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Re: Why is recursion so slow in Mathematica?


Let me tell you straight away that you won't get efficiency of OCaml
in Mathematica,
the latter really being a CAS, research and prototyping  tool, but not
an efficiency-oriented production language.

Regarding your question, the short answer is that recursion is slow
because lists are implemented as arrays in Mathematica. When you
pattern-match, the whole array <tail> is copied every time. This
explains both the slowdown - the time becomes effectively quadratic
(assuming that the complexity of array copying is linear), and the
(stack) memory use explosion.

So, IMO, indeed Mathematica is unfortunately less suited for the
straightforward recursive solutions like yours. The closest thing you
can probably do to stay in the recursive mindset is to use linked
lists. Here is an implementation that does just that:

myselectAux[{}, predicate_] = {};
myselectAux[{head_, tail_List}, predicate_] :=
  If[predicate[head], {head, myselectAux[tail, predicate]},
   myselectAux[tail, predicate]];

toLinkedList[x_List] := Fold[{#2, #1} &, {}, Reverse[x]];

mySelectLL[x_List, predicate_] :=
  Flatten@myselectAux[toLinkedList[x], predicate];

You can find more info on linked lists in Mathematica in the book of
David Wagner,
in the notes by Daniel Lichtblau (see Wolfram library archive, should
be named
something like "efficient data structures in Mathematica"), and also
in my book

For example:

In[1] = test = RandomInteger[{1, 10}, 10]

Out[1] = {6, 10, 1, 9, 4, 6, 1, 9, 10, 4}

In[2] = toLinkedList@test

Out[2] = {6, {10, {1, {9, {4, {6, {1, {9, {10, {4, {}}}}}}}}}}}

In[3] = mySelectLL[test, # > 5 &]

Out[3] = {6, 10, 9, 6, 9, 10}

Now, the benchmarks: I use yours, on my 5 years old 1.8 GHz P IV, and
I get:

In[4] =
data = Table[Random[], {20000}];
Block[{$RecursionLimit = Infinity},
 Timing[data2 = mySelectLL[data, # > 0.5 &];]]

Out[4] = {0.2, Null}

Now, this is not the most efficient implementation. David Wagner in
his excellent book
discussed in great detail this sort of problems with the mergesort
algorithm as an archetypal example. I believe that you can squeeze
another factor of 2-5 by further optimizing the implementation
(keeping it recursive) with several performance-tuning techniques
available in Mathematica.

By the way, I tested the above code with larger lists and it seems to
continue be linear complexity  up to 100 000 elements, but the kernel
(v.6) just crashes after that - this
behavior I can not explain, since the memory usage seems quite decent.

So, to summarize my opinion:  if you prefer to stay in a recursive
mode, you *can* use recursion in Mathematica rather efficiently by
switching to linked lists (and this will require some extra work but
not so much of it). This may be enough to move you from the domain of
toy problems to that of reasonably complex prototypes.  But don't
expect efficiency of the highly optimized production language with a
native-code compiler (or even byte code interpreter) from a rewrite
system such as Mathematica. Also, I'd say that other programming
styles such as structural operations (APL - like) will bring you
closer to efficiency of Ocaml etc. since then you can leverage the
highly-optimized vectorized operations built in Mathematica.

Hope this helps.


On Jun 6, 12:46 am, Daniel <jan... at> 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) :: randlis=
> (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 Mathematic=
> *)

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