# An elegant codes on finding all permutations of a list

EDIT 2023/06/03

I have compared robustness of the `permutations/2` with `all_perm/2. The `all_perm/2` works on my experimetal ZDD library in Pack/pac. ( Note that the ZDD library can be used with some care as if the atoms were not ordered.)

According to the experiment below, `all_perm` could be at leaset about

``````12 x 13 x 14 x 15
``````

times more robust than the `permutations/2`. The `permutions/2` does not care about any structure sharing but only stylish aspect of programming. On the other hand, `all_perm` works on the systematic structure sharing. My quick words for impression after this experiment, ZDD version seemed more easy to write than that on the standard list processing. I am not sure why. To find all the permutations might be an exceptional case.

Statistics of experiment
``````% ?- time((N = 11, numlist(1, N, Ns), permutations(Ns, Ps), length(Ps, Len))),
%	 writeln(pass(N)), fail; writeln("\nDone.\n").
%@ % 309,470,824 inferences, 14.952 CPU in 17.814 seconds (84% CPU, 20697790 Lips)
%@ pass(11)
%@
%@ Done.

% ?- call_with_time_limit(120, time((N = 12, numlist(1, N, Ns), permutations(Ns, Ps), length(Ps, Len)))),
%	 writeln(pass(N)), fail; writeln("\nDone.\n").
%@ % 878,420,042 inferences, 85.383 CPU in 268.075 seconds (32% CPU, 10288036 Lips)
%@ ERROR: Unhandled exception: Time limit exceeded\
``````
``````% ?- N=15, time((numlist(1, N, Ns),  (zdd all_perm(Ns, X), card(X, C)))).
%@ % 84,073,484 inferences, 47.282 CPU in 47.647 seconds (99% CPU, 1778120 Lips)
%@ N = 15,
%@ X = 1105921,
%@ C = 1,307,674,368,000.

all_perm(D, P, S):-  zdd_sort(D, D0, S),
findall_perm(D0, P, S).
%
findall_perm([], 1, _):-!.
findall_perm(D, X, S):- memo(perm(D)-X, S),
( nonvar(X) -> true		%	, write(.) % many hits.
;
findall(I-D0, select(I, D, D0), U),
join_perm_for_selection(U, 0, X, S)).
%
join_perm_for_selection([], X, X, _).
join_perm_for_selection([I-D|U], X, Y, S):-
findall_perm(D, X0, S),
zdd_cons(X1, I, X0, S),
zdd_join(X1, X, X2, S),
join_perm_for_selection(U, X2, Y, S).
``````

EDIT End

How do you write codes on finding all permutations of a list ? So far I had several chances to write codes for that, but this is my first time to write successfully an “elegant” one from my point of view:

• `append` is not used.
• ! (cut) is not used.
• right recursive form (Tail Rcursion form ?).

However I have to use reverse/3 with stack. I would like to see more elegant one which does not use the reverse.

``````% ?- permutations([a,b,c],X), length(X, Len).
% ?- between(1, 11, N), numlist(1, N, Ns), permutations(Ns, Ps),
%	length(Ps, Len),
%	factorial(N, Len), writeln(pass(N)), fail; writeln("\nDone.\n").

reverse([], X, X).
reverse([A|As], X, Y):- reverse(As, [A|X], Y).

permutations(As, Ps):- permutations(As, [[]], Ps).
%
permutations([], Ps, Ps).
permutations([X|Xs], Ps, Qs):- permutations(Ps, X, Rs, []),
permutations(Xs, Rs, Qs).
%
permutations([], _, Ps, Ps).
permutations([P|Ps], X, Qs, Rs):-
permutations(P, X, [], Qs, Us),
permutations(Ps, X, Us, Rs).
%
permutations([], X, Bs, [U|Ps], Ps):- reverse(Bs, [X], U).
permutations([A|As], X, Bs, [U|Ps], Qs):-
reverse(Bs, [X, A|As], U),
permutations(As, X, [A|Bs], Ps, Qs).
``````