Sieve of Eratosthenes in Erlang [closed]
https://stackoverflow.com/questions/146622
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02-07-2019 - |
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I'm in the process of learning Erlang. As an exercise I picked up the Sieve of Eratosthenes algorithm of generating prime numbers. Here is my code:
-module(seed2).
-export([get/1]).
get(N) -> WorkList = lists:duplicate(N, empty),
get(2, N, WorkList, []).
get(thats_the_end, _N, _WorkList, ResultList) -> lists:reverse(ResultList);
get(CurrentPrime, N, WorkList, ResultList) -> ModWorkList = markAsPrime(CurrentPrime, N, WorkList),
NextPrime = findNextPrime(CurrentPrime + 1, N, WorkList),
get(NextPrime, N, ModWorkList, [CurrentPrime|ResultList]).
markAsPrime(CurrentPrime, N, WorkList) when CurrentPrime =< N -> WorkListMod = replace(CurrentPrime, WorkList, prime),
markAllMultiples(CurrentPrime, N, 2*CurrentPrime, WorkListMod).
markAllMultiples(_ThePrime, N, TheCurentMark, WorkList) when TheCurentMark > N -> WorkList;
markAllMultiples(ThePrime, N, TheCurrentMark, WorkList) -> WorkListMod = replace(TheCurrentMark, WorkList, marked),
markAllMultiples(ThePrime, N, TheCurrentMark + ThePrime, WorkListMod).
findNextPrime(Iterator, N, _WorkList) when Iterator > N -> thats_the_end;
findNextPrime(Iterator, N, WorkList) -> I = lists:nth(Iterator, WorkList),
if
I =:= empty -> Iterator;
true -> findNextPrime(Iterator + 1, N, WorkList)
end.
replace(N, L, New)-> {L1, [_H|L2]} = lists:split(N - 1, L),
lists:append(L1, [New|L2]).
This code actually works :) . The problem is that I have this feeling that it is not the best possible implementation.
My question is what would be the "erlangish" way of implementing the "Sieve of Eratosthenes"
EDIT: OK, Andreas solution is very good but it is slow. Any ideas how to improve that?
Solution
Here's a simple (but not terribly fast) sieve implementation:
-module(primes).
-export([sieve/1]).
-include_lib("eunit/include/eunit.hrl").
sieve([]) ->
[];
sieve([H|T]) ->
List = lists:filter(fun(N) -> N rem H /= 0 end, T),
[H|sieve(List)];
sieve(N) ->
sieve(lists:seq(2,N)).
OTHER TIPS
Here's my sieve implementation which uses list comprehensions and tries to be tail recursive. I reverse the list at the end so the primes are sorted:
primes(Prime, Max, Primes,Integers) when Prime > Max ->
lists:reverse([Prime|Primes]) ++ Integers;
primes(Prime, Max, Primes, Integers) ->
[NewPrime|NewIntegers] = [ X || X <- Integers, X rem Prime =/= 0 ],
primes(NewPrime, Max, [Prime|Primes], NewIntegers).
primes(N) ->
primes(2, round(math:sqrt(N)), [], lists:seq(3,N,2)). % skip odds
Takes approx 2.8 ms to calculate primes up to 2 mil on my 2ghz mac.
I approached the problem by using concurrent processing.
My previous post did not get formatted correctly. Here is a repost of the code. Sorry for spamming...
-module(test).
%%-export([sum_primes/1]).
