5c7fa27bc85fdeb1cb88490660fcdca725b6fe35
2 * Library code to divide up a rectangle into a number of equally
3 * sized ominoes, in a random fashion.
5 * Could use this for generating solved grids of
6 * http://www.nikoli.co.jp/ja/puzzles/block_puzzle/
7 * or for generating the playfield for Jigsaw Sudoku.
11 * Possible improvements which might cut the fail rate:
13 * - instead of picking one omino to extend in an iteration, try
14 * them all in succession (in a randomised order)
16 * - (for real rigour) instead of bfsing over ominoes, bfs over
17 * the space of possible _removed squares_. That way we aren't
18 * limited to randomly choosing a single square to remove from
19 * an omino and failing if that particular square doesn't
22 * However, I don't currently think it's necessary to do either of
23 * these, because the failure rate is already low enough to be
24 * easily tolerable, under all circumstances I've been able to
36 * Subroutine which implements a function used in computing both
37 * whether a square can safely be added to an omino, and whether
38 * it can safely be removed.
40 * We enumerate the eight squares 8-adjacent to this one, in
41 * cyclic order. We go round that loop and count the number of
42 * times we find a square owned by the target omino next to one
43 * not owned by it. We then return success iff that count is 2.
45 * When adding a square to an omino, this is precisely the
46 * criterion which tells us that adding the square won't leave a
47 * hole in the middle of the omino. (There's no explicit
48 * requirement in the statement of our problem that the ominoes be
49 * simply connected, but we do know they must be all of equal size
50 * and so it's clear that we must avoid leaving holes, since a
51 * hole would necessarily be smaller than the maximum omino size.)
53 * When removing a square from an omino, the _same_ criterion
54 * tells us that removing the square won't disconnect the omino.
56 static int addremcommon(int w
, int h
, int x
, int y
, int *own
, int val
)
61 for (dir
= 0; dir
< 8; dir
++) {
62 int dx
= ((dir
& 3) == 2 ?
0 : dir
> 2 && dir
< 6 ?
+1 : -1);
63 int dy
= ((dir
& 3) == 0 ?
0 : dir
< 4 ?
-1 : +1);
64 int sx
= x
+dx
, sy
= y
+dy
;
66 if (sx
< 0 || sx
>= w
|| sy
< 0 || sy
>= h
)
67 neighbours
[dir
] = -1; /* outside the grid */
69 neighbours
[dir
] = own
[sy
*w
+sx
];
73 * To begin with, check 4-adjacency.
75 if (neighbours
[0] != val
&& neighbours
[2] != val
&&
76 neighbours
[4] != val
&& neighbours
[6] != val
)
81 for (dir
= 0; dir
< 8; dir
++) {
82 int next
= (dir
+ 1) & 7;
83 int gotthis
= (neighbours
[dir
] == val
);
84 int gotnext
= (neighbours
[next
] == val
);
86 if (gotthis
!= gotnext
)
94 * w and h are the dimensions of the rectangle.
96 * k is the size of the required ominoes. (So k must divide w*h,
99 * The returned result is a w*h-sized dsf.
101 * In both of the above suggested use cases, the user would
102 * probably want w==h==k, but that isn't a requirement.
104 static int *divvy_internal(int w
, int h
, int k
, random_state
*rs
)
106 int *order
, *queue
, *tmp
, *own
, *sizes
, *addable
, *removable
, *retdsf
;
108 int i
, j
, n
, x
, y
, qhead
, qtail
;
113 order
= snewn(wh
, int);
114 tmp
= snewn(wh
, int);
115 own
= snewn(wh
, int);
116 sizes
= snewn(n
, int);
117 queue
= snewn(n
, int);
118 addable
= snewn(wh
*4, int);
119 removable
= snewn(wh
, int);
122 * Permute the grid squares into a random order, which will be
123 * used for iterating over the grid whenever we need to search
124 * for something. This prevents directional bias and arranges
125 * for the answer to be non-deterministic.
127 for (i
= 0; i
< wh
; i
++)
129 shuffle(order
, wh
, sizeof(*order
), rs
);
132 * Begin by choosing a starting square at random for each
135 for (i
= 0; i
< wh
; i
++) {
138 for (i
= 0; i
< n
; i
++) {
144 * Now repeatedly pick a random omino which isn't already at
145 * the target size, and find a way to expand it by one. This
146 * may involve stealing a square from another omino, in which
147 * case we then re-expand that omino, forming a chain of
148 * square-stealing which terminates in an as yet unclaimed
149 * square. Hence every successful iteration around this loop
150 * causes the number of unclaimed squares to drop by one, and
151 * so the process is bounded in duration.
155 #ifdef DIVVY_DIAGNOSTICS
158 printf("Top of loop. Current grid:\n");
159 for (y
= 0; y
< h
; y
++) {
160 for (x
= 0; x
< w
; x
++)
161 printf("%3d", own
[y
*w
+x
]);
168 * Go over the grid and figure out which squares can
169 * safely be added to, or removed from, each omino. We
170 * don't take account of other ominoes in this process, so
171 * we will often end up knowing that a square can be
172 * poached from one omino by another.
