Is there a better way to add a "key" for the list of GatherBy?

Question

In ruby ,

1.upto(10).to_a.group_by{|x| x%3}

gives

{1=>[1, 4, 7, 10], 2=>[2, 5, 8], 0=>[3, 6, 9]}

I would like use patten replace as a hash table in Mathematica. For example:

{1 -> {1, 4, 7, 10}, 2 -> {2, 5, 8}, 0 -> {3, 6, 9}}

I know I can use GatherBy, but it only contains the value. I tried to solve and I feel it isn't efficient for a large list.

fun = Mod[#, 3] &;
Thread[fun /@ #[[All, 1]] -> #] &@GatherBy[Range[10], fun]

Answer 1

You can use MapIndexed:

With[{fun = Mod[#, 3] &}, MapIndexed[fun@First@#2 -> # &, GatherBy[Range@10, fun]]]
(* {1 -> {1, 4, 7, 10}, 2 -> {2, 5, 8}, 0 -> {3, 6, 9}} *)

On my machine, this runs for $10^7$ integers in under a second, roughly on the same order as your own approach.

---

It can be made faster by noting that if you're using Mod[#, n]& as the function, your keys will be Range[0, n-1]. Hence, we don't really need to Thread or use MapIndexed when we can simply generate this list, do a ragged transpose using Flatten and apply Rule at the very end:

Rule @@@ With[{n = 3}, 
    {RotateLeft@Range[0, n - 1], GatherBy[Range[10], Mod[#, n] &]} ~Flatten~ {2}
]
(* {1 -> {1, 4, 7, 10}, 2 -> {2, 5, 8}, 0 -> {3, 6, 9}} *) 

Answer 2

I would use Reap and Sow for this, or more specifically a re-packaging of them, originally posted here:

SelectEquivalents[x_List,f_:Identity, g_:Identity, h_:(#2&)]:=
 Reap[Sow[g[#],{f[#]}]&/@x, _, h][[2]];

where the second argument is the (f) is the equivalence function, the third argument (g) transforms the data prior to being gathered, and the fourth argument (h) outputs the collection. Note, two arguments are passed to h, the first is the equivalence key and the second is the list of elements that match that key.

Then,

SelectEquivalents[Range@10, Mod[#,3]&, #&, #1 -> #2&]
(*{1 -> {1, 4, 7, 10}, 2 -> {2, 5, 8}, 0 -> {3, 6, 9}}*)

It should be noted, that this isn't a speed demon by any stretch of the imagination; clocking in at 24 secs on my machine, while RM's suggestion takes under a second.

Answer 3

 fun = Mod[#, 3] &;
 selector = fun /@ Range[10];
 (# -> Pick[Range@10, selector, #]) & /@ DeleteDuplicates[selector]
 (* {1 -> {1, 4, 7, 10}, 2 -> {2, 5, 8}, 0 -> {3, 6, 9}}*) 

More generally,

 ClearAll[pickEquivalents];
 pickEquivalents[domain_, fun_] :=  With[{selector = fun /@ domain},
  With[{range = DeleteDuplicates[selector]},
 (# -> Pick[domain, selector, #]) & /@  range]]
 pickEquivalents[Range[10], Mod[#, 4] &]
 (* {1 -> {1, 5, 9}, 2 -> {2, 6, 10}, 3 -> {3, 7}, 0 -> {4, 8}} *)

Answer 4

Mathematica 10 introduced GroupBy which does exactly what you want:

a = GroupBy[Range@10, Mod[#, 3] &]

<|1 -> {1, 4, 7, 10}, 2 -> {2, 5, 8}, 0 -> {3, 6, 9}|>

The result is an Association which among other things can be used for replacement:

foo[2, bar[0, 1]] /. a

foo[{2, 5, 8}, bar[{3, 6, 9}, {1, 4, 7, 10}]]

Also:

a[2]

{2, 5, 8}

Lookup[a, {2, 1}]

{{2, 5, 8}, {1, 4, 7, 10}}

Answer 5

With[{fun = Mod[#, 3] &},
   Rule[fun[#1], {##}] & @@@ GatherBy[Range[10^6], fun]
   ]; // Timing

A neat way,but not fast.

GatherBy[Range[10^6], Mod[#, 3] &][[All, 1]] // Timing
(*{0.093, {1, 2, 3}}*)
#1 & @@@ GatherBy[Range[10^6], Mod[#, 3] &] // Timing
{0.234, {1, 2, 3}}

Answer 6

Not very efficient, but I like it:

Reap[# ~Sow~ fun@# & ~Scan~ Range@10, _, Rule][[2]]

{1 -> {1, 4, 7, 10}, 2 -> {2, 5, 8}, 0 -> {3, 6, 9}}