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# Wave function collapse

Wave function collapse is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
This task has been flagged for clarification. Code on this page in its current state may be flagged incorrect once this task has been clarified. See this page's Talk page for discussion.

The Wave Function Collapse algorithm is a heuristic for generating tiled images.

The algorithm begins with a collection of equal sized image blocks and randomly places them, one at a time, within a grid subject to the tiling constraint and an entropy constraint, and it wraps (the top row of blocks in the grid is treated as adjacent to the bottom row of blocks, and similarly the left and right columns of blocks are treated as adjacent to each other).

The blocks are tiled within the grid. Tiled means they are placed with a one pixel overlap and the tiling constraint requires that the pixels overlapping border between two adjacent blocks match.

Entropy, here, means the number of blocks eligible to be placed in an unassigned grid location. The entropy constraint here is that each block is placed in a grid location with minimum entropy. (Placing a block may constrain the entropy of its four nearest neighbors -- up, down, left, right.)

For this task, we start with five blocks of 3x3 pixels and place them in an 8x8 grid to form a 17x17 tile. A tile is a block which may be tiled with itself. Here, we show these five blocks adjacent but not tiled:

Note that this algorithm sometimes does not succeed. If an unassigned grid location has an entropy of 0, the algorithm fails and returns an empty or null result. We'll ignore those failure cases for this task.

Reference WFC explained and another WFC explained

## C

Translation of: J
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <time.h>

#define XY(row, col, width) ((col)+(row)*(width))
#define XYZ(page, row, col, height, width) XY(XY(page, row, height), col, width)

char blocks[5][3][3]= {
{
{0, 0, 0},
{0, 0, 0},
{0, 0, 0}
},{
{0, 0, 0},
{1, 1, 1},
{0, 1, 0}
},{
{0, 1, 0},
{0, 1, 1},
{0, 1, 0}
},{
{0, 1, 0},
{1, 1, 1},
{0, 0, 0}
},{
{0, 1, 0},
{1, 1, 0},
{0, 1, 0}
}
};

/* avoid problems with slightly negative numbers and C's X%Y */
#define MOD(X,Y) ((Y)+(X))%(Y)

char *wfc(char *blocks, int *bdim /* 5,3,3 */, int *tdim /* 8,8 */) {
int N= tdim[0]*tdim[1], td0= tdim[0], td1= tdim[1];
int *adj= calloc(N*4, sizeof (int)); /* indices in R of the four adjacent blocks */
for (int i= 0; i<td0; i++) {
for (int j=0; j<td1; j++) {
adj[XYZ(i,j,0,td1,4)]= XY(MOD(i-1, td0), MOD(j, td1), td1); /* above (index 1 in a 3x3 grid) */
adj[XYZ(i,j,1,td1,4)]= XY(MOD(i, td0), MOD(j-1, td1), td1); /* left (index 3 in a 3x3 grid) */
adj[XYZ(i,j,2,td1,4)]= XY(MOD(i, td0), MOD(j+1, td1), td1); /* right (index 5 in a 3x3 grid) */
adj[XYZ(i,j,3,td1,4)]= XY(MOD(i+1, td0), MOD(j, td1), td1); /* below (index 7 in a 3x3 grid) */
}
}
int bd0= bdim[0], bd1= bdim[1], bd2= bdim[2];
char *horz= malloc(bd0*bd0); /* blocks which can sit next to each other horizontally */
for (int i= 0; i<bd0; i++) {
for (int j= 0; j<bd0; j++) {
horz[XY(i,j,bd0)]= 1;
for (int k= 0; k<bd1; k++) {
if (blocks[XYZ(i, k, 0, bd1, bd2)] != blocks[XYZ(j, k, bd2-1, bd1, bd2)]) {
horz[XY(i, j, bd0)]= 0;
}
}
}
}
char *vert= malloc(bd0*bd0); /* blocks which can sit next to each other vertically */
for (int i= 0; i<bd0; i++) {
for (int j= 0; j<bd0; j++) {
vert[XY(i,j,bd0)]= 1;
for (int k= 0; k<bd2; k++) {
if (blocks[XYZ(i, 0, k, bd1, bd2)] != blocks[XYZ(j, bd1-1, k, bd1, bd2)]) {
vert[XY(i, j, bd0)]= 0;
break;
}
}
}
}
char *allow= malloc(4*(bd0+1)*bd0); /* all block constraints, based on neighbors */
memset(allow, 1, 4*(bd0+1)*bd0);
for (int i= 0; i<bd0; i++) {
for (int j= 0; j<bd0; j++) {
allow[XYZ(0, i, j, bd0+1, bd0)]= vert[XY(j, i, bd0)]; /* above (north) */
allow[XYZ(1, i, j, bd0+1, bd0)]= horz[XY(j, i, bd0)]; /* left (west) */
allow[XYZ(2, i, j, bd0+1, bd0)]= horz[XY(i, j, bd0)]; /* right (east) */
allow[XYZ(3, i, j, bd0+1, bd0)]= vert[XY(i, j, bd0)]; /* below (south) */
}
}
free(horz);
free(vert);
int *todo= calloc(N, sizeof (int));
char *wave= malloc(N*bd0);
int *entropy= calloc(N, sizeof (int));
int *indices= calloc(N, sizeof (int));
int min;
int *possible= calloc(bd0, sizeof (int));
int *R= calloc(N, sizeof (int)); /* tile expressed as list of block indices */
for (int i= 0; i<N; i++) R[i]= bd0;
while (1) {
int c= 0;
for (int i= 0; i<N; i++)
if (bd0==R[i])
todo[c++]= i;
if (!c) break;
min= bd0;
for (int i= 0; i<c; i++) {
entropy[i]= 0;
for (int j= 0; j<bd0; j++) {
int K= 4*todo[i];
entropy[i]+=
wave[XY(i, j, bd0)]=
allow[XYZ(0, R[adj[XY(todo[i],0,4)]], j, bd0+1, bd0)] &
allow[XYZ(1, R[adj[XY(todo[i],1,4)]], j, bd0+1, bd0)] &
allow[XYZ(2, R[adj[XY(todo[i],2,4)]], j, bd0+1, bd0)] &
allow[XYZ(3, R[adj[XY(todo[i],3,4)]], j, bd0+1, bd0)];
}
if (entropy[i] < min) min= entropy[i];
}
if (!min) {
free(R);
R= NULL;
break;
}
int d= 0;
for (int i= 0; i<c; i++) {
if (min==entropy[i]) indices[d++]= i;
}
int ndx= indices[random()%d];
int ind= ndx*bd0;
d= 0;
for (int i= 0; i<bd0; i++) {
if (wave[ind+i]) possible[d++]= i;
}
R[todo[ndx]]= possible[random()%d];
}
free(allow);
free(todo);
free(wave);
free(entropy);
free(indices);
free(possible);
if (!R) return NULL;
char *tile= malloc((1+td0*(bd1-1))*(1+td1*(bd2-1)));
for (int i0= 0; i0<td0; i0++)
for (int i1= 0; i1<bd1; i1++)
for (int j0= 0; j0<td1; j0++)
for (int j1= 0; j1<bd2; j1++)
tile[XY(XY(j0, j1, bd2-1), XY(i0, i1, bd1-1), 1+td1*(bd2-1))]=
blocks[XYZ(R[XY(i0, j0, td1)], i1, j1, bd1, bd2)];
free(R);
return tile;
}

