Computer Science 3675
Fall 1999
Programming Assignment 1

Arithmetic on large integers can be done by storing an integer in an array, in binary notation. So each member of the array is either 0 or 1. Functions can be written to perform operations such as addition, subtraction and multiplication.

Functions

Write an implementation of addition and multiplication of nonnegative integers, stored in binary, in C++. Make your functions have the following contracts and prototypes.

  // sum(A,m,B,n,C) stores the sum of integers A and B into array C.
  //
  //   All integers are stored in binary as arrays of bits, with the least
  //   significant bit stored at index 0.
  //
  //   Array A has length m.
  //   Array B has length n.
  //
  //   If k = max(m,n), then array C must have at least k+1
  //   cells available.  Those k+1 cells are set to hold the
  //   binary sum of A and B.

  void sum(BIT A[], int m, BIT B[], int n, BIT C[]);

  // product(A,m,B,n,C) stores the product of integers A and B into
  // array C.
  //
  //   All integers are stored in binary as arrays of bits, with the least
  //   significant bit stored at index 0.
  //
  //   Array A has length m.
  //   Array B has length n.
  //
  //   Array C must have at least m+n cells available.  Those m+n
  //   bits are set to the product of A and B.

  void product(BIT A[], int m, BIT B[], int n, BIT C[]);

  typedef char BIT;

Requirements

Your program should meet the following requirements. (And yes, I will grade down for failure to meet these requirements, even if you have not been asked to meet such stringent requirements in the past.)

1. Your functions must not make any assumptions about how large the arrays are. For example, it is unacceptable to assume that the numbers have no more than 100 bits in them. It must be possible to compile your functions and put them into a library, and find that they can be used for arbitrarily large integers without recompiling them.

2. Your functions must not have requirements that are not stated in the contracts. For example, it is unacceptable to insist that array C be set to all zeros by the caller before calling sum or product.

3. Your functions must implement something close to the standard addition and multiplication algorithms. It is not acceptable, for example, to add x and y by starting at x and doing y increments. That is extremely slow. Similarly, it is not acceptable to multiply x and y by adding y to itself x times. That is also too slow.

   The multiplication algorithm that you learned in grade shool has you write down all of the intermediate products before adding them all up. It is not necessary to do that. Just accumulate the sum as you generate each intermediate product. Be sure to shift over an appropriate amount. You will find it convenient if you design a helper addition function that computes x*(2^z) + y, where x and y are integers stored in arrays and z is a nonnegative integer (just an int) giving the amount to shift x to the left before adding. That function can be used to perform addition (by setting z = 0) and as a tool for doing multiplication.

   (By 2^z I mean 2 to the z power. Multiplying by 2^z is the same as adding z zeros to the right end of a binary number, just as multiplying by 10^z is the same as adding z zeros to the end of a decimal number.)

4. Strive for simplicity and elegance in your program.

5. Comment your program well. Make it clear and readable. Literate programming is a form of programming where programs are written in a way that is intended to be read, as a textbook would be read. Get as close to literate programming as you can. Write your program for other people to read.

Testing the functions

IF YOU DO NOT TEST YOUR FUNCTIONS, YOU CAN REST ASSURED THAT THEY DO NOT WORK, AND THEY WILL FARE POORLY WHEN GRADED.

Here are some recommended tests. You would be well advised to do others as well. All numbers are in standard binary notation (high order bit first). You might get leading zeros, which you can ignore. For example, you might find that the sum of 1 and 1 is 01.

1. 0 + 0 = 0
2. 0 x 0 = 0
3. 1 + 1 = 10
4. 1 x 1 = 1
5. 10 + 11 = 101
6. 10 x 11 = 110
7. 101001 + 101111100 = 110100101
8. 101111100 + 101001 = 110100101
9. 101111100 x 101001 = 11110011011100
10. 111111 + 111111111111 = 1000000111110
11. 111111111111 + 111111 = 1000000111110
12. 111111 x 111111111111 = 111110111111000001
13. 111111111111 x 111111 = 111110111111000001

An easy way to do these tests is just to initialize some arrays in your program. For example, write

    BIT A3[] = {1,0,0,1,0,1};
    BIT B3[] = {0,0,1,1,1,1,1,0,1};

Keeping track of time

1. Time spent learning the language. If you are familiar with C++, this will probably be 0.

2. Time spent developing and writing the program. Count time spent thinking about how to write it.

3. Time spent testing and debugging. Include time spent getting out compile errors, checking test results, finding errors in your program, etc.

Report time spent actually working on the problem, not counting time in between work sessions where you are doing something else.

What to turn in

3675 Assn 1 your name

Be sure your functions have the correct prototypes. If they do not, they will not link with my tester.

The body of the email message should report your account of time spent, in hours. For example,

           Learning language: 0 hr
           Development:       2.5 hr
           Debugging/testing: 1.3 hr