174 * For each square, there may be up to four ominoes to
175 * which it can be added (those to which it is
178 for (y
= 0; y
< h
; y
++) {
179 for (x
= 0; x
< w
; x
++) {
185 removable
[yx
] = FALSE
; /* can't remove if not owned! */
186 } else if (sizes
[curr
] == 1) {
187 removable
[yx
] = TRUE
; /* can always remove a singleton */
190 * See if this square can be removed from its
191 * omino without disconnecting it.
193 removable
[yx
] = addremcommon(w
, h
, x
, y
, own
, curr
);
196 for (dir
= 0; dir
< 4; dir
++) {
197 int dx
= (dir
== 0 ?
-1 : dir
== 1 ?
+1 : 0);
198 int dy
= (dir
== 2 ?
-1 : dir
== 3 ?
+1 : 0);
199 int sx
= x
+ dx
, sy
= y
+ dy
;
202 addable
[yx
*4+dir
] = -1;
204 if (sx
< 0 || sx
>= w
|| sy
< 0 || sy
>= h
)
205 continue; /* no omino here! */
207 continue; /* also no omino here */
208 if (own
[syx
] == own
[yx
])
209 continue; /* we already got one */
210 if (!addremcommon(w
, h
, x
, y
, own
, own
[syx
]))
211 continue; /* would non-simply connect the omino */
213 addable
[yx
*4+dir
] = own
[syx
];
218 for (i
= j
= 0; i
< n
; i
++)
222 break; /* all ominoes are complete! */
223 j
= tmp
[random_upto(rs
, j
)];
224 #ifdef DIVVY_DIAGNOSTICS
225 printf("Trying to extend %d\n", j
);
229 * So we're trying to expand omino j. We breadth-first
230 * search out from j across the space of ominoes.
232 * For bfs purposes, we use two elements of tmp per omino:
233 * tmp[2*i+0] tells us which omino we got to i from, and
234 * tmp[2*i+1] numbers the grid square that omino stole
237 * This requires that wh (the size of tmp) is at least 2n,
238 * i.e. k is at least 2. There would have been nothing to
239 * stop a user calling this function with k=1, but if they
240 * did then we wouldn't have got to _here_ in the code -
241 * we would have noticed above that all ominoes were
242 * already at their target sizes, and terminated :-)
245 for (i
= 0; i
< n
; i
++)
246 tmp
[2*i
] = tmp
[2*i
+1] = -1;
249 tmp
[2*j
] = tmp
[2*j
+1] = -2; /* special value: `starting point' */
251 while (qhead
< qtail
) {
257 * We wish to expand omino j. However, we might have
258 * got here by omino j having a square stolen from it,
259 * so first of all we must temporarily mark that
260 * square as not belonging to j, so that our adjacency
261 * calculations don't assume j _does_ belong to us.
265 assert(own
[tmpsq
] == j
);
270 * OK. Now begin by seeing if we can find any
271 * unclaimed square into which we can expand omino j.
272 * If we find one, the entire bfs terminates.
274 for (i
= 0; i
< wh
; i
++) {
277 if (own
[order
[i
]] != -1)
278 continue; /* this square is claimed */
281 * Special case: if our current omino was size 1
282 * and then had a square stolen from it, it's now
283 * size zero, which means it's valid to `expand'
284 * it into _any_ unclaimed square.
286 if (sizes
[j
] == 1 && tmpsq
>= 0)
290 * Failing that, we must do the full test for
293 for (dir
= 0; dir
< 4; dir
++)
294 if (addable
[order
[i
]*4+dir
] == j
) {
296 * We know this square is addable to this
297 * omino with the grid in the state it had
298 * at the top of the loop. However, we
299 * must now check that it's _still_
300 * addable to this omino when the omino is
301 * missing a square. To do this it's only
302 * necessary to re-check addremcommon.
304 if (!addremcommon(w
, h
, order
[i
]%w
, order
[i
]/w
,
310 continue; /* we can't add this square to j */
312 break; /* got one! */
318 * Restore the temporarily removed square _before_
319 * we start shifting ownerships about.
325 * We are done. We can add square i to omino j,
326 * and then backtrack along the trail in tmp
327 * moving squares between ominoes, ending up
328 * expanding our starting omino by one.
330 #ifdef DIVVY_DIAGNOSTICS
331 printf("(%d,%d)", i
%w
, i
/w
);
335 #ifdef DIVVY_DIAGNOSTICS
342 #ifdef DIVVY_DIAGNOSTICS
343 printf("; (%d,%d)", i
%w
, i
/w
);
346 #ifdef DIVVY_DIAGNOSTICS
351 * Increment the size of the starting omino.
356 * Terminate the bfs loop.
362 * If we get here, we haven't been able to expand
363 * omino j into an unclaimed square. So now we begin
364 * to investigate expanding it into squares which are
365 * claimed by ominoes the bfs has not yet visited.
367 for (i
= 0; i
< wh
; i
++) {
371 if (nj
< 0 || tmp
[2*nj
] != -1)
372 continue; /* unclaimed, or owned by wrong omino */
373 if (!removable
[order
[i
]])
374 continue; /* its omino won't let it go */
376 for (dir
= 0; dir
< 4; dir
++)
377 if (addable
[order
[i
]*4+dir
] == j
) {
379 * As above, re-check addremcommon.