int main() {
int bdims[3]= {5,3,3};
int size[2]= {8,8};
time_t t;
srandom((unsigned) time(&t));
char *tile= wfc((char*)blocks, bdims, size);
if (!tile) exit(0);
for (int i= 0; i<17; i++) {
for (int j= 0; j<17; j++) {
printf("%c ", " #"[tile[XY(i, j, 17)]]);
}
printf("\n");
}
free(tile);
exit(0);
}

Note: here we use R where J used i, because we use i as an index/loop counter (other than m, y and i, the comments on the j implementation should be directly relevant here). Also, when assembling the result at the end, it was convenient to treat the block overlap issue during indexing.

For simplicity, we use char as our pixel datatype (and for truth values), and int for indices (C offers a variety of similar datatypes but nothing we are doing here is big enough for that to be a concern).

Output:
#               #   #   #
# # # #               # # # # # #
#   #               #
# # # #               # # # # # #
#               #   #   #
# # # # # # # # # # # #   # # # #
#       #   #   #   #   #
# # # # # # # # # #   # # # # # #
#           #   #       #
# # # # # # #   # # # # #
#   #   #   #   #   #   #
# # # # #   # # #   # # #
#       #   #   #   #   #
# # # # # # # #   # # # # #   # #
#       #   #   #       #   #
# # # # # # # # # # # # # #   # #
#               #   #   #

## J

Implementation:
blocks=: 0,(|[email protected]|:)^:(i.4)0,1 1 1,:0 1 0
wfc=: {{
adj=: y#.y|"1(y#:,i.y)+"1/<:3 3#:1 3 5 7
horz=: ({."1 -:"1/ {:"1) m NB. horizontal tile pairs
vert=: ({."2 -:"1/ {:"2) m NB. vertical tile pairs
north=: 1,~|:vert NB. adj 1 constraint
south=: 1,~vert NB. adj 7 constraint
west=: 1,~|:horz NB. adj 3 constrint
east=: 1,~horz NB. adj 5 constraint
allow=: north,west,east,:south
i=: ,y\$_1
while. #todo=: I._1=i do.
wave=: */"2 ((todo{adj){i){"0 2"1 3 allow
entropy=: +/"1 wave
min=: <./ entropy
if. 0=min do. EMPTY return. end.
ndx=: ({~ [email protected]#) I.min=entropy
i=: (({[email protected]#)I.ndx{wave) (ndx{todo)} i
end.
lap=. {{ y#~(+0[email protected]#)-.;m\$<n{.1 }}
({:y)lap({:\$m)"1 ({.y)lap({:\$m),/"2,/0 2 1 3|:(y\$i){m
}}

We work with the 3x3 partial tiles, and the larger 17x17 tile which we are randomly generating. (17x17 because every 3x3 block contributes 2x2 pixels to the result and along a horizontal and vertical edge row and column of the tile, the 3x3 blocks contribute an additional row and column of pixels.)

Here, m is the list of tiles, and i represents an 8x8 list of indexes into that list (or, conceptually whatever dimensions were specified by y, the right argument to wfc -- but for this task y will always be 8 8), with _1 being a placeholder for the case where the index hasn't been choosen -- initially, we pick a random location in i and assign an arbitrarily picked tile to that location.

adj indexes into i -- for each item in i, adj selects that item, the item "above" it, the item to the "left" of it, the item to the "right" of it and the item "below" it (with scare quotes because the tile represented by i "wraps around" on all sides). And, allow lists the allowable tiles corresponding to each of those adj constraints.

