prologpath-findingconstraint-programmingclpfdsicstus-prolog
Optimizing pathfinding in Constraint Logic Programming with Prolog
I am working on a small prolog application to solve the Skyscrapers and Fences puzzle.
An unsolved puzzle:
A solved puzzle:
When I pass the program already solved puzzles it is quick, almost instantaneous, to validate it for me. When I pass the program really small puzzles (2x2, for example, with modified rules, of course), it is also quite fast to find a solution.
The problem is on computing puzzles with the "native" size of 6x6. I've left it running for 5 or so hours before aborting it. Way too much time.
I've found that the part that takes the longest is the "fences" one, not the "skyscrapers". Running "skyscrapers" separately results in a fast solution.
Here's my algorithm for fences:
- Vertices are represented by numbers, 0 means the path doesn't pass through that particular vertex, > 1 represents that vertex's order in the path.
- Constrain each cell to have the appropriate amount of lines surrounding it.
- That means that two vertexes are connected if they have sequential numbers, e.g., 1 -> 2, 2 -> 1, 1 ->
Max,Max-> 1 (Maxis the number for the last vertex in the path. computed viamaximum/2)
- That means that two vertexes are connected if they have sequential numbers, e.g., 1 -> 2, 2 -> 1, 1 ->
- Make sure each non-zero vertex has at least two neighboring vertices with sequential numbers
- Constrain
Maxto be equal to(BoardWidth + 1)^2 - NumberOfZeros(BoardWidth+1is the number of vertices along the edge andNumberOfZerosis computed viacount/4). - Use
nvalue(Vertices, Max + 1)to make sure the number of distinct values inVerticesisMax(i.e. the number of vertices in the path) plus1(zero values) - Find the first cell containing a
3and force the path to begin and end there, for efficiency purposes
What can I do to increase efficiency? Code is included below for reference.
skyscrapersinfences.pro
:-use_module(library(clpfd)).
:-use_module(library(lists)).
:-ensure_loaded('utils.pro').
:-ensure_loaded('s1.pro').
print_row([]).
print_row([Head|Tail]) :-
write(Head), write(' '),
print_row(Tail).
print_board(Board, BoardWidth) :-
print_board(Board, BoardWidth, 0).
print_board(_, BoardWidth, BoardWidth).
print_board(Board, BoardWidth, Index) :-
make_segment(Board, BoardWidth, Index, row, Row),
print_row(Row), nl,
NewIndex is Index + 1,
print_board(Board, BoardWidth, NewIndex).
print_boards([], _).
print_boards([Head|Tail], BoardWidth) :-
print_board(Head, BoardWidth), nl,
print_boards(Tail, BoardWidth).
get_board_element(Board, BoardWidth, X, Y, Element) :-
Index is BoardWidth*Y + X,
get_element_at(Board, Index, Element).
make_column([], _, _, []).
make_column(Board, BoardWidth, Index, Segment) :-
get_element_at(Board, Index, Element),
munch(Board, BoardWidth, MunchedBoard),
make_column(MunchedBoard, BoardWidth, Index, ColumnTail),
append([Element], ColumnTail, Segment).
make_segment(Board, BoardWidth, Index, row, Segment) :-
NIrrelevantElements is BoardWidth*Index,
munch(Board, NIrrelevantElements, MunchedBoard),
select_n_elements(MunchedBoard, BoardWidth, Segment).
make_segment(Board, BoardWidth, Index, column, Segment) :-
make_column(Board, BoardWidth, Index, Segment).
verify_segment(_, 0).
verify_segment(Segment, Value) :-
verify_segment(Segment, Value, 0).
verify_segment([], 0, _).
verify_segment([Head|Tail], Value, Max) :-
Head #> Max #<=> B,
Value #= M+B,
maximum(NewMax, [Head, Max]),
verify_segment(Tail, M, NewMax).
exactly(_, [], 0).
exactly(X, [Y|L], N) :-
X #= Y #<=> B,
N #= M +B,
exactly(X, L, M).
constrain_numbers(Vars) :-
exactly(3, Vars, 1),
exactly(2, Vars, 1),
exactly(1, Vars, 1).
iteration_values(BoardWidth, Index, row, 0, column) :-
Index is BoardWidth - 1.
iteration_values(BoardWidth, Index, Type, NewIndex, Type) :-
\+((Type = row, Index is BoardWidth - 1)),
NewIndex is Index + 1.
solve_skyscrapers(Board, BoardWidth) :-
solve_skyscrapers(Board, BoardWidth, 0, row).
solve_skyscrapers(_, BoardWidth, BoardWidth, column).
