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Re: Re: Why is recursion so slow in Mathematica?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg100553] Re: [mg100545] Re: Why is recursion so slow in Mathematica?
*From*: DrMajorBob <btreat1 at austin.rr.com>
*Date*: Mon, 8 Jun 2009 02:04:17 -0400 (EDT)
*References*: <h0d6s8$spq$1@smc.vnet.net> <200906070903.FAA28150@smc.vnet.net>
*Reply-to*: drmajorbob at bigfoot.com
As Szabolcs demonstrated in this case, using the right algorithms and
structures makes Mathematica far more competitive with compiled languages
like OCaml.
But the difference in speed between languages is also overwhelmed by time
itself.
IBM Assembler was my first favorite language (1971 or so), and I hauled
around card decks a tenth the size of my classmates; I was good at it. But
there's no way I'd go back to that, just to get more speed in trivial
calculations. Mathematica today is a hundred times faster (or more) than
Assembler in 1971, and it does a thousand times more FOR me.
If I had spent 40 years writing Assembler code, I'd have very little to
show for it.
I hope that doesn't become anyone's experience with OCaml.
Bobby
On Sun, 07 Jun 2009 04:03:11 -0500, Szabolcs Horvát <szhorvat at gmail.com>
wrote:
> Daniel wrote:
>> This post is about functional programming in Mathematica versus other
>> functional languages such as OCaml, SML or Haskell. At least a naive
>> use of functional constructs in Mathematica is horrendously slow. Am I
>> doing something wrong? Or isn't Mathematica really suitable for
>> functional programming beyond toy programs? Couldn't the Wolfram team
>> make a more efficient implementation for recursion, as other
>> functional languages has done? (Instead of the naive C-like behavior
>> for recursively defined functions.)
>>
>> As grounds for my question/argument, I wrote my own version of select,
>> as below
>>
>> myselect[{}, predicate_] = {}
>> myselect[{head_, tail___}, predicate_] := If[predicate[head],
>> Join[{head}, myselect[{tail}, predicate]],
>> myselect[{tail}, predicate]
>> ]
>>
>> Then I tried this function on a 20.000 element vector with machine
>> size floats:
>>
>> data = Table[Random[], {20000}];
>> $RecursionLimit = 100000;
>> Timing[data2 = myselect[data, # > 0.5 &];]
>>
>> The result is {7.05644, Null}, and hundreds of MB of system memory are
>> allocated. On 1.7 GHZ dual core Intel machine with 1 GB of RAM. For
>> 20.000 floats! It's just a megabyte!
>>
>> The following OCaml program executes in apparently no-time. It is not
>> compiled and does the same thing as the above Mathematica code. After
>> increasing the list by a factor of ten to 200.000 elements, it still
>> executes in a blink. (But with 2.000.000 elements my OCaml interpreter
>> says Stack overflow.)
>>
>> let rec randlist n = if n=0 then [] else Random.float(1.0) :: randlist
>> (n-1);;
>>
>> let rec myselect = function
>> [],predicate -> []
>> | x::xs,predicate -> if predicate(x) then x::myselect(xs,predicate)
>> else myselect(xs,predicate);;
>>
>> let mypred x = x>0.5;;
>>
>> let l=randlist(20000);;
>> let l2=myselect(l,mypred);; (* lightning-fast compared to Mathematica
>> *)
>>
>
> So you discovered that appending to arrays is slow while appending to
> linked lists is fast :)
>
> Actually you are not comparing the exact same algorithms. The data
> structure you used in those languages was a linked list. Removing or
> appending elements takes a constant time for linked lists.
> Mathematica's List is a vector-type data structure, so
> removing/appending elements takes a time proportional to the vector's
> length.
>
> One data structure is not better than the other, of course, they're just
> useful for different purposes. For example, a linked list is unsuitable
> for applications where the elements need to be accessed randomly instead
> of sequentially. Random access is often needed for numerical
> computations/simulations.
>
> For a fair comparison, use a linked list in Mathematica too. This could
> look like {1,{2,{3,{}}}}
>
> myselect[{}, test_] = {};
> myselect[{head_, tail_}, test_] :=
> If[test[head], {head, myselect[tail, test]}, myselect[tail, test]]
>
> toLinkedList[list_List] := Fold[{#2, #1} &, {}, list]
>
> data = toLinkedList@Table[Random[], {20000}];
>
> Block[{$RecursionLimit = \[Infinity]},
> Timing[myselect[data, # > 0.5 &];]]
>
> This runs in 0.2 sec on my (not very fast) system.
>
> And you can check that the function works correctly:
>
> Block[{$RecursionLimit = \[Infinity]},
> Flatten@myselect[data, # > 0.5 &] ===
> Select[Flatten[data], # > 0.5 &]]
>
> About the thing that you call "naive C-like behavior": you probably mean
> that Mathematica doesn't automatically perform the tail-call
> optimization. That is correct, you need to transform your recursions
> explicitly.
>
> Here's an example:
>
> myselect2[dest_, {head_, tail_}, test_] :=
> If[test[head],
> myselect2[{head, dest}, tail, test],
> myselect2[dest, tail, test]]
>
> myselect2[dest_, {}, test_] := dest
>
> Note that it is $IterationLimit that we need to increase now and this
> version doesn't fill up the evaluation stack.
>
> Also note that this produces the result in the reverse order:
>
> Block[{$IterationLimit = \[Infinity]},
> Reverse@Flatten@myselect2[{}, data, # > 0.5 &] ===
> Select[Flatten[data], # > 0.5 &]]
>
> We could produce the result in the same order, but reverse linking (like
> {{{{},1},2},3}).
>
> I couldn't do any better than this using a singly-linked list. It looks
> like OCaml (which I don't know BTW) uses a singly linked list too, and
> can't do the tail-call optimization, so its stack gets filled up.
>
> One more thing: even when using the same data structure and same
> algorithm, it is of course expected that a very high level interpreted
> language like Mathematica is not going to perform nearly as well as a
> low level compiled language like OCaml. In high level languages the
> usual solution to this performance problem is built-in functions: don't
> implement Select in Mathematica itself. Use the built-in one, which is
> written in C and much faster than a Mathematica implementation could
> ever be.
>
> Hope this helps,
> Szabolcs
>
--
DrMajorBob at bigfoot.com
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