To build allow we first matched the left side of each tile with the right side of each tile (cartesian product) forming horz and similarly matched the tops and bottoms of the tiles forming vert. Then we build north which limits tiles based on the tile above it, and similarly for west, east, and south (when the adjacent tile is a _1 tile, no limit is imposed).

Once we're set up, we drop into a loop: todo selects the unchosen tile locations, wavelists for each of the unchoosen tiles (for each todo value in i we select the tiles allowed by each of its adjacent locations and find the set intersection of all of those), entropy counts how many tiles are eligible for each of those location, and min is the smallest value in entropy. ndx is a randomly picked index into todo with minimal entropy and for that location we randomly pick one of the options and update i with it. (When there's only one option remaining, "randomly pick" here means we pick that option.)

Once we've assigned a tile to every location in i, we use those indices to assemble the result (note that the 3x3 tiles overlap at their borders so we introduce a mechanism to discard the redundant pixels).

For task purposes here, we will use space to represent a white pixel and "#" to represent a black pixel. Also, because characters are narrow, we will insert a space between each of these "pixels" to better approximate a square aspect ratio.

Task example (the initial tiles and three runs of wave function collapse (three, to illustrate randomness):
(<"2) 1j1#"1 ' #'{~ tiles
โโโโโโโโฌโโโโโโโฌโโโโโโโฌโโโโโโโฌโโโโโโโ
โ โ โ # โ # โ # โ
โ โ# # # โ # # โ# # # โ# # โ
โ โ # โ # โ โ # โ
โโโโโโโโดโโโโโโโดโโโโโโโดโโโโโโโดโโโโโโโ

task=: {{ 1j1#"1 ' #'{~ blocks wfc 8 8}}
โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโฌโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโฌโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ
โ # # # # โ # # # # # # โ # # # # # โ
โ# # # # # # # # # # # # # # โ # # # # # # # # # # # # # # โ # # # # # # # # # # # # โ
โ # # # # # โ # # # # # # โ # # # # # # โ
โ # # # # # # # # # # # # โ # # # # # # # # # # # # # # โ# # # # # # # # # # # # # # # โ
โ # # # # # # โ # # # # # # โ # # # # # # โ
โ# # # # # # # # # # # # # # # โ# # # # # # # # # # # โ# # # # # # # # # # # # # # # โ
โ # # # # # # โ # # # # โ # # # # # # โ
โ# # # # # # # # # # # # # # # โ # # # # # # # # # # # โ # # # # # # # # # # # โ
โ # # # # # # โ # # # # โ # # # # # # # โ
โ# # # # # # # # # # # # # # # โ# # # # # # # # # # # โ# # # # # # # # # # # # # โ
โ # # # # # # โ # # # # # # โ # # # # โ
โ # # # # # # # # # # # # โ# # # # # # # # # # โ # # # # # # # # โ
โ # # # # # โ # # # # # # โ # # # # # โ
โ# # # # # # # # # # # # # # โ# # # # # # # # # # # # # # # โ# # # # # # # # # # # โ
โ # # # # โ # # # # # # โ # # # # # โ
โ # # # # # # โ# # # # # # # # # # # # # # # โ# # # # # # # # # # # โ
โ # # # # โ # # # # # # โ # # # # # โ
โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโดโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโดโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ

## Perl

Translation of: Raku
use v5.36;
use experimental 'for_list';

my @Blocks = ( [ <0 0 0>, <0 0 0>, <0 0 0> ],
[ <0 0 0>, <1 1 1>, <0 1 0> ],
[ <0 1 0>, <0 1 1>, <0 1 0> ],
[ <0 1 0>, <1 1 1>, <0 0 0> ],
[ <0 1 0>, <1 1 0>, <0 1 0> ],
);

sub X(\$a,\$b) { my @c; for my \$aa (0..\$a-1) { map { push @c, \$aa, \$_ } 0..\$b-1 } @c }

sub XY( \$row, \$col, \$width) { \$col + \$row * \$width }
sub XYZ(\$page, \$row, \$col, \$height, \$width) { XY( XY(\$page, \$row, \$height), \$col, \$width) }

sub wfc(\$B, \$bdim, \$tdim) {
my (\$td0,\$td1) = @\$tdim;
my \$N = \$td0 * \$td1;
my @blocks = map @\$_, @\$B; # flatten

my @adj; # indices in R of the four adjacent blocks
for my(\$i,\$j) (X \$td0, \$td1) {
\$adj[XYZ(\$i, \$j, 0, \$td1, 4)] = XY((\$i-1)%\$td0, \$j %\$td1, \$td1); # above (index 1)
\$adj[XYZ(\$i, \$j, 1, \$td1, 4)] = XY( \$i %\$td0, (\$j-1)%\$td1, \$td1); # left (index 3)
\$adj[XYZ(\$i, \$j, 2, \$td1, 4)] = XY( \$i %\$td0, (\$j+1)%\$td1, \$td1); # right (index 5)
\$adj[XYZ(\$i, \$j, 3, \$td1, 4)] = XY((\$i+1)%\$td0, \$j %\$td1, \$td1); # below (index 7)
}