solve_skyscrapers(Board, BoardWidth, Index, Type) :-
make_segment(Board, BoardWidth, Index, Type, Segment),
domain(Segment, 0, 3),
constrain_numbers(Segment),
observer(Type, Index, forward, ForwardObserver),
verify_segment(Segment, ForwardObserver),
observer(Type, Index, reverse, ReverseObserver),
reverse(Segment, ReversedSegment),
verify_segment(ReversedSegment, ReverseObserver),
iteration_values(BoardWidth, Index, Type, NewIndex, NewType),
solve_skyscrapers(Board, BoardWidth, NewIndex, NewType).
build_vertex_list(_, Vertices, BoardWidth, X, Y, List) :-
V1X is X, V1Y is Y, V1Index is V1X + V1Y*(BoardWidth+1),
V2X is X+1, V2Y is Y, V2Index is V2X + V2Y*(BoardWidth+1),
V3X is X+1, V3Y is Y+1, V3Index is V3X + V3Y*(BoardWidth+1),
V4X is X, V4Y is Y+1, V4Index is V4X + V4Y*(BoardWidth+1),
get_element_at(Vertices, V1Index, V1),
get_element_at(Vertices, V2Index, V2),
get_element_at(Vertices, V3Index, V3),
get_element_at(Vertices, V4Index, V4),
List = [V1, V2, V3, V4].
build_neighbors_list(Vertices, VertexWidth, X, Y, [NorthMask, EastMask, SouthMask, WestMask], [NorthNeighbor, EastNeighbor, SouthNeighbor, WestNeighbor]) :-
NorthY is Y - 1,
EastX is X + 1,
SouthY is Y + 1,
WestX is X - 1,
NorthNeighborIndex is (NorthY)*VertexWidth + X,
EastNeighborIndex is Y*VertexWidth + EastX,
SouthNeighborIndex is (SouthY)*VertexWidth + X,
WestNeighborIndex is Y*VertexWidth + WestX,
(NorthY >= 0, get_element_at(Vertices, NorthNeighborIndex, NorthNeighbor) -> NorthMask = 1 ; NorthMask = 0),
(EastX < VertexWidth, get_element_at(Vertices, EastNeighborIndex, EastNeighbor) -> EastMask = 1 ; EastMask = 0),
(SouthY < VertexWidth, get_element_at(Vertices, SouthNeighborIndex, SouthNeighbor) -> SouthMask = 1 ; SouthMask = 0),
(WestX >= 0, get_element_at(Vertices, WestNeighborIndex, WestNeighbor) -> WestMask = 1 ; WestMask = 0).
solve_path(_, VertexWidth, 0, VertexWidth) :-
write('end'),nl.
solve_path(Vertices, VertexWidth, VertexWidth, Y) :-
write('switch row'),nl,
Y \= VertexWidth,
NewY is Y + 1,
solve_path(Vertices, VertexWidth, 0, NewY).
solve_path(Vertices, VertexWidth, X, Y) :-
X >= 0, X < VertexWidth, Y >= 0, Y < VertexWidth,
write('Path: '), nl,
write('Vertex width: '), write(VertexWidth), nl,
write('X: '), write(X), write(' Y: '), write(Y), nl,
VertexIndex is X + Y*VertexWidth,
write('1'),nl,
get_element_at(Vertices, VertexIndex, Vertex),
write('2'),nl,
build_neighbors_list(Vertices, VertexWidth, X, Y, [NorthMask, EastMask, SouthMask, WestMask], [NorthNeighbor, EastNeighbor, SouthNeighbor, WestNeighbor]),
L1 = [NorthMask, EastMask, SouthMask, WestMask],
L2 = [NorthNeighbor, EastNeighbor, SouthNeighbor, WestNeighbor],
write(L1),nl,
write(L2),nl,
write('3'),nl,
maximum(Max, Vertices),
write('4'),nl,
write('Max: '), write(Max),nl,
write('Vertex: '), write(Vertex),nl,
(Vertex #> 1 #/\ Vertex #\= Max) #=> (
((NorthMask #> 0 #/\ NorthNeighbor #> 0) #/\ (NorthNeighbor #= Vertex - 1)) #\
((EastMask #> 0 #/\ EastNeighbor #> 0) #/\ (EastNeighbor #= Vertex - 1)) #\
((SouthMask #> 0 #/\ SouthNeighbor #> 0) #/\ (SouthNeighbor #= Vertex - 1)) #\
