/* This code implements Red-Black trees as described in: "Introduction to Algorithms", Second Edition Cormen, Leiserson, Rivest, and Stein, MIT Press */ :- module(rbtrees, [new/1, lookup/3, insert/4, delete/3]). % create an empty tree. new(black([],[],[],[])). new(K,V,black(Nil,K,V,Nil)) :- Nil = black([],[],[],[]). lookup(Key, Val, black([],_,_,[])) :- !, fail. lookup(Key, Val, Tree) :- arg(Tree,2,KA), compare(Cmp,KA,Key,Tree), lookup(Cmp,Key,Val,Tree). lookup(<, K, V, Tree) :- arg(1,Tree,NTree), lookup(K, V, NTree). lookup(>, K, V, Tree) :- arg(4,Tree,NTree), lookup(K, V, NTree). lookup(>, K, V, Tree) :- arg(3,Tree,V). % % Tree insertion % % We don't use parent nodes, so we may have to fix the root. % insert(Tree0,Key,Val,Tree) :- insert2(Tree0,Key,Val,TreeI,_), fix_root(TreeI,Tree). % % make sure the root is always black. % fix_root(black(L,K,V,R),black(L,K,V,R)). fix_root(red(L,K,V,R),black(L,K,V,R)). % % Cormen et al present the algorithm as % (1) standard tree insertion; % (2) from the viewpoint of the newly inserted node: % partially fix the tree; % move upwards % until reaching the root. % % We do it a little bit different: % % (1) standard tree insertion; % (2) move upwards: % when reaching a black node; % if the tree below may be broken, fix it. % We take advantage of Prolog unification % to do several operations in a single go. % % % actual insertion % insert2(black([],[],[],[]), K, V, T, Status) :- !, Nil = black([],[],[],[]), T = red(Nil,K,V,Nil), Status = not_done. insert2(red(L,K0,V0,R), K, V, red(NL,K0,V0,R), Flag) :- K @< K0, !, insert2(L, K, V, NL, Flag). insert2(red(L,K0,V0,R), K, V, red(L,K0,V0,NR), Flag) :- insert2(R, K, V, NR, Flag). insert2(black(L,K0,V0,R), K, V, NT, Flag) :- K @< K0, !, insert2(L, K, V, IL, Flag0), fix_left(Flag0, black(IL,K0,V0,R), NT, Flag). insert2(black(L,K0,V0,R), K, V, NT, Flag) :- insert2(R, K, V, IR, Flag0), fix_right(Flag0, black(L,K0,V0,IR), NT, Flag). % % How to fix if we have inserted on the left % fix_left(done,T,T,done) :- !. fix_left(not_done,Tmp,Final,Done) :- fix_left(Tmp,Final,Done). % % case 1 of RB: just need to change colors. % fix_left(black(red(Al,AK,AV,red(Be,BK,BV,Ga)),KC,VC,red(De,KD,VD,Ep)), red(black(Al,AK,AV,red(Be,BK,BV,Ga)),KC,VC,black(De,KD,VD,Ep)), not_done) :- !. fix_left(black(red(red(Al,KA,VA,Be),KB,VB,Ga),KC,VC,red(De,KD,VD,Ep)), red(black(red(Al,KA,VA,Be),KB,VB,Ga),KC,VC,black(De,KD,VD,Ep)), not_done) :- !. % % case 2 of RB: got a knee so need to do rotations % fix_left(black(red(Al,KA,VA,red(Be,KB,VB,Ga)),KC,VC,De), black(red(Al,KA,VA,Be),KB,VB,red(Ga,KC,VC,De)), done) :- !. % % case 3 of RB: got a line % fix_left(black(red(red(Al,KA,VA,Be),KB,VB,Ga),KC,VC,De), black(red(Al,KA,VA,Be),KB,VB,red(Ga,KC,VC,De)), done) :- !. % % case 4 of RB: nothig to do % fix_left(T,T,done). % % How to fix if we have inserted on the right % fix_right(done,T,T,done) :- !. fix_right(not_done,Tmp,Final,Done) :- fix_right(Tmp,Final,Done). % % case 1 of RB: just need to change