Corning Community College
CSCS1320 C/C++ Programming
With any increasingly complex piece of code or environment, we must find effective means of organizing various processes or communications. This “Errata” section and related updates are one such instance of that; intended as a means of disseminating information on changes to the project, you can be informed what modifications have taken place, along with any unique actions you need to take to compensate.
Any typos, bugs, or other updates/changes to the project will be documented here.
Some changes may involve updates being made available to the project, in which case you'll be prompted with such notification and can run the available updating commands to synchronize your copy of the project with the changes.
To apply your skills in the implementation of prime number calculating algorithms.
In mathematics, a prime number is a value that is only evenly divisible by 1 and itself; it has just that one pair of factors, no others. Numbers that have divisibility/factors are classified as composite numbers.
The number 6 is a composite number, as in addition to 1 and 6, it also has the factors of 2 and 3.
The number 17, however, is a prime number, as no numbers other than 1 and 17 can be evenly divided into it.
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 ever-expanding 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 a prime number program using the straightforward, unoptimized brute-force algorithm, which determines the primality of a number in a “trial by division” approach.
The brute force approach is the simplest to implement (although at some cost).
As we will be looking to do some time/performance analysis and comparisons, it is often good to have a baseline. This program will be it.
To perform the process of computing the primality of a number, we simply attempt to evenly divide all the values between 2 and one less than the number in question. If any one of them divides evenly, the number is NOT prime, but instead composite.
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) 119 % 8 = 7 119 % 9 = 2 119 % 10 = 9 119 % 11 = 9 119 % 12 = 11 119 % 13 = 2 ...
Because, during our range of testing every value from 2-118, we find that 7 evenly divides into 119, it failed the test: 119 is not prime, but is instead a composite number.
Please NOTE: Even once a number is identified as composite, your primereg MUST CONTINUE evaluating the remainder of the values (up to 119-1, or 118). It might seem pointless (and it is for a production program), but I want you to see the performance implications this creates.
Some things to keep in mind on your implementation:
For simplicity, I have encoded important implementation information into the file name (and therefore resulting executable/binary) that will correspond to the indicated algorithm plus any optimizations.
To break it down, all prime programs will be of the form:
The optimizations we will be implementing in this project (and their naming values) include:
Unless specified in the encoded name, your algorithm should only implement the algorithm and optimization(s) specified.
That is, if your program to implement is primerego, that means you are ONLY to implement the brute force algorithm and odds-only checking. NO break on composite, NO sqrt() trick, etc. We are establishing separate data points for analytical comparison.
Some of these optimizations can co-exist easily (break + map, odd + sqrt()), others are partially compatible (map + odd can coexist in a certain form), while others are mutually exclusive (sqrt() and approximated square root conflict). So there are definitely a few combinations that are possible using this scheme.
Here are the variants you'll be implementing for this project:
This optimization to primereg will make but one algorithmic change, and that takes place at the moment of identifying a number as composite. So, if we had our 119 example above, and discovered that 7 was a factor:
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 (how does this impact overall performance???).
Make no other optimizations- this first project is to set up some important base line values that we can use for algorithmic comparison later on.
This optimization will check only the numbers that fall on either side of a factor of 6 for primality.
NOTE: If applicable, just display the initial “2” and “3” as hardcoded values.
This optimization will check only the odd numbers for primality, skipping the evens entirely.
NOTE: If applicable, just display the initial “2” as a hardcoded value.
This optimization employs the square root trick utilizing the C library's sqrt() function.
This optimization employs the approximated square root trick (NOT utilizing an existing square root function, but using simpler logic you implement to approximate the square root point).
An optimization to the previous process, which used sqrt(), this variation will do the exact same thing, but without using the sqrt() function. It will approximate the square root.