-compile(export_all).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%Sum of all primes below Max. Will use sieve of Eratosthenes
sum_primes(Max) ->
LastCheck = round(math:sqrt(Max)),
All = lists:seq(3, Max, 2), %note are creating odd-only array
%%Primes = sieve(noref,All, LastCheck),
Primes = spawn_sieve(All, LastCheck),
lists:sum(Primes) + 2. %adding back the number 2 to the list
%%sieve of Eratosthenes
sieve(Ref,All, LastCheck) ->
sieve(Ref,[], All, LastCheck).
sieve(noref,Primes, All = [Cur|_], LastCheck) when Cur > LastCheck ->
lists:reverse(Primes, All); %all known primes and all remaining from list (not sieved) are prime
sieve({Pid,Ref},Primes, All=[Cur|_], LastCheck) when Cur > LastCheck ->
Pid ! {Ref,lists:reverse(Primes, All)};
sieve(Ref,Primes, [Cur|All2], LastCheck) ->
%%All3 = lists:filter(fun(X) -> X rem Cur =/= 0 end, All2),
All3 = lists_filter(Cur,All2),
sieve(Ref,[Cur|Primes], All3, LastCheck).
lists_filter(Cur,All2) ->
lists_filter(Cur,All2,[]).
lists_filter(V,[H|T],L) ->
case H rem V of
0 ->
lists_filter(V,T,L);
_ ->
lists_filter(V,T,[H|L])
end;
lists_filter(_,[],L) ->
lists:reverse(L).
%% This is a sloppy implementation ;)
spawn_sieve(All,Last) ->
%% split the job
{L1,L2} = lists:split(round(length(All)/2),All),
Filters = filters(All,Last),
L3 = lists:append(Filters,L2),
Pid = self(),
Ref1=make_ref(),
Ref2=make_ref(),
erlang:spawn(?MODULE,sieve,[{Pid,Ref1},L1,Last]),
erlang:spawn(?MODULE,sieve,[{Pid,Ref2},L3,Last]),
Res1=receive
{Ref1,R1} ->
{1,R1};
{Ref2,R1} ->
{2,R1}
end,
Res2= receive
{Ref1,R2} ->
{1,R2};
{Ref2,R2} ->
{2,R2}
end,
apnd(Filters,Res1,Res2).
filters([H|T],Last) when H
[H|filters(T,Last)];
filters([H|_],_) ->
[H];
filters(_,_) ->
[].
apnd(Filters,{1,N1},{2,N2}) ->
lists:append(N1,subtract(N2,Filters));
apnd(Filters,{2,N2},{1,N1}) ->
lists:append(N1,subtract(N2,Filters)).
subtract([H|L],[H|T]) ->
subtract(L,T);
subtract(L=[A|_],[B|_]) when A > B ->
L;
subtract(L,[_|T]) ->
subtract(L,T);
subtract(L,[]) ->
L.
I haven't studied these in detail, but I've tested my implementation below (that I wrote for a Project Euler challenge) and it's orders of magnitude faster than the above two implementations. It was excruciatingly slow until I eliminated some custom functions and instead looked for lists: functions that would do the same. It's good to learn the lesson to always see if there's a library implementation of something you need to do - it'll usually be faster! This calculates the sum of primes up to 2 million in 3.6 seconds on a 2.8GHz iMac...
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Sum of all primes below Max. Will use sieve of Eratosthenes
sum_primes(Max) ->
LastCheck = round(math:sqrt(Max)),
All = lists:seq(3, Max, 2), %note are creating odd-only array
Primes = sieve(All, Max, LastCheck),
%io:format("Primes: ~p~n", [Primes]),
lists:sum(Primes) + 2. %adding back the number 2 to the list
%sieve of Eratosthenes
sieve(All, Max, LastCheck) ->
sieve([], All, Max, LastCheck).
sieve(Primes, All, Max, LastCheck) ->
%swap the first element of All onto Primes
[Cur|All2] = All,
Primes2 = [Cur|Primes],
case Cur > LastCheck of
true ->
lists:append(Primes2, All2); %all known primes and all remaining from list (not sieved) are prime
false ->
All3 = lists:filter(fun(X) -> X rem Cur =/= 0 end, All2),
sieve(Primes2, All3, Max, LastCheck)
end.
I kind of like this subject, primes that is, so I started to modify BarryE's code a bit and I manged to make it about 70% faster by making my own lists_filter function and made it possible to utilize both of my CPUs. I also made it easy to swap between to two version. A test run shows:
61> timer:tc(test,sum_primes,[2000000]).
