Learning Prolog via Examples - Combinatorics

Part of INTERACTIVE PROLOG GUIDE

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This lecture covers basic combinatorial algorithms which generate successively all permutations, combinations and variations respectively. Refresh your memory! How many permutations, combinations and variations can be generated from set of N elements? And what about if repeated elements are allowed?

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permutations

Permutation of the list L is a list containing all elements of list L in some order. Guess which permutation is generated first usign following procedure. And what about second?

perm(List,[H|Perm]):-delete(H,List,Rest),perm(Rest,Perm).
perm([],[]).

delete(X,[X|T],T).
delete(X,[H|T],[H|NT]):-delete(X,T,NT).

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combinations

Combination is an arbitrary subset of the set containing given number of elements. The order of elements is irrelevant.

comb(0,_,[]).
comb(N,[X|T],[X|Comb]):-N>0,N1 is N-1,comb(N1,T,Comb).
comb(N,[_|T],Comb):-N>0,comb(N,T,Comb).

It is possible to program generator of combinations without arithmetics. The following procedure comb2 assumes the list with N free variables as its second argument and it binds these variables. So, use ?-comb2([1,2,3,4],[X,Y]) to generate combinations with two elements.

comb2(_,[]).
comb2([X|T],[X|Comb]):-comb2(T,Comb).
comb2([_|T],[X|Comb]):-comb2(T,[X|Comb]).

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combinations with repeated elements

This type of combination can contain an element more times. Thus, it is not a set but a multi-set.

comb_rep(0,_,[]).
comb_rep(N,[X|T],[X|RComb]):-N>0,N1 is N-1,comb_rep(N1,[X|T],RComb).
comb_rep(N,[_|T],RComb):-N>0,comb_rep(N,T,RComb).

Try to program combinations with repeted elements as well as followingcombinatorial algorithms without aritmetics, i.e., without numerical counter.

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variations

Variation is a subset with given number of elements. The order of elements in variation is significant.

varia(0,_,[]).
varia(N,L,[H|Varia]):-N>0,N1 is N-1,delete(H,L,Rest),varia(N1,Rest,Varia).

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variations with repeated elements

Again, this type of variation can contain repeated elements.

varia_rep(0,_,[]).
varia_rep(N,L,[H|RVaria]):-N>0,N1 is N-1,delete(H,L,_),varia_rep(N1,L,RVaria).