my (\$bd0,\$bd1,\$bd2) = @\$bdim;
my @horz; # blocks which can sit next to each other horizontally
for my(\$i,\$j) (X \$bd0, \$bd0) {
@horz[XY(\$i,\$j,\$bd0)] = 1;
for my \$k (0..\$bd1-1) {
\$horz[XY(\$i, \$j, \$bd0)]= 0 if \$blocks[XYZ(\$i, \$k, 0, \$bd1, \$bd2)]
!= \$blocks[XYZ(\$j, \$k, \$bd2-1, \$bd1, \$bd2)]
}
}

my @vert; # blocks which can sit next to each other vertically */
for my(\$i,\$j) (X \$bd0, \$bd0) {
\$vert[XY(\$i,\$j,\$bd0)] = 1;
for my \$k (0..\$bd2-1) {
if (\$blocks[XYZ(\$i, 0, \$k, \$bd1, \$bd2)] != \$blocks[XYZ(\$j, \$bd1-1, \$k, \$bd1, \$bd2)]) {
\$vert[XY(\$i, \$j, \$bd0)] = 0;
last
}
}
}

my @allow = (1) x (4*(\$bd0+1)*\$bd0); # all block constraints, based on neighbors
for my(\$i,\$j) (X \$bd0, \$bd0) {
\$allow[XYZ(0, \$i, \$j, \$bd0+1, \$bd0)] = \$vert[XY(\$j, \$i, \$bd0)]; # above (north)
\$allow[XYZ(1, \$i, \$j, \$bd0+1, \$bd0)] = \$horz[XY(\$j, \$i, \$bd0)]; # left (west)
\$allow[XYZ(2, \$i, \$j, \$bd0+1, \$bd0)] = \$horz[XY(\$i, \$j, \$bd0)]; # right (east)
\$allow[XYZ(3, \$i, \$j, \$bd0+1, \$bd0)] = \$vert[XY(\$i, \$j, \$bd0)]; # below (south)
}

my @R = (\$bd0) x \$N;
my (@todo, @wave, @entropy, @indices, \$min, @possible);

while () {
my \$c;
for (0..\$N-1) { \$todo[\$c++] = \$_ if \$bd0 == \$R[\$_] }
last unless \$c;
\$min = \$bd0;
for my \$i (0..\$c-1) {
\$entropy[\$i] = 0;
for my \$j (0..\$bd0-1) {
\$entropy[\$i] +=
\$wave[XY(\$i, \$j, \$bd0)] =
\$allow[XYZ(0, \$R[ \$adj[XY(\$todo[\$i],0,4)] ], \$j, \$bd0+1, \$bd0)] &
\$allow[XYZ(1, \$R[ \$adj[XY(\$todo[\$i],1,4)] ], \$j, \$bd0+1, \$bd0)] &
\$allow[XYZ(2, \$R[ \$adj[XY(\$todo[\$i],2,4)] ], \$j, \$bd0+1, \$bd0)] &
\$allow[XYZ(3, \$R[ \$adj[XY(\$todo[\$i],3,4)] ], \$j, \$bd0+1, \$bd0)]
}
\$min = \$entropy[\$i] if \$entropy[\$i] < \$min
}

@R=[] and last unless \$min;

my \$d = 0;
for (0..\$c-1) { \$indices[\$d++] = \$_ if \$min == \$entropy[\$_] }
my \$ind = \$bd0 * (my \$ndx = \$indices[ int rand \$d ]);
\$d = 0;
for (0..\$bd0-1) { \$possible[\$d++] = \$_ if \$wave[\$ind+\$_] }
\$R[\$todo[\$ndx]] = \$possible[ int rand \$d ];
}

return "DOES NOT COMPUTE" unless @R > 1;

my @tile;
for my(\$i0,\$i1)(X \$td0, \$bd1) {
for my(\$j0,\$j1) (X \$td1, \$bd2) {
\$tile[XY(XY(\$j0, \$j1, \$bd2-1), XY(\$i0, \$i1, \$bd1-1), 1+\$td1*(\$bd2-1))] =
(' ','#')[ \$blocks[XYZ(\$R[XY(\$i0, \$j0, \$td1)], \$i1, \$j1, \$bd1, \$bd2)] ]
}
}
my \$width = 2 * sqrt scalar @tile;
join(' ', @tile) =~ s/.{\$width}\K(?=.)/\n/gr;
}

my @bdims = (5,3,3);
my @size = (8,8);
say wfc(\@Blocks, \@bdims, \@size);
Output:
#               #   #   #
# # # #               # # # # # #
#   #               #
# # # #               # # # # # #
#               #   #   #
# # # # # # # # #   # # #
#   #   #   #   #   #   #
# # # # # # # # # # # # #
#                       #
# # # # # # # # # # # # #
#   #   #   #   #   #   #
# # # # # #   # # #   # # #   # #
#       #   #   #   #   #   #
# #       # # #   # # # # # # # #
#       #   #   #
# # # # # # # # # # # # # # # # #
#               #   #   #

## Phix

Library: Phix/pGUI
Library: Phix/online

You can run this online here.