((WestMask #> 0 #/\ WestNeighbor #> 0) #/\ (WestNeighbor #= Vertex - 1))
) #/\ (
((NorthMask #> 0 #/\ NorthNeighbor #> 0) #/\ (NorthNeighbor #= Vertex + 1)) #\
((EastMask #> 0 #/\ EastNeighbor #> 0) #/\ (EastNeighbor #= Vertex + 1)) #\
((SouthMask #> 0 #/\ SouthNeighbor #> 0) #/\ (SouthNeighbor #= Vertex + 1)) #\
((WestMask #> 0 #/\ WestNeighbor #> 0) #/\ (WestNeighbor #= Vertex + 1))
),
write('5'),nl,
Vertex #= 1 #=> (
((NorthMask #> 0 #/\ NorthNeighbor #> 0) #/\ (NorthNeighbor #= Max)) #\
((EastMask #> 0 #/\ EastNeighbor #> 0) #/\ (EastNeighbor #= Max)) #\
((SouthMask #> 0 #/\ SouthNeighbor #> 0) #/\ (SouthNeighbor #= Max)) #\
((WestMask #> 0 #/\ WestNeighbor #> 0) #/\ (WestNeighbor #= Max))
) #/\ (
((NorthMask #> 0 #/\ NorthNeighbor #> 0) #/\ (NorthNeighbor #= 2)) #\
((EastMask #> 0 #/\ EastNeighbor #> 0) #/\ (EastNeighbor #= 2)) #\
((SouthMask #> 0 #/\ SouthNeighbor #> 0) #/\ (SouthNeighbor #= 2)) #\
((WestMask #> 0 #/\ WestNeighbor #> 0) #/\ (WestNeighbor #= 2))
),
write('6'),nl,
Vertex #= Max #=> (
((NorthMask #> 0 #/\ NorthNeighbor #> 0) #/\ (NorthNeighbor #= 1)) #\
((EastMask #> 0 #/\ EastNeighbor #> 0) #/\ (EastNeighbor #= 1)) #\
((SouthMask #> 0 #/\ SouthNeighbor #> 0) #/\ (SouthNeighbor #= 1)) #\
((WestMask #> 0 #/\ WestNeighbor #> 0) #/\ (WestNeighbor #= 1))
) #/\ (
((NorthMask #> 0 #/\ NorthNeighbor #> 0) #/\ (NorthNeighbor #= Max - 1)) #\
((EastMask #> 0 #/\ EastNeighbor #> 0) #/\ (EastNeighbor #= Max - 1)) #\
((SouthMask #> 0 #/\ SouthNeighbor #> 0) #/\ (SouthNeighbor #= Max - 1)) #\
((WestMask #> 0 #/\ WestNeighbor #> 0) #/\ (WestNeighbor #= Max - 1))
),
write('7'),nl,
NewX is X + 1,
solve_path(Vertices, VertexWidth, NewX, Y).
solve_fences(Board, Vertices, BoardWidth) :-
VertexWidth is BoardWidth + 1,
write('- Solving vertices'),nl,
solve_vertices(Board, Vertices, BoardWidth, 0, 0),
write('- Solving path'),nl,
solve_path(Vertices, VertexWidth, 0, 0).
solve_vertices(_, _, BoardWidth, 0, BoardWidth).
solve_vertices(Board, Vertices, BoardWidth, BoardWidth, Y) :-
Y \= BoardWidth,
NewY is Y + 1,
solve_vertices(Board, Vertices, BoardWidth, 0, NewY).
solve_vertices(Board, Vertices, BoardWidth, X, Y) :-
X >= 0, X < BoardWidth, Y >= 0, Y < BoardWidth,
write('process'),nl,
write('X: '), write(X), write(' Y: '), write(Y), nl,
build_vertex_list(Board, Vertices, BoardWidth, X, Y, [V1, V2, V3, V4]),
write('1'),nl,
get_board_element(Board, BoardWidth, X, Y, Element),
write('2'),nl,
maximum(Max, Vertices),
(V1 #> 0 #/\ V2 #> 0 #/\
(
(V1 + 1 #= V2) #\
(V1 - 1 #= V2) #\
(V1 #= Max #/\ V2 #= 1) #\
(V1 #= 1 #/\ V2 #= Max)
)
) #<=> B1,
(V2 #> 0 #/\ V3 #> 0 #/\
(
(V2 + 1 #= V3) #\
(V2 - 1 #= V3) #\
(V2 #= Max #/\ V3 #= 1) #\
(V2 #= 1 #/\ V3 #= Max)
)
) #<=> B2,
(V3 #> 0 #/\ V4 #> 0 #/\
(
(V3 + 1 #= V4) #\
(V3 - 1 #= V4) #\
(V3 #= Max #/\ V4 #= 1) #\
(V3 #= 1 #/\ V4 #= Max)
)
) #<=> B3,
(V4 #> 0 #/\ V1 #> 0 #/\
(
(V4 + 1 #= V1) #\
(V4 - 1 #= V1) #\
(V4 #= Max #/\ V1 #= 1) #\
(V4 #= 1 #/\ V1 #= Max)
)
) #<=> B4,
write('3'),nl,
sum([B1, B2, B3, B4], #= , C),
write('4'),nl,
Element #> 0 #=> C #= Element,
write('5'),nl,
NewX is X + 1,
solve_vertices(Board, Vertices, BoardWidth, NewX, Y).
