How to transpose a matrix in prolog - list

Question

How can I transpose a list like [[1,2,3][4,5,6][6,7,8]] to [[1,4,6],[2,7,8],[3,6,9]]?
To depict it: I'd like to flip the matrix 90 degree to the left. How can I do that?

Answer 1

Not sure your example is correct, but I get the idea.
If using SWI-PROLOG, you can use the CLPFD module, like so:
:- use_module(library(clpfd)).
Allowing you to use the transpose/2 predicate, like this:
1 ?- transpose([[1,2,3],[4,5,6],[6,7,8]], X).
X = [[1, 4, 6], [2, 5, 7], [3, 6, 8]].
Otherwise (if no SWI-PROLOG), you could simply use this implementation (which happened to be an old one in SWI's clpfd):
transpose([], []).
transpose([F|Fs], Ts) :-
transpose(F, [F|Fs], Ts).
transpose([], _, []).
transpose([_|Rs], Ms, [Ts|Tss]) :-
lists_firsts_rests(Ms, Ts, Ms1),
transpose(Rs, Ms1, Tss).
lists_firsts_rests([], [], []).
lists_firsts_rests([[F|Os]|Rest], [F|Fs], [Os|Oss]) :-
lists_firsts_rests(Rest, Fs, Oss).
For an updated version which uses foldl and maplist built-ins, see clpfd.pl.

Answer 2

This is the smallest solution I could come up with.
Code
transpose([[]|_], []).
transpose(Matrix, [Row|Rows]) :- transpose_1st_col(Matrix, Row, RestMatrix),
transpose(RestMatrix, Rows).
transpose_1st_col([], [], []).
transpose_1st_col([[H|T]|Rows], [H|Hs], [T|Ts]) :- transpose_1st_col(Rows, Hs, Ts).
Test
:- transpose([[1,2,3],
[4,5,6],
[7,8,9]], R),
print(R).
Prints:
[[1,4,7],
[2,5,8],
[3,6,9]]
Explanation
The way it works is that transpose will recursively call transpose_1st_col which extracts and transposes the first column of the matrix. For example:
:- transpose_1st_col([[1,2,3],
[4,5,6],
[7,8,9]], Row, RestMatrix),
print(Row),
print(RestMatrix).
will print
[1,4,7]
and
[[2,3],
[5,6],
[8,9]]
This is repeated until the input matrix is empty, at which point all columns have been transposed. The transposed columns are then joined into the transposed matrix.

Answer 3

Here's a fragment of a larger answer:
% transposed(+A, ?B) iff matrix B is transposed matrix A
transposed(A, B) :- transposed(A, [], B).
transposed(M, X, X) :- empty(M), !.
transposed(M, A, X) :- columns(M, Hs, Ts), transposed(Ts, [Hs|A], X).
% empty(+A) iff A is empty list or a list of empty lists
empty([[]|A]) :- empty(A).
empty([]).
% columns(+M, ?Hs, ?Ts) iff Hs is the first column
% of matrix M and Ts is the rest of matrix M
columns([[Rh|Rt]|Rs], [Rh|Hs], [Rt|Ts]) :- columns(Rs, Hs, Ts).
columns([[]], [], []).
columns([], [], []).

Answer 4

simpler approach:
trans(M, [P|T]):- first(M, P, A), trans(A, T).
trans(Empty, []):- empty(Empty).
empty([[]|T]):- empty(T).
empty([[]]).
first([[P|A]|R], [P|Ps], [A|As]):- first(R, Ps, As).
first([], [], []).
efficient also
[debug] 36 ?- time(trans([[1,2,3],[4,5,6],[7,8,9]],A)).
% 21 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips)
A = [[1,4,7],[2,5,8],[3,6,9]] ;
% 12 inferences, 0.000 CPU in 0.000 seconds (?% CPU, Infinite Lips)
false.

Answer 5

Another simple approach:
transpose(M0, M) :-
nonvar(M0),
findall(L, maplist(nth1(_), M0, L), M).
?- transpose([[1,2,3],[4,5,6],[7,8,9]], M).
M = [[1, 4, 7], [2, 5, 8], [3, 6, 9]]. `

Answer 6

An iterative approach:
trans([H|R],[H1|R1]):-trans2([H|R],[H|R],[],[H1|R1],0),!.
trans2([A|_],_,_,[],N):-length(A,N).
trans2(M,[],H1,[H1|R1],N):-N1 is N+1, trans2(M,M,[],R1,N1).
trans2(M,[H|R],L,[H1|R1],N):-nth0(N,H,X),
append(L,[X],L1),trans2(M,R,L1,[H1|R1],N).

Answer 7

My solution with full names for a better understanding:
% emptyMatrix(Line, EmptyMatrix)
emptyMatrix([],[]).
emptyMatrix([_|T1],[[]|T2]):-emptyMatrix(T1,T2).
% only length of parameter 'Line' is interesting. It ignores its content.
% appendElement(Element, InputList, OutputList)
appendElement(E,[],[E]).
appendElement(E,[H|T],[H|L]):-appendElement(E,T,L).
% appendTransposed(NestedList, InputMatrix, OutputMatrix)
appendTransposed([],[],[]).
appendTransposed([X|T1],[],[[X]|T3]):-appendTransposed(T1,[],T3).
appendTransposed([X|T1],[R|T2],[C|T3]):-appendElement(X,R,C),appendTransposed(T1,T2,T3).
% transposeMatrix(InputMatrix, TransposedMatrix)
transposeMatrix([L|M],T):-emptyMatrix(L,A),transpose([L|M],T,A).
transpose([],T,T).
transpose([L|M],T,A):-appendTransposed(L,A,B),transpose(M,T,B).
A 'line' can be a col or a row.
The idea lies in appending the elements into the lists of an empty matrix.
(e.g. all elements of the first row = the first elements of all cols
=> all elements of the first i-nth row = the i-nth elements of all cols)
It works on my machine as this session protocol shows to me:
5 ?- transposeMatrix([[1,2],[3,4]],T).
T = [[1, 3], [2, 4]] ;
false.
6 ?- transposeMatrix([[1],[2]],T).
T = [[1, 2]] ;
false.
7 ?- transposeMatrix([[1,2,3],[4,5,6]],T).
T = [[1, 4], [2, 5], [3, 6]] ;
false.
8 ?- transposeMatrix([[1]],T).
T = [[1]] ;
false.

Answer 8

Another approach:
delete_one_list([], []).
delete_one_list([[_|L]|LLs], [L|Ls]) :-
delete_one_list(LLs, Ls).
transpose_helper([], []).
transpose_helper([[X|_]|Xs], [X|Ys]) :-
transpose_helper(Xs, Ys).
transpose([[]|_], []).
transpose(List, [L|Ls]) :-
transpose_helper(List, L),
delete_one_list(List, NewList),
transpose(NewList, Ls).