(tye)Re3: (Golf): Sieve of Eratosthenes

[tye]

tilly's comment about things being "nearly linear" threw me for a bit. Then I realized that the quadratic nature is countered by the outer loop only needing to run to sqrt(N) and the inner loop being somewhat similarly restricted.

Which made me realize that my solution was suboptimal. Here is a faster one at the same number of characters [ thanks to MeowChow noting that I'd stupidly left in a trailing semicolon in my previous one ;) ]:

sub sieve3 {
    grep{@_[map$a*$_,$_..@_/($a=$_)]=0if$_[$_]>1}@_=0..pop
} #                  ^^

for( @ARGV ) {
    print "$_: ",join(" ",sieve3($_)),$/;
}

In playing with this and verifying that I didn't break it, I noticed something interesting and expanded on it. For how long of a stretch can you go without hitting any prime numbers? Well, stopping at 0.5million (because of memory limits), here are the results. "xN" means there were N ties before a new "winner" appeared:

2=5-3(x2)    # 3..5, 5..7
4=11-7(x3)   # 7..11, 13..17, 19..23
6=29-23(x7)
8=97-89
14=127-113(x3)
18=541-523
20=907-887
22=1151-1129
34=1361-1327(x2)
36=9587-9551(x3)
44=15727-15683
52=19661-19609(x2)
72=31469-31397
86=156007-155921(x2)
96=360749-360653
112=370373-370261
114=492227-492113

        - tye (but my friends call me "Tye")

[tilly]

In the Big-O analysis being able to stop the outer loop early turns out to be a red herring. Being able to stop the inner loop after 1/p operations is key, as is the density of the primes. It means that the work is O(n*(sum of 1/p)). The sum of 1/i scales like log(n), the density of the primes scales as 1/log(n), and between them they cancel out for O(n*log(log(n))).

log(log(n)) is essentially constant in the range of numbers I can test before hitting memory limitations. Also there is a theoretical log(n) that we are missing from the overhead of addressing and representing ever larger numbers. While we tend to call that constant, in reality it is not.

BTW if you are interested, longer tables of maximal gaps have been compiled...

[tilly]

And almost a decade later I was randomly looking at this, and realized that it could be made shorter still.

sub sieve3 {
    grep{@_[map$a*$_,2..@_/($a=$_)]=0if$_[$_]>1}@_=0..pop
}

Never give up on that last optimization!