We know that a square root (especially a whole numbered square root), is when we have whole number factors that are squared. But in addition, only considering the whole number aspect of the square root, we start seeing series of values with the same whole square root value:
lab46:~$ count=0; for ((i=2; i<152; i++)); do printf "[%3d] %2d " "${i}" `echo "sqrt($i)" | bc -q`; let count=count+1; if [ "${count}" -eq 10 ]; then echo; count=0; fi; done; echo [ 2] 1 [ 3] 1 [ 4] 2 [ 5] 2 [ 6] 2 [ 7] 2 [ 8] 2 [ 9] 3 [ 10] 3 [ 11] 3 [ 12] 3 [ 13] 3 [ 14] 3 [ 15] 3 [ 16] 4 [ 17] 4 [ 18] 4 [ 19] 4 [ 20] 4 [ 21] 4 [ 22] 4 [ 23] 4 [ 24] 4 [ 25] 5 [ 26] 5 [ 27] 5 [ 28] 5 [ 29] 5 [ 30] 5 [ 31] 5 [ 32] 5 [ 33] 5 [ 34] 5 [ 35] 5 [ 36] 6 [ 37] 6 [ 38] 6 [ 39] 6 [ 40] 6 [ 41] 6 [ 42] 6 [ 43] 6 [ 44] 6 [ 45] 6 [ 46] 6 [ 47] 6 [ 48] 6 [ 49] 7 [ 50] 7 [ 51] 7 [ 52] 7 [ 53] 7 [ 54] 7 [ 55] 7 [ 56] 7 [ 57] 7 [ 58] 7 [ 59] 7 [ 60] 7 [ 61] 7 [ 62] 7 [ 63] 7 [ 64] 8 [ 65] 8 [ 66] 8 [ 67] 8 [ 68] 8 [ 69] 8 [ 70] 8 [ 71] 8 [ 72] 8 [ 73] 8 [ 74] 8 [ 75] 8 [ 76] 8 [ 77] 8 [ 78] 8 [ 79] 8 [ 80] 8 [ 81] 9 [ 82] 9 [ 83] 9 [ 84] 9 [ 85] 9 [ 86] 9 [ 87] 9 [ 88] 9 [ 89] 9 [ 90] 9 [ 91] 9 [ 92] 9 [ 93] 9 [ 94] 9 [ 95] 9 [ 96] 9 [ 97] 9 [ 98] 9 [ 99] 9 [100] 10 [101] 10 [102] 10 [103] 10 [104] 10 [105] 10 [106] 10 [107] 10 [108] 10 [109] 10 [110] 10 [111] 10 [112] 10 [113] 10 [114] 10 [115] 10 [116] 10 [117] 10 [118] 10 [119] 10 [120] 10 [121] 11 [122] 11 [123] 11 [124] 11 [125] 11 [126] 11 [127] 11 [128] 11 [129] 11 [130] 11 [131] 11 [132] 11 [133] 11 [134] 11 [135] 11 [136] 11 [137] 11 [138] 11 [139] 11 [140] 11 [141] 11 [142] 11 [143] 11 [144] 12 [145] 12 [146] 12 [147] 12 [148] 12 [149] 12 [150] 12 [151] 12
Or, if perhaps we instead order by square root value:
lab46:~$ oldsqrt=$(echo "sqrt(2)" | bc -q); for ((i=2; i<49; i++)); do newsqrt=$(echo "sqrt($i)" | bc -q); if [ "${newsqrt}" -ne "${oldsqrt}" ]; then echo; fi; printf "[%3d] %2d " "${i}" "${newsqrt}"; oldsqrt="${newsqrt}"; done; echo [ 2] 1 [ 3] 1 [ 4] 2 [ 5] 2 [ 6] 2 [ 7] 2 [ 8] 2 [ 9] 3 [ 10] 3 [ 11] 3 [ 12] 3 [ 13] 3 [ 14] 3 [ 15] 3 [ 16] 4 [ 17] 4 [ 18] 4 [ 19] 4 [ 20] 4 [ 21] 4 [ 22] 4 [ 23] 4 [ 24] 4 [ 25] 5 [ 26] 5 [ 27] 5 [ 28] 5 [ 29] 5 [ 30] 5 [ 31] 5 [ 32] 5 [ 33] 5 [ 34] 5 [ 35] 5 [ 36] 6 [ 37] 6 [ 38] 6 [ 39] 6 [ 40] 6 [ 41] 6 [ 42] 6 [ 43] 6 [ 44] 6 [ 45] 6 [ 46] 6 [ 47] 6 [ 48] 6
We see that the square root of 36 is 6, but so is the square root of 37, 38, 39… etc. up until we hit 49 (where the whole number square root increments to 7).
