* some confusing conditions.
*
* TODO:
- *
+ *
* - it might be nice to make setter-provided tent/nontent clues
* inviolable?
* * on the other hand, this would introduce considerable extra
COL_TENT,
COL_ERROR,
COL_ERRTEXT,
+ COL_ERRTRUNK,
NCOLOURS
};
ret[COL_ERRTEXT * 3 + 1] = 1.0F;
ret[COL_ERRTEXT * 3 + 2] = 1.0F;
+ ret[COL_ERRTRUNK * 3 + 0] = 0.6F;
+ ret[COL_ERRTRUNK * 3 + 1] = 0.0F;
+ ret[COL_ERRTRUNK * 3 + 2] = 0.0F;
+
*ncolours = NCOLOURS;
return ret;
}
int x, y;
/*
+ * This function goes through a grid and works out where to
+ * highlight play errors in red. The aim is that it should
+ * produce at least one error highlight for any complete grid
+ * (or complete piece of grid) violating a puzzle constraint, so
+ * that a grid containing no BLANK squares is either a win or is
+ * marked up in some way that indicates why not.
+ *
+ * So it's easy enough to highlight errors in the numeric clues
+ * - just light up any row or column number which is not
+ * fulfilled - and it's just as easy to highlight adjacent
+ * tents. The difficult bit is highlighting failures in the
+ * tent/tree matching criterion.
+ *
+ * A natural approach would seem to be to apply the maxflow
+ * algorithm to find the tent/tree matching; if this fails, it
+ * must necessarily terminate with a min-cut which can be
+ * reinterpreted as some set of trees which have too few tents
+ * between them (or vice versa). However, it's bad for
+ * localising errors, because it's not easy to make the
+ * algorithm narrow down to the _smallest_ such set of trees: if
+ * trees A and B have only one tent between them, for instance,
+ * it might perfectly well highlight not only A and B but also
+ * trees C and D which are correctly matched on the far side of
+ * the grid, on the grounds that those four trees between them
+ * have only three tents.
+ *
+ * Also, that approach fares badly when you introduce the
+ * additional requirement that incomplete grids should have
+ * errors highlighted only when they can be proved to be errors
+ * - so that a tree surrounded by BLANK squares should not be
+ * marked as erroneous (it would be patronising, since the
+ * overwhelming likelihood is not that the player has forgotten
+ * to put a tree there but that they have merely not put one
+ * there _yet), but one surrounded by NONTENTs should.
+ *
+ * So I adopt an alternative approach, which is to consider the
+ * bipartite adjacency graph between trees and tents
+ * ('bipartite' in the sense that for these purposes I
+ * deliberately ignore two adjacent trees or two adjacent
+ * tents), divide that graph up into its connected components
+ * using a dsf, and look for components which contain different
+ * numbers of trees and tents. This allows me to highlight
+ * groups of tents with too few trees between them immediately,
+ * and then in order to find groups of trees with too few tents
+ * I redo the same process but counting BLANKs as potential
+ * tents (so that the only trees highlighted are those
+ * surrounded by enough NONTENTs to make it impossible to give
+ * them enough tents).
+ *
+ * However, this technique is incomplete: it is not a sufficient
+ * condition for the existence of a perfect matching that every
+ * connected component of the graph has the same number of tents
+ * and trees. An example of a graph which satisfies the latter
+ * condition but still has no perfect matching is
+ *
+ * A B C
+ * | / ,/|
+ * | / ,'/ |
+ * | / ,' / |
+ * |/,' / |
+ * 1 2 3
+ *
+ * which can be realised in Tents as
+ *
+ * B
+ * A 1 C 2
+ * 3
+ *
+ * The matching-error highlighter described above will not mark
+ * this construction as erroneous. However, something else will:
+ * the three tents in the above diagram (let us suppose A,B,C
+ * are the tents, though it doesn't matter which) contain two
+ * diagonally adjacent pairs. So there will be _an_ error
+ * highlighted for the above layout, even though not all types
+ * of error will be highlighted.
+ *
+ * And in fact we can prove that this will always be the case:
+ * that the shortcomings of the matching-error highlighter will
+ * always be made up for by the easy tent adjacency highlighter.
+ *
+ * Lemma: Let G be a bipartite graph between n trees and n
+ * tents, which is connected, and in which no tree has degree
+ * more than two (but a tent may). Then G has a perfect matching.
+ *
+ * (Note: in the statement and proof of the Lemma I will
+ * consistently use 'tree' to indicate a type of graph vertex as
+ * opposed to a tent, and not to indicate a tree in the graph-
+ * theoretic sense.)
+ *
+ * Proof:
+ *
+ * If we can find a tent of degree 1 joined to a tree of degree
+ * 2, then any perfect matching must pair that tent with that
+ * tree. Hence, we can remove both, leaving a smaller graph G'
+ * which still satisfies all the conditions of the Lemma, and
+ * which has a perfect matching iff G does.
+ *
+ * So, wlog, we may assume G contains no tent of degree 1 joined
+ * to a tree of degree 2; if it does, we can reduce it as above.
+ *
+ * If G has no tent of degree 1 at all, then every tent has
+ * degree at least two, so there are at least 2n edges in the
+ * graph. But every tree has degree at most two, so there are at
+ * most 2n edges. Hence there must be exactly 2n edges, so every
+ * tree and every tent must have degree exactly two, which means
+ * that the whole graph consists of a single loop (by
+ * connectedness), and therefore certainly has a perfect
+ * matching.
+ *
+ * Alternatively, if G does have a tent of degree 1 but it is
+ * not connected to a tree of degree 2, then the tree it is
+ * connected to must have degree 1 - and, by connectedness, that
+ * must mean that that tent and that tree between them form the
+ * entire graph. This trivial graph has a trivial perfect
+ * matching. []
+ *
+ * That proves the lemma. Hence, in any case where the matching-
+ * error highlighter fails to highlight an erroneous component
+ * (because it has the same number of tents as trees, but they
+ * cannot be matched up), the above lemma tells us that there
+ * must be a tree with degree more than 2, i.e. a tree
+ * orthogonally adjacent to at least three tents. But in that
+ * case, there must be some pair of those three tents which are
+ * diagonally adjacent to each other, so the tent-adjacency
+ * highlighter will necessarily show an error. So any filled
+ * layout in Tents which is not a correct solution to the puzzle
+ * must have _some_ error highlighted by the subroutine below.
+ *
+ * (Of course it would be nicer if we could highlight all
+ * errors: in the above example layout, we would like to
+ * highlight tents A,B as having too few trees between them, and
+ * trees 2,3 as having too few tents, in addition to marking the
+ * adjacency problems. But I can't immediately think of any way
+ * to find the smallest sets of such tents and trees without an
+ * O(2^N) loop over all subsets of a given component.)
+ */
+
+ /*
* ret[0] through to ret[w*h-1] give error markers for the grid
* squares. After that, ret[w*h] to ret[w*h+w-1] give error
* markers for the column numbers, and ret[w*h+w] to
(printing ? draw_rect_outline : draw_rect)
(dr, cx-TILESIZE/15, ty+TILESIZE*3/10,
2*(TILESIZE/15)+1, (TILESIZE*9/10 - TILESIZE*3/10),
- (err & (1<<ERR_OVERCOMMITTED) ? COL_ERROR : COL_TREETRUNK));
+ (err & (1<<ERR_OVERCOMMITTED) ? COL_ERRTRUNK : COL_TREETRUNK));
for (i = 0; i < (printing ? 2 : 1); i++) {
int col = (i == 1 ? COL_BACKGROUND :
~mdw / sgt / puzzles / blobdiff