### Sieve of Eratosthenes in C#

In my personal quest to find out if the programming language really matters, I present to you my first piece of code. This is my reference implementation of Sieve of Eratosthenes in my preferred language, C#. For this and all future language implementations, I'll put the code at the top (without comments) and then narrate my experience below. Now, to the code!

IList<int> findPrimes(int max) { var vals = new List<int>((int)(max/(Math.Log(max)-1.08366))); var maxSquareRoot = Math.Sqrt(max); var eliminated = new System.Collections.BitArray(max + 1); vals.Add(2); for (int i = 3; i <= max; i+=2) { if (!eliminated[i]) { if (i < maxSquareRoot) { for (int j = i * i; j <= max; j+=2*i) eliminated[j] = true; } vals.Add(i); } } return vals; }

I started by following the wikipedia definition and then optimized from there.

**Algorithm Optimizations**

I cut my work in half by treating the special case of '2'. We know that 2 is prime and all even numbers thereafter are not. So, we'll add two immediately and then start looping at 3 only checking odd numbers from there forward.

After we've found a prime, we only need to eliminate numbers from it's square and forward. Let's say we want to find all prime numbers up to 100 and we've just identified 7 as a prime. Per the algorithm, I'll need to eliminate 2*7, 3*7 ,4*7, 5*7, 6*7, 7*7 ,8*7 ,9*7, 10*7 ,11*7, 12*7 ,13*7 and 14*7. None of the even multiples matter (even times an odd is always even) and none of the multiples up to the square of the prime matter since we've already done those multiples in previous loops. So really we only have to eliminate 7*7, 9*7, 11*7 and 13*7. That's a 9 fewer iterations and those savings become more fruitful the deeper you go!

The last optimization is the square root calculation and check. We know from above that we only need to start eliminating beginning at the square of the current prime. Therefore it also makes sense that we can stop even trying once we get past the to square root of the max. This saves a bunch more iterations.

**Language Optimizations**

Originally I had started by returning an `IEnumerable<int>`

. I wasn't using the list you see above and instead I was using `yield return i`

. I really like that syntax, but once I got to the VB.net version (Coming Soon!), I didn't have a direct translation for the yield keyword. I took the lazy route in the VB version and just stuffed it all into a list and returned that. To my surprise it was faster! I went back and changed the C# version above and it performed better. I'm not sure why, but I'm going with it.

What do you think that you get when do a `sizeof(bool)`

in C#? I was surprised to find out that my trusty booleans actually take up a whole byte instead of a single bit. I speculate that there is a performance benefit that all of your types fit into a byte level offset in memory. I was thrilled to find out that we have a `BitArray`

class that is useful for situations above when you need to store a lot of booleans and you need them to only take up a bit in memory. I'm not sure it helped anything, but I feel better knowing I'm using the least amount of memory possible. :)

**Conclusion**

Despite the fact that I know C# really well, I'm very thrilled that I was able to learn a few things about the language. Also, I'm really happy with the performance of this reference implementation. On my machine (2.66 GHz Core2 Duo and 2 GB of RAM) I can find all of the primes under 1,000,000 in 19ms. I think I've squeezed all I can out of this version. Please let me know if you see something I missed or did wrong and I'll make adjustments.

**EDIT:** I just added one more optimization that's worth noting. Instead of constructing my list with an empty constructor, I can save a several milliseconds off the larger sets by specifying a start size of the internal array structure behind the list. If I set this size at or slightly above the end count of prime numbers, then I avoid a lot of costly array copying as the array bounds keep getting hit. It turns out that there is quite a bit of math involved in accurately predicting the number of primes underneath a given number. I chose to cheat and just use Legendre's constant with the Prime Number Theorem which is close enough for my purposes. I can now calculate all primes under 1,000,000 in 10ms on my machine. Neat!