General cleanups patch from James H:

[sgt/puzzles] / latin.c

#include <assert.h>
#include <string.h>
#include <stdarg.h>

#include "puzzles.h"
#include "tree234.h"
#include "maxflow.h"

#ifdef STANDALONE_LATIN_TEST
#define STANDALONE_SOLVER
#endif

#include "latin.h"

/* --------------------------------------------------------
 * Solver.
 */

/*
 * Function called when we are certain that a particular square has
 * a particular number in it. The y-coordinate passed in here is
 * transformed.
 */
void latin_solver_place(struct latin_solver *solver, int x, int y, int n)
{
    int i, o = solver->o;

    assert(n <= o);
    assert(cube(x,y,n));

    /*
     * Rule out all other numbers in this square.
     */
    for (i = 1; i <= o; i++)
        if (i != n)
            cube(x,y,i) = FALSE;

    /*
     * Rule out this number in all other positions in the row.
     */
    for (i = 0; i < o; i++)
        if (i != y)
            cube(x,i,n) = FALSE;

    /*
     * Rule out this number in all other positions in the column.
     */
    for (i = 0; i < o; i++)
        if (i != x)
            cube(i,y,n) = FALSE;

    /*
     * Enter the number in the result grid.
     */
    solver->grid[YUNTRANS(y)*o+x] = n;

    /*
     * Cross out this number from the list of numbers left to place
     * in its row, its column and its block.
     */
    solver->row[y*o+n-1] = solver->col[x*o+n-1] = TRUE;
}

int latin_solver_elim(struct latin_solver *solver, int start, int step
#ifdef STANDALONE_SOLVER
                      , char *fmt, ...
#endif
                      )
{
    int o = solver->o;
    int fpos, m, i;

    /*
     * Count the number of set bits within this section of the
     * cube.
     */
    m = 0;
    fpos = -1;
    for (i = 0; i < o; i++)
        if (solver->cube[start+i*step]) {
            fpos = start+i*step;
            m++;
        }

    if (m == 1) {
        int x, y, n;
        assert(fpos >= 0);

        n = 1 + fpos % o;
        y = fpos / o;
        x = y / o;
        y %= o;

        if (!solver->grid[YUNTRANS(y)*o+x]) {
#ifdef STANDALONE_SOLVER
            if (solver_show_working) {
                va_list ap;
                printf("%*s", solver_recurse_depth*4, "");
                va_start(ap, fmt);
                vprintf(fmt, ap);
                va_end(ap);
                printf(":\n%*s  placing %d at (%d,%d)\n",
                       solver_recurse_depth*4, "", n, x, YUNTRANS(y));
            }
#endif
            latin_solver_place(solver, x, y, n);
            return +1;
        }
    } else if (m == 0) {
#ifdef STANDALONE_SOLVER
        if (solver_show_working) {
            va_list ap;
            printf("%*s", solver_recurse_depth*4, "");
            va_start(ap, fmt);
            vprintf(fmt, ap);
            va_end(ap);
            printf(":\n%*s  no possibilities available\n",
                   solver_recurse_depth*4, "");
        }
#endif
        return -1;
    }

    return 0;
}

struct latin_solver_scratch {
    unsigned char *grid, *rowidx, *colidx, *set;
    int *neighbours, *bfsqueue;
#ifdef STANDALONE_SOLVER
    int *bfsprev;
#endif
};

int latin_solver_set(struct latin_solver *solver,
                     struct latin_solver_scratch *scratch,
                     int start, int step1, int step2
#ifdef STANDALONE_SOLVER
                     , char *fmt, ...
#endif
                     )
{
    int o = solver->o;
    int i, j, n, count;
    unsigned char *grid = scratch->grid;
    unsigned char *rowidx = scratch->rowidx;
    unsigned char *colidx = scratch->colidx;
    unsigned char *set = scratch->set;

    /*
     * We are passed a o-by-o matrix of booleans. Our first job
     * is to winnow it by finding any definite placements - i.e.
     * any row with a solitary 1 - and discarding that row and the
     * column containing the 1.
     */
    memset(rowidx, TRUE, o);
    memset(colidx, TRUE, o);
    for (i = 0; i < o; i++) {
        int count = 0, first = -1;
        for (j = 0; j < o; j++)
            if (solver->cube[start+i*step1+j*step2])
                first = j, count++;

        if (count == 0) return -1;
        if (count == 1)
            rowidx[i] = colidx[first] = FALSE;
    }

