To apply your skills in the implementation of prime number calculating algorithms.

In addition to the new skills required on previous projects, to successfully accomplish/perform this project, the listed resources/experiences need to be consulted/achieved:

• ability to make decisions (if statement)
• ability to iterate sections of code (for/while statement)

A concept in Computer Science curriculum is the notion of computational/algorithmic complexity.

Basically, a solution to a problem exists on a spectrum of efficiency (typically constrained by time vs. space): if optimizing for time, the code size tends to grow.

Additionally, if optimizing for time (specifically to reduce the amount of time taken), strategic approaches are taken to reduce unnecessary or redundant operations (yet still achieving the desired end results).

This project will endeavor to introduce you to the notion that the algorithms and constructs you use in coding your solution can and do make a difference to the overall runtime of your code.

In mathematics, a prime number is a value that is only evenly divisible by 1 and itself; it has no other factors. Numbers that have divisibility/factors are known as composite numbers.

The number 6 is a composite value, as in addition to 1 and 6, it also has the factors of 2 and 3.

The number 17 is a prime number, as no numbers other than 1 and 17 can be evenly divided.

As of yet, there is no quick and direct way of determining the primality of a given number. Instead, we must perform a series of tests to determine if it fails primality (typically by proving it is composite).

This process incurs a considerable amount of processing overhead on the task, so much so that increasingly large values take increasing amounts of time. Often, approaches to prime number calculation involve various algorithms, which offer various benefits (less time) and drawback (more complex code).

Your task for this project is to implement 3 prime number programs:

1. brute force prime calculation
2. square root-optimized brute force calculation
3. a further optimized solution

The brute force approach is the simplest to implement (and likely also the worst-performing). We will use it as our baseline (it is nice to have something to compare against).

To perform it, we simply attempt to evenly divide all the values between 1 and the number in question. If any one of them divides evenly, the number is NOT prime, but instead a composite value.

Checking the remainder of a division indicates whether or not a division was clean (having 0 remainder indicates such a state).

For example, the number 11:

11 % 2 = 1 (2 is not a factor of 11)
11 % 3 = 2 (3 is not a factor of 11)
11 % 4 = 3 (4 is not a factor of 11)
11 % 5 = 1 (5 is not a factor of 11)
11 % 6 = 5 (6 is not a factor of 11)
11 % 7 = 4 (7 is not a factor of 11)
11 % 8 = 3 (8 is not a factor of 11)
11 % 9 = 2 (9 is not a factor of 11)
11 % 10 = 1 (10 is not a factor of 11)

Because none of the values 2-10 evenly divided into 11, we can say it passed the test: 11 is a prime number

On the other hand, take 119:

119 % 2 = 1 (2 is not a factor of 119)
119 % 3 = 2 (3 is not a factor of 119)
119 % 4 = 3 (4 is not a factor of 119)
119 % 5 = 4 (5 is not a factor of 119)
119 % 6 = 5 (6 is not a factor of 119)
119 % 7 = 0 (7 is a factor of 119)

Because 7 evenly divided into 119, it failed the test: 119 is not a prime, but instead a composite number.

There is no further need to check the remaining values, as once we have proven the non-primality of a number, the state is set: it is composite. So be sure to use a break statement to terminate the computation loop (will also be a nice boost to runtime).

An optimization to the computation of prime numbers is the square root trick. Basically, if we've processed numbers up to the square root of the number we're testing, and none have proven to be evenly divisible, we can also assume primality and bail out.

The C library has a sqrt() function available through including the math.h header file, and linking against the math library at compile time (add -lm to your gcc line).

To use sqrt(), we pass in the value we wish to obtain the square root of, and assign the result to an int:

int x = 25;
int y = 0;
 
y = sqrt(x);
 
// y should be 5 as a result

For instance, the number 37 (using the square root optimization), we find the square root (whole number) of 37 is 6, so we only need to check 2-6:

37 % 2 = 1 (2 is not a factor of 37)
37 % 3 = 1 (3 is not a factor of 37)
37 % 4 = 1 (4 is not a factor of 37)
37 % 5 = 2 (5 is not a factor of 37)
37 % 6 = 1 (6 is not a factor of 37)

Because none of these values evenly divides, we can give 37 a pass: it is a prime

This will dramatically improve the runtime, and offers a nice comparison against our brute force baseline.