{2458537,142913970581}
Code:
-module(test).
%%-export([sum_primes/1]).
-compile(export_all).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%Sum of all primes below Max. Will use sieve of Eratosthenes
sum_primes(Max) ->
LastCheck = round(math:sqrt(Max)),
All = lists:seq(3, Max, 2), %note are creating odd-only array
%%Primes = sieve(noref,All, LastCheck),
Primes = spawn_sieve(All, LastCheck),
lists:sum(Primes) + 2. %adding back the number 2 to the list
%%sieve of Eratosthenes
sieve(Ref,All, LastCheck) ->
sieve(Ref,[], All, LastCheck).
sieve(noref,Primes, All = [Cur|_], LastCheck) when Cur > LastCheck ->
lists:reverse(Primes, All); %all known primes and all remaining from list (not sieved) are prime
sieve({Pid,Ref},Primes, All=[Cur|_], LastCheck) when Cur > LastCheck ->
Pid ! {Ref,lists:reverse(Primes, All)};
sieve(Ref,Primes, [Cur|All2], LastCheck) ->
%%All3 = lists:filter(fun(X) -> X rem Cur =/= 0 end, All2),
All3 = lists_filter(Cur,All2),
sieve(Ref,[Cur|Primes], All3, LastCheck).
lists_filter(Cur,All2) ->
lists_filter(Cur,All2,[]).
lists_filter(V,[H|T],L) ->
case H rem V of
0 ->
lists_filter(V,T,L);
_ ->
lists_filter(V,T,[H|L])
end;
lists_filter(_,[],L) ->
lists:reverse(L).
%% This is a sloppy implementation ;)
spawn_sieve(All,Last) ->
%% split the job
{L1,L2} = lists:split(round(length(All)/2),All),
Filters = filters(All,Last),
%%io:format("F:~p~n",[Filters]),
L3 = lists:append(Filters,L2),
%%io:format("L1:~w~n",[L1]),
%% io:format("L2:~w~n",[L3]),
%%lists_filter(Cur,All2,[]).
Pid = self(),
Ref1=make_ref(),
Ref2=make_ref(),
erlang:spawn(?MODULE,sieve,[{Pid,Ref1},L1,Last]),
erlang:spawn(?MODULE,sieve,[{Pid,Ref2},L3,Last]),
Res1=receive
{Ref1,R1} ->
{1,R1};
{Ref2,R1} ->
{2,R1}
end,
Res2= receive
{Ref1,R2} ->
{1,R2};
{Ref2,R2} ->
{2,R2}
end,
apnd(Filters,Res1,Res2).
filters([H|T],Last) when H
[H|filters(T,Last)];
filters([H|_],_) ->
[H];
filters(_,_) ->
[].
apnd(Filters,{1,N1},{2,N2}) ->
lists:append(N1,subtract(N2,Filters));
apnd(Filters,{2,N2},{1,N1}) ->
lists:append(N1,subtract(N2,Filters)).
subtract([H|L],[H|T]) ->
subtract(L,T);
subtract(L=[A|_],[B|_]) when A > B ->
L;
subtract(L,[_|T]) ->
subtract(L,T);
subtract(L,[]) ->
L.
you could show your boss this: http://www.sics.se/~joe/apachevsyaws.html. And some other (classic?) erlang arguments are:
-nonstop operation, new code can be loaded on the fly.
-easy to debug, no more core dumps to analyse.
-easy to utilize multi core/CPUs
-easy to utilize clusters maybe?
-who wants to deal with pointers and stuff? Is this not the 21 century? ;)
Some pifalls: - it might look easy and fast to write something, but the performance can suck. If I want to make something fast I usually end up writing 2-4 different versions of the same function. And often you need to take a hawk eye aproach to problems which might be a little bit different from what one is used too.
looking up things in lists > about 1000 elements is slow, try using ets tables.
the string "abc" takes a lot more space than 3 bytes. So try to use binaries, (which is a pain).