--
-- demo\rosetta\WaveFunctionCollapse.exw
-- =====================================
--
with javascript_semantics
requires("1.0.2") -- (do until, and a switch <atom> bugfix)
include pGUI.e
Ihandle dlg, canvas
cdCanvas cddbuffer

bool bOverlap = true, -- (debug aids)
bSpat = false -- (show space as '@')

integer N = 8 -- board size (nb must be even)
constant title = "Wave Function Collapse",
help_text = """
Press 'o' to toggle overlap (see note below).
Press '@' to toggle display spaces as '@'.
Press '-' to decrease board size (min 2x2).
Press '+' to increase board size (max 40x40).
Press ' ' to start afresh.

Note that it is not really possible to visually verify
that a pattern is correct unless overlap is turned off.
"""
-- space,     T,    -|,    iT,    |-
constant tilem = {0b0000,0b0111,0b1011,0b1101,0b1110},
--       L,      R,      U,      D
valid = {{0b00101,0b10001,0b01001,0b00011},
{0b11010,0b01110,0b01001,0b11100},
{0b11010,0b10001,0b10110,0b11100},
{0b11010,0b01110,0b10110,0b00011},
{0b00101,0b01110,0b10110,0b11100}}
-- eg valid[1=space][4=D] means space or T can go below it,
--    with bits of each valid[][] being read right-to-left.

sequence grid,      -- -1 if unknown, else one of tilem
allowed,   -- initially 0b11111 (all possible) -> 1 bit set
entropy    -- count matching allowed (speedwise/simplicity)

integer left    -- N*N..0, with 0=finished, -1=FAIL, -2=REDO

function lowest_entropy()
-- returns a random tile from those with the lowest entropy
integer row, col, me = 5, count = 0
for r=1 to N do
for c=1 to N do
if grid[r][c]=-1 then -- ignoring any already done
integer e = entropy[r][c]
if e<=me then
if e<me then
me = e
count = 0
end if
count += 1
if rand(count)=1 then
{row,col} = {r,c}
end if
end if
end if
end for
end for
return {row,col}
end function

function pop_count(integer p)
-- Kernigans bit counter:
integer e = 0
while p do
p &&= p-1
e += 1
end while
return e
end function

function permitted(integer p, d)
--
-- Given p, 0b00001..0b11111, a 1-5 bitmask,
-- calculate the permitted tiles in direction
-- d (1..4 for LRUD), eg a T(2) can have 2|3|4
-- on the right, and a 3 can have 1|5, so if
-- p is 0b01100 the result is 0b11111 (all),
-- that is, when d is 2 (ie right).
--
integer nm = 0
for i=1 to 5 do
integer m = power(2,i-1)
if and_bits(p,m) then
nm = or_bits(nm,valid[i][d])
end if
end for
return nm
end function

function propagate(integer r,c,p)
--
-- Propagate the permitted tiles, given that only
-- those in p (0b00001..0b11111, a 1..5 bitmask)
-- are now allowed at [r][c]. Note this can fail,
-- especially for some ~2x3 enclosed spaces, and
-- in that case you want to undo everything, and
-- clear some initial permitted bit setting.
--
for j,d in {{0,-1},{0,1},{-1,0},{1,0}} do --LRUD
integer {dr,dc} = d,
nr = r+dr,
nc = c+dc
if nr>=1 and nr<=N and nc>=1 and nc<=N then
integer nm = permitted(p,j),
op = allowed[nr][nc],
np = and_bits(op,nm)
if np=0 then return false end if
if op!=np then
allowed[nr][nc] = np
entropy[nr][nc] = pop_count(np)
if not propagate(nr,nc,np) then return false end if
end if
end if
end for
return true
end function

procedure wfc(object f=0)
--
-- wave function collapse: (iterative/one cell at a time, because
--      this was once on a timer, but now wfc_init() just loops.)
--
-- There is, I guess, around a 1 in 8000 chance of this failing,
-- which means 1 in 10 40x40 boards fail, presumbably because it
-- has surrounded an area and none of the edges will work out.
-- Setting left to -2 triggers the outer retry in wfc_init().
--
integer {r,c} = lowest_entropy(),
g = grid[r][c],
p = allowed[r][c]
assert(g=-1)
assert(p!=0)
-- pick a random but valid tile:
for i in shuffle(tagset(5)) do
integer m = power(2,i-1)
if and_bits(p,m) then
-- in case propagation fails, make a backup
sequence saveae = deep_copy({allowed,entropy})
grid[r][c] = tilem[i]
allowed[r][c] = m
entropy[r][c] = 1
left -= 1
if not propagate(r,c,m) then
grid[r][c] = g
{allowed,entropy} = saveae
saveae = {} -- kill refcounts
p -= m -- don't try this again!
if p=0 then
--                  printf(1,"panic: allowed[%d][%d] := 0!\n",{r,c})
left = -2   -- trigger a restart
return
end if
integer e = pop_count(p)
assert(p!=0 and e!=0 and e==entropy[r][c]-1)
allowed[r][c] = p
entropy[r][c] = e
left += 1
end if
return
end if
end for
end procedure

procedure wfc_init()
do
grid = repeat(repeat(-1,N),N)
allowed = repeat(repeat(0b11111,N),N)
entropy = repeat(repeat(5,N),N)
left = N*N
do
wfc()
until left<=0
until left!=-2
end procedure