sel_next_variable_for_path(Vars,Sel,Rest) :-
% write(Vars), nl,
findall(Idx-Cost, (nth1(Idx, Vars,V), fd_set(V,S), fdset_size(S,Size), fdset_min(S,Min), var_cost(Min,Size, Cost)), L),
min_member(comp, BestIdx-_MinCost, L),
nth1(BestIdx, Vars, Sel, Rest),!.
var_cost(0, _, 1000000) :- !.
var_cost(_, 1, 1000000) :- !.
var_cost(X, _, X).
%build_vertex_list(_, Vertices, BoardWidth, X, Y, List)
constrain_starting_and_ending_vertices(Vertices, [V1,V2,V3,V4]) :-
maximum(Max, Vertices),
(V1 #= 1 #/\ V2 #= Max #/\ V3 #= Max - 1 #/\ V4 #= 2 ) #\
(V1 #= Max #/\ V2 #= 1 #/\ V3 #= 2 #/\ V4 #= Max - 1 ) #\
(V1 #= Max - 1 #/\ V2 #= Max #/\ V3 #= 1 #/\ V4 #= 2 ) #\
(V1 #= 2 #/\ V2 #= 1 #/\ V3 #= Max #/\ V4 #= Max - 1 ) #\
(V1 #= 1 #/\ V2 #= 2 #/\ V3 #= Max - 1 #/\ V4 #= Max ) #\
(V1 #= Max #/\ V2 #= Max - 1 #/\ V3 #= 2 #/\ V4 #= 1 ) #\
(V1 #= Max - 1 #/\ V2 #= 2 #/\ V3 #= 1 #/\ V4 #= Max ) #\
(V1 #= 2 #/\ V2 #= Max - 1 #/\ V3 #= Max #/\ V4 #= 1 ).
set_starting_and_ending_vertices(Board, Vertices, BoardWidth) :-
set_starting_and_ending_vertices(Board, Vertices, BoardWidth, 0, 0).
set_starting_and_ending_vertices(Board, Vertices, BoardWidth, BoardWidth, Y) :-
Y \= BoardWidth,
NewY is Y + 1,
solve_path(Board, Vertices, BoardWidth, 0, NewY).
set_starting_and_ending_vertices(Board, Vertices, BoardWidth, X, Y) :-
X >= 0, X < BoardWidth, Y >= 0, Y < BoardWidth,
build_vertex_list(_, Vertices, BoardWidth, X, Y, List),
get_board_element(Board, BoardWidth, X, Y, Element),
(Element = 3 ->
constrain_starting_and_ending_vertices(Vertices, List)
;
NewX is X + 1,
set_starting_and_ending_vertices(Board, Vertices, BoardWidth, NewX, Y)).
solve(Board, Vertices, BoardWidth) :-
write('Skyscrapers'), nl,
solve_skyscrapers(Board, BoardWidth),
write('Labeling'), nl,
labeling([ff], Board), !,
write('Setting domain'), nl,
NVertices is (BoardWidth+1)*(BoardWidth+1),
domain(Vertices, 0, NVertices),
write('Starting and ending vertices'), nl,
set_starting_and_ending_vertices(Board, Vertices, BoardWidth),
write('Setting maximum'), nl,
maximum(Max, Vertices),
write('1'),nl,
Max #> BoardWidth + 1,
write('2'),nl,
Max #< NVertices,
count(0, Vertices, #=, NZeros),
Max #= NVertices - NZeros,
write('3'),nl,
write('Calling nvalue'), nl,
ValueCount #= Max + 1,
nvalue(ValueCount, Vertices),
write('Solving fences'), nl,
solve_fences(Board, Vertices, BoardWidth),
write('Labeling'), nl,
labeling([ff], Vertices).
