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 Seive of Eratosthenes optimizing
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Insectoid




PostPosted: Thu Apr 15, 2010 4:51 pm   Post subject: Seive of Eratosthenes optimizing

I found the Sieve of Eratosthenes algorithm, and decided to write one myself. It works, but it's really, really slow calculating primes up to anything higher than 100000. I want to make it as fast as possible. Anyway, here's my code:

Ruby:

#Seive of Eratosthenes
#Finds Prime Numbers

print "Enter the last number to check (greater than 1): "
max_int = gets.to_i
list = (2..max_int).to_a
#(2..Math.sqrt(max_int).to_i).to_a.each do |n|
list.each do |n| #n will only ever be a prime number
puts "Prime Found: #{n}\n Deleting multiples of #{n}..."
        if n**2 > max_int then #Quits once n squared is greater than max_int (only primes will be left)
                puts "Sieve complete..."
                break
        end
       
        list.each_index do |i|
       
                if (list[i] % n == 0) && (list[i] != n) then #Checks if list[i] is divisable by n (but not equal to it, n is prime)
                        #puts "Deleting: #{list[i]}"
                        list.slice!(i) #deletes list[i]
                 end

        end
end

#This won't be necessary later. Once I delete extra puts above I'll delete this.
puts "Primes up to #{max_int}:"
list.each do |i|
        puts i
end


I figure, if I can change
code:
list.each_index do |i|

to start at element [i]n
+1 (all previous elements are prime) I can drop the 2nd condition out of
code:
if (list[i] % n == 0) && (list[i] != n) then

Which should significantly speed up the script (that conditional is run 849117 of times when max_int = 100000, whereas that loop is only tested 65 times)

I tried changing the loop to
code:
list.slice(1..-1).each_index do |i|

but that causes a syntax error (dunno if -1 is valid in that scenario...). I also tried
code:
list.slice(1..list.length-1).each_index do |i|

but that throws the same error.

If anyone could help me execute that loop without passing the first element in list (in a non-hackish manner), please do!
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Brightguy




PostPosted: Thu Apr 15, 2010 6:55 pm   Post subject: Re: Seive of Eratosthenes optimizing

Insectoid @ Thu Apr 15, 2010 4:51 pm wrote:
Which should significantly speed up the script (that conditional is run 849117 of times when max_int = 100000, whereas that loop is only tested 65 times)

That conditional isn't of much concern: it's only O(1). (And BTW, you could actually use a list[i] >= n^2 condition). If you want to be efficient, don't use Ruby's split inside the loop! Have you seen the code for even delete_at? It's O(N)! Also checking every element in the list for divisibility is quite needlessly inefficient.
Tony




PostPosted: Thu Apr 15, 2010 7:06 pm   Post subject: RE:Seive of Eratosthenes optimizing

Quote:

list.each_index do |i|

if (list[i] % n == 0) && (list[i] != n) then

There's a _much faster_ way of getting the same list of values, without doing any comparisons. (modulo is expensive in a loop)

Quote:

list.slice!(i) #deletes list[i]

There's a _much faster_ way to mark a number as prime or not, than taking it out of a list.


Currently, your list.include? is O(N). You want to be able to check if the number is prime in O(1), once you build that list.
Latest from compsci.ca/blog: Tony's programming blog. DWITE - a programming contest.
Insectoid




PostPosted: Thu Apr 15, 2010 7:13 pm   Post subject: RE:Seive of Eratosthenes optimizing

Brightguy, I did try delete_at after posting this, but figured out something else that might be faster, which your post moots anyway. So delete_at might be the thing.

Tony, I don't see what you're getting at. I'll take your bolded 'mark' as a hint.
Brightguy




PostPosted: Thu Apr 15, 2010 7:29 pm   Post subject: Re: RE:Seive of Eratosthenes optimizing

Insectoid @ Thu Apr 15, 2010 7:13 pm wrote:
Brightguy, I did try delete_at after posting this, but figured out something else that might be faster, which your post moots anyway. So delete_at might be the thing.

Exclamation I was pointing out how bad it is. The O(N) call inside your loop is terrible to your efficiency.
Insectoid




PostPosted: Thu Apr 15, 2010 7:30 pm   Post subject: RE:Seive of Eratosthenes optimizing

ah, heh.

*Insectoid hasn't looked up Big-O notation and so is just assuming that O(N) is significantly worse than O(1).
chrisbrown




PostPosted: Thu Apr 15, 2010 7:51 pm   Post subject: RE:Seive of Eratosthenes optimizing

To put it loosely, if N is the size of your input, runtime can be estimated by counting the number of operations as a function of N. O(1) means as N grows large, the time it takes to perform whatever you're looking at is constant - a good thing. O(N) is a linear relationship. O(N^2) gets large faster than O(N), so it is worse and is very poor for large inputs.
Brightguy




PostPosted: Thu Apr 15, 2010 7:51 pm   Post subject: Re: RE:Seive of Eratosthenes optimizing

This is abusing the notation, but if you want a better feeling than "significantly worse", here it means that computing the primes to 100,000 takes you about 100,000 times longer.
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Insectoid




PostPosted: Thu Apr 15, 2010 7:57 pm   Post subject: RE:Seive of Eratosthenes optimizing

Ah. Well. That makes sense. I did notice incredible slowdowns at higher numbers (500 000 took over 100 times as long as 100 000). Damn.
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