ref: 5bd02f982a878312065b4d81d05a90bc41a98c6c
dir: /findloop.c/
/* * Routine for finding loops in graphs, reusable across multiple * puzzles. * * The strategy is Tarjan's bridge-finding algorithm, which is * designed to list all edges whose removal would disconnect a * previously connected component of the graph. We're interested in * exactly the reverse - edges that are part of a loop in the graph * are precisely those which _wouldn't_ disconnect anything if removed * (individually) - but of course flipping the sense of the output is * easy. */ #include "puzzles.h" struct findloopstate { int parent, child, sibling, component_root; bool visited; int index, minindex, maxindex; int minreachable, maxreachable; int bridge; }; struct findloopstate *findloop_new_state(int nvertices) { /* * Allocate a findloopstate structure for each vertex, and one * extra one at the end which will be the overall root of a * 'super-tree', which links the whole graph together to make it * as easy as possible to iterate over all the connected * components. */ return snewn(nvertices + 1, struct findloopstate); } void findloop_free_state(struct findloopstate *state) { sfree(state); } bool findloop_is_loop_edge(struct findloopstate *pv, int u, int v) { /* * Since the algorithm is intended for finding bridges, and a * bridge must be part of any spanning tree, it follows that there * is at most one bridge per vertex. * * Furthermore, by finding a _rooted_ spanning tree (so that each * bridge is a parent->child link), you can find an injection from * bridges to vertices (namely, map each bridge to the vertex at * its child end). * * So if the u-v edge is a bridge, then either v was u's parent * when the algorithm ran and we set pv[u].bridge = v, or vice * versa. */ return !(pv[u].bridge == v || pv[v].bridge == u); } static bool findloop_is_bridge_oneway( struct findloopstate *pv, int u, int v, int *u_vertices, int *v_vertices) { int r, total, below; if (pv[u].bridge != v) return false; r = pv[u].component_root; total = pv[r].maxindex - pv[r].minindex + 1; below = pv[u].maxindex - pv[u].minindex + 1; if (u_vertices) *u_vertices = below; if (v_vertices) *v_vertices = total - below; return true; } bool findloop_is_bridge( struct findloopstate *pv, int u, int v, int *u_vertices, int *v_vertices) { return (findloop_is_bridge_oneway(pv, u, v, u_vertices, v_vertices) || findloop_is_bridge_oneway(pv, v, u, v_vertices, u_vertices)); } bool findloop_run(struct findloopstate *pv, int nvertices, neighbour_fn_t neighbour, void *ctx) { int u, v, w, root, index; int nbridges, nedges; root = nvertices; /* * First pass: organise the graph into a rooted spanning forest. * That is, a tree structure with a clear up/down orientation - * every node has exactly one parent (which may be 'root') and * zero or more children, and every parent-child link corresponds * to a graph edge. * * (A side effect of this is to find all the connected components, * which of course we could do less confusingly with a dsf - but * then we'd have to do that *and* build the tree, so it's less * effort to do it all at once.) */ for (v = 0; v <= nvertices; v++) { pv[v].parent = root; pv[v].child = -2; pv[v].sibling = -1; pv[v].visited = false; } pv[root].child = -1; nedges = 0; debug(("------------- new find_loops, nvertices=%d\n", nvertices)); for (v = 0; v < nvertices; v++) { if (pv[v].parent == root) { /* * Found a new connected component. Enumerate and treeify * it. */ pv[v].sibling = pv[root].child; pv[root].child = v; pv[v].component_root = v; debug(("%d is new child of root\n", v)); u = v; while (1) { if (!pv[u].visited) { pv[u].visited = true; /* * Enumerate the neighbours of u, and any that are * as yet not in the tree structure (indicated by * child==-2, and distinct from the 'visited' * flag) become children of u. */ debug((" component pass: processing %d\n", u)); for (w = neighbour(u, ctx); w >= 0; w = neighbour(-1, ctx)) { debug((" edge %d-%d\n", u, w)); if (pv[w].child == -2) { debug((" -> new child\n")); pv[w].child = -1; pv[w].sibling = pv[u].child; pv[w].parent = u; pv[w].component_root = pv[u].component_root; pv[u].child = w; } /* While we're here, count the edges in the whole * graph, so that we can easily check at the end * whether all of them are bridges, i.e. whether * no loop exists at all. */ if (w > u) /* count each edge only in one direction */ nedges++; } /* * Now descend