Therefore, if we were checking 42 to be prime, we'd only have to check up to 6.
We don't need a sqrt() function to tell us this, we can determine the approximate square root point ourselves- by squaring the current factor being tested, and so long as it hasn't exceeded the value we're checking, we know to continue.
There are some important lessons at play here:
Depending on how you implement this and the original sqrt() algorithms, this version may have a noticeable performance difference. If, on the other hand, you were really optimal in both implementations, the performance difference may be narrower (if negligible).
It is your task to write the following prime number variants:
Your program should:
I have made some skeleton files and a custom Makefile available for this project. Since we've amassed considerable experience manually compiling our files, it is time to start experiencing some of the other development tools that can automate or facilitate various processes.
I have written a tool, known as grabit, which will let you obtain the files I have put together for this project. To “grab” it:
lab46:~/src/cprog$ grabit cprog pnc1 make: Entering directory '/var/public/SEMESTER/cprog/pnc1' Commencing copy process for SEMESTER cprog project pnc1: -> Creating project pnc1 directory tree ... OK -> Copying pnc1 project files ... OK -> Synchronizing pnc1 project revision level ... OK -> Establishing sane file permissions for pnc1 ... OK *** Copy COMPLETE! You may now go to the '/home/USER/src/cprog/pnc1' directory *** make: Leaving directory '/var/public/SEMESTER/cprog/pnc1' lab46:~/src/cprog$
NOTE: You do NOT want to do this on a populated pnc1 project directory– it will overwrite files.
And, of course, your basic compile and clean-up operations via the Makefile.
Makefiles provide a build automation system for our programs, instructing the computer on how to compile files, so we don't have to constantly type compiler command-lines ourselves. I've also integration some other useful, value-added features that will help you with overall administration of the project.
Basic operation of the Makefile is invoked by running the command “make” by itself. The default action is to compile everything in the project directory.
Additional options are available, and they are provided as an argument to the make command. You can see the available options by running “make help”:
lab46:~/src/cprog/pnc1$ make help *******************[ C/C++ Programming pnc1 Project ]******************* ** make - build everything ** ** make showerrors - display compiler warnings/errors ** ** ** ** make debug - build everything with debug symbols ** ** make checkqty - runtime evaluation for qty ** ** make checkrange - runtime evaluation for range ** ** ** ** make verifyqty - check implementation for qty validity ** ** make verifyrange - check implementation for range validity ** ** make verifyall - verify project specifications ** ** ** ** make save - create a backup archive ** ** make submit - submit assignment (based on dirname) ** ** ** ** make update - check for and apply updates ** ** make reupdate - re-apply last revision ** ** make reupdate-all - re-apply all revisions ** ** ** ** make clean - clean; remove all objects/compiled code ** ** make help - this information ** ************************************************************************ lab46:~/src/cprog/pnc1$
A description of some available commands include:
The various “check” options do a runtime performance grid, allowing you to compare timings between your implementations.
The various “verify” options do more aggressive checks, helping to ensure your project falls within stated project specifications.
Just another “nice thing” we deserve.
To automate our comparisons, we will be making use of command-line arguments in our programs.
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)
There are two very important variables involved here (the types are actually what are important, the names given to the variables are actually quite, variable; you may see other references refer to them as things like “ac” and “av”):
The arguments are accessible via the argv array, in the order they were specified:
Additionally, let's not forget the argc variable, an integer, which contains a count of arguments (argc == argument count). If we provided argv[0] through argv[4], argc would contain a 5.
For example, if we were to execute the primereg program:
lab46:~/src/cprog/pnc1$ ./primeregb 128 1 2 2048
We'd have:
and let's not forget:
With the conditionally optional arguments as part of the program spec, for a valid execution of the program, argc could be a value anywhere from 3 to 5.
