Seive of Eratosthenes optimizing

[Insectoid]

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

[Brightguy]

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]

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.

[Insectoid]

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]

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.

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

[Insectoid]

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]

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]

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.

[Insectoid]

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.