There are many other methods, approaches, and tweaks that can be employed to further improve runtime (while maintaining accuracy– all your solutions must match: the same prime numbers should be identified no matter which program is run).

So I'd like you to explore other optimizations that can be made, be it using other prime number algorithms, further refining existing ones, or playing off patterns in numbers.

One assumption I will allow you to make for your optimized solution is that the single-digit primes (2, 3, 5, 7) can be assumed prime, and just printed out if having them be calculated would otherwise break your algorithm (might be helpful to some people; I certainly found it useful in some of my solutions).

• Prime patterns of 6
• Visualization of Primes in a web of 6 divisions
• Visualization of Primes in a web of 12 divisions
• Ulam spiral
• Sieve of Eratosthenes

Of particular note: the sieve algorithms take advantage of a increased storage space, where others (like brute force) are predominantly time-based. The sieve is also more detailed… even if you don't decide to implement a sieve, take a look and compare the algorithm to what you've done to see the differences in approaches.

It is your task to write 3 separate prime number calculating programs:

1. primebrute.c: for your brute force implementation
2. primesqrt.c: for your square root-optimization of the brute force
3. primeopt.c: for your optimized solution, be it basing off the existing ones, or taking a different approach

Your program should:

• obtain 2 parameters from the command-line (see command-line arguments section below):
  • argv[1]: maximum value to calculate to (your program should run from (approximately) 2 through that number (inclusive of that number)
  • argv[2]: visibility. If a 1 is provided, print out the prime numbers in a space separated list; if a 0 is provided, run silent: only display the runtime information.
  • these values should be positive integer values; you can make the assumption that the user will always do the right thing.
• start your stopwatch (see timing section below):
• perform the algorithm against the value
• if enabled, display the prime numbers found in the range
• output the processing run-time to STDERR (do this always).
• your output MUST be conformant to the example output in the execution section below. This is also a test to see how well you can implement to specifications. Basically:
  • if primes are being displayed, they are space-separated (first prime hugs the left margin), and when all said and done, a newline is issued.
  • the timing information will be displayed in accordance to code I will provide (in the timing section).

All your programs MUST perform the calculations to determine primality- you may not always be printing it out (depending on argv[2]), but work must be done to ensure the value is identified as a prime/composite value.

For example:

if (show == 1)
{
	work to determine if it is prime
	if prime
		print number
}

will actually skip the core processing, and you’ll see some amazing runtimes as a result. They may be amazing, but they’re not real, because you’re not actually doing anything.

What you want instead:

work to determine if it is prime
if (show == 1)
{
	if prime
		print number
}

there are many ways to express the above, through compound if statements and other arrangements, but notice how nothing is holding back “work to determine if it is prime”.

That also isn’t to say you can’t avoid doing a work run if you’re able to determine its non-primality with a simple pretest (even value, factor of 3, etc.), but that’s actually considered more of the core “work”, so it is more than okay (and encouraged in the primeopt).

To automate our comparisons, we will be making use of command-line arguments in our programs. As we have yet to really get into arrays, I will provide you same code that you can use that will allow you to utilize them for the purposes of this project.

We don't need any extra header files to use command-line arguments, but we will need an additional header file to use the atoi(3) function, which we'll use to quickly turn the command-line parameter into an integer, and that header file is stdlib.h, so be sure to include it with the others:

#include <stdio.h>
#include <stdlib.h>

To accept (or rather, to gain access) to arguments given to your program at runtime, we need to specify two parameters to the main() function. While the names don't matter, the types do.. I like the traditional argc and argv names, although it is also common to see them abbreviated as ac and av.

