School of Computing and Information Systemscomp10002 Foundations of AlgorithmsSemester 2, 2023Assignment 1
1. Learning Outcomes
In this assignment you will demonstrate your understanding of arrays, strings, functions, and thetypedef facility. You may also (but are not required to) use structs (see Chapter 8) if you wish.You must not make any use of malloc() (Chapter 10) or file operations (Chapter 11)in this project.
2. Big Numbers
It is sometimes necessary to work with very big numbers. For example, application areas such ascryptography require exact manipulation of integers containing hundreds of digits. And by now youwill be aware that the standard int and double data types do not possess that ability.In this project you will develop a high-precision numeric calculator. You’ll be using C arraysto store the digits of the numbers, so it won’t quite be arbitrary precision; but it will certainly bepossible to have 500-digit values manipulated exactly within reasonable computation times, which isbig enough for many purposes.Before doing anything else, you should copy the skeleton program ass1-skel.c from the LMSAssignment 1 page, and spend an houror two reading through the code, understanding how it fitstogether. Check that you can compile it via either Grok or aterminal shell and gcc. Once you have itcompiled, try this sequence:
mac: ./ass1-skel
> a=12345
> a+23456
> a?
register a: 35801
> a+a
> a?
register a: 71602
> ^D
ta daa!!!The “>”s are a prompt printed by the program, with the remainder of each of those lines the commandstyped by the user. There are also two output lines in response to the “?” commands, and the final (whatelse?!) “ta daa!!!” message printed by the program as it exits. (Notethat there is some fiddly codethat makes use of the isatty() function to decide where to write each output line, so that the rogramworks sensibly when input commands come from a file, and also when the output is directed to anothertext file. You do not need to understand how all that code works.)The calculator maintains 26 one-letter “variables” (or registers),each of which has an initial valueof zero, and can store one integer value. Each “command” operates onone of those 26 variables,applying a single operator to it, using (except for the “?” operator) one further non-negativeintegeroperand. So, in the example shown above, register “a” is firstassigned the value 12345; then it has23456 added to it; then it is printed; and then register “a” has register “a” added to it; andthen it isprinted a second time.The skeleton program uses a simple array of intvariables to storethe 26 registers. This skeletonis provided as a starting point, so that you don’t have todevelopawhole lot of uninteresting functions.The only operators provided in the skeleton program are “=” (assignment); “+” (addition); and “?”(printing). And the arithmetic will suffer from integer overflow, ofcourse.
3. Stage 1 – Find Your Own Type (12/20 marks)
The first change in the program is to employ a new definition for the type longint_t so that itbecomes an array of INTSIZE decimal digits, each stored as an int in an array, plus a matching buddyvariable stored in the first array position. That is, each register will be anarray of int (rather than asingle int), with the digits stored inreverse order, and with the first element in the array storing thearray’s buddy variable. (You’ll understand why the digits are to bestored in reverse order once youstart coding your solution.) Forexample, the number 12,345,542 has eight digits, and would bestoredin a variable longint_t v as:v[0] v[1] v[2] v[3] v[4] v[5] v[6] v[7] v[8] v[9] v[10] ...8 2 4 5 5 4 3 2 1 ? ? ...where v[0] stores the (current) number of digits in the register’s value; where “?” represents a valuethat isn’t currently in use; and where thearray v is declared to contain a total of INTSIZE+1 elements.All input numbers will be non-negative integers, and nor is a subtractionoperator required. Thatmeans that you do not need to store a sign, nor worry about negative numbers.If you wish to read Chapter 8 early, you may instead define and use a suitable struct for each ofthe registers (rather than the array option that is illustrated in the example), and then maintain the setof registers in an array of struct (rather than as an array of arrays). Note that the use of struct typesis not required in this project, and you can get fullmarks without them.As part of this first stage you’ll need to rewrite the function do plus() so that two variables ofyour new type longint_t are added correctly, and you’ll also need to modify severalother functions(zero_vars() and do_assign(), plus others) so that they operate correctly after you have changedthe underlying type. You should carefully plan the required changes before you jump in and starttyping, because you’ll need to do quite a bit of editing before getting the program back to “compilable/testable” state again.As part of your changes, your program should check for overflow beyond INTSIZE digits bothwhen constants are being input, and also when addition is being performed. If an overflow situationarises your program should print a suitable error message (your choice of wording) and then exit.Successful completion of this stage means that your program should be able to handle additionsinvolving numbers of up to INTSIZE (currently 500) decimal digits. Your program should detect anyoperations that overflow beyond INTSIZE digits, and writean error message of your choice and thenexit, returning EXIT FAILURE.
4.Stage 2 – Go Forth and Multiply (16/20 marks)
You surely knew it was coming, right? Well, now is the time to addmultiplication operator:
> b=123456789
> b*987654321
> b?
register b: 121,932,631,112,635,269
Yes, you need to sit down with pen and paper and remember your long multiplications from primary
school, and then turn that process into a C function in your program. If you get stuck, ask your parents
(or grandparents!), they might still remember how to do this kind of multiplication.