    /*
     * Convert each of rowidx/colidx from a list of 0s and 1s to a
     * list of the indices of the 1s.
     */
    for (i = j = 0; i < o; i++)
        if (rowidx[i])
            rowidx[j++] = i;
    n = j;
    for (i = j = 0; i < o; i++)
        if (colidx[i])
            colidx[j++] = i;
    assert(n == j);

    /*
     * And create the smaller matrix.
     */
    for (i = 0; i < n; i++)
        for (j = 0; j < n; j++)
            grid[i*o+j] = solver->cube[start+rowidx[i]*step1+colidx[j]*step2];

    /*
     * Having done that, we now have a matrix in which every row
     * has at least two 1s in. Now we search to see if we can find
     * a rectangle of zeroes (in the set-theoretic sense of
     * `rectangle', i.e. a subset of rows crossed with a subset of
     * columns) whose width and height add up to n.
     */

    memset(set, 0, n);
    count = 0;
    while (1) {
        /*
         * We have a candidate set. If its size is <=1 or >=n-1
         * then we move on immediately.
         */
        if (count > 1 && count < n-1) {
            /*
             * The number of rows we need is n-count. See if we can
             * find that many rows which each have a zero in all
             * the positions listed in `set'.
             */
            int rows = 0;
            for (i = 0; i < n; i++) {
                int ok = TRUE;
                for (j = 0; j < n; j++)
                    if (set[j] && grid[i*o+j]) {
                        ok = FALSE;
                        break;
                    }
                if (ok)
                    rows++;
            }

            /*
             * We expect never to be able to get _more_ than
             * n-count suitable rows: this would imply that (for
             * example) there are four numbers which between them
             * have at most three possible positions, and hence it
             * indicates a faulty deduction before this point or
             * even a bogus clue.
             */
            if (rows > n - count) {
#ifdef STANDALONE_SOLVER
                if (solver_show_working) {
                    va_list ap;
                    printf("%*s", solver_recurse_depth*4,
                           "");
                    va_start(ap, fmt);
                    vprintf(fmt, ap);
                    va_end(ap);
                    printf(":\n%*s  contradiction reached\n",
                           solver_recurse_depth*4, "");
                }
#endif
                return -1;
            }

            if (rows >= n - count) {
                int progress = FALSE;

                /*
                 * We've got one! Now, for each row which _doesn't_
                 * satisfy the criterion, eliminate all its set
                 * bits in the positions _not_ listed in `set'.
                 * Return +1 (meaning progress has been made) if we
                 * successfully eliminated anything at all.
                 *
                 * This involves referring back through
                 * rowidx/colidx in order to work out which actual
                 * positions in the cube to meddle with.
                 */
                for (i = 0; i < n; i++) {
                    int ok = TRUE;
                    for (j = 0; j < n; j++)
                        if (set[j] && grid[i*o+j]) {
                            ok = FALSE;
                            break;
                        }
                    if (!ok) {
                        for (j = 0; j < n; j++)
                            if (!set[j] && grid[i*o+j]) {
                                int fpos = (start+rowidx[i]*step1+
                                            colidx[j]*step2);
#ifdef STANDALONE_SOLVER
                                if (solver_show_working) {
                                    int px, py, pn;

                                    if (!progress) {
                                        va_list ap;
                                        printf("%*s", solver_recurse_depth*4,
                                               "");
                                        va_start(ap, fmt);
                                        vprintf(fmt, ap);
                                        va_end(ap);
                                        printf(":\n");
                                    }

                                    pn = 1 + fpos % o;
                                    py = fpos / o;
                                    px = py / o;
                                    py %= o;

                                    printf("%*s  ruling out %d at (%d,%d)\n",
                                           solver_recurse_depth*4, "",
                                           pn, px, YUNTRANS(py));
                                }
#endif
                                progress = TRUE;
                                solver->cube[fpos] = FALSE;
                            }
                    }
                }

                if (progress) {
                    return +1;
                }
            }
        }

