Sunday, 6 May 2012

Tree data structure in Mathematica


I have used mathematica mostly as a mathematics workbench and for writing relatively small ad-hoc programs.
Mathematica really excels at this.
What approach have you used with regard to data structures? Gradually developing your own util package?
I avoid creating my own data structures in Mathematica because it cannot handle them efficiently. Specifically, general data structures tend to be 10-1,000× slower in Mathematica than elsewhere which greatly limits their practical usefulness. For example, Mathematica is 100× slower than F# at computing the range of depths in a red-black tree.
Logic programming with lists is one example where Mathematica is typically orders of magnitude slower than other compiled languages. The following Mathematica program uses linked lists to solve the n-queens problem:
safe[{x0_, y0_}][{x1_, y1_}] := 
 x0 != x1 && y0 != y1 && x0 - y0 != x1 - y1 && x0 + y0 != x1 + y1

filter[_, {}] := {}
filter[p_, {h_, t_}] := If[p[h], {h, filter[p, t]}, filter[p, t]]

search[n_, nqs_, qs_, {}, a_] := If[nqs == n, a + 1, a]
search[n_, nqs_, qs_, {q_, ps_}, a_] := 
 search[n, nqs, qs, ps, 
  search[n, nqs + 1, {q, qs}, filter[safe[q], ps], a]]

ps[n_] := 
 Fold[{#2, #1} &, {}, Flatten[Table[{i, j}, {i, n}, {j, n}], 1]]

solve[n_] := search[n, 0, {}, ps[n], 0]
Here is the equivalent F#:
let safe (x0, y0) (x1, y1) =
  x0<>x1 && y0<>y1 && x0-y0<>x1-y1 && x0+y0<>x1+y1

let rec filter f = function
  | [] -> []
  | x::xs -> if f x then x::filter f xs else filter f xs

let rec search n nqs qs ps a =
  match ps with
  | [] -> if nqs=n then a+1 else a
  | q::ps ->
      search n (nqs+1) (q::qs) (filter (safe q) ps) a
      |> search n nqs qs ps

let ps n =
  [ for i in 1..n do
      for j in 1..n do
        yield i, j ]

let solve n = search n 0 [] (ps n) 0

solve 8
Solving the 8-queens problem takes 10.5s with Mathematica and 0.07s with F#. So F# is 150× faster than Mathematica in this case.
The Stack Overflow question Mathematica "linked lists" and performance gives a more extreme example. Naive translation of that Mathematica code into F# gives an equivalent program that runs between 4,000 and 200,000× faster than the Mathematica:
let rand = System.Random()
let xs = List.init 10000 (fun _ -> rand.Next 100)
Array.init 100 (fun _ ->
  let t = System.Diagnostics.Stopwatch.StartNew()
  ignore(List.length xs)
  t.Elapsed.TotalSeconds)
Specifically, Mathematica takes 0.156s to 16s to perform a single iteration whereas the F# takes 42µs to 86µs.
If I really want to stay in Mathematica then I shoehorn everything I'm doing into Mathematica's handful of built-in data structures, e.g. Dispatch.

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