colors. % fix_right(black(red(Ep,KD,VD,De),KC,VC,red(red(Ga,KB,VB,Be),KA,VA,Al)), red(black(Ep,KD,VD,De),KC,VC,black(red(Ga,KB,VB,Be),KA,VA,Al)), not_done) :- !. fix_right(black(red(Ep,KD,VD,De),KC,VC,red(Ga,Ka,Va,red(Be,KB,VB,Al))), red(black(Ep,KD,VD,De),KC,VC,black(Ga,Ka,Va,red(Be,KB,VB,Al))), not_done) :- !. % % case 2 of RB: got a knee so need to do rotations % fix_right(black(De,KC,VC,red(red(Ga,KB,VB,Be),KA,VA,Al)), black(red(De,KC,VC,Ga),KB,VB,red(Be,KA,VA,Al)), done) :- !. % % case 3 of RB: got a line % fix_right(black(De,KC,VC,red(Ga,KB,VB,red(Be,KA,VA,Al))), black(red(De,KC,VC,Ga),KB,VB,red(Be,KA,VA,Al)), done) :- !. % % case 4 of RB: nothing to do. % fix_right(T,T,done). % % simplified processor % % pretty_print(T) :- pretty_print(T,6). pretty_print(black([],[],[],[]),_) :- !. pretty_print(red(L,K,_,R),D) :- DN is D+6, pretty_print(L,DN), format("~t~a:~d~*|~n",[r,K,D]), pretty_print(R,DN). pretty_print(black(L,K,_,R),D) :- DN is D+6, pretty_print(L,DN), format("~t~a:~d~*|~n",[b,K,D]), pretty_print(R,DN). delete(T, K, NT) :- delete(T, K, NT, _). % % I am afraid our representation is not as nice for delete % delete(red(L,K0,V0,R), K, NT, Flag) :- K @< K0, !, delete(L, K, NL, Flag0), fixup_left(Flag0,red(NL,K0,V0,R),NT, Flag). delete(red(L,K0,V0,R), K, NT, Flag) :- K @> K0, !, delete(R, K, NR, Flag0), fixup_right(Flag0,red(L,K0,V0,NR),NT, Flag). delete(red(L,_,_,R), _, OUT, Flag) :- % K == K0, delete_red_node(L,R,OUT,Flag). delete(black(L,K0,V0,R), K, NT, Flag) :- K @< K0, !, delete(L, K, NL, Flag0), fixup_left(Flag0,black(NL,K0,V0,R),NT, Flag). delete(black(L,K0,V0,R), K, NT, Flag) :- K @> K0, !, delete(R, K, NR, Flag0), fixup_right(Flag0,black(L,K0,V0,NR),NT, Flag). delete(black(L,_,_,R), _, OUT, Flag) :- % K == K0, delete_black_node(L,R,OUT,Flag). delete_red_node(L,L,L,done) :- !. delete_red_node(black([],[],[],[]),R,R,done) :- !. delete_red_node(L,black([],[],[],[]),L,done) :- !. delete_red_node(L,R,OUT,Done) :- delete_next(R,NK,NV,NR,Done0), fixup_right(Done0,red(L,NK,NV,NR),OUT,Done). delete_black_node(L,L,L,not_done) :- !. delete_black_node(black([],[],[],[]),red(L,K,V,R),black(L,K,V,R),done) :- !. delete_black_node(black([],[],[],[]),R,R,not_done) :- !. delete_black_node(red(L,K,V,R),black([],[],[],[]),black(L,K,V,R),done) :- !. delete_black_node(L,black([],[],[],[]),L,not_done) :- !. delete_black_node(L,R,OUT,Done) :- delete_next(R,NK,NV,NR,Done0), fixup_right(Done0,black(L,NK,NV,NR),OUT,Done). delete_next(red(black([],[],[],[]),K,V,R),K,V,R,done) :- !. delete_next(black(black([],[],[],[]),K,V,red(L1,K1,V1,R1)), K,V,black(L1,K1,V1,R1),done) :- !. delete_next(black(black([],[],[],[]),K,V,R),K,V,R,not_done) :- !. delete_next(red(L,K,V,R),K0,V0,OUT,Done) :- delete_next(L,K0,V0,NL,Done0), fixup_left(Done0,red(NL,K,V,R),OUT,Done). delete_next(black(L,K,V,R),K0,V0,OUT,Done) :- delete_next(L,K0,V0,NL,Done0), fixup_left(Done0,black(NL,K,V,R),OUT,Done). fixup_left(done,T,T,done). fixup_left(not_done,T,NT,Done) :- fixup2(T,NT,Done). % % case 1: x moves down, so we have to try to fix it again. % case 1 -> 2,3,4 -> done % fixup2(black(black(Al,KA,VA,Be),KB,VB,red(black(Ga,KC,VC,De),KD,VD,black(Ep,KE,VE,Fi))), black(T1,KD,VD,black(Ep,KE,VE,Fi)),done) :- !, fixup2(red(black(Al,KA,VA,Be),KB,VB,black(Ga,KC,VC,De)), T1, _). % % case 