        /*
         * Binary increment: change the rightmost 0 to a 1, and
         * change all 1s to the right of it to 0s.
         */
        i = n;
        while (i > 0 && set[i-1])
            set[--i] = 0, count--;
        if (i > 0)
            set[--i] = 1, count++;
        else
            break;                     /* done */
    }

    return 0;
}

/*
 * Look for forcing chains. A forcing chain is a path of
 * pairwise-exclusive squares (i.e. each pair of adjacent squares
 * in the path are in the same row, column or block) with the
 * following properties:
 *
 *  (a) Each square on the path has precisely two possible numbers.
 *
 *  (b) Each pair of squares which are adjacent on the path share
 *      at least one possible number in common.
 *
 *  (c) Each square in the middle of the path shares _both_ of its
 *      numbers with at least one of its neighbours (not the same
 *      one with both neighbours).
 *
 * These together imply that at least one of the possible number
 * choices at one end of the path forces _all_ the rest of the
 * numbers along the path. In order to make real use of this, we
 * need further properties:
 *
 *  (c) Ruling out some number N from the square at one end
 *      of the path forces the square at the other end to
 *      take number N.
 *
 *  (d) The two end squares are both in line with some third
 *      square.
 *
 *  (e) That third square currently has N as a possibility.
 *
 * If we can find all of that lot, we can deduce that at least one
 * of the two ends of the forcing chain has number N, and that
 * therefore the mutually adjacent third square does not.
 *
 * To find forcing chains, we're going to start a bfs at each
 * suitable square, once for each of its two possible numbers.
 */
int latin_solver_forcing(struct latin_solver *solver,
                         struct latin_solver_scratch *scratch)
{
    int o = solver->o;
    int *bfsqueue = scratch->bfsqueue;
#ifdef STANDALONE_SOLVER
    int *bfsprev = scratch->bfsprev;
#endif
    unsigned char *number = scratch->grid;
    int *neighbours = scratch->neighbours;
    int x, y;

    for (y = 0; y < o; y++)
        for (x = 0; x < o; x++) {
            int count, t, n;

            /*
             * If this square doesn't have exactly two candidate
             * numbers, don't try it.
             *
             * In this loop we also sum the candidate numbers,
             * which is a nasty hack to allow us to quickly find
             * `the other one' (since we will shortly know there
             * are exactly two).
             */
            for (count = t = 0, n = 1; n <= o; n++)
                if (cube(x, y, n))
                    count++, t += n;
            if (count != 2)
                continue;

            /*
             * Now attempt a bfs for each candidate.
             */
            for (n = 1; n <= o; n++)
                if (cube(x, y, n)) {
                    int orign, currn, head, tail;

                    /*
                     * Begin a bfs.
                     */
                    orign = n;

                    memset(number, o+1, o*o);
                    head = tail = 0;
                    bfsqueue[tail++] = y*o+x;
#ifdef STANDALONE_SOLVER
                    bfsprev[y*o+x] = -1;
#endif
                    number[y*o+x] = t - n;

                    while (head < tail) {
                        int xx, yy, nneighbours, xt, yt, i;

                        xx = bfsqueue[head++];
                        yy = xx / o;
                        xx %= o;

                        currn = number[yy*o+xx];

                        /*
                         * Find neighbours of yy,xx.
                         */
                        nneighbours = 0;
                        for (yt = 0; yt < o; yt++)
                            neighbours[nneighbours++] = yt*o+xx;
                        for (xt = 0; xt < o; xt++)
                            neighbours[nneighbours++] = yy*o+xt;

                        /*
                         * Try visiting each of those neighbours.
                         */
                        for (i = 0; i < nneighbours; i++) {
                            int cc, tt, nn;

                            xt = neighbours[i] % o;
                            yt = neighbours[i] / o;

                            /*
                             * We need this square to not be
                             * already visited, and to include
                             * currn as a possible number.
                             */
                            if (number[yt*o+xt] <= o)
                                continue;
                            if (!cube(xt, yt, currn))
                                continue;

                            /*
                             * Don't visit _this_ square a second
                             * time!
                             */
                            if (xt == xx && yt == yy)
                                continue;

                            /*
                             * To continue with the bfs, we need
                             * this square to have exactly two
                             * possible numbers.
                             */
                            for (cc = tt = 0, nn = 1; nn <= o; nn++)
                                if (cube(xt, yt, nn))
                                    cc++, tt += nn;
                            if (cc == 2) {
                                bfsqueue[tail++] = yt*o+xt;
#ifdef STANDALONE_SOLVER
                                bfsprev[yt*o+xt] = yy*o+xx;
#endif
                                number[yt*o+xt] = tt - currn;
                            }

