Published at Jul 13 2018
·
2 comments

Instructions

Test suite

Solution

Use the Sieve of Eratosthenes to find all the primes from 2 up to a given number.

The Sieve of Eratosthenes is a simple, ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite (i.e. not prime) the multiples of each prime, starting with the multiples of 2.

Create your range, starting at two and continuing up to and including the given limit. (i.e. [2, limit])

The algorithm consists of repeating the following over and over:

- take the next available unmarked number in your list (it is prime)
- mark all the multiples of that number (they are not prime)

Repeat until you have processed each number in your range.

When the algorithm terminates, all the numbers in the list that have not been marked are prime.

The wikipedia article has a useful graphic that explains the algorithm: https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes

Notice that this is a very specific algorithm, and the tests don't check that you've implemented the algorithm, only that you've come up with the correct list of primes.

For installation and learning resources, refer to the exercism help page.

For running the tests provided, you will need the Minitest gem. Open a terminal window and run the following command to install minitest:

```
gem install minitest
```

If you would like color output, you can `require 'minitest/pride'`

in
the test file, or note the alternative instruction, below, for running
the test file.

Run the tests from the exercise directory using the following command:

```
ruby sieve_test.rb
```

To include color from the command line:

```
ruby -r minitest/pride sieve_test.rb
```

Sieve of Eratosthenes at Wikipedia http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes

It's possible to submit an incomplete solution so you can see how others have completed the exercise.

```
require 'minitest/autorun'
require_relative 'sieve'
# Common test data version: 1.0.0 f2b2693
class SieveTest < Minitest::Test
def test_no_primes_under_two
# skip
expected = []
assert_equal expected, Sieve.new(1).primes
end
def test_find_first_prime
skip
expected = [2]
assert_equal expected, Sieve.new(2).primes
end
def test_find_primes_up_to_10
skip
expected = [2, 3, 5, 7]
assert_equal expected, Sieve.new(10).primes
end
def test_limit_is_prime
skip
expected = [2, 3, 5, 7, 11, 13]
assert_equal expected, Sieve.new(13).primes
end
def test_find_primes_up_to_1000
skip
expected = [
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59,
61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139,
149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233,
239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337,
347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439,
443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557,
563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653,
659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769,
773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883,
887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997
]
assert_equal expected, Sieve.new(1000).primes
end
# Problems in exercism evolve over time, as we find better ways to ask
# questions.
# The version number refers to the version of the problem you solved,
# not your solution.
#
# Define a constant named VERSION inside of the top level BookKeeping
# module, which may be placed near the end of your file.
#
# In your file, it will look like this:
#
# module BookKeeping
# VERSION = 1 # Where the version number matches the one in the test.
# end
#
# If you are curious, read more about constants on RubyDoc:
# http://ruby-doc.org/docs/ruby-doc-bundle/UsersGuide/rg/constants.html
def test_bookkeeping
skip
assert_equal 1, BookKeeping::VERSION
end
end
```

```
require "Set"
class Sieve
def initialize(number)
@number = number
end
# "mark" them by *removing* them.
# that's why a set, as removal is so much faster than an array.
# other optimizations:
# - stop checking for primes after [floor of] square root of limit;
# any higher composites are also multiples of same or lower prime.
# - start multiple-marking with prime's square;
# any lower multiples are multiples of some previous prime.
def primes
possible_primes = Set.new(2..@number)
(2..Math.sqrt(@number)).each do |prime|
next unless possible_primes.include? prime
factors = prime..(@number/prime).floor
multiples = factors.map { |factor| factor * prime }
possible_primes.subtract multiples
end
possible_primes.to_a
end
end
```

A huge amount can be learned from reading other people’s code. This is why we wanted to give exercism users the option of making their solutions public.

Here are some questions to help you reflect on this solution and learn the most from it.

- What compromises have been made?
- Are there new concepts here that you could read more about to improve your understanding?

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## Community comments

Any kind of multi-item deletion inside a loop is going to be slow. I think that's why it was specifically termed "marking" because they're meant to be removed after the loop.

At least with a Set (or Hash), we don't have to shuffle all the higher keys down to new indices. :-) I had also considered adding them to a "marked" Set/Hash, but at least this way what's left is more easily extracted than subtracting the marked ones from all possibilities....