2: x moves up, change one to red % fixup2(red(black(Al,KA,VA,Be),KB,VB,black(black(Ga,KC,VC,De),KD,VD,black(Ep,KE,VE,Fi))), black(black(Al,KA,VA,Be),KB,VB,red(black(Ga,KC,VC,De),KD,VD,black(Ep,KE,VE,Fi))),done) :- !. fixup2(black(black(Al,KA,VA,Be),KB,VB,black(black(Ga,KC,VC,De),KD,VD,black(Ep,KE,VE,Fi))), black(black(Al,KA,VA,Be),KB,VB,red(black(Ga,KC,VC,De),KD,VD,black(Ep,KE,VE,Fi))),not_done) :- !. % % case 3: x stays put, shift left and do a 4 % fixup2(red(black(Al,KA,VA,Be),KB,VB,black(red(Ga,KC,VC,De),KD,VD,black(Ep,KE,VE,Fi))), red(black(black(Al,KA,VA,Be),KB,VB,Ga),KC,VC,black(De,KD,VD,black(Ep,KE,VE,Fi))), done) :- !. fixup2(black(black(Al,KA,VA,Be),KB,VB,black(red(Ga,KC,VC,De),KD,VD,black(Ep,KE,VE,Fi))), black(black(black(Al,KA,VA,Be),KB,VB,Ga),KC,VC,black(De,KD,VD,black(Ep,KE,VE,Fi))), done) :- !. % % case 4: rotate left, get rid of red % fixup2(red(black(Al,KA,VA,Be),KB,VB,black(C,KD,VD,red(Ep,KE,VE,Fi))), red(black(black(Al,KA,VA,Be),KB,VB,C),KD,VD,black(Ep,KE,VE,Fi)), done). fixup2(black(black(Al,KA,VA,Be),KB,VB,black(C,KD,VD,red(Ep,KE,VE,Fi))), black(black(black(Al,KA,VA,Be),KB,VB,C),KD,VD,black(Ep,KE,VE,Fi)), done). fixup_right(done,T,T,done). fixup_right(not_done,T,NT,Done) :- fixup3(T,NT,Done). % % case 1: x moves down, so we have to try to fix it again. % case 1 -> 2,3,4 -> done % fixup3(black(red(black(Fi,KE,VE,Ep),KD,VD,black(De,KC,VC,Ga)),KB,VB,black(Be,KA,VA,Al)), black(black(Fi,KE,VE,Ep),KD,VD,T1),done) :- !, fixup3(red(black(De,KC,VC,Ga),KB,VB,black(Be,KA,VA,Al)),T1,_). % % case 2: x moves up, change one to red % fixup3(red(black(black(Fi,KE,VE,Ep),KD,VD,black(De,KC,VC,Ga)),KB,VB,black(Be,KA,VA,Al)), black(red(black(Fi,KE,VE,Ep),KD,VD,black(De,KC,VC,Ga)),KB,VB,black(Be,KA,VA,Al)), done) :- !. fixup3(black(black(black(Fi,KE,VE,Ep),KD,VD,black(De,KC,VC,Ga)),KB,VB,black(Be,KA,VA,Al)), black(red(black(Fi,KE,VE,Ep),KD,VD,black(De,KC,VC,Ga)),KB,VB,black(Be,KA,VA,Al)), not_done):- !. % % case 3: x stays put, shift left and do a 4 % fixup3(red(black(black(Fi,KE,VE,Ep),KD,VD,red(De,KC,VC,Ga)),KB,VB,black(Be,KA,VA,Al)), red(black(black(Fi,KE,VE,Ep),KD,VD,De),KC,VC,black(Ga,KB,VB,black(Be,KA,VA,Al))), done) :- !. fixup3(black(black(black(Fi,KE,VE,Ep),KD,VD,red(De,KC,VC,Ga)),KB,VB,black(Be,KA,VA,Al)), black(black(black(Fi,KE,VE,Ep),KD,VD,De),KC,VC,black(Ga,KB,VB,black(Be,KA,VA,Al))), done) :- !. % % case 4: rotate right, get rid of red % fixup3(red(black(red(Fi,KE,VE,Ep),KD,VD,C),KB,VB,black(Be,KA,VA,Al)), red(black(Fi,KE,VE,Ep),KD,VD,black(C,KB,VB,black(Be,KA,VA,Al))), done). fixup3(black(black(red(Fi,KE,VE,Ep),KD,VD,C),KB,VB,black(Be,KA,VA,Al)), black(black(Fi,KE,VE,Ep),KD,VD,black(C,KB,VB,black(Be,KA,VA,Al))), done). % % This code checks if a tree is ordered and a rbtree % % rbtree(black([],[],[],[])) :- !. rbtree(T) :- catch(rbtree1(T),msg(S,Args),format(S,Args)). rbtree1(black(L,K,_,R)) :- find_path_blacks(L, 0, Bls), check_rbtree(L,-1000000,K,Bls), check_rbtree(R,K,1000000,Bls). rbtree1(red(_,_,_,_)) :- throw(msg("root should be black",[])). find_path_blacks(black([],[],[],[]), Bls, Bls) :- !. find_path_blacks(black(L,_,_,_), Bls0, Bls) :- Bls1 