                            /*
                             * One other possibility is that this
                             * might be the square in which we can
                             * make a real deduction: if it's
                             * adjacent to x,y, and currn is equal
                             * to the original number we ruled out.
                             */
                            if (currn == orign &&
                                (xt == x || yt == y)) {
#ifdef STANDALONE_SOLVER
                                if (solver_show_working) {
                                    char *sep = "";
                                    int xl, yl;
                                    printf("%*sforcing chain, %d at ends of ",
                                           solver_recurse_depth*4, "", orign);
                                    xl = xx;
                                    yl = yy;
                                    while (1) {
                                        printf("%s(%d,%d)", sep, xl,
                                               YUNTRANS(yl));
                                        xl = bfsprev[yl*o+xl];
                                        if (xl < 0)
                                            break;
                                        yl = xl / o;
                                        xl %= o;
                                        sep = "-";
                                    }
                                    printf("\n%*s  ruling out %d at (%d,%d)\n",
                                           solver_recurse_depth*4, "",
                                           orign, xt, YUNTRANS(yt));
                                }
#endif
                                cube(xt, yt, orign) = FALSE;
                                return 1;
                            }
                        }
                    }
                }
        }

    return 0;
}

struct latin_solver_scratch *latin_solver_new_scratch(struct latin_solver *solver)
{
    struct latin_solver_scratch *scratch = snew(struct latin_solver_scratch);
    int o = solver->o;
    scratch->grid = snewn(o*o, unsigned char);
    scratch->rowidx = snewn(o, unsigned char);
    scratch->colidx = snewn(o, unsigned char);
    scratch->set = snewn(o, unsigned char);
    scratch->neighbours = snewn(3*o, int);
    scratch->bfsqueue = snewn(o*o, int);
#ifdef STANDALONE_SOLVER
    scratch->bfsprev = snewn(o*o, int);
#endif
    return scratch;
}

void latin_solver_free_scratch(struct latin_solver_scratch *scratch)
{
#ifdef STANDALONE_SOLVER
    sfree(scratch->bfsprev);
#endif
    sfree(scratch->bfsqueue);
    sfree(scratch->neighbours);
    sfree(scratch->set);
    sfree(scratch->colidx);
    sfree(scratch->rowidx);
    sfree(scratch->grid);
    sfree(scratch);
}

void latin_solver_alloc(struct latin_solver *solver, digit *grid, int o)
{
    int x, y;

    solver->o = o;
    solver->cube = snewn(o*o*o, unsigned char);
    solver->grid = grid;                /* write straight back to the input */
    memset(solver->cube, TRUE, o*o*o);

    solver->row = snewn(o*o, unsigned char);
    solver->col = snewn(o*o, unsigned char);
    memset(solver->row, FALSE, o*o);
    memset(solver->col, FALSE, o*o);

    for (x = 0; x < o; x++)
        for (y = 0; y < o; y++)
            if (grid[y*o+x])
                latin_solver_place(solver, x, YTRANS(y), grid[y*o+x]);
}

void latin_solver_free(struct latin_solver *solver)
{
    sfree(solver->cube);
    sfree(solver->row);
    sfree(solver->col);
}

int latin_solver_diff_simple(struct latin_solver *solver)
{
    int x, y, n, ret, o = solver->o;
    /*
     * Row-wise positional elimination.
     */
    for (y = 0; y < o; y++)
        for (n = 1; n <= o; n++)
            if (!solver->row[y*o+n-1]) {
                ret = latin_solver_elim(solver, cubepos(0,y,n), o*o
#ifdef STANDALONE_SOLVER
                                        , "positional elimination,"
                                        " %d in row %d", n, YUNTRANS(y)
#endif
                                        );
                if (ret != 0) return ret;
            }
    /*
     * Column-wise positional elimination.
     */
    for (x = 0; x < o; x++)
        for (n = 1; n <= o; n++)
            if (!solver->col[x*o+n-1]) {
                ret = latin_solver_elim(solver, cubepos(x,0,n), o
#ifdef STANDALONE_SOLVER
                                        , "positional elimination,"
                                        " %d in column %d", n, x
#endif
                                        );
                if (ret != 0) return ret;
            }