main :-
board(Board),
board_width(BoardWidth),
vertices(Vertices),
solve(Board, Vertices, BoardWidth),
%findall(Board,
% labeling([ff], Board),
% Boards
%),
%append(Board, Vertices, Final),
write('done.'),nl,
print_board(Board, 6), nl,
print_board(Vertices, 7).
utils.pro
get_element_at([Head|_], 0, Head).
get_element_at([_|Tail], Index, Element) :-
Index \= 0,
NewIndex is Index - 1,
get_element_at(Tail, NewIndex, Element).
reverse([], []).
reverse([Head|Tail], Inv) :-
reverse(Tail, Aux),
append(Aux, [Head], Inv).
munch(List, 0, List).
munch([_|Tail], Count, FinalList) :-
Count > 0,
NewCount is Count - 1,
munch(Tail, NewCount, FinalList).
select_n_elements(_, 0, []).
select_n_elements([Head|Tail], Count, FinalList) :-
Count > 0,
NewCount is Count - 1,
select_n_elements(Tail, NewCount, Result),
append([Head], Result, FinalList).
generate_list(Element, NElements, [Element|Result]) :-
NElements > 0,
NewNElements is NElements - 1,
generate_list(Element, NewNElements, Result).
generate_list(_, 0, []).
s1.pro
% Skyscrapers and Fences puzzle S1
board_width(6).
%observer(Type, Index, Orientation, Observer),
observer(row, 0, forward, 2).
observer(row, 1, forward, 2).
observer(row, 2, forward, 2).
observer(row, 3, forward, 1).
observer(row, 4, forward, 2).
observer(row, 5, forward, 1).
observer(row, 0, reverse, 1).
observer(row, 1, reverse, 1).
observer(row, 2, reverse, 2).
observer(row, 3, reverse, 3).
observer(row, 4, reverse, 2).
observer(row, 5, reverse, 2).
observer(column, 0, forward, 2).
observer(column, 1, forward, 3).
observer(column, 2, forward, 0).
observer(column, 3, forward, 2).
observer(column, 4, forward, 2).
observer(column, 5, forward, 1).
observer(column, 0, reverse, 1).
observer(column, 1, reverse, 1).
observer(column, 2, reverse, 2).
observer(column, 3, reverse, 2).
observer(column, 4, reverse, 2).
observer(column, 5, reverse, 2).
board(
[
_, _, 2, _, _, _,
_, _, _, _, _, _,
_, 2, _, _, _, _,
_, _, _, 2, _, _,
_, _, _, _, _, _,
_, _, _, _, _, _
]
).
vertices(
[
_, _, _, _, _, _, _,
_, _, _, _, _, _, _,
_, _, _, _, _, _, _,
_, _, _, _, _, _, _,
_, _, _, _, _, _, _,
_, _, _, _, _, _, _,
_, _, _, _, _, _, _
]
).
Solution
I also, like twinterer, enjoyed this puzzle. But being a principiant, I had first to discover an appropriate strategy, for both skyscrapes and fences part, and then deeply debugging the latter, cause a copy variables problem that locked me many hours.
Once solved the bug, I faced the inefficiency of my first attempt. I reworked in plain Prolog a similar schema, just to verify how inefficient it was.
At least, I understood how use CLP(FD) more effectively to model the problem (with help from the twinterer' answer), and now the program is fast (0,2 sec). So now I can hint you about your code: the required constraints are far simpler than those you coded: for the fences part, i.e. with a buildings placement fixed, we have 2 constraints: number of edges where height > 0, and linking the edges together: when an edge is used, the sum of adjacents must be 1 (on both sides).
Here is the last version of my code, developed with SWI-Prolog.
/* File: skys.pl
Author: Carlo,,,
Created: Dec 11 2011
Purpose: questions/8458945 on http://stackoverflow.com
http://stackoverflow.com/questions/8458945/optimizing-pathfinding-in-constraint-logic-programming-with-prolog
*/
:- module(skys, [skys/0, fences/2, draw_path/2]).
:- [index_square,
lambda,
library(clpfd),
library(aggregate)].
puzzle(1,
[[-,2,3,-,2,2,1,-],
[2,-,-,2,-,-,-,1],
[2,-,-,-,-,-,-,1],
[2,-,2,-,-,-,-,2],
[1,-,-,-,2,-,-,3],
[2,-,-,-,-,-,-,2],
[1,-,-,-,-,-,-,2],
[-,1,1,2,2,2,2,-]]).
skys :-
puzzle(1, P),
skyscrapes(P, Rows),
flatten(Rows, Flat),
label(Flat),
maplist(writeln, Rows),
fences(Rows, Loop),
writeln(Loop),
draw_path(7, Loop).