is Bls0+1, find_path_blacks(L, Bls1, Bls). find_path_blacks(red(L,_,_,_), Bls0, Bls) :- find_path_blacks(L, Bls0, Bls). check_rbtree(black([],[],[],[]),Min,Max,Bls0) :- !, check_height(Bls0,Min,Max). check_rbtree(red(L,K,_,R),Min,Max,Bls) :- check_val(K,Min,Max), check_red_child(L), check_red_child(R), check_rbtree(L,Min,K,Bls), check_rbtree(R,K,Max,Bls). check_rbtree(black(L,K,_,R),Min,Max,Bls0) :- check_val(K,Min,Max), Bls is Bls0-1, check_rbtree(L,Min,K,Bls), check_rbtree(R,K,Max,Bls). check_height(0,_,_) :- !. check_height(Bls0,Min,Max) :- throw(msg("Unbalance ~d between ~w and ~w~n",[Bls0,Min,Max])). check_val(K, Min, Max) :- K > Min, K < Max, !. check_val(K, Min, Max) :- throw(msg("not ordered: ~w not between ~w and ~w~n",[K,Min,Max])). check_red_child(black(_,_,_,_)). check_red_child(red(_,K,_,_)) :- throw(msg("must be red: ~w~n",[K])). %count(1,16,X), format("deleting ~d~n",[X]), new(1,a,T0), insert(T0,2,b,T1), insert(T1,3,c,T2), insert(T2,4,c,T3), insert(T3,5,c,T4), insert(T4,6,c,T5), insert(T5,7,c,T6), insert(T6,8,c,T7), insert(T7,9,c,T8), insert(T8,10,c,T9),insert(T9,11,c,T10), insert(T10,12,c,T11),insert(T11,13,c,T12),insert(T12,14,c,T13),insert(T13,15,c,T14), insert(T14,16,c,T15),delete(T15,X,T16),pretty_print(T16),rbtree(T16),fail. % count(1,16,X0), X is -X0, format("deleting ~d~n",[X]), new(-1,a,T0), insert(T0,-2,b,T1), insert(T1,-3,c,T2), insert(T2,-4,c,T3), insert(T3,-5,c,T4), insert(T4,-6,c,T5), insert(T5,-7,c,T6), insert(T6,-8,c,T7), insert(T7,-9,c,T8), insert(T8,-10,c,T9),insert(T9,-11,c,T10), insert(T10,-12,c,T11),insert(T11,-13,c,T12),insert(T12,-14,c,T13),insert(T13,-15,c,T14), insert(T14,-16,c,T15),delete(T15,X,T16),pretty_print(T16),rbtree(T16),fail. count(I,_,I). count(I,M,L) :- I < M, I1 is I+1, count(I1,M,L). test_pos :- new(1,a,T0), N = 10000, build_ptree(2,N,T0,T), % pretty_print(T), rbtree(T), clean_tree(1,N,T,_), bclean_tree(N,1,T,_), count(1,N,X), ( delete(T,X,TF) -> true ; abort ), % pretty_print(TF), rbtree(TF), format("done ~d~n",[X]), fail. test_pos. build_ptree(X,X,T0,TF) :- !, insert(T0,X,X,TF). build_ptree(X1,X,T0,TF) :- insert(T0,X1,X1,TI), X2 is X1+1, build_ptree(X2,X,TI,TF). clean_tree(X,X,T0,TF) :- !, format("cleaning ~d~n", [X]), delete(T0,X,TF), ( rbtree(TF) -> true ; abort). clean_tree(X1,X,T0,TF) :- format("cleaning ~d~n", [X1]), delete(T0,X1,TI), X2 is X1+1, ( rbtree(TI) -> true ; abort), clean_tree(X2,X,TI,TF). bclean_tree(X,X,T0,TF) :- !, format("cleaning ~d~n", [X]), delete(T0,X,TF), ( rbtree(TF) -> true ; abort). bclean_tree(X1,X,T0,TF) :- format("cleaning ~d~n", [X1]), delete(T0,X1,TI), X2 is X1-1, ( rbtree(TI) -> true ; abort), bclean_tree(X2,X,TI,TF). tree(X2,X,TI,TF). test_neg :- Size = 10000, new(-1,a,T0), build_ntree(2,Size,T0,T), % pretty_print(T), rbtree(T), clean_tree(1,Size,T,_), bclean_tree(Size,1,T,_), count(1,Size,X), NX is -X, ( delete(T,NX,TF) -> true ; abort ), % pretty_print(TF), rbtree(TF), format("done ~d~n",[X]), fail. test_neg. build_ntree(X,X,T0,TF) :- !, X1 is -X, insert(T0,X1,X1,TF). build_ntree(X1,X,T0,TF) :- NX1 is -X1, insert(T0,NX1,NX1,TI), X2 is X1+1, build_ntree(X2,X,TI,TF).