-- (the rest of this is all fairly standard code)

function redraw_cb(Ihandle ih)
integer {cw,ch} = IupGetIntInt(ih, "DRAWSIZE"),
N2 = N/2,   -- (nb forces N to be even)
N3 = N2+1,
d = floor(min(cw,ch)/N),
d9 = floor(min(cw,ch)/(2*N+1)),
d2 = floor(d/2), d4 = N2*d
cw = floor(cw/2)
ch = floor(ch/2)

cdCanvasActivate(cddbuffer)
cdCanvasClear(cddbuffer)
cdCanvasSetForeground(cddbuffer,CD_BLUE)
cdCanvasSetLineWidth(cddbuffer,3)
for row=1 to N do
integer ry = iff(bOverlap?ch-(row*2-N-1)*d9
:ch-(row-N3)*d-d2)
for col=1 to N do
integer rx = iff(bOverlap?cw+(col*2-N-1)*d9
:cw+(col-N3)*d+d2)
integer g = grid[row][col],
e = entropy[row][col]
if g=0b1111 then
cdCanvasSetForeground(cddbuffer,CD_RED)
cdCanvasSetTextAlignment(cddbuffer, CD_CENTER)
cdCanvasText(cddbuffer,rx,ry,"?")
cdCanvasSetForeground(cddbuffer,CD_BLUE)
elsif bSpat and g=0b0000 then
cdCanvasSetTextAlignment(cddbuffer, CD_CENTER)
cdCanvasText(cddbuffer,rx,ry,"@")
elsif g!=-1 then
assert(e=1)
for i,dxy in {{0,-1},{-1,0},{0,1},{1,0}} do -- LURD
if and_bits(g,power(2,i-1)) then
integer {dr,dc} = sq_mul(dxy,iff(bOverlap?d9*2:d2))
cdCanvasLine(cddbuffer,rx,ry,rx+dc,ry+dr)
end if
end for
end if
end for
end for
cdCanvasFlush(cddbuffer)
string o = iff(bOverlap?"":" (no overlap)"),
f = iff(left!=-1?"":" FAIL")
IupSetStrAttribute(dlg,"TITLE","%s [%dx%d] %s%s",{title,N,N,o,f})
return IUP_DEFAULT
end function

function map_cb(Ihandle ih)
cdCanvas cdcanvas = cdCreateCanvas(CD_IUP, ih)
cddbuffer = cdCreateCanvas(CD_DBUFFER, cdcanvas)
return IUP_DEFAULT
end function

function help_cb(Ihandln /*ih*/)
IupMessage(title,help_text,bWrap:=false)
return IUP_IGNORE -- (don't open browser help!)
end function

function key_cb(Ihandle /*dlg*/, atom c)
if c=K_ESC then return IUP_CLOSE end if -- (standard practice for me)
if c=K_F5 then return IUP_DEFAULT end if -- (let browser reload work)
if c=K_F1 then return help_cb(NULL) end if
switch lower(c)
case 'c': wfc()
case 'o': bOverlap = not bOverlap
case '-': N = max(N-4,0)
fallthrough
case '+': N = min(N+2,40)
fallthrough
case ' ': wfc_init()
case '@': bSpat = not bSpat
end switch
IupUpdate(canvas)
return IUP_IGNORE
end function

procedure main()
IupOpen()
canvas = IupGLCanvas("RASTERSIZE=440x440")
dlg = IupDialog(canvas,`TITLE="%s"`,{title})
IupSetCallbacks(canvas,{"MAP_CB",Icallback("map_cb"),
"ACTION",Icallback("redraw_cb")})
IupSetCallback(dlg, "KEY_CB", Icallback("key_cb"))
IupSetAttributeHandle(NULL,"PARENTDIALOG",dlg)
wfc_init()
IupShow(dlg)
IupSetAttribute(canvas,"RASTERSIZE",NULL)
if platform()!=JS then
IupMainLoop()
IupClose()
end if
end procedure

main()

### trivial ditty

Translation of the Python ditty on the talk page:
This trivial solution works because we're only ever adding left to right,
and therefore only verifying L/U vs R/D, and it is not possible to have
any two R/D for which no tile is valid. Were this to fill in each row/line
more randomly, it would soon fail with no tile matching >=3 neighbours.
(lowest entropy would also fail b/c it'd favour 2 no conn over 3 with)

with javascript_semantics
include builtins/unicode_console.e
{} = unicode_console()

with trace
procedure make_rows(integer w)
sequence conn = split("0000.1101.1110.0111.1011","."), -- RULD flags
tiles = split("  ... โ ...โโฉ...โโฃ...โโฆ","..."), --[...aligned]
res = {},
r = {}, p, t
for n=1 to w do
{p, r} = {r, {}}
for i=1 to w do
t = {}
for x=1 to 5 do
if  (length(r)=0 or conn[x][3]=conn[r[\$]][1])       -- L=R
and (length(p)=0 or conn[x][2]=conn[p[i]][4]) then  -- U=D
t &= x
end if
end for
r &= t[rand(length(t))]
end for
res = append(res,join(extract(tiles,r),""))
end for
puts(1,join(res,"\n"))
wait_key()
end procedure

make_rows(20)

