const unsigned char *tiles,
const unsigned char *barriers)
{
- int W = w + 1, H = h + 1;
int *loops, *dsf;
int x, y;
/*
- * Construct a dsf covering _vertices_ of the grid, so we have one
- * more in each direction than we do squares.
- */
- dsf = snew_dsf(W*H);
-
- /*
- * For each grid square, unify adjacent vertices of that square
- * unless there's a connection separating them. (We only need to
- * check the connection in _this_ square, without bothering to
- * look for one on the other side of the grid line, because the
- * loop will do that anyway when it gets to the other square.)
+ * The loop-detecting algorithm I use here is not quite the same
+ * one as I've used in Slant and Loopy. Those two puzzles use a
+ * very similar algorithm which works by finding connected
+ * components, not of the graph _vertices_, but of the pieces of
+ * space in between them. You divide the plane into maximal areas
+ * that can't be intersected by a grid edge (faces in Loopy,
+ * diamond shapes centred on a grid edge in Slant); you form a dsf
+ * over those areas, and unify any pair _not_ separated by a graph
+ * edge; then you've identified the connected components of the
+ * space, and can now immediately tell whether an edge is part of
+ * a loop or not by checking whether the pieces of space on either
+ * side of it are in the same component.
*
- * Barriers break loops, so we disallow any connection which
- * terminates in a barrier.
- */
+ * In Net, this doesn't work reliably, because of the toroidal
+ * wrapping mode. A torus has non-trivial homology, which is to
+ * say, there can exist a closed loop on its surface which is not
+ * the boundary of any proper subset of the torus's area. For
+ * example, consider the 'loop' consisting of a straight vertical
+ * line going off the top of the grid and coming back on the
+ * bottom to join up with itself. This certainly wants to be
+ * marked as a loop, but it won't be detected as one by the above
+ * algorithm, because all the area of the grid is still connected
+ * via the left- and right-hand edges, so the two sides of the
+ * loop _are_ in the same equivalence class.
+ *
+ * The replacement algorithm I use here is also dsf-based, but the
+ * dsf is now over _sides of edges_. That is to say, on a general
+ * graph, you would have two dsf elements per edge of the graph.
+ * The unification rule is: for each vertex, iterate round the
+ * edges leaving that vertex in cyclic order, and dsf-unify the
+ * _near sides_ of each pair of adjacent edges. The effect of this
+ * is to trace round the outside edge of each connected component
+ * of the graph (this time of the actual graph, not the space
+ * between), so that the outline of each component becomes its own
+ * equivalence class. And now, just as before, an edge is part of
+ * a loop iff its two sides are not in the same component.
+ *
+ * This correctly detects even homologically nontrivial loops on a
+ * torus, because a torus is still _orientable_ - there's no way
+ * that a loop can join back up with itself with the two sides
+ * swapped. It would stop working, however, on a Mobius strip or a
+ * Klein bottle - so if I ever implement either of those modes for
+ * Net, I'll have to revisit this algorithm yet again and probably
+ * replace it with a completely general and much more fiddly
+ * approach such as Tarjan's bridge-finding algorithm (which is
+ * linear-time, but looks to me as if it's going to take more
+ * effort to get it working, especially when the graph is
+ * represented so unlike an ordinary graph).
+ *
+ * In Net, the algorithm as I describe it above has to be fiddled
+ * with just a little, to deal with the fact that there are two
+ * kinds of 'vertex' in the graph - one set at face-centres, and
+ * another set at edge-midpoints where two wires either do or do
+ * not join. Since those two vertex classes have very different
+ * representations in the Net data structure, separate code is
+ * needed for them.
+ */
+
+ /* Four potential edges per grid cell; one dsf node for each side
+ * of each one makes 8 per cell. */
+ dsf = snew_dsf(w*h*8);
+
+ /* Encode the dsf nodes. We imagine going round anticlockwise, so
+ * BEFORE(dir) indicates the clockwise side of an edge, e.g. the
+ * underside of R or the right-hand side of U. AFTER is the other
+ * side. */
+#define BEFORE(dir) ((dir)==R?7:(dir)==U?1:(dir)==L?3:5)
+#define AFTER(dir) ((dir)==R?0:(dir)==U?2:(dir)==L?4:6)
+
+#if 0
+ printf("--- begin\n");
+#endif
for (y = 0; y < h; y++) {
for (x = 0; x < w; x++) {
- int t = tiles[y*w+x];
- if (barriers)
- t &= ~barriers[y*w+x];
- if (!(t & L))
- dsf_merge(dsf, y*W+x, (y+1)*W+x);
- if (!(t & R))
- dsf_merge(dsf, y*W+(x+1), (y+1)*W+(x+1));
- if (!(t & U))
- dsf_merge(dsf, y*W+x, y*W+(x+1));
- if (!(t & D))
- dsf_merge(dsf, (y+1)*W+x, (y+1)*W+(x+1));
+ int tile = tiles[y*w+x];
+ int dir;
+ for (dir = 1; dir < 0x10; dir <<= 1) {
+ /*
+ * To unify dsf nodes around a face-centre vertex,
+ * it's easiest to do it _unconditionally_ - e.g. just
+ * unify the top side of R with the right side of U
+ * regardless of whether there's an edge in either
+ * place. Later we'll also unify the top and bottom
+ * sides of any nonexistent edge, which will e.g.