While there are a number of checks we should perform, one of the first should be a check to see if the minimal number of arguments has been provided:
if (argc < 3) // if less than 3 arguments (program_name + quantity + argv[2] == 3) have been provided { fprintf(stderr, "%s: insufficient number of arguments!\n", argv[0]); exit(1); }
Since argv[3] (lower bound) and argv[4] (upper bound) are conditionally optional, it wouldn't make sense to check for them in the overall count. But we can and do still want to stategically utilize argc to determine if an argv[3] or argv[4] is present.
Finally, we need to put the argument representing the maximum quantity into a variable.
I'd recommend declaring a variable of type int.
We will use the atoi(3) function to quickly convert the command-line arguments into int values:
max = atoi (argv[1]);
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 have the starting and ending times, we can display this to the timing file pointer. You'll want this line:
fprintf(stderr, "%8.4lf\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 “%8.4lf”, 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.
Your program output should be as follows (given the specified quantity):
lab46:~/src/cprog/pnc1$ ./primeregm 24 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 0.0001 lab46:~/src/cprog/pnc1$
The execution of the programs is short and simple- grab the parameters, do the processing, produce the output, and then terminate.
Here's an example that should generate an error upon running (based on project specifications):
lab46:~/src/cprog/pnc1$ ./primerego 32 1 0 ./primereg: invalid lower bound lab46:~/src/cprog/pnc1$
In this case, the program logic should have detected an invalid condition and bailed out before prime computations even began. No timing data is displayed, because exiting should occur even prior to that.
As indicated above, there is potential interplay with an active quantity and upper bound values. Here is an example where upper bound overrides quantity, resulting in an early termination (ie upper bound is hit before quantity):
lab46:~/src/cprog/pnc1$ ./primeregs 128 1 7 23 7 11 13 17 19 23 0.0001 lab46:~/src/cprog/pnc1$
Also for fun, I set the lower bound to 7, so you'll see computation starts at 7 (vs. the usual 2).
If you'd like to compare your implementations, I rigged up a Makefile checking rule called “make checkqty” and “make checkrange” which you can run to get a nice side-by-side runtime comparisons of your implementations.
In order to work, you MUST be in the directory where your pnc1 binaries reside, and must be named as such (which occurs if you ran make to compile them).
For instance (running on my implementation of the pnc1 programs):
lab46:~/src/cprog/pnc1$ make checkqty ========================================================= qty reg regm rego regb regs rega ========================================================= 32 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 64 0.0003 0.0001 0.0001 0.0001 0.0001 0.0001 128 0.0012 0.0004 0.0003 0.0002 0.0001 0.0001 256 0.0057 0.0020 0.0014 0.0009 0.0003 0.0003 512 0.0278 0.0098 0.0066 0.0038 0.0009 0.0009 1024 0.1348 0.0476 0.0318 0.0166 0.0025 0.0025 2048 0.6416 0.2317 0.1510 0.0727 0.0073 0.0073 4096 3.0281 1.0707 0.7096 0.3144 0.0218 0.0217 8192 ------ ------ 3.2926 1.3627 0.0649 0.0649 16384 ------ ------ ------ ------ 0.1955 0.1954 32768 ------ ------ ------ ------ 0.5910 0.5905 65536 ------ ------ ------ ------ 1.7891 1.7864 131072 ------ ------ ------ ------ ------ ------ ========================================================= verify: OK OK OK OK OK OK ========================================================= lab46:~/src/cprog/pnc1$
Or check range runtimes:
lab46:~/src/cprog/pnc1$ make checkrange ========================================================= range reg regm rego regb regs rega ========================================================= 32 0.0001 0.0000 0.0000 0.0000 0.0001 0.0000 64 0.0001 0.0001 0.0000 0.0000 0.0000 0.0000 128 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 256 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 512 0.0006 0.0003 0.0002 0.0002 0.0001 0.0001 1024 0.0023 0.0008 0.0006 0.0004 0.0002 0.0002 2048 0.0088 0.0032 0.0021 0.0013 0.0004 0.0004 4096 0.0344 0.0125 0.0082 0.0047 0.0010 0.0010 8192 0.1358 0.0495 0.0322 0.0167 0.0025 0.0025 16384 0.5402 0.1968 0.1270 0.0616 0.0065 0.0065 32768 2.1530 0.7857 0.5050 0.2271 0.0170 0.0171 65536 ------ 3.1395 2.0088 0.8468 0.0454 0.0455 131072 ------ ------ ------ 3.1817 0.1230 0.1230 262144 ------ ------ ------ ------ 0.3359 0.3359 524288 ------ ------ ------ ------ 0.9245 0.9240 1048576 ------ ------ ------ ------ 2.5601 2.5585 2097152 ------ ------ ------ ------ ------ ------ ========================================================= verify: OK OK OK OK OK OK ========================================================= lab46:~/src/cprog/pnc1$
If the runtime of a particular prime variant exceeds an upper runtime threshold (likely to be set at 1 second), 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 (maybe a couple of times) and the script will terminate.