## Raku

Translation of: C
# 20220728 Raku programming solution

my @Blocks = ( [ <0 0 0>,
<0 0 0>,
<0 0 0> ],
[ <0 0 0>,
<1 1 1>,
<0 1 0> ],
[ <0 1 0>,
<0 1 1>,
<0 1 0> ],
[ <0 1 0>,
<1 1 1>,
<0 0 0> ],
[ <0 1 0>,
<1 1 0>,
<0 1 0> ], );

sub XY(\row, \col, \width) { col+row*width }
sub XYZ(\page, \row, \col, \height, \width) {
XY( XY(page, row, height), col, width)
}

sub wfc(@blocks, @bdim, @tdim) {

my \N = [*] my (\td0,\td1) = @tdim[0,1];
my @adj; # indices in R of the four adjacent blocks
for ^td0 X ^td1 -> (\i,\j) { # in a 3x3 grid
@adj[XYZ(i,j,0,td1,4)]= XY((i-1)%td0,j%td1,td1); # above (index 1)
@adj[XYZ(i,j,1,td1,4)]= XY(i%td0,(j-1)%td1,td1); # left (index 3)
@adj[XYZ(i,j,2,td1,4)]= XY(i%td0,(j+1)%td1,td1); # right (index 5)
@adj[XYZ(i,j,3,td1,4)]= XY((i+1)%td0,j%td1,td1); # below (index 7)
}

my (\bd0,\bd1,\bd2) = @bdim[0..2];
my @horz; # blocks which can sit next to each other horizontally
for ^bd0 X ^bd0 -> (\i,\j) {
@horz[XY(i,j,bd0)] = 1;
for ^bd1 -> \k {
@horz[XY(i, j, bd0)]= 0 if @blocks[XYZ(i, k, 0, bd1, bd2)] !=
@blocks[XYZ(j, k, bd2-1, bd1, bd2)]
}
}

my @vert; # blocks which can sit next to each other vertically */
for ^bd0 X ^bd0 -> (\i,\j) {
@vert[XY(i,j,bd0)] = 1;
for ^bd2 -> \k {
if @blocks[XYZ(i, 0, k, bd1, bd2)] !=
@blocks[XYZ(j, bd1-1, k, bd1, bd2)] {
@vert[XY(i, j, bd0)]= 0 andthen last;
}
}
}

my @allow = 1 xx 4*(bd0+1)*bd0; # all block constraints, based on neighbors
for ^bd0 X ^bd0 -> (\i,\j) {
@allow[XYZ(0, i, j, bd0+1, bd0)] = @vert[XY(j, i, bd0)]; # above (north)
@allow[XYZ(1, i, j, bd0+1, bd0)] = @horz[XY(j, i, bd0)]; # left (west)
@allow[XYZ(2, i, j, bd0+1, bd0)] = @horz[XY(i, j, bd0)]; # right (east)
@allow[XYZ(3, i, j, bd0+1, bd0)] = @vert[XY(i, j, bd0)]; # below (south)
}

my (@R, @todo, @wave, @entropy, @indices, \$min, @possible) = bd0 xx N;
loop {
my \$c = 0;
for ^N { @todo[\$c++] = \$_ if bd0 == @R[\$_] }
last unless \$c;
\$min = bd0;
for ^\$c -> \i {
@entropy[i]= 0;
for ^bd0 -> \j {
@entropy[i] +=
@wave[XY(i, j, bd0)] =
@allow[XYZ(0, @R[@adj[XY(@todo[i],0,4)]], j, bd0+1, bd0)] +&
@allow[XYZ(1, @R[@adj[XY(@todo[i],1,4)]], j, bd0+1, bd0)] +&
@allow[XYZ(2, @R[@adj[XY(@todo[i],2,4)]], j, bd0+1, bd0)] +&
@allow[XYZ(3, @R[@adj[XY(@todo[i],3,4)]], j, bd0+1, bd0)]
}
\$min = @entropy[i] if @entropy[i] < \$min
}

unless \$min { @R=[] andthen last } # original behaviour
#unless \$min { @R = bd0 xx N andthen redo } # if failure is not an option

my \$d = 0;
for ^\$c { @indices[\$d++] = \$_ if \$min == @entropy[\$_] }
my \ind = bd0 * my \ndx = @indices[ ^\$d .roll ];
\$d = 0;
for ^bd0 { @possible[\$d++] = \$_ if @wave[ind+\$_] }
@R[@todo[ndx]] = @possible[ ^\$d .roll ];
}

exit unless @R.Bool;

my @tile;
for ^td0 X ^bd1 X ^td1 X ^bd2 -> (\i0,\i1,\j0,\j1) {
@tile[XY(XY(j0, j1, bd2-1), XY(i0, i1, bd1-1), 1+td1*(bd2-1))] =
@blocks[XYZ(@R[XY(i0, j0, td1)], i1, j1, bd1, bd2)]
}

return @tile
}

my (@bdims,@size) := (5,3,3), (8,8);

my @tile = wfc @Blocksยป.List.flat, @bdims, @size ;

say .join.trans( [ '0', '1' ] => [ ' ', '# ' ] ) for @tile.rotor(17)
Output:
#           #   #
# # # # # # # # # # # # # # # # #
#   #       #   #           #
# # # # # # #   # # # # # # #
#       #   #   #   #   #   #
# # # # #   # # #   # # # # #
#   #   #   #   #   #       #
# #   # # # # #   # # #       # #
#   #       #   #   #       #
# #   # # # # # # # # #       # #
#   #   #           #       #
# #   # # #           # # # # # #
#   #   #           #   #
# #   # # #           # # # # # #
#   #   #           #       #
# # # # # #           # # # # # #
#           #   #

## Wren

Translation of: C

Well, I don't know whether this task is going to be deleted or not though, given the effort Rdm has put into understanding it, I'd be in favor now of letting it stand. It is after all an interesting application of an algorithm inspired by quantum mechanics to generating images.