+ * complete a connection BEFORE(U) - AFTER(R) -
+ * BEFORE(R) - AFTER(D) in the absence of an R edge.
+ *
+ * This is a safe optimisation because these extra dsf
+ * nodes unified into our equivalence class can't get
+ * out of control - they are never unified with
+ * anything _else_ elsewhere in the algorithm.
+ */
+#if 0
+ printf("tile centre %d,%d: merge %d,%d\n",
+ x, y,
+ (y*w+x)*8+AFTER(C(dir)),
+ (y*w+x)*8+BEFORE(dir));
+#endif
+ dsf_merge(dsf,
+ (y*w+x)*8+AFTER(C(dir)),
+ (y*w+x)*8+BEFORE(dir));
+
+ if (tile & dir) {
+ int x1, y1;
+
+ OFFSETWH(x1, y1, x, y, dir, w, h);
+
+ /*
+ * If the tile does have an edge going out in this
+ * direction, we must check whether it joins up
+ * (without being blocked by a barrier) to an edge
+ * in the next cell along. If so, we unify around
+ * the edge-centre vertex by joining each side of
+ * this edge to the appropriate side of the next
+ * cell's edge; otherwise, the edge is a stub (the
+ * only one reaching the edge-centre vertex) and
+ * so we join its own two sides together.
+ */
+ if ((barriers && barriers[y*w+x] & dir) ||
+ !(tiles[y1*w+x1] & F(dir))) {
+#if 0
+ printf("tile edge stub %d,%d -> %c: merge %d,%d\n",
+ x, y, (dir==L?'L':dir==U?'U':dir==R?'R':'D'),
+ (y*w+x)*8+BEFORE(dir),
+ (y*w+x)*8+AFTER(dir));
+#endif
+ dsf_merge(dsf,
+ (y*w+x)*8+BEFORE(dir),
+ (y*w+x)*8+AFTER(dir));
+ } else {
+#if 0
+ printf("tile edge conn %d,%d -> %c: merge %d,%d\n",
+ x, y, (dir==L?'L':dir==U?'U':dir==R?'R':'D'),
+ (y*w+x)*8+BEFORE(dir),
+ (y*w+x)*8+AFTER(F(dir)));
+#endif
+ dsf_merge(dsf,
+ (y*w+x)*8+BEFORE(dir),
+ (y1*w+x1)*8+AFTER(F(dir)));
+#if 0
+ printf("tile edge conn %d,%d -> %c: merge %d,%d\n",
+ x, y, (dir==L?'L':dir==U?'U':dir==R?'R':'D'),
+ (y*w+x)*8+AFTER(dir),
+ (y*w+x)*8+BEFORE(F(dir)));
+#endif
+ dsf_merge(dsf,
+ (y*w+x)*8+AFTER(dir),
+ (y1*w+x1)*8+BEFORE(F(dir)));
+ }
+ } else {
+ /*
+ * As discussed above, if this edge doesn't even
+ * exist, we unify its two sides anyway to
+ * complete the unification of whatever edges do
+ * exist in this cell.
+ */
+#if 0
+ printf("tile edge missing %d,%d -> %c: merge %d,%d\n",
+ x, y, (dir==L?'L':dir==U?'U':dir==R?'R':'D'),
+ (y*w+x)*8+BEFORE(dir),
+ (y*w+x)*8+AFTER(dir));
+#endif
+ dsf_merge(dsf,
+ (y*w+x)*8+BEFORE(dir),
+ (y*w+x)*8+AFTER(dir));
+ }
+ }
}
}
- /*
- * If the game is in wrapping mode, unify each edge vertex with
- * its opposite.
- */
- if (wrapping) {
- for (y = 0; y < H; y++)
- dsf_merge(dsf, y*W+0, y*W+w);
- for (x = 0; x < W; x++)
- dsf_merge(dsf, 0*W+x, h*W+x);
- }
-
+#if 0
+ printf("--- end\n");
+#endif
loops = snewn(w*h, int);
/*
*/
for (y = 0; y < h; y++) {
for (x = 0; x < w; x++) {
- int t = tiles[y*w+x];
+ int dir;
+ int tile = tiles[y*w+x];
int flags = 0;
- if ((t & L) && (dsf_canonify(dsf, y*W+x) !=
- dsf_canonify(dsf, (y+1)*W+x)))
- flags |= LLOOP;
- if ((t & R) && (dsf_canonify(dsf, y*W+(x+1)) !=
- dsf_canonify(dsf, (y+1)*W+(x+1))))
- flags |= RLOOP;
- if ((t & U) && (dsf_canonify(dsf, y*W+x) !=
- dsf_canonify(dsf, y*W+(x+1))))
- flags |= ULOOP;
- if ((t & D) && (dsf_canonify(dsf, (y+1)*W+x) !=
- dsf_canonify(dsf, (y+1)*W+(x+1))))
- flags |= DLOOP;
+ for (dir = 1; dir < 0x10; dir <<= 1) {
+ if ((tile & dir) &&
+ (dsf_canonify(dsf, (y*w+x)*8+BEFORE(dir)) !=
+ dsf_canonify(dsf, (y*w+x)*8+AFTER(dir)))) {
+ flags |= LOOP(dir);
+ }
+ }
loops[y*w+x] = flags;
}
}
~ian / sgt-puzzles.git / commitdiff