All in all i think the performance issue is something to keep in mind at all times when writing something in erlang. The Erlang dudes need to work that out, and I think they will.
Have a look here to find 4 different implementations for finding prime numbers in Erlang (two of which are "real" sieves) and for performance measurement results:
http://caylespandon.blogspot.com/2009/01/in-euler-problem-10-we-are-asked-to.html
Simple enough, implements exactly the algorithm, and uses no library functions (only pattern matching and list comprehension). Not very powerful, indeed. I only tried to make it as simple as possible.
-module(primes).
-export([primes/1, primes/2]).
primes(X) -> sieve(range(2, X)).
primes(X, Y) -> remove(primes(X), primes(Y)).
range(X, X) -> [X];
range(X, Y) -> [X | range(X + 1, Y)].
sieve([X]) -> [X];
sieve([H | T]) -> [H | sieve(remove([H * X || X <-[H | T]], T))].
remove(_, []) -> [];
remove([H | X], [H | Y]) -> remove(X, Y);
remove(X, [H | Y]) -> [H | remove(X, Y)].
Here is my sieve of eratophenes implementation C&C please:
-module(sieve).
-export([find/2,mark/2,primes/1]).
primes(N) -> [2|lists:reverse(primes(lists:seq(2,N),2,[]))].
primes(_,0,[_|T]) -> T;
primes(L,P,Primes) -> NewList = mark(L,P),
NewP = find(NewList,P),
primes(NewList,NewP,[NewP|Primes]).
find([],_) -> 0;
find([H|_],P) when H > P -> H;
find([_|T],P) -> find(T,P).
mark(L,P) -> lists:reverse(mark(L,P,2,[])).
mark([],_,_,NewList) -> NewList;
mark([_|T],P,Counter,NewList) when Counter rem P =:= 0 -> mark(T,P,Counter+1,[P|NewList]);
mark([H|T],P,Counter,NewList) -> mark(T,P,Counter+1,[H|NewList]).
Here is my sample
S = lists:seq(2,100),
lists:foldl(fun(A,X) -> X--[A] end,S,[Y||X<-S,Y<-S,X<math:sqrt(Y)+1,Y rem X==0]).
:-)
my fastest code so far (faster than Andrea's) is with using array:
-module(seed4).
-export([get/1]).
get(N) -> WorkList = array:new([{size, N}, {default, empty}]),
get(2, N, WorkList, []).
get(thats_the_end, _N, _WorkList, ResultList) -> lists:reverse(ResultList);
get(CurrentPrime, N, WorkList, ResultList) -> ModWorkList = markAsPrime(CurrentPrime, N, WorkList),
NextPrime = findNextPrime(CurrentPrime + 1, N, WorkList),
get(NextPrime, N, ModWorkList, [CurrentPrime|ResultList]).
markAsPrime(CurrentPrime, N, WorkList) when CurrentPrime =< N -> WorkListMod = replace(CurrentPrime, WorkList, prime),
markAllMultiples(CurrentPrime, N, 2*CurrentPrime, WorkListMod).
markAllMultiples(_ThePrime, N, TheCurentMark, WorkList) when TheCurentMark > N -> WorkList;
markAllMultiples(ThePrime, N, TheCurrentMark, WorkList) -> WorkListMod = replace(TheCurrentMark, WorkList, marked),
markAllMultiples(ThePrime, N, TheCurrentMark + ThePrime, WorkListMod).
findNextPrime(Iterator, N, _WorkList) when Iterator > N -> thats_the_end;
findNextPrime(Iterator, N, WorkList) -> I = array:get(Iterator - 1, WorkList),
if
I =:= empty -> Iterator;
true -> findNextPrime(Iterator + 1, N, WorkList)
end.
replace(N, L, New) -> array:set(N - 1, New, L).