The following is a translation of his C version before macros were added. Wren doesn't support macros and, whilst I could use functions instead, I decided on efficiency grounds to leave it as it is.

import "random" for Random

var rand = Random.new()

var blocks = [
0, 0, 0,
0, 0, 0,
0, 0, 0,
0, 0, 0,
1, 1, 1,
0, 1, 0,
0, 1, 0,
0, 1, 1,
0, 1, 0,
0, 1, 0,
1, 1, 1,
0, 0, 0,
0, 1, 0,
1, 1, 0,
0, 1, 0
]

var wfc = Fn.new { |blocks, tdim, target|
var N = target[0] * target[1]
var t0 = target[0]
var t1 = target[1]
var adj = List.filled(4*N, 0)
for (i in 0...t0) {
for (j in 0...t1) {
var k = j + t1*i
var m = 4 * k
adj[m ] = j + t1*((t0+i-1)%t0) /* above (1) */
adj[m+1] = (t1+j-1)%t1 + t1* i /* left (3) */
adj[m+2] = ( j+1)%t1 + t1* i /* right (5) */
adj[m+3] = j + t1*(( i+1)%t0) /* below (7) */
}
}
var td0 = tdim[0]
var td1 = tdim[1]
var td2 = tdim[2]
var horz = List.filled(td0*td0, 0)
for (i in 0...td0) {
for (j in 0...td0) {
horz[j+i*td0] = 1
for (k in 0...td1) {
if (blocks[0+td2*(k+td1*i)] != blocks[(td2-1)+td2*(k+td1*j)]) {
horz[j+i*td0] = 0
break
}
}
}
}
var vert = List.filled(td0*td0, 0)
for (i in 0...td0) {
for (j in 0...td0) {
vert[j+i*td0]= 1
for (k in 0...td2) {
if (blocks[k+td2*(0+td1*i)] != blocks[k+td2*((td2-1)+td1*j)]) {
vert[j+i*td0]= 0
break
}
}
}
}
var stride = (td0+1) * td0
var allow = List.filled(4 * stride, 1)
for (i in 0...td0) {
for (j in 0...td0) {
allow[ (i*td0)+j] = vert[(j*td0)+i] /* above (north) */
allow[ stride +(i*td0)+j] = horz[(j*td0)+i] /* left (west) */
allow[(2*stride)+(i*td0)+j] = horz[(i*td0)+j] /* right (east) */
allow[(3*stride)+(i*td0)+j] = vert[(i*td0)+j] /* below (south) */
}
}
var R = List.filled(N, td0)
var todo = List.filled(N, 0)
var wave = List.filled(N*td0, 0)
var entropy = List.filled(N, 0)
var indices = List.filled(N, 0)
var min = 0
var possible = List.filled(td0, 0)
while (true) {
var c = 0
for (i in 0...N) {
if (td0 == R[i]) {
todo[c] = i
c = c + 1
}
}
if (c == 0) break
min = td0
for (i in 0...c) {
entropy[i] = 0
for (j in 0...td0) {
var K = 4*todo[i]
wave[i*td0 + j] = allow[ td0*R[adj[K ]]+j] & /* above */
allow[ stride +td0*R[adj[K+1]]+j] & /* left */
allow[(2*stride)+td0*R[adj[K+2]]+j] & /* right */
allow[(3*stride)+td0*R[adj[K+3]]+j] /* below */
entropy[i] = entropy[i] + wave[i*td0 + j]
}
if (entropy[i] < min) min = entropy[i]
}
if (min == 0) {
R = null
break
}
var d = 0
for (i in 0...c) {
if (min == entropy[i]) {
indices[d] = i
d = d + 1
}
}
var ndx = indices[rand.int(0, d)]
var ind = ndx * td0
d = 0
for (i in 0...td0) {
if (wave[ind+i] != 0) {
possible[d] = i
d = d + 1
}
}
R[todo[ndx]] = possible[rand.int(0, d)]
}
if (!R) return null
var tile = List.filled((1+t0*(td1-1))*(1+t1*(td2-1)), 0)
for (i0 in 0...t0) {
for (i1 in 0...td1) {
for (j0 in 0...t1) {
for (j1 in 0...td2) {
var t = j1 + (td2-1)*j0 + (1+t1*(td2-1))*(i1 + (td1-1)*i0)
tile[t] = blocks[j1 + td2*(i1 + td1*R[j0+t1*i0])]
}
}
}
}
return tile
}

var tdims = [5, 3, 3]
var size = [8, 8]
var tile = wfc.call(blocks, tdims, size)
if (!tile) return
for (i in 0..16) {
for (j in 0..16) {
System.write("%(" #"[tile[j+i*17]]) ")
}
System.print()
}
Output:

Sample output:

#       #       #   #       #
# # # # # #       # # #       # #
#   #       #   #       #
# # # # # #       # # # # # # # #
#       #       #       #
# # # # #       # # # # #
#   #   #       #   #   #
# # # # # #       # # #   # # # #
#       #   #   #   #
# # # # # # # # # # # #   # # # #
#   #       #       #   #
# # # # # # # # # # # # #
#       #       #       #
# # # # # #       # # # # # # # #
#   #       #   #       #
# # # # # #       # # #       # #
#       #       #   #       #