Please declare your main() function as follows:

int main(int argc, char **argv)

The arguments are accessible via the argv array, in the order they were specified:

• argv[0]: program invocation (path + program name)
• argv[1]: our maximum / upper bound
• argv[2]: visibility (1 to show primes, 0 to be silent)

There are ways to do flexible argument parsing, and even to have dashed options as we have on various commands. But such things are beyond the scope of our current endeavors, so we will stick to this basic functionality for now.

Although I'm not going to require extensive argument checking for this project, here's how we would check to see if the minimal number of arguments has been provided:

    if (argc < 3)  // if less than 3 arguments have been provided
    {
        fprintf(stderr, "Not enough arguments!\n");
        exit(1);
    }

If you're wondering, “why 3? I thought we only had 2.”, C includes the program's name as the first argument, so we want program + max + visibility, or 3.

Finally, we need to put the arguments representing the maximum value and visibility settings into variables.

I'd recommend declaring two variables of type int.

We will use the atoi(3) function to quickly convert the command-line arguments into int values:

    max  = atoi(argv[1]);
    show = atoi(argv[2]);

And now we can proceed with the rest of our prime implementation.

Often times, when checking the efficiency of a solution, a good measurement (especially for comparison), is to time how long the processing takes.

In order to do that in our prime number programs, we are going to use C library functions that obtain the current time, and use it as a stopwatch: we'll grab the time just before starting processing, and then once more when done. The total time will then be the difference between the two (end_time - start_time).

We are going to use the gettimeofday(2) function to aid us in this, and to use it, we'll need to do the following:

In order to use the gettimeofday(2) function in our program, we'll need to include the sys/time.h header file, so be sure to add it in with the existing ones:

#include <stdio.h>
#include <stdlib.h>
#include <sys/time.h>

gettimeofday(2) uses a struct timeval data type, of which we'll need to declare two variables in our programs (one for storing the starting time, and the other for the ending time).

Please declare these with your other variables, up at the top of main() (but still WITHIN main()– you do not need to declare global variables).

    struct timeval time_start; // starting time
    struct timeval time_end;   // ending time

To use gettimeofday(2), we merely place it at the point in our code we wish to take the time.

For our prime number programs, you'll want to grab the start time AFTER you've declared variables and processed arguments, but JUST BEFORE starting the driving loop doing the processing.

That call will look something like this:

    gettimeofday(&time_start, 0);

The ending time should be taken immediately after all processing (and prime number output) is completed, and right before we display the timing information to STDERR:

    gettimeofday(&time_end, 0);

Once we having the starting and ending times, we can display this to STDERR. You'll want this line:

    fprintf(stderr, "%10.6lf\n", time_end.tv_sec - time_start.tv_sec + ((time_end.tv_usec - time_start.tv_usec) / 1000000.0));

For clarity sake, that format specifier is “%10.6lf”, where the “lf” is “long float”, that is NOT a number one but a lowercase letter 'ell'.

And with that, we can compute an approximate run-time of our programs. The timing won't necessarily be accurate down to that level of precision, but it will be informative enough for our purposes.

Several operating behaviors are shown as examples.

Brute force showing primes:

lab46:~/src/cprog/pnc0$ ./primebrute 90 1
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 
  0.000088
lab46:~/src/cprog/pnc0$ 

Brute force not showing primes:

lab46:~/src/cprog/pnc0$ ./primebrute 90 0
  0.000008
lab46:~/src/cprog/pnc0$ 

Similarly, for the square root version (showing primes):

lab46:~/src/cprog/pnc0$ ./primesqrt 90 1
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 
  0.000089
lab46:~/src/cprog/pnc0$ 

And, without showing primes:

lab46:~/src/cprog/pnc0$ ./primesqrt 90 0
  0.000006
lab46:~/src/cprog/pnc0$ 

Don't be alarmed by the visible square root actually seeming to take MORE time; we have to consider the range as well: 90 is barely anything, and there is overhead incurred from the sqrt() function call. The real savings will start to be seen once we get into the thousands (and beyond).

