PROJECT: Prime Number Computation (PNC1)

OBJECTIVE

Taking your naive/brute force trial-by-division prime number program from pnc0, implement some algorithmic optimizations and analysis its improvements to performance over similar workloads.

TASK

Implement program in Vircon32 C, that:

performs the “trial-by-division” process on a range of values, 2-N
- in addition to the naive/brute force implementation from pnc0, you should implement optimizations for:
  - break on composite
  - odds-only
  - sqrt trick
  - and all the combinations of these optimizations thereof. With brute, there should be 8 total variations you are testing.
- the values for N are some sufficient quantity still small enough to fit within an integer
- the values for N will have some relationship (powers of 2, powers of 10/magnitudes) that ideally can be computed via some loop/equation (ie 1024, 2048, 4098, 8192, 16384, etc.)
  - the upper bound should be fairly easy to change, even if it requires an edit and recompile.
- the values for N have some sufficient quantity large enough where its upper set values will take some amount of time to compute (fast enough to have some relatable value, not to exceed 16 seconds)
- for each value of N:
  - display that N/upper bound
  - tally: display the number of primes identified (2-N)
  - display the amount of time taken to do the total computation for that value of N, out to at least 4 decimal places
  - display each N value and result in an arrangement on the screen that can be clearly identified and read by the viewer
timing should go out, as reasonable, to a few decimal places, and should be consistent across all attempts.
- timing is on the computational process only, not the display of results.
create a graph (using some external tool) that plots the performance of the C implementation working on various workloads of this brute force algorithm according to the various N's and the time it took. Share your graph of your results on the class discord and on the project documentation page.
- a line graph is the suggested best candidate
this will not be an interactive program: it starts up, does its thing, outputs it results, then halts.
the brute force implementation is meant as our baseline. As such, it should not contain any optimizations or attempted improvements. Here in pnc1, this base implementation should be the least efficient. This is important, to allow us to realize the impact of various improvements we will be making in those upcoming projects.

REFERENCE

The following are reference screenshots of what your implementations should approximate.

PNC1

C implementation

EDIT

You will want to go here to edit and fill in the various sections of the document:

https://lab46.g7n.org/notes/fall2024/projects/pnc1

PNCX

algorithm: brute force / trial-by-division

variant: naive

The naive implementation is our baseline: implement with no awareness of potential tweaks, improvements, or optimizations. This should be the worst performing when compared to any optimization.

START TIMEKEEPING
NUMBER: FROM 2 THROUGH UPPERBOUND:
    ISPRIME <- YES
    FACTOR: FROM 2 THROUGH NUMBER-1:
        SHOULD FACTOR DIVIDE EVENLY INTO NUMBER:
            ISPRIME <- NO
    PROCEED TO NEXT FACTOR
    SHOULD ISPRIME STILL BE YES:
        INCREMENT OUR PRIME TALLY
PROCEED TO NEXT NUMBER
STOP TIMEKEEPING

variant: break on composite (BOC)

just add a break; statement within your brute loop like so:

START TIMEKEEPING
NUMBER: FROM 2 THROUGH UPPERBOUND:
    ISPRIME <- YES
    FACTOR: FROM 2 THROUGH NUMBER-1:
        SHOULD FACTOR DIVIDE EVENLY INTO NUMBER:
            ISPRIME <- NO
            BREAK
    PROCEED TO NEXT FACTOR
    SHOULD ISPRIME STILL BE YES:
        INCREMENT OUR PRIME TALLY
PROCEED TO NEXT NUMBER
STOP TIMEKEEPING

variant: odds-only processing

Start at 3 and increment by two to get only odd numbers. Then add one to tally count to account for 2 like so:

START TIMEKEEPING
NUMBER: FROM 3 THROUGH UPPERBOUND:
    ISPRIME <- YES
    FACTOR: FROM 3 THROUGH NUMBER-1:
        SHOULD FACTOR DIVIDE EVENLY INTO NUMBER:
            ISPRIME <- NO
    PROCEED TO NEXT FACTOR BY TWO
    SHOULD ISPRIME STILL BE YES:
        INCREMENT OUR PRIME TALLY
PROCEED TO NEXT NUMBER BY TWO
ONCE UPPERBOUND IS REACHED ADD A ONE TO YOUR PRIME TALLY TO ACCOUNT FOR NOT STARTING AT TWO
STOP TIMEKEEPING

variant: sqrt point

Say you're using i for the outer loop and j for the inner loop, now rather that j < i you want j * j < = i