in depth-first search. */ if (pv[u].child >= 0) { u = pv[u].child; debug((" descending to %d\n", u)); continue; } } if (u == v) { debug((" back at %d, done this component\n", u)); break; } else if (pv[u].sibling >= 0) { u = pv[u].sibling; debug((" sideways to %d\n", u)); } else { u = pv[u].parent; debug((" ascending to %d\n", u)); } } } } /* * Second pass: index all the vertices in such a way that every * subtree has a contiguous range of indices. (Easily enough done, * by iterating through the tree structure we just built and * numbering its elements as if they were those of a sorted list.) * * For each vertex, we compute the min and max index of the * subtree starting there. * * (We index the vertices in preorder, per Tarjan's original * description, so that each vertex's min subtree index is its own * index; but that doesn't actually matter; either way round would * do. The important thing is that we have a simple arithmetic * criterion that tells us whether a vertex is in a given subtree * or not.) */ debug(("--- begin indexing pass\n")); index = 0; for (v = 0; v < nvertices; v++) pv[v].visited = false; pv[root].visited = true; u = pv[root].child; while (1) { if (!pv[u].visited) { pv[u].visited = true; /* * Index this node. */ pv[u].minindex = pv[u].index = index; debug((" vertex %d <- index %d\n", u, index)); index++; /* * Now descend in depth-first search. */ if (pv[u].child >= 0) { u = pv[u].child; debug((" descending to %d\n", u)); continue; } } if (u == root) { debug((" back at %d, done indexing\n", u)); break; } /* * As we re-ascend to here from its children (or find that we * had no children to descend to in the first place), fill in * its maxindex field. */ pv[u].maxindex = index-1; debug((" vertex %d <- maxindex %d\n", u, pv[u].maxindex)); if (pv[u].sibling >= 0) { u = pv[u].sibling; debug((" sideways to %d\n", u)); } else { u = pv[u].parent; debug((" ascending to %d\n", u)); } } /* * We're ready to generate output now, so initialise the output * fields. */ for (v = 0; v < nvertices; v++) pv[v].bridge = -1; /* * Final pass: determine the min and max index of the vertices * reachable from every subtree, not counting the link back to * each vertex's parent. Then our criterion is: given a vertex u, * defining a subtree consisting of u and all its descendants, we * compare the range of vertex indices _in_ that subtree (which is * just the minindex and maxindex of u) with the range of vertex * indices in the _neighbourhood_ of the subtree (computed in this * final pass, and not counting u's own edge to its parent), and * if the latter includes anything outside the former, then there * must be some path from u to outside its subtree which does not * go through the parent edge - i.e. the edge from u to its parent * is part of a loop. */ debug(("--- begin min-max pass\n")); nbridges = 0; for (v = 0; v < nvertices; v++) pv[v].visited = false; u = pv[root].child; pv[root].visited = true; while (1) { if (!pv[u].visited) { pv[u].visited = true; /* * Look for vertices reachable directly from u, including * u itself. */ debug((" processing vertex %d\n", u)); pv[u].minreachable = pv[u].maxreachable = pv[u].minindex; for (w = neighbour(u, ctx); w >= 0; w = neighbour(-1, ctx)) { debug((" edge %d-%d\n", u, w)); if (w != pv[u].parent) { int i = pv[w].index; if (pv[u].minreachable > i) pv[u].minreachable = i; if (pv[u].maxreachable < i) pv[u].maxreachable = i; } } debug((" initial min=%d max=%d\n", pv[u].minreachable, pv[u].maxreachable)); /* * Now descend in depth-first search. */ if (pv[u].child >= 0) { u = pv[u].child; debug((" descending to %d\n", u)); continue; } } if (u == root) { debug((" back at %d, done min-maxing\n", u)); break; } /* * As we re-ascend to this vertex, go back through its * immediate children and do a post-update of its min/max. */ for (v = pv[u].child; v >= 0; v = pv[v].sibling) { if (pv[u].minreachable > pv[v].minreachable) pv[u].minreachable = pv[v].minreachable; if (pv[u].maxreachable < pv[v].maxreachable) pv[u].maxreachable = pv[v].maxreachable; } debug((" postorder update of %d: min=%d max=%d (indices %d-%d)\n", u, pv[u].minreachable, pv[u].maxreachable, pv[u].minindex, pv[u].maxindex)); /* * And now we know whether each to our own parent is a bridge. */ if ((v = pv[u].parent) != root) { if (pv[u].minreachable >= pv[u].minindex && pv[u].maxreachable <= pv[u].maxindex) { /* Yes, it's a bridge. */ pv[u].bridge = v; nbridges++; debug((" %d-%d is a bridge\n", v, u)); } else { debug((" %d-%d is not a bridge\n", v, u)); } } if (pv[u].sibling >= 0) { u = pv[u].sibling; debug((" sideways to %d\n", u)); } else { u = pv[u].parent; debug((" ascending to %d\n", u)); } } debug(("finished, nedges=%d nbridges=%d\n", nedges, nbridges)); /* * Done. */ return nbridges < nedges; } /* * Appendix: the long and painful history of loop detection in these puzzles * ========================================================================= * * For interest, I thought I'd write up the five loop-finding methods * I've gone through before getting to this algorithm. It's a case * study in all the ways you can solve this particular problem * wrongly, and also how much effort you can waste by not managing to * find the existing solution in the literature :-( * * Vertex dsf * ---------- * * Initially, in puzzles where you need to not have any loops in the * solution graph, I detected them by using a dsf to track connected * components of vertices. Iterate over each edge unifying the two * vertices it connects; but before that, check if the two vertices * are _already_ known to be connected. If so, then the new edge is * providing a second path between them, i.e. a loop exists. * * That's adequate for automated solvers, where you just need to know * _whether_ a loop exists, so as to rule out that move and do * something else. But during play, you want to do better than that: * you want to _point out_ the loops with error highlighting. * * Graph pruning * ------------- * * So my second attempt worked by iteratively pruning the graph. Find * a vertex with degree 1; remove that edge; repeat until you can't * find such a vertex any more. This procedure will remove *every* * edge of the graph if and only if there were no loops; so if there * are any edges remaining, highlight them. * * This successfully highlights loops, but not _only_ loops. If the * graph contains a 'dumb-bell' shaped subgraph consisting of two * loops connected by a path, then we'll end up highlighting the * connecting path as well as the loops. That's not what we wanted. * * Vertex dsf with ad-hoc loop tracing * ----------------------------------- * * So my third attempt was to go back to the dsf strategy, only this * time, when you detect that a particular edge connects two * already-connected vertices (and hence is part of a loop), you try * to trace round that loop to highlight it - before adding the new * edge, search for a path between its endpoints among the edges the * algorithm has already visited, and when you find one (which you * must), highlight the loop consisting of that path plus the new * edge. * * This solves the dumb-bell problem - we definitely now cannot * accidentally highlight any edge that is *not* part of a loop. But * it's far from clear that we'll highlight *every* edge that *is* * part of a loop - what if there were multiple paths between the two * vertices? It would be difficult to guarantee that we'd always catch * every single one. * * On the other hand, it is at least guaranteed that we'll highlight * _something_ if any loop exists, and in other error highlighting * situations (see in particular the Tents connected component * analysis) I've been known to consider that sufficient. So this * version hung around for quite a while, until I had a better idea. * * Face dsf * -------- * * Round about the time Loopy was being revamped to include non-square * grids, I had a much cuter idea, making use of the fact that the * graph is planar, and hence has a concept of faces. * * In Loopy, there are really two graphs: the 'grid', consisting of * all the edges that the player *might* fill in, and the solution * graph of the edges the player actually *has* filled in. The * algorithm is: set up a dsf on the *faces* of the grid. Iterate over * each edge of the grid which is _not_ marked by the player as an * edge of the solution graph, unifying the faces on either side of * that edge. This groups the faces into connected components. Now, * there is more than one connected component iff a loop exists, and * moreover, an edge of the solution graph is part of a loop iff the * faces on either side of it are in different connected components! * * This is the first algorithm I came up with that I was confident * would successfully highlight exactly the correct set of