%% %%%%%%%%%%
% skyscrapes part
% %%%%%%%%%%
skyscrapes(Puzzle, Rows) :-
% massaging definition: separe external 'visibility' counters
first_and_last(Puzzle, Fpt, Lpt, Wpt),
first_and_last(Fpt, -, -, Fp),
first_and_last(Lpt, -, -, Lp),
maplist(first_and_last, Wpt, Lc, Rc, InnerData),
% InnerData it's the actual 'playground', Fp, Lp, Lc, Rc are list of counters
maplist(make_vars, InnerData, Rows),
% exploit symmetry wrt rows/cols
transpose(Rows, Cols),
% each row or col contains once 1,2,3
Occurs = [0-_, 1-1, 2-1, 3-1], % allows any grid size leaving unspecified 0s
maplist(\Vs^global_cardinality(Vs, Occurs), Rows),
maplist(\Vs^global_cardinality(Vs, Occurs), Cols),
% apply 'external visibility' constraint
constraint_views(Lc, Rows),
constraint_views(Fp, Cols),
maplist(reverse, Rows, RRows),
constraint_views(Rc, RRows),
maplist(reverse, Cols, RCols),
constraint_views(Lp, RCols).
first_and_last(List, First, Last, Without) :-
append([[First], Without, [Last]], List).
make_vars(Data, Vars) :-
maplist(\C^V^(C \= (-) -> V #= C ; V in 0..3), Data, Vars).
constraint_views(Ns, Ls) :-
maplist(\N^L^
( N \= (-)
-> constraint_view(0, L, Rs),
sum(Rs, #=, N)
; true
), Ns, Ls).
constraint_view(_, [], []).
constraint_view(Top, [V|Vs], [R|Rs]) :-
R #<==> V #> 0 #/\ V #> Top,
Max #= max(Top, V),
constraint_view(Max, Vs, Rs).
%% %%%%%%%%%%%%%%%
% fences part
% %%%%%%%%%%%%%%%
fences(SkyS, Ps) :-
length(SkyS, D),
% allocate edges
max_dimensions(D, _,_,_,_, N),
N1 is N + 1,
length(Edges, N1),
Edges ins 0..1,
findall((R, C, V),
(nth0(R, SkyS, Row), nth0(C, Row, V), V > 0),
Buildings),
maplist(count_edges(D, Edges), Buildings),
findall((I, Adj1, Adj2),
(between(0, N, I), edge_adjacents(D, I, Adj1, Adj2)),
Path),
maplist(make_path(Edges), Path, Vs),
flatten([Edges, Vs], Gs),
label(Gs),
used_edges_to_path_coords(D, Edges, Ps).
count_edges(D, Edges, (R, C, V)) :-
cell_edges(D, (R, C), Is),
idxs0_to_elems(Is, Edges, Es),
sum(Es, #=, V).
make_path(Edges, (Index, G1, G2), [S1, S2]) :-
idxs0_to_elems(G1, Edges, Adj1),
idxs0_to_elems(G2, Edges, Adj2),
nth0(Index, Edges, Edge),
[S1, S2] ins 0..3,
sum(Adj1, #=, S1),
sum(Adj2, #=, S2),
Edge #= 1 #<==> S1 #= 1 #/\ S2 #= 1.
%% %%%%%%%%%%%%%%
% utility: draw a path with arrows
% %%%%%%%%%%%%%%
draw_path(D, P) :-
forall(between(1, D, R),
( forall(between(1, D, C),
( V is (R - 1) * D + C - 1,
U is (R - 2) * D + C - 1,
( append(_, [V, U|_], P)
-> write(' ^ ')
; append(_, [U, V|_], P)
-> write(' v ')
; write(' ')
)
)),
nl,
forall(between(1, D, C),
( V is (R - 1) * D + C - 1,
( V < 10
-> write(' ') ; true
),
write(V),
U is V + 1,
( append(_, [V, U|_], P)
-> write(' > ')
; append(_, [U, V|_], P)
-> write(' < ')
; write(' ')
)
)),
nl
)
).
% convert from 'edge used flags' to vertex indexes
%
used_edges_to_path_coords(D, EdgeUsedFlags, PathCoords) :-
findall((X, Y),
(nth0(Used, EdgeUsedFlags, 1), edge_verts(D, Used, X, Y)),
Path),
Path = [(First, _)|_],
edge_follower(First, Path, PathCoords).
edge_follower(C, Path, [C|Rest]) :-
( select(E, Path, Path1),
( E = (C, D) ; E = (D, C) )
-> edge_follower(D, Path1, Rest)
; Rest = []
).