And that's another neat thing with algorithm comparison: a “better” algorithm may have a sweet spot or power band: they may actually perform worse until (especially at the beginning).

The same goes for your optimized solution (same parameters).

The execution of the programs is short and simple- grab the parameters, do the processing, produce the output, and then terminate.

If you'd like to compare your implementations, I rigged up a script called primerun which you can run.

In order to work, you MUST be in the directory where your primebrute, primesqrt, and primeopt binaries reside, and they must be named as such.

For instance (running on my implementations):

lab46:~/src/cprog/pnc0$ primerun
============================================
   range       brute        sqrt         opt 
============================================
       8    0.000002    0.000002    0.000002  
      16    0.000002    0.000002    0.000002  
      32    0.000003    0.000004    0.000002  
      64    0.000005    0.000020    0.000003  
     128    0.000012    0.000023    0.000003  
     256    0.000037    0.000029    0.000006  
     512    0.000165    0.000036    0.000014  
    1024    0.000540    0.000080    0.000033  
    2048    0.001761    0.000187    0.000078  
    4096    0.006115    0.000438    0.000189  
    8192    0.021259    0.001036    0.000458  
   16384    0.077184    0.002520    0.001153  
   32768    0.281958    0.006156    0.002826  
   65536    1.046501    0.015234    0.007135  
  131072    5.160141    0.045482    0.021810  
  262144    --------    0.119042    0.057520  
  524288    --------    0.301531    0.146561  
 1048576    --------    0.758027    0.370700  
 2097152    --------    1.921014    0.943986  
 4194304    --------    4.914725    2.423202  
 8388608    --------    --------    --------  
============================================
 verify:       OK          OK          OK     
============================================
lab46:~/src/cprog/pnc0$ 

For evaluation, each test is run 4 times, and the resulting time is averaged. During development, I have it set to only run each test once.

If the runtime of a particular prime variant exceeds an upper threshold (likely to be set at 2 seconds), it will be omitted from further tests, and a series of dashes will instead appear in the output.

If you don't feel like waiting, simply hit CTRL-c and the script will terminate.

In the example output above, my primeopt is playing with an implementation of the 6a+/-1 algorithm.

I also include a validation check- to ensure your prime programs are actually producing the correct list of prime numbers. If the check is successful, you will see “OK” displayed beneath in the appropriate column; if unsuccessful, you will be “MISMATCH”.

If you'd like to experiment with other variations, the script also recognizes prime variants of the following names:

• primeopt0 (for an additional optimization)
• primeopt1 (and another)
• primeopt2 (if you'd like another entry for another optimization)
• primeopt3 (for yet another optimization)
• primeopt4 (and one more; hey, I want you to have nice things)

There will be an additional bonus point opportunity with this project, based on processing run-time of your optimized solution.

To successfully complete this project, the following criteria must be met:

• Code must compile cleanly (no warnings or errors)
• Output must be correct, and match the form given in the sample output above.
• Code must be nicely and consistently indented (you may use the indent tool)
• Code must utilize the algorithm(s) presented above:
  • primebrute.c must do the unoptimized brute force method
  • primesqrt.c must utilize the brute force method along with the square root trick (no other tricks)
  • primeopt.c is your attempt at optimizing the code, using whatever additional tricks (or enhancements to tricks) to further improve runtime.
• Code must be commented
  • have a properly filled-out comment banner at the top
    • be sure to include any compiling instructions
  • have at least 20% of your program consist of //-style descriptive comments
• Output Formatting (including spacing) of program must conform to the provided output (see above).
• Track/version the source code in a repository
• Submit a copy of your source code to me using the submit tool.

To submit this program to me using the submit tool, run the following command at your lab46 prompt:

$ submit cprog pnc0 primebrute.c primesqrt.c primeopt.c
Submitting cprog project "pnc0":
    -> primebrute.c(OK)
    -> primesqrt.c(OK)
    -> primeopt.c(OK)

SUCCESSFULLY SUBMITTED

You should get some sort of confirmation indicating successful submission if all went according to plan. If not, check for typos and or locational mismatches.