    /*
     * Numeric elimination.
     */
    for (x = 0; x < o; x++)
        for (y = 0; y < o; y++)
            if (!solver->grid[YUNTRANS(y)*o+x]) {
                ret = latin_solver_elim(solver, cubepos(x,y,1), 1
#ifdef STANDALONE_SOLVER
                                        , "numeric elimination at (%d,%d)", x,
                                        YUNTRANS(y)
#endif
                                        );
                if (ret != 0) return ret;
            }
    return 0;
}

int latin_solver_diff_set(struct latin_solver *solver,
                          struct latin_solver_scratch *scratch,
                          int extreme)
{
    int x, y, n, ret, o = solver->o;

    if (!extreme) {
        /*
         * Row-wise set elimination.
         */
        for (y = 0; y < o; y++) {
            ret = latin_solver_set(solver, scratch, cubepos(0,y,1), o*o, 1
#ifdef STANDALONE_SOLVER
                                   , "set elimination, row %d", YUNTRANS(y)
#endif
                                  );
            if (ret != 0) return ret;
        }
        /*
         * Column-wise set elimination.
         */
        for (x = 0; x < o; x++) {
            ret = latin_solver_set(solver, scratch, cubepos(x,0,1), o, 1
#ifdef STANDALONE_SOLVER
                                   , "set elimination, column %d", x
#endif
                                  );
            if (ret != 0) return ret;
        }
    } else {
        /*
         * Row-vs-column set elimination on a single number
         * (much tricker for a human to do!)
         */
        for (n = 1; n <= o; n++) {
            ret = latin_solver_set(solver, scratch, cubepos(0,0,n), o*o, o
#ifdef STANDALONE_SOLVER
                                   , "positional set elimination, number %d", n
#endif
                                  );
            if (ret != 0) return ret;
        }
    }
    return 0;
}

/* This uses our own diff_* internally, but doesn't require callers
 * to; this is so it can be used by games that want to rewrite
 * the solver so as to use a different set of difficulties.
 *
 * It returns:
 * 0 for 'didn't do anything' implying it was already solved.
 * -1 for 'impossible' (no solution)
 * 1 for 'single solution'
 * >1 for 'multiple solutions' (you don't get to know how many, and
 *     the first such solution found will be set.
 *
 * and this function may well assert if given an impossible board.
 */
int latin_solver_recurse(struct latin_solver *solver, int recdiff,
                         latin_solver_callback cb, void *ctx)
{
    int best, bestcount;
    int o = solver->o, x, y, n;

    best = -1;
    bestcount = o+1;

    for (y = 0; y < o; y++)
        for (x = 0; x < o; x++)
            if (!solver->grid[y*o+x]) {
                int count;

                /*
                 * An unfilled square. Count the number of
                 * possible digits in it.
                 */
                count = 0;
                for (n = 1; n <= o; n++)
                    if (cube(x,YTRANS(y),n))
                        count++;

                /*
                 * We should have found any impossibilities
                 * already, so this can safely be an assert.
                 */
                assert(count > 1);

                if (count < bestcount) {
                    bestcount = count;
                    best = y*o+x;
                }
            }

    if (best == -1)
        /* we were complete already. */
        return 0;
    else {
        int i, j;
        digit *list, *ingrid, *outgrid;
        int diff = diff_impossible;    /* no solution found yet */

        /*
         * Attempt recursion.
         */
        y = best / o;
        x = best % o;

        list = snewn(o, digit);
        ingrid = snewn(o*o, digit);
        outgrid = snewn(o*o, digit);
        memcpy(ingrid, solver->grid, o*o);

        /* Make a list of the possible digits. */
        for (j = 0, n = 1; n <= o; n++)
            if (cube(x,YTRANS(y),n))
                list[j++] = n;

#ifdef STANDALONE_SOLVER
        if (solver_show_working) {
            char *sep = "";
            printf("%*srecursing on (%d,%d) [",
                   solver_recurse_depth*4, "", x, y);
            for (i = 0; i < j; i++) {
                printf("%s%d", sep, list[i]);
                sep = " or ";
            }
            printf("]\n");
        }
#endif

        /*
         * And step along the list, recursing back into the
         * main solver at every stage.
         */
        for (i = 0; i < j; i++) {
            int ret;

            memcpy(outgrid, ingrid, o*o);
            outgrid[y*o+x] = list[i];

#ifdef STANDALONE_SOLVER
            if (solver_show_working)
                printf("%*sguessing %d at (%d,%d)\n",
                       solver_recurse_depth*4, "", list[i], x, y);
            solver_recurse_depth++;
#endif

            ret = cb(outgrid, o, recdiff, ctx);

