How to find the number of inversions in a list in Prolog

Question

As someone who\'s new to Prolog, I\'m looking to find out what a good way to count the number of inversions in a list.

I know how to flatten a matrix using flatten

Answer 1

First, I didn't quite get the meaning of what you were calling "inversion", so I'll stick to the quasi-canonical interpretation that @CapelliC used in his answer to this question.

Let's assume that all list items are integers, so we can use clpfd.

:- use_module(library(clpfd)).

z_z_order(X,Y,Op) :-
   zcompare(Op,X,Y).

To count the number of inversions (up-down direction changes), we do the following four steps:

1. compare adjacent items (using mapadj/3, as defined at the very end of this answer)

   ?- Zs = [1,2,4,3,2,3,3,4,5,6,7,6,6,6,5,8], mapadj(z_z_order,Zs,Cs0).
   Zs  = [1,2,4,3,2,3,3,4,5,6,7,6,6,6,5,8],
   Cs0 = [ <,<,>,>,<,=,<,<,<,<,>,=,=,>,< ].
   

2. eliminate all occurrences of = in Cs0 (using tfilter/3 and dif/3)

   ?- Cs0 = [<,<,>,>,<,=,<,<,<,<,>,=,=,>,<,<], tfilter(dif(=),Cs0,Cs1).
   Cs0 = [<,<,>,>,<,=,<,<,<,<,>,=,=,>,<,<],
   Cs1 = [<,<,>,>,<,  <,<,<,<,>,    >,<,<].
   

3. get runs of equal items in Cs1 (using splitlistIfAdj/3 and dif/3)

   ?- Cs1 = [<,<,>,>,<,<,<,<,<,>,>,<,<], splitlistIfAdj(dif,Cs1,Cs).
   Cs1 = [ <,< , >,> , <,<,<,<,< , >,> , <,< ],
   Cs  = [[<,<],[>,>],[<,<,<,<,<],[>,>],[<,<]].
   

4. the number of inversions is one less than the number of runs (using length/2 and (#=)/2)

   ?- Cs = [[<,<],[>,>],[<,<,<,<,<],[>,>],[<,<]], length(Cs,L), N #= max(0,L-1).
   Cs = [[<,<],[>,>],[<,<,<,<,<],[>,>],[<,<]], L = 5, N = 4.
   

That's it. Let's put it all together!

zs_invcount(Zs,N) :-
   mapadj(z_z_order,Zs,Cs0),
   tfilter(dif(=),Cs0,Cs1),
   splitlistIfAdj(dif,Cs1,Cs),
   length(Cs,L),
   N #= max(0,L-1).

Sample uses:

?- zs_invcount([1,2,3],0),    
   zs_invcount([1,2,3,2],1),    
   zs_invcount([1,2,3,3,2],1),               % works with duplicate items, too
   zs_invcount([1,2,3,3,2,1,1,1],1),
   zs_invcount([1,2,3,3,2,1,1,1,4,6],2),
   zs_invcount([1,2,3,3,2,1,1,1,4,6,9,1],3),
   zs_invcount([1,2,3,3,2,1,1,1,4,6,9,1,1],3).
true.

---

Implementation of meta-predicate mapadj/3

:- meta_predicate mapadj(3,?,?), list_prev_mapadj_list(?,?,3,?).
mapadj(P_3,[A|As],Bs) :-
   list_prev_mapadj_list(As,A,P_3,Bs).

list_prev_mapadj_list([]     ,_ , _ ,[]).
list_prev_mapadj_list([A1|As],A0,P_3,[B|Bs]) :-
   call(P_3,A0,A1,B),
   list_prev_mapadj_list(As,A1,P_3,Bs).