The output:
[0,0,2,1,0,3]
[2,1,3,0,0,0]
[0,2,0,3,1,0]
[0,3,0,2,0,1]
[1,0,0,0,3,2]
[3,0,1,0,2,0]
[1,2,3,4,5,6,13,12,19,20,27,34,41,48,47,40,33,32,39,46,45,38,31,24,25,18,17,10,9,16,23,
22,29,30,37,36,43,42,35,28,21,14,7,8,1]
0 1 > 2 > 3 > 4 > 5 > 6
^ v
7 > 8 9 < 10 11 12 < 13
^ v ^ v
14 15 16 17 < 18 19 > 20
^ v ^ v
21 22 < 23 24 > 25 26 27
^ v ^ v
28 29 > 30 31 32 < 33 34
^ v ^ v ^ v
35 36 < 37 38 39 40 41
^ v ^ v ^ v
42 < 43 44 45 < 46 47 < 48
As I mentioned, my first attempt was more 'procedural': it draws a loop, but the problem I was unable to solve is basically that the cardinality of vertices subset must be known before, being based on the global constraint all_different. It painfully works on a reduced 4*4 puzzle, but I stopped it after some hours on the 6*6 original. Anyway, learning from scratch how to draw a path with CLP(FD) has been rewarding.
t :-
time(fences([[0,0,2,1,0,3],
[2,1,3,0,0,0],
[0,2,0,3,1,0],
[0,3,0,2,0,1],
[1,0,0,0,3,2],
[3,0,1,0,2,0]
],L)),
writeln(L).
fences(SkyS, Ps) :-
length(SkyS, Dt),
D is Dt + 1,
Sq is D * D - 1,
% min/max num. of vertices
aggregate_all(sum(V), (member(R, SkyS), member(V, R)), MinVertsT),
MinVerts is max(4, MinVertsT),
MaxVerts is D * D,
% find first cell with heigth 3, for sure start vertex
nth0(R, SkyS, Row), nth0(C, Row, 3),
% search a path with at least MinVerts
between(MinVerts, MaxVerts, NVerts),
length(Vs, NVerts),
Vs ins 0 .. Sq,
all_distinct(Vs),
% make a loop
Vs = [O|_],
O is R * D + C,
append(Vs, [O], Ps),
% apply #edges check
findall(rc(Ri, Ci, V),
(nth0(Ri, SkyS, Rowi),
nth0(Ci, Rowi, V),
V > 0), VRCs),
maplist(count_edges(Ps, D), VRCs),
connect_path(D, Ps),
label(Vs).
count_edges(Ps, D, rc(R, C, V)) :-
V0 is R * D + C,
V1 is R * D + C + 1,
V2 is (R + 1) * D + C,
V3 is (R + 1) * D + C + 1,
place_edges(Ps, [V0-V1, V0-V2, V1-V3, V2-V3], Ts),
flatten(Ts, Tsf),
sum(Tsf, #=, V).
place_edges([A,B|Ps], L, [R|Rs]) :-
place_edge(L, A-B, R),
place_edges([B|Ps], L, Rs).
place_edges([_], _L, []).
place_edge([M-N | L], A-B, [Y|R]) :-
Y #<==> (A #= M #/\ B #= N) #\/ (A #= N #/\ B #= M),
place_edge(L, A-B, R).
place_edge([], _, []).
connect(X, D, Y) :-
D1 is D - 1,
[R, C] ins 0 .. D1,
X #= R * D + C,
( C #< D - 1, Y #= R * D + C + 1
; R #< D - 1, Y #= (R + 1) * D + C
; C #> 0, Y #= R * D + C - 1
; R #> 0, Y #= (R - 1) * D + C
).
connect_path(D, [X, Y | R]) :-
connect(X, D, Y),
connect_path(D, [Y | R]).
connect_path(_, [_]).
Thanks you for such interesting question.
MORE EDIT:here the main miss piece of code for the complete solution (index_square.pl)
/* File: index_square.pl
Author: Carlo,,,
Created: Dec 15 2011
Purpose: indexing square grid for FD mapping
*/
:- module(index_square,
[max_dimensions/6,
idxs0_to_elems/3,
edge_verts/4,
edge_is_horiz/3,
cell_verts/3,
cell_edges/3,
edge_adjacents/4,
edge_verts_all/2
]).
