every time you compute a result you save it for possible reuse. Because I was curious myself about how the iterative and recursive versions compared, and how much memoization might help, I wrote a program that defines functions for all three approaches (using long long rather than long for the elements of the sequence, to allow computing larger values) and times them. For me only the simple recursive function took more than trivial time, so I added code to also count the number of recursive calls. You can find the result -- minus the actual computation of values! -- in fibonacci-starter.c. Sample output for an input of 46:
   

   which index?
   computing fibonacci(46)
   
   iterative version:
   result 2971215073 (time 0.00000000)
   
   recursive version:
   result 2971215073 (time 46.40543509, count 5942430145)
   
   memoized recursive version:
   result 2971215073 (time 0.00000095, count 91)
   

   Your mission is to fill in the blanks in this starter program so that it performs the desired computation. (Look for comments with the word FIXME.) For the iterative and simple recursive approaches, you can get the code from the course ``sample programs'' page; you will just need to change some data types from long to long long.

   For the memoized recursive approach, a simple way to use this technique is to have an array for saving previously-computed results, with the n-th element of the Fibonacci sequence stored as the n-th element of the array, and a value of 0 meaning ``has not been previously computed''. This array could be an additional argument for the function, or it could be a global variable. (Usually global variables are discouraged, but for this problem they might make sense.) The program should do something reasonable if this array is not big enough, perhaps just rejecting any input that would overflow it.

   The only changes you should need to make in the starter program are:

   • Fill in the body of functions fibonacci_iterate() and fibonacci_recurse(). It's okay to use code from the ``sample programs'' page (edited so it computes a long long), and in fact that is what I strongly recommend. (I didn't just do this for you myself because I hope that copying and pasting will encourage you to look at the copied/pasted code.)
   • Fill in the body of function fibonacci_recurse_m(). You can add additional parameters if you like, or use global variables for the saved values.
   Turn in the resulting code. Notice that all three functions should return the same value for the n-th element of the Fibonacci sequence; the only difference should be execution time and count of recursive calls. I'm not asking you (at this point?) to formally collect results that would show how the count of recursive calls, and the execution time, increases as the input variable increases, but you may find it interesting to try some different values and observe!

Footnotes

... apply).¹

Credit where credit is due: I based the wording of this list on a posting to a SIGCSE mailing list. SIGCSE is the ACM's Special Interest Group on CS Education.

...memoization²

No, that's not a typo -- there really is no ``r'' in this word!