performance - Projecting a list of lists efficiently in F#

Question

I have to do projection of a list of lists which returns all combinations with each element from each list. For example:

projection([[1]; [2; 3]]) = [[1; 2]; [1; 3]].
projection([[1]; [2; 3]; [4; 5]]) = [[1; 2; 4]; [1; 2; 5]; [1; 3; 4]; [1; 3; 5]].

I come up with a function:

let projection lss0 =
    let rec projectionUtil lss accs =
        match lss with
        | []        ->  accs
        | ls::lss'  ->  projectionUtil lss' (List.fold (fun accs' l -> 
                                                        accs' @ List.map (fun acc -> acc @ [l]) accs) 
                                                        [] ls)
match lss0 with
| [] -> []
| ls::lss' ->         
    projectionUtil lss' (List.map (fun l -> [l]) ls)

and a testcase:

#time "on";;
let N = 10
let fss0 = List.init N (fun i -> List.init (i+1) (fun j -> j+i*i+i));;
let fss1 = projection fss0;;

The function is quite slow now, with N = 10 it takes more than 10 seconds to complete. Moreover, I think the solution is unnatural because I have to breakdown the same list in two different ways. Any suggestion how I can improve performance and readability of the function?

Answer 1

First of all, try to avoid list concatenation (@) whenever possible, since it's O(N) instead of O(1) prepend.

I'd start with a (relatively) easy to follow plan of how to compute the cartesian outer product of lists.

• Prepend each element of the first list to each sublist in the cartesian product of the remaining lists.
• Take care of the base case.

First version:

let rec cartesian = function
  | [] -> [[]]
  | L::Ls -> [for C in cartesian Ls do yield! [for x in L do yield x::C]]

This is the direct translation of the sentences above to code.

Now speed this up: instead of list comprehensions, use list concatenations and maps:

let rec cartesian2 = function
  | [] -> [[]]
  | L::Ls -> cartesian2 Ls |> List.collect (fun C -> L |> List.map (fun x->x::C))

This can be made faster still by computing the lists on demand via a sequence:

let rec cartesian3 = function
  | [] -> Seq.singleton []
  | L::Ls -> cartesian3 Ls |> Seq.collect (fun C -> L |> Seq.map (fun x->x::C))

This last form is what I use myself, since I most often just need to iterate over the results instead of having them all at once.

Some benchmarks on my machine: Test code:

let test f N = 
  let fss0 = List.init N (fun i -> List.init (i+1) (fun j -> j+i*i+i))
  f fss0 |> Seq.length

Results in FSI:

> test projection 10;;
Real: 00:00:18.066, CPU: 00:00:18.062, GC gen0: 168, gen1: 157, gen2: 7
val it : int = 3628800
> test cartesian 10;;
Real: 00:00:19.822, CPU: 00:00:19.828, GC gen0: 244, gen1: 121, gen2: 3
val it : int = 3628800
> test cartesian2 10;;
Real: 00:00:09.247, CPU: 00:00:09.250, GC gen0: 94, gen1: 52, gen2: 2
val it : int = 3628800
> test cartesian3 10;;
Real: 00:00:04.254, CPU: 00:00:04.250, GC gen0: 359, gen1: 1, gen2: 0
val it : int = 3628800