Help speed up this algorithm? Sieve of Eratosthenes

Question

I've written an algorithm that I believe to be correct for computing prime numbers up to n with the Sieve of Eratosthenes. Unfortunately, this program hangs on really large values of n (try 10 million). Here is what I've written...

Protected Function Eratosthenes(ByVal n As Integer) As String
    Dim maxValue As Integer = Math.Sqrt(n)
    Dim values As Generic.List(Of Integer) = New Generic.List(Of Integer)
    Dim i As Integer
    ''//create list.
    For i = 2 To n
        values.Add(i)
    Next

    For i = 2 To maxValue
        If values.Contains(i) Then
            Dim k As Integer
            For k = i + 1 To n
                If values.Contains(k) Then
                    If (k Mod i) = 0 Then
                        values.Remove(k)
                    End If
                End If
            Next
        End If
    Next

    Dim result As String = ""
    For i = 0 To values.Count - 1
        result = result & " " & values(i)
    Next

    Return result
End Function

How might I speed this algorithm up? Where are my bottlenecks?

Answer 1

Removing elements from a large list is slow.

Why not create an array of Booleans values instead and set a value to "True" when you know that it's non-prime?

When you've found a new prime, you don't need to go through all higher values, just multiple of that one, setting the array element to True.

You can keep a separate list for primes you've found so far, if you want to return them.

Here's a C# implementation which just prints them out as it goes. (In C# if I wanted to return the values I'd return IEnumerable<T> and use an iterator block.)

using System;

public class ShowPrimes
{
    static void Main(string[] args)
    {
        ShowPrimes(10000000);
    }

    static void ShowPrimes(int max)
    {        
        bool[] composite = new bool[max+1];
        int maxFactor = (int) Math.Sqrt(max);

        for (int i=2; i <= maxFactor; i++)
        {
            if (composite[i])
            {
                continue;
            }
            Console.WriteLine("Found {0}", i);
            // This is probably as quick as only
            // multiplying by primes.
            for (int multiples = i * i;
                 multiples <= max;
                 multiples += i)
            {
                composite[multiples] = true;
            }
        }
        // Anything left is a prime, but not
        // worth sieving
        for (int i = maxFactor + 1; i <= max; i++)
        {
            if (composite[i])
            {
                continue;
            }
            Console.WriteLine("Found {0}", i);
        }
    }
}

Answer 2

I am not a VB guy, but I think:

values.Remove(k)

is a waste of time, just set the place to 0, then extract the prime numbers to another list or sort the same list and remove all zeros at once.