381 if (!addremcommon(w
, h
, order
[i
]%w
, order
[i
]/w
,
386 * We have found a square we can use to
387 * expand omino j, at the expense of the
388 * as-yet unvisited omino nj. So add this
394 tmp
[2*nj
+1] = order
[i
];
397 * Now terminate the loop over dir, to
398 * ensure we don't accidentally add the
399 * same omino twice to the queue.
406 * Restore the temporarily removed square.
412 * Advance the queue head.
417 if (qhead
== qtail
) {
419 * We have finished the bfs and not found any way to
420 * expand omino j. Panic, and return failure.
422 * FIXME: or should we loop over all ominoes before we
425 #ifdef DIVVY_DIAGNOSTICS
433 #ifdef DIVVY_DIAGNOSTICS
436 printf("SUCCESS! Final grid:\n");
437 for (y
= 0; y
< h
; y
++) {
438 for (x
= 0; x
< w
; x
++)
439 printf("%3d", own
[y
*w
+x
]);
446 * Construct the output dsf.
448 for (i
= 0; i
< wh
; i
++) {
449 assert(own
[i
] >= 0 && own
[i
] < n
);
452 retdsf
= snew_dsf(wh
);
453 for (i
= 0; i
< wh
; i
++) {
454 dsf_merge(retdsf
, i
, tmp
[own
[i
]]);
458 * Construct the output dsf a different way, to verify that
459 * the ominoes really are k-ominoes and we haven't
460 * accidentally split one into two disconnected pieces.
463 for (y
= 0; y
< h
; y
++)
464 for (x
= 0; x
+1 < w
; x
++)
465 if (own
[y
*w
+x
] == own
[y
*w
+(x
+1)])
466 dsf_merge(tmp
, y
*w
+x
, y
*w
+(x
+1));
467 for (x
= 0; x
< w
; x
++)
468 for (y
= 0; y
+1 < h
; y
++)
469 if (own
[y
*w
+x
] == own
[(y
+1)*w
+x
])
470 dsf_merge(tmp
, y
*w
+x
, (y
+1)*w
+x
);
471 for (i
= 0; i
< wh
; i
++) {
472 j
= dsf_canonify(retdsf
, i
);
473 assert(dsf_canonify(tmp
, j
) == dsf_canonify(tmp
, i
));
479 * Free our temporary working space.
496 static int fail_counter
= 0;
499 int *divvy_rectangle(int w
, int h
, int k
, random_state
*rs
)
504 ret
= divvy_internal(w
, h
, k
, rs
);
519 * gcc -g -O0 -DTESTMODE -I.. -o divvy divvy.c ../random.c ../malloc.c ../dsf.c ../misc.c ../nullfe.c
523 * gcc -g -O0 -DDIVVY_DIAGNOSTICS -DTESTMODE -I.. -o divvy divvy.c ../random.c ../malloc.c ../dsf.c ../misc.c ../nullfe.c
526 int main(int argc
, char **argv
)
530 int w
= 9, h
= 4, k
= 6, tries
= 100;
533 rs
= random_new("123456", 6);
542 tries
= atoi(argv
[4]);
544 for (i
= 0; i
< tries
; i
++) {
547 dsf
= divvy_rectangle(w
, h
, k
, rs
);
550 for (y
= 0; y
<= 2*h
; y
++) {
551 for (x
= 0; x
<= 2*w
; x
++) {
552 int miny
= y
/2 - 1, maxy
= y
/2;
553 int minx
= x
/2 - 1, maxx
= x
/2;
554 int classes
[4], tx
, ty
;
555 for (ty
= 0; ty
< 2; ty
++)
556 for (tx
= 0; tx
< 2; tx
++) {
557 int cx
= minx
+tx
, cy
= miny
+ty
;
558 if (cx
< 0 || cx
>= w
|| cy
< 0 || cy
>= h
)
559 classes
[ty
*2+tx
] = -1;
561 classes
[ty
*2+tx
] = dsf_canonify(dsf
, cy
*w
+cx
);
563 switch (y
%2 * 2 + x
%2) {
566 * Cases for the corner:
568 * - if all four surrounding squares belong
569 * to the same omino, we print a space.
571 * - if the top two are the same and the
572 * bottom two are the same, we print a
575 * - if the left two are the same and the
576 * right two are the same, we print a
579 * - otherwise, we print a cross.
581 if (classes
[0] == classes
[1] &&
582 classes
[1] == classes
[2] &&
583 classes
[2] == classes
[3])
585 else if (classes
[0] == classes
[1] &&
586 classes
[2] == classes
[3])
588 else if (classes
[0] == classes
[2] &&
589 classes
[1] == classes
[3])
594 case 1: /* horiz edge */
595 if (classes
[1] == classes
[3])
600 case 2: /* vert edge */
601 if (classes
[2] == classes
[3])
606 case 3: /* square centre */
617 printf("%d retries needed for %d successes\n", fail_counter
, tries
);