START TIMEKEEPING
NUMBER: FROM 2 THROUGH UPPERBOUND:
    ISPRIME <- YES
    FACTOR: FROM 2 * 2 THROUGH NUMBER-1:
        SHOULD FACTOR DIVIDE EVENLY INTO NUMBER:
            ISPRIME <- NO
    PROCEED TO NEXT FACTOR BY TWO
    SHOULD ISPRIME STILL BE YES:
        INCREMENT OUR PRIME TALLY
PROCEED TO NEXT NUMBER BY TWO
ONCE UPPERBOUND IS REACHED ADD A ONE TO YOUR PRIME TALLY TO ACCOUNT FOR NOT STARTING AT TWO
STOP TIMEKEEPING

variant: break+odds

START TIMEKEEPING
NUMBER: FROM 3 THROUGH UPPERBOUND:
    ISPRIME <- YES
    FACTOR: FROM 3 THROUGH NUMBER-1:
        SHOULD FACTOR DIVIDE EVENLY INTO NUMBER:
            ISPRIME <- NO
            BREAK
    PROCEED TO NEXT FACTOR BY TWO
    SHOULD ISPRIME STILL BE YES:
        INCREMENT OUR PRIME TALLY
PROCEED TO NEXT NUMBER BY TWO
ONCE UPPERBOUND IS REACHED ADD A ONE TO YOUR PRIME TALLY TO ACCOUNT FOR NOT STARTING AT TWO
STOP TIMEKEEPING

variant: break+sqrt

Same as sqrt but add a break

START TIMEKEEPING
NUMBER: FROM 2 THROUGH UPPERBOUND:
    ISPRIME <- YES
    FACTOR: FROM 2 * 2 THROUGH NUMBER-1:
        SHOULD FACTOR DIVIDE EVENLY INTO NUMBER:
            ISPRIME <- NO
            BREAK
    PROCEED TO NEXT FACTOR BY TWO
    SHOULD ISPRIME STILL BE YES:
        INCREMENT OUR PRIME TALLY
PROCEED TO NEXT NUMBER BY TWO
ONCE UPPERBOUND IS REACHED ADD A ONE TO YOUR PRIME TALLY TO ACCOUNT FOR NOT STARTING AT TWO
STOP TIMEKEEPING

variant: break+odds+sqrt

For this version, you will combine all three of the above into one process!

ALGORITHM: sieve of eratosthenes

variant: baseline soe

The sieve of Eratosthenes is one of the best algorithms for finding prime numbers, you may have noticed that up to this point all the code we have written has a complexity of O(n^2). The soe takes the next step and goes to O(nlog(log(n)).

Here is how the Sieve of Eratosthenes works:

First, you start with 2, and count up to your upper bound. For this example, let's say it is 40:

    2  3  4  5  6  7  8  9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40

Then, you go through the list and remove multiples of 2. After that, you go to the next remaining number, which you now know is prime. Then, you remove multiples of that number, and so on.

To continue from above, 2 is a prime number, so you leave it alone, and remove any multiples of 2:

    2  3     5     7     9  
11    13    15    17    19  
21    23    25    27    29  
31    33    35    37    39

Then, you go to the next number: 3. Now you know 3 is a prime number, so you can remove multiples of 3:

    2  3     5     7        
11    13          17    19  
      23    25          29  
31          35    37

You go through the entire list, and when you get to the end, you are only left with prime numbers:

    2  3     5     7        
11    13          17    19  
      23                29  
31                37

START TIMEKEEPING
NUMBER: FROM 2 THROUGH UPPERBOUND:
    SHOULD THE NUMBER SLOT BE TRUE:
        VALUE AT NUMBER IS PRIME, INCREMENT TALLY
        MULTIPLE: FROM NUMBER+NUMBER THROUGH UPPERBOUND:
            VALUE AT MULTIPLE IS NOT PRIME
            MULTIPLE IS MULTIPLE PLUS NUMBER
        PROCEED TO NEXT MULTIPLE
    INCREMENT NUMBER
PROCEED TO NEXT NUMBER
STOP TIMEKEEPING

variant: sieve of eratosthenes with sqrt trick (soes)

START TIMEKEEPING
NUMBER: FROM 2 THROUGH NUMBER*NUMBER<UPPERBOUND:
    SHOULD THE NUMBER SLOT BE TRUE:
        VALUE AT NUMBER IS PRIME, INCREMENT TALLY
        MULTIPLE: FROM NUMBER*NUMBER THROUGH UPPERBOUND:
            VALUE AT MULTIPLE IS NOT PRIME
            MULTIPLE IS MULTIPLE PLUS NUMBER
        PROCEED TO NEXT MULTIPLE
    INCREMENT NUMBER
PROCEED TO NEXT NUMBER
STOP TIMEKEEPING

timing

wedge pnc1 runtimes

cgrant9 pnc1 runtimes

VerbalGnat48's pnc1 runtimes

MrVengeance's pnc1 runtimes

XaViEr'S pnc1 runtimes

Cburling's pnc1 runtimes

Blaize Patricelli pnc1 runtimes

SUBMISSION

To be successful in this project, the following criteria (or their equivalent) must be met:

Project must be submit on time, by the deadline.
- Late submissions will lose 33% credit per day, with the submission window closing on the 3rd day following the deadline.
Executed programs must display in a manner similar to provided output
- output formatted, where applicable, must match that of project requirements
Processing must be correct based on input given and output requested
Output, if applicable, must be correct based on values input
Code must be nicely and consistently indented
Code must be consistently written, to strive for readability from having a consistent style throughout
Code must be commented
- Any “to be implemented” comments MUST be removed
  - these “to be implemented” comments, if still present at evaluation time, will result in points being deducted.
  - Sufficient comments explaining the point of provided logic MUST be present
No global variables (without instructor approval), no goto statements, no calling of main()!
Track/version the source code in your lab46 semester repository
Submit a copy of your source code to me using the submit tool by the deadline.

Submit Tool Usage

Let's say you have completed work on the project, and are ready to submit, you would do the following:

lab46:~/src/SEMESTER/DESIG/PROJECT$ submit DESIG PROJECT file1 file2 file3 ... fileN

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.

RUBRIC

I'll be evaluating the project based on the following criteria:

208:pnc1:final tally of results (208/208)
*:pnc1:code, XML, build script, and cartridge submitted [26/26]
*:pnc1:processing and output is correct, and to specifications [26/26]
*:pnc1:break on composite optimization present and functional [26/26]
*:pnc1:odds only processing optimization present and functional [26/26]
*:pnc1:sqrt trick processing optimization present and functional [26/26]
*:pnc1:graph produced from timing data produced of all 8 variants [26/26]
*:pnc1:graph posted to discord and documentation page [26/26]
*:pnc1:timing data is the taken out to at least 4 decimal places [26/26]

Pertaining to the collaborative authoring of project documentation

each class member is to participate in the contribution of relevant information and formatting of the documentation
- minimal member contributions consist of:
  - near the class average edits (a value of at least four productive edits)
  - near the average class content change average (a value of at least 1024 bytes (absolute value of data content change))
  - no zero-sum commits (adding in one commit then later removing in its entirety for the sake of satisfying edit requirements)
- adding and formatting data in an organized fashion, aiming to create an informative and readable document that anyone in the class can reference
- content contributions will be factored into a documentation coefficient, a value multiplied against your actual project submission to influence the end result:
  - no contributions, co-efficient is 0.50
  - less than minimum contributions is 0.75
  - met minimum contribution threshold is 1.00

Additionally

Solutions not abiding by spirit of project will be subject to a 50% overall deduction
Solutions not utilizing descriptive why and how comments will be subject to a 25% overall deduction
Solutions not utilizing indentation to promote scope and clarity or otherwise maintaining consistency in code style and presentation will be subject to a 25% overall deduction
Solutions not organized and easy to read (assume a terminal at least 90 characters wide, 40 characters tall) are subject to a 25% overall deduction

Lab46 Wiki

Sidebar

Table of Contents

PROJECT: Prime Number Computation (PNC1)

OBJECTIVE

TASK

REFERENCE

PNC1

C implementation

EDIT

PNCX

algorithm: brute force / trial-by-division

variant: naive

variant: break on composite (BOC)

variant: odds-only processing

variant: sqrt point

variant: break+odds

variant: break+sqrt

variant: break+odds+sqrt

ALGORITHM: sieve of eratosthenes

variant: baseline soe

variant: sieve of eratosthenes with sqrt trick (soes)

timing

wedge pnc1 runtimes

cgrant9 pnc1 runtimes

VerbalGnat48's pnc1 runtimes

MrVengeance's pnc1 runtimes

XaViEr'S pnc1 runtimes

Cburling's pnc1 runtimes

Blaize Patricelli pnc1 runtimes

SUBMISSION

Submit Tool Usage

RUBRIC

Pertaining to the collaborative authoring of project documentation

Additionally

Lab46 Wiki

User Tools

Site Tools

Sidebar

Table of Contents

PROJECT: Prime Number Computation (PNC1)

OBJECTIVE

TASK

REFERENCE

PNC1

C implementation

EDIT

PNCX

algorithm: brute force / trial-by-division

variant: naive

variant: break on composite (BOC)

variant: odds-only processing

variant: sqrt point

variant: break+odds

variant: break+sqrt

variant: break+odds+sqrt

ALGORITHM: sieve of eratosthenes

variant: baseline soe

variant: sieve of eratosthenes with sqrt trick (soes)

timing

wedge pnc1 runtimes

cgrant9 pnc1 runtimes

VerbalGnat48's pnc1 runtimes

MrVengeance's pnc1 runtimes

XaViEr'S pnc1 runtimes

Cburling's pnc1 runtimes

Blaize Patricelli pnc1 runtimes

SUBMISSION

Submit Tool Usage

RUBRIC

Pertaining to the collaborative authoring of project documentation

Additionally

Page Tools