You should not implement multiplication via repeated addition, that will be frighteningly inefficient. At the other end of the spectrum, nor are you required to investigate efficient integer multiplication algorithms. It is expected that your process for multiplying two n-digit numbers will involve As a second change required in this stage, note the commas in the output values, following theAustralian convention of placing them every three digits, counting from the right. (Other cultureshave different conventions.)Stage 3a – Seize the Power (19/20 marks)
And what comes after multiplication? Exponentiation, of course.
> c=2
> c^50
> c?
register c: 1,125,899,906,842,624Just as multiplication shouldn’t be done via repeated addition,exponentiation shouldn’t really be doneby repeated multiplication – it isn’t an efficient algorithm. But in the context of this project you maycarry out exponentiation via repeated multiplication (and that is certainly what you should get workingfirst) since the largest (interesting, non-overflowing)exponent that can arise within 500-digit numbers(that is, generating values less than 10500) is log2 10500 = 1661, andexecuting a few thousand manydigit multiplications should still only take atiny fraction of a second. Have fun!
5.Stage 3b – Divide, and then Conquer (20/20 marks)
[Note: This last operator involves a great deal of further work forone measly little mark. Thinkcarefully about your other subjects and assessment priorities before embarking on this stage, becauseit might become a black hole that consumes a lot of your precious time. There is no shame at all insubmitting a program that handles Stages 1 to 3a only; and for many (perhaps even most) of you, thatwill be the correct decision to make.]For an exciting adventure, add in integer division:
> d=b
> d/c
> d?register d: 108with b and c having the values assigned in the two previous examples. Hopefully your grandparents haven’t forgotten howto do long division and can teach it to you (and I bet your parent; orlook at https://en.wikipedia.org/wiki/Long_division. You may not implement division viarepeated subtraction, but probably will need some subtraction-based looping for each digit in thequotientthat is generated.General Tips...You will probably find it helpful to include a DEBUG mode in your program that prints out intermediate data and variable values. Use #if (DEBUG) and #endif around such blocks of code, and then#define DEBUG 1 or #define DEBUG 0 at the top. Turn off the debug mode when making yourfinal submission, but leave the debug code in place. The FAQ page has more information about this.The sequence of stages described in this handout is deliberate – it represents a sensible paththrough to the final program. You can, of course, ignore the advice and try and write final programin a single effort, without developing it incrementally and testing it in phases. You might even getaway with it, this time and at this somewhat limited scale, and develop a program that works. But ingeneral, one of the key things that makes some people better at programming than others is the abilityto see a design path through simple programs, to more comprehensive programs, to finalprograms,hat keeps the complexity under control at all times. That is one of the skills this subject is intended toteach you. And if you submit each of the stages as you complete it, you’ll know that you are accumulating evidence should you need to demonstrate your progress in the event of a special considerationapplication becoming necessary.Boring But Important...This project is worth 20% of your final mark, and is due at 6:00pm onFriday 15 September.Submissions that are made after the deadline will incur penalty marks at the rate of two marks perday or part day late. Students seeking extensions for medical or other “outside my control” reasonsshould [email protected] as soon as possible after those circumstances arise. If youattend a GP or other health care service as a result of illness, be sure to obtain a letter from themthat describes your illness and their recommendation for treatment. Suitable documentation should beattached to all extension requests.Multiple submissions may be made; only the last submission that you make before the deadlinewill be marked. If you make any late submission at all, your on-time submissions will be ignored, andif you have not been granted an extension, the late penalty will be applied.A rubric explaining the marking expectations is linked from the LMS, and you should study itcarefully. Marks and feedback will be provided approximately two weeks after submissions close.Academic Honesty: You may discuss your work during your workshop, and with others in the class,but what gets typed into your program must be individual work, not copied from anyone else. So, donot give hard copy or soft copy of your work to anyone else; do not have any “accidents” that allowothers to access your work; and do not ask others to give you their programs “just so that I can take alook and get some ideas, I won’t copy, honest”. The best way to help your friends in this regard is tosay a very firm “no” if they ask to see your program, pointing out that your “no”, and their acceptanceof that decision, are the only way to preserve your friendship. See https://academicintegrity.unimelb.edu.au for more information. Note also that solicitation of solutions via posts to onlineforums, whether or not there is payment involved, is also Academic Misconduct. In the past studentshave had their enrolment terminated for such behavior.The skeleton program includes an Authorship Declaration that you must “sign” and include at thetop of your submitted program. Marks will be deducted if you do not include the declaration, or donot sign it, or do not comply with its expectations. A sophisticated program that undertakes deepstructural analysis of C code identifying regions of similarity will be run over all submissions.Students whose programs are identified as containing significant overlaps will have substantialmark penalties applied, or be referred to the Student Center for possible disciplinary action.Nor should you post your code to any public location (github, codeshare.io, etc) while the assignment is active or prior to the release of the assignment marks.And remember, algorithms are fun!
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