edges in * all cases. It's also conceptually elegant, and very easy to * implement and to be confident you've got it right (since it just * consists of two very simple loops over the edge set, one building * the dsf and one reading it off). I was very pleased with it. * * Doing the same thing in Slant is slightly more difficult because * the set of edges the user can fill in do not form a planar graph * (the two potential edges in each square cross in the middle). But * you can still apply the same principle by considering the 'faces' * to be diamond-shaped regions of space around each horizontal or * vertical grid line. Equivalently, pretend each edge added by the * player is really divided into two edges, each from a square-centre * to one of the square's corners, and now the grid graph is planar * again. * * However, it fell down when - much later - I tried to implement the * same algorithm in Net. * * Net doesn't *absolutely need* loop detection, because of its system * of highlighting squares connected to the source square: an argument * involving counting vertex degrees shows that if any loop exists, * then it must be counterbalanced by some disconnected square, so * there will be _some_ error highlight in any invalid grid even * without loop detection. However, in large complicated cases, it's * still nice to highlight the loop itself, so that once the player is * clued in to its existence by a disconnected square elsewhere, they * don't have to spend forever trying to find it. * * The new wrinkle in Net, compared to other loop-disallowing puzzles, * is that it can be played with wrapping walls, or - topologically * speaking - on a torus. And a torus has a property that algebraic * topologists would know of as a 'non-trivial H_1 homology group', * which essentially means that there can exist a loop on a torus * which *doesn't* separate the surface into two regions disconnected * from each other. * * In other words, using this algorithm in Net will do fine at finding * _small_ localised loops, but a large-scale loop that goes (say) off * the top of the grid, back on at the bottom, and meets up in the * middle again will not be detected. * * Footpath dsf * ------------ * * To solve this homology problem in Net, I hastily thought up another * dsf-based algorithm. * * This time, let's consider each edge of the graph to be a road, with * a separate pedestrian footpath down each side. We'll form a dsf on * those imaginary segments of footpath. * * At each vertex of the graph, we go round the edges leaving that * vertex, in order around the vertex. For each pair of edges adjacent * in this order, we unify their facing pair of footpaths (e.g. if * edge E appears anticlockwise of F, then we unify the anticlockwise * footpath of F with the clockwise one of E) . In particular, if a * vertex has degree 1, then the two footpaths on either side of its * single edge are unified. * * Then, an edge is part of a loop iff its two footpaths are not * reachable from one another. * * This algorithm is almost as simple to implement as the face dsf, * and it works on a wider class of graphs embedded in plane-like * surfaces; in particular, it fixes the torus bug in the face-dsf * approach. However, it still depends on the graph having _some_ sort * of embedding in a 2-manifold, because it relies on there being a * meaningful notion of 'order of edges around a vertex' in the first * place, so you couldn't use it on a wildly nonplanar graph like the * diamond lattice. Also, more subtly, it depends on the graph being * embedded in an _orientable_ surface - and that's a thing that might * much more plausibly change in future puzzles, because it's not at * all unlikely that at some point I might feel moved to implement a * puzzle that can be played on the surface of a Mobius strip or a * Klein bottle. And then even this algorithm won't work. * * Tarjan's bridge-finding algorithm * --------------------------------- * * And so, finally, we come to the algorithm above. This one is pure * graph theory: it doesn't depend on any concept of 'faces', or 'edge * ordering around a vertex', or any other trapping of a planar or * quasi-planar graph embedding. It should work on any graph * whatsoever, and reliably identify precisely the set of edges that * form part of some loop. So *hopefully* this long string of failures * has finally come to an end... */