#ifdef STANDALONE_SOLVER
            solver_recurse_depth--;
            if (solver_show_working) {
                printf("%*sretracting %d at (%d,%d)\n",
                       solver_recurse_depth*4, "", list[i], x, y);
            }
#endif
            /* we recurse as deep as we can, so we should never find
             * find ourselves giving up on a puzzle without declaring it
             * impossible.  */
            assert(ret != diff_unfinished);

            /*
             * If we have our first solution, copy it into the
             * grid we will return.
             */
            if (diff == diff_impossible && ret != diff_impossible)
                memcpy(solver->grid, outgrid, o*o);

            if (ret == diff_ambiguous)
                diff = diff_ambiguous;
            else if (ret == diff_impossible)
                /* do not change our return value */;
            else {
                /* the recursion turned up exactly one solution */
                if (diff == diff_impossible)
                    diff = recdiff;
                else
                    diff = diff_ambiguous;
            }

            /*
             * As soon as we've found more than one solution,
             * give up immediately.
             */
            if (diff == diff_ambiguous)
                break;
        }

        sfree(outgrid);
        sfree(ingrid);
        sfree(list);

        if (diff == diff_impossible)
            return -1;
        else if (diff == diff_ambiguous)
            return 2;
        else {
            assert(diff == recdiff);
            return 1;
        }
    }
}

enum { diff_simple = 1, diff_set, diff_extreme, diff_recursive };

static int latin_solver_sub(struct latin_solver *solver, int maxdiff, void *ctx)
{
    struct latin_solver_scratch *scratch = latin_solver_new_scratch(solver);
    int ret, diff = diff_simple;

    assert(maxdiff <= diff_recursive);
    /*
     * Now loop over the grid repeatedly trying all permitted modes
     * of reasoning. The loop terminates if we complete an
     * iteration without making any progress; we then return
     * failure or success depending on whether the grid is full or
     * not.
     */
    while (1) {
        /*
         * I'd like to write `continue;' inside each of the
         * following loops, so that the solver returns here after
         * making some progress. However, I can't specify that I
         * want to continue an outer loop rather than the innermost
         * one, so I'm apologetically resorting to a goto.
         */
        cont:
        latin_solver_debug(solver->cube, solver->o);

        ret = latin_solver_diff_simple(solver);
        if (ret < 0) {
            diff = diff_impossible;
            goto got_result;
        } else if (ret > 0) {
            diff = max(diff, diff_simple);
            goto cont;
        }

        if (maxdiff <= diff_simple)
            break;

        ret = latin_solver_diff_set(solver, scratch, 0);
        if (ret < 0) {
            diff = diff_impossible;
            goto got_result;
        } else if (ret > 0) {
            diff = max(diff, diff_set);
            goto cont;
        }

        if (maxdiff <= diff_set)
            break;

        ret = latin_solver_diff_set(solver, scratch, 1);
        if (ret < 0) {
            diff = diff_impossible;
            goto got_result;
        } else if (ret > 0) {
            diff = max(diff, diff_extreme);
            goto cont;
        }

        /*
         * Forcing chains.
         */
        if (latin_solver_forcing(solver, scratch)) {
            diff = max(diff, diff_extreme);
            goto cont;
        }

        /*
         * If we reach here, we have made no deductions in this
         * iteration, so the algorithm terminates.
         */
        break;
    }

    /*
     * Last chance: if we haven't fully solved the puzzle yet, try
     * recursing based on guesses for a particular square. We pick
     * one of the most constrained empty squares we can find, which
     * has the effect of pruning the search tree as much as
     * possible.
     */
    if (maxdiff == diff_recursive) {
        int nsol = latin_solver_recurse(solver, diff_recursive, latin_solver, ctx);
        if (nsol < 0) diff = diff_impossible;
        else if (nsol == 1) diff = diff_recursive;
        else if (nsol > 1) diff = diff_ambiguous;
        /* if nsol == 0 then we were complete anyway
         * (and thus don't need to change diff) */
    } else {
        /*
         * We're forbidden to use recursion, so we just see whether
         * our grid is fully solved, and return diff_unfinished
         * otherwise.
         */
        int x, y, o = solver->o;

        for (y = 0; y < o; y++)
            for (x = 0; x < o; x++)
                if (!solver->grid[y*o+x])
                    diff = diff_unfinished;
    }

    got_result:

#ifdef STANDALONE_SOLVER
    if (solver_show_working)
        printf("%*s%s found\n",
               solver_recurse_depth*4, "",
               diff == diff_impossible ? "no solution (impossible)" :
               diff == diff_unfinished ? "no solution (unfinished)" :
               diff == diff_ambiguous ? "multiple solutions" :
               "one solution");
#endif

    latin_solver_free_scratch(scratch);

    return diff;
}

int latin_solver(digit *grid, int o, int maxdiff, void *ctx)
{
    struct latin_solver solver;
    int diff;

    latin_solver_alloc(&solver, grid, o);
    diff = latin_solver_sub(&solver, maxdiff, ctx);
    latin_solver_free(&solver);
    return diff;
}

void latin_solver_debug(unsigned char *cube, int o)
{
#ifdef STANDALONE_SOLVER
    if (solver_show_working) {
        struct latin_solver ls, *solver = &ls;
        char *dbg;
        int x, y, i, c = 0;

        ls.cube = cube; ls.o = o; /* for cube() to work */

        dbg = snewn(3*o*o*o, unsigned char);
        for (y = 0; y < o; y++) {
            for (x = 0; x < o; x++) {
                for (i = 1; i <= o; i++) {
                    if (cube(x,y,i))
                        dbg[c++] = i + '0';
                    else
                        dbg[c++] = '.';
                }
                dbg[c++] = ' ';
            }
            dbg[c++] = '\n';
        }
        dbg[c++] = '\n';
        dbg[c++] = '\0';

        printf("%s", dbg);
        sfree(dbg);
    }
#endif
}

void latin_debug(digit *sq, int o)
{
#ifdef STANDALONE_SOLVER
    if (solver_show_working) {
        int x, y;

        for (y = 0; y < o; y++) {
            for (x = 0; x < o; x++) {
                printf("%2d ", sq[y*o+x]);
            }
            printf("\n");
        }
        printf("\n");
    }
#endif
}

/* --------------------------------------------------------
 * Generation.
 */

digit *latin_generate(int o, random_state *rs)
{
    digit *sq;
    int *edges, *backedges, *capacity, *flow;
    void *scratch;
    int ne, scratchsize;
    int i, j, k;
    digit *row, *col, *numinv, *num;

    /*
     * To efficiently generate a latin square in such a way that
     * all possible squares are possible outputs from the function,
     * we make use of a theorem which states that any r x n latin
     * rectangle, with r < n, can be extended into an (r+1) x n
     * latin rectangle. In other words, we can reliably generate a
     * latin square row by row, by at every stage writing down any
     * row at all which doesn't conflict with previous rows, and
     * the theorem guarantees that we will never have to backtrack.
     *
     * To find a viable row at each stage, we can make use of the
     * support functions in maxflow.c.
     */

    sq = snewn(o*o, digit);

    /*
     * In case this method of generation introduces a really subtle
     * top-to-bottom directional bias, we'll generate the rows in
     * random order.
     */
    row = snewn(o, digit);
    col = snewn(o, digit);
    numinv = snewn(o, digit);
    num = snewn(o, digit);
    for (i = 0; i < o; i++)
        row[i] = i;
    shuffle(row, i, sizeof(*row), rs);

    /*
     * Set up the infrastructure for the maxflow algorithm.
     */
    scratchsize = maxflow_scratch_size(o * 2 + 2);
    scratch = smalloc(scratchsize);
    backedges = snewn(o*o + 2*o, int);
    edges = snewn((o*o + 2*o) * 2, int);
    capacity = snewn(o*o + 2*o, int);
    flow = snewn(o*o + 2*o, int);
    /* Set up the edge array, and the initial capacities. */
    ne = 0;
    for (i = 0; i < o; i++) {
        /* Each LHS vertex is connected to all RHS vertices. */
        for (j = 0; j < o; j++) {
            edges[ne*2] = i;
            edges[ne*2+1] = j+o;
            /* capacity for this edge is set later on */
            ne++;
        }
    }
    for (i = 0; i < o; i++) {
        /* Each RHS vertex is connected to the distinguished sink vertex. */
        edges[ne*2] = i+o;
        edges[ne*2+1] = o*2+1;
        capacity[ne] = 1;
        ne++;
    }
    for (i = 0; i < o; i++) {
        /* And the distinguished source vertex connects to each LHS vertex. */
        edges[ne*2] = o*2;
        edges[ne*2+1] = i;
        capacity[ne] = 1;
        ne++;
    }
    assert(ne == o*o + 2*o);
    /* Now set up backedges. */
    maxflow_setup_backedges(ne, edges, backedges);
    
