This program will display a multiplication table of N prime numbers to the console.

CD into the project directory and run the following command:

$ ./run_program

If you want to determine N yourself, just supply an integer argument after the command:

$ ./run_program 20

I immediately knew that I would need to create classes Prime and MultiplicationTable to abstract away these two objects. I also knew that the Prime class wouldn't need to be instantiated because storing Prime objects wasn't necessary. Therefore, the class only contains factory methods one of which allows you to calculate the Nth prime number.

The MultiplicationTable class is used to represent the multiplication table. When this class is instantiated, it will invoke Prime::calc_nth_prime(N) where N is the argument provided (N is defaulted to 10). Instances of the MultiplicationTable class will hold values of the multiplication table and a MultiplicationTable#render method will be available to STDOUT the table to the console.

I took the BDD/TDD approach and wrote out the tests before coding the classes. This allowed me to fully understand what I was building and what functionalities needed to be implemented upfront. These tests also allowed me to perform regression tests as I was changing the code to make sure nothing got broken.

Lastly, I built the run_program script so the program can be run from the command line. It has some basic error handling to make sure the supplied input is valid otherwise an exception is raised.

• Byebug - preferred debugger
• RSpec - write test scripts
• Colorize - beautify the boring console output

Design

• The Prime class was created with the idea that its sole purpose is to calculate N prime numbers and provide access to retrieve them. It also exposes the Prime::is_prime? factory method to provide on the spot prime number checking. When invoking the Prime::calc_nth_prime factory method, the prime numbers are cached in a class variable such that subsequent calculations of N prime numbers can be done by leveraging the primes that have already been determined.

Performance

• Space complexity is O(N) because the program caches N prime numbers once they are calculated. Time complexity is O(m * sqrt(m)) for completely calculating N prime numbers during the worst case scenario (first time calculation, cache is empty) where m is the Nth prime number.

Reasoning

• N = 10, m = Nth prime number = 29

• We calculate prime numbers with the factory method Prime::calc_nth_prime(N). In here, numbers 2 - 29 need to be checked with Prime::is_prime?(m) so this brings us to O(m) time complexity.

• Within the Prime::is_prime?(m) factory method, in the worst case we need to check up to the sqrt(m). Checking any number after the sqrt is actually a duplication of effort (100 divides evenly by 2 and 50; checking 2 is the same as checking 50).

• Prime::calc_nth_prime(N) and the embedded Prime::is_prime?(m) check brings our overall time complexity to O(m * sqrt(m)).

Benchmarking

• To run the benchmark tests for the Prime class type in the following command at the project root:

  $ ruby benchmark/prime_benchmark_tests.rb
  

• Here were my results of the benchmark tests:

• These tests loosely confirm that the big O for Prime::calc_nth_prime is O(m * sqrt(m))

  test case 1: n = 10000, m = 104729
  t1 = 104729 * Math.sqrt(104729) = 33892252.64
  
  test case 2: n = 100000, m = 1299709
  t2 = 1299709 * Math.sqrt(1299709) = 1481730393.91
  
  test case 3: n = 1000000, m = 15485863
  t3 = 15485863 * Math.sqrt(15485863) = 60940093925.69
  
  test case 4: n = 10000000, m = 179424673
  t4 = 179424673 * Math.sqrt(179424673) = 2403384439866.17
  
  t4 / t3 = t3 / t2 = t2 / t1 = approx. 40
  

  As you can see, in each case the total CPU time is expected to increase by a factor of about 40. The data from the benchmark tests show that each run is increasing by a factor of about 32.

Design

• The MultiplicationTable class was created such that a multiplication table can be represented as an object. When instantiated, it can take an argument N (10 by default) which will determine how many prime numbers get calculated. Also upon instantiation, the MultiplicationTable#fill_grid method is invoked to store values of the table in the @grid instance variable. Lastly, the MultiplicationTable#render method STDOUTs the table to the console using several helper methods to build the display strings and a neat little gem called colorize to sprinkle in some color.

Performance

• Space complexity is O(N^2) because the @grid instance variable will grow in both x and y dimensions as the input N gets larger. Time complexity to fill @grid is also O(N^2). (See reasoning below).

• Overall, the total time to instantiate a MultiplicationTable object in the worst case (Prime cache is empty, calculating Primes for the first time) is O(N^2) + O(m * sqrt(m)) where N is the number of primes and m is the Nth prime. However, at scale O(N^2) will overshadow O(m * sqrt(m)) therefore the time complexity is just O(N^2).

Reasoning

• Without the known expressions optimization, the MultiplicationTable#fill_grid is simply a double nested loop which has a time complexity of O(N^2). All double nested loops share this same time complexity. I knew this was no good so I attempted to look for a pattern and noticed that known expressions can be stored and retrieved once the algorithm comes to those expressions:

The cells highlighted in orange are retrieved from the known expressions hash while the cells highlighted in blue are the ones that are actually searched and calculated.

• This eliminates almost half of the expressions that need to be searched and evaluated. However, as mentioned earlier the time complexity of the algorithm is still growing exponentially.

• Despite attempts to optimize the MultiplicationTable#fill_grid method, the algorithm still averages out to O(N^2) because the optimization is overshadowed by the exponential at scale. The optimization brings the time complexity to O((N^2) / 2) which becomes O(N^2) at scale.

Benchmarking

• To run the benchmark tests for the MultiplicationTable class type in the following command at the project root:

  $ ruby benchmark/mult_table_benchmark_tests.rb
  

• Here were my results of the benchmark tests:

• You can clearly see that the run times are increasing exponentially after each test case.