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 see “MISMATCH”.
There's also a more rigorous verification step you can take, which runs your programs through a series to tests to see if they conform to project specifications:
lab46:~/src/cprog/pnc1$ make verifyall ========================================================= reg regm rego regb regs rega ========================================================= qtynorm: OK OK OK OK OK OK qtypart: OK OK OK OK OK OK rngnorm: OK OK OK OK OK OK rngpart: OK OK OK OK OK OK coop: OK OK OK OK OK OK coop2: OK OK OK OK OK OK coop3: OK OK OK OK OK OK noargs: OK OK OK OK OK OK invargs: OK OK OK OK OK OK invqty: OK OK OK OK OK OK invnary: OK OK OK OK OK OK invlow: OK OK OK OK OK OK invhigh: OK OK OK OK OK OK ========================================================= lab46:~/src/cprog/pnc1$
The “verifyall” is an industrial grade verification; there are 13 specific tests performed, they are:
If you'd actually to see the output your program's output is being tested against, that can be found in the /usr/local/etc directory in the file primeTEST, where “TEST” is the name of the verify test mentioned above.
For example, if you wanted to see the intended output of the invnary test, that would be found in:
You could easily run your program with the stated arguments for the test, then use cat to display the test results and do a visual comparison.
Analyze the times you see… do they make sense, especially when comparing the algorithm used and the quantity being processed? These are related to some very important core Computer Science considerations we need to be increasingly mindful of as we design our programs and implement our solutions. Algorithmic complexity and algorithmic efficiency will be common themes in all we do.
To successfully complete this project, the following criteria must be met:
To submit this program to me using the submit tool, run the following command at your lab46 prompt:
lab46:~/src/cprog/pnc1$ make submit removed ‘primeregb’ removed ‘primeregm’ removed ‘primerego’ removed ‘primeregs’ removed ‘primerega’ removed ‘errors’ Project backup process commencing Taking snapshot of current project (pnc1) ... OK Compressing snapshot of pnc1 project archive ... OK Setting secure permissions on pnc1 archive ... OK Project backup process complete Submitting cprog project "pnc1": -> ../pnc1-DATESTRING-HR.tar.gz(OK) SUCCESSFULLY SUBMITTED
You should get that final “SUCCESSFULLY SUBMITTED” with no error messages occurring. If not, check for typos and or locational mismatches.
Grand total points:
80:pnc1:final tally of results (80/80)
What I will be looking for (for each file):
*:pnc1:primeALGO.c compiles cleanly, no compiler messages [1/1] *:pnc1:primeALGO.c implements only specified algorithm [4/4] *:pnc1:primeALGO.c consistent indentation throughout code [1/1] *:pnc1:primeALGO.c relevant comments throughout code [1/1] *:pnc1:primeALGO.c code conforms to project specifications [2/2] *:pnc1:primeALGO.c runtime output conforms to specifications [1/1] *:pnc1:primeALGO.c make checkqty test times within reason [1/1] *:pnc1:primeALGO.c make checkrange test times within reason [1/1] *:pnc1:primeALGO.c make verifyall tests succeed [4/4]
As the optimizations improve upon others, some evaluations will be based upon differences between a baseline (in some cases, primereg) and the optimization.