%
% index row : {D}, left to right
% index col : {D}, top to bottom
% index cell : same as top edge or row,col
% index vert : {(D + 1) * 2}
% index edge : {(D * (D + 1)) * 2}, first all horiz, then vert
%
% {N} denote range 0 .. N-1
%
% on a 2*2 grid, the numbering schema is
%
% 0 1
% 0-- 0 --1-- 1 --2
% | | |
% 0 6 0,0 7 0,1 8
% | | |
% 3-- 2 --4-- 3 --5
% | | |
% 1 9 1,0 10 1,1 11
% | | |
% 6-- 4 --7-- 5 --8
%
% while on a 4*4 grid:
%
% 0 1 2 3
% 0-- 0 --1-- 1 --2-- 2 --3-- 3 --4
% | | | | |
% 0 20 21 22 23 24
% | | | | |
% 5-- 4 --6-- 5 --7-- 6 --8-- 7 --9
% | | | | |
% 1 25 26 27 28 29
% | | | | |
% 10--8 --11- 9 --12--10--13--11--14
% | | | | |
% 2 30 31 32 33 34
% | | | | |
% 15--12--16--13--17--14--18--15--19
% | | | | |
% 3 35 36 37 38 39
% | | | | |
% 20--16--21--17--22--18--23--19--24
%
% | |
% --+-- N --+--
% | |
% W R,C E
% | |
% --+-- S --+--
% | |
%
% get range upper value for interesting quantities
%
max_dimensions(D, MaxRow, MaxCol, MaxCell, MaxVert, MaxEdge) :-
MaxRow is D - 1,
MaxCol is D - 1,
MaxCell is D * D - 1,
MaxVert is ((D + 1) * 2) - 1,
MaxEdge is (D * (D + 1) * 2) - 1.
% map indexes to elements
%
idxs0_to_elems(Is, Edges, Es) :-
maplist(nth0_(Edges), Is, Es).
nth0_(Edges, I, E) :-
nth0(I, Edges, E).
% get vertices of edge
%
edge_verts(D, E, X, Y) :-
S is D + 1,
edge_is_horiz(D, E, H),
( H
-> X is (E // D) * S + E mod D,
Y is X + 1
; X is E - (D * S),
Y is X + S
).
% qualify edge as horizontal (never fail!)
%
edge_is_horiz(D, E, H) :-
E >= (D * (D + 1)) -> H = false ; H = true.
% get 4 vertices of cell
%
cell_verts(D, (R, C), [TL, TR, BL, BR]) :-
TL is R * (D + 1) + C,
TR is TL + 1,
BL is TR + D,
BR is BL + 1.
% get 4 edges of cell
%
cell_edges(D, (R, C), [N, S, W, E]) :-
N is R * D + C,
S is N + D,
W is (D * (D + 1)) + R * (D + 1) + C,
E is W + 1.
% get adjacents at two extremities of edge I
%
edge_adjacents(D, I, G1, G2) :-
edge_verts(D, I, X, Y),
edge_verts_all(D, EVs),
setof(E, U^V^(member(E - (U, V), EVs), E \= I, (U == X ; V == X)), G1),
setof(E, U^V^(member(E - (U, V), EVs), E \= I, (U == Y ; V == Y)), G2).
% get all edge_verts/4 for grid D
%
edge_verts_all(D, L) :-
( edge_verts_all_(D, L)
-> true
; max_dimensions(D, _,_,_,_, S), %S is (D + 1) * (D + 2) - 1,
findall(E - (X, Y),
( between(0, S, E),
edge_verts(D, E, X, Y)
), L),
assert(edge_verts_all_(D, L))
).
:- dynamic edge_verts_all_/2.
%% %%%%%%%%%%%%%%%%%%%%
:- begin_tests(index_square).
test(1) :-
cell_edges(2, (0,1), [1, 3, 7, 8]),
cell_edges(2, (1,1), [3, 5, 10, 11]).
test(2) :-
cell_verts(2, (0,1), [1, 2, 4, 5]),
cell_verts(2, (1,1), [4, 5, 7, 8]).
test(3) :-
edge_is_horiz(2, 0, true),
edge_is_horiz(2, 5, true),
edge_is_horiz(2, 6, false),
edge_is_horiz(2, 9, false),
edge_is_horiz(2, 11, false).
test(4) :-
edge_verts(2, 0, 0, 1),
edge_verts(2, 3, 4, 5),
edge_verts(2, 5, 7, 8),
edge_verts(2, 6, 0, 3),
edge_verts(2, 11, 5, 8).
test(5) :-
edge_adjacents(2, 0, A, B), A = [6], B = [1, 7],
edge_adjacents(2, 9, [2, 6], [4]),
edge_adjacents(2, 10, [2, 3, 7], [4, 5]).
test(6) :-
cell_edges(4, (2,1), [9, 13, 31, 32]).
:- end_tests(index_square).
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