    /*
     * Now generate each row of the latin square.
     */
    for (i = 0; i < o; i++) {
        /*
         * To prevent maxflow from behaving deterministically, we
         * separately permute the columns and the digits for the
         * purposes of the algorithm, differently for every row.
         */
        for (j = 0; j < o; j++)
            col[j] = num[j] = j;
        shuffle(col, j, sizeof(*col), rs);
        shuffle(num, j, sizeof(*num), rs);
        /* We need the num permutation in both forward and inverse forms. */
        for (j = 0; j < o; j++)
            numinv[num[j]] = j;

        /*
         * Set up the capacities for the maxflow run, by examining
         * the existing latin square.
         */
        for (j = 0; j < o*o; j++)
            capacity[j] = 1;
        for (j = 0; j < i; j++)
            for (k = 0; k < o; k++) {
                int n = num[sq[row[j]*o + col[k]] - 1];
                capacity[k*o+n] = 0;
            }

        /*
         * Run maxflow.
         */
        j = maxflow_with_scratch(scratch, o*2+2, 2*o, 2*o+1, ne,
                                 edges, backedges, capacity, flow, NULL);
        assert(j == o);   /* by the above theorem, this must have succeeded */

        /*
         * And examine the flow array to pick out the new row of
         * the latin square.
         */
        for (j = 0; j < o; j++) {
            for (k = 0; k < o; k++) {
                if (flow[j*o+k])
                    break;
            }
            assert(k < o);
            sq[row[i]*o + col[j]] = numinv[k] + 1;
        }
    }

    /*
     * Done. Free our internal workspaces...
     */
    sfree(flow);
    sfree(capacity);
    sfree(edges);
    sfree(backedges);
    sfree(scratch);
    sfree(numinv);
    sfree(num);
    sfree(col);
    sfree(row);

    /*
     * ... and return our completed latin square.
     */
    return sq;
}

/* --------------------------------------------------------
 * Checking.
 */

typedef struct lcparams {
    digit elt;
    int count;
} lcparams;

static int latin_check_cmp(void *v1, void *v2)
{
    lcparams *lc1 = (lcparams *)v1;
    lcparams *lc2 = (lcparams *)v2;

    if (lc1->elt < lc2->elt) return -1;
    if (lc1->elt > lc2->elt) return 1;
    return 0;
}

#define ELT(sq,x,y) (sq[((y)*order)+(x)])

/* returns non-zero if sq is not a latin square. */
int latin_check(digit *sq, int order)
{
    tree234 *dict = newtree234(latin_check_cmp);
    int c, r;
    int ret = 0;
    lcparams *lcp, lc, *aret;

    /* Use a tree234 as a simple hash table, go through the square
     * adding elements as we go or incrementing their counts. */
    for (c = 0; c < order; c++) {
        for (r = 0; r < order; r++) {
            lc.elt = ELT(sq, c, r); lc.count = 0;
            lcp = find234(dict, &lc, NULL);
            if (!lcp) {
                lcp = snew(lcparams);
                lcp->elt = ELT(sq, c, r);
                lcp->count = 1;
                aret = add234(dict, lcp);
                assert(aret == lcp);
            } else {
                lcp->count++;
            }
        }
    }

    /* There should be precisely 'order' letters in the alphabet,
     * each occurring 'order' times (making the OxO tree) */
    if (count234(dict) != order) ret = 1;
    else {
        for (c = 0; (lcp = index234(dict, c)) != NULL; c++) {
            if (lcp->count != order) ret = 1;
        }
    }
    for (c = 0; (lcp = index234(dict, c)) != NULL; c++)
        sfree(lcp);
    freetree234(dict);

    return ret;
}


/* --------------------------------------------------------
 * Testing (and printing).
 */

#ifdef STANDALONE_LATIN_TEST

#include <stdio.h>
#include <time.h>

const char *quis;

static void latin_print(digit *sq, int order)
{
    int x, y;

    for (y = 0; y < order; y++) {
        for (x = 0; x < order; x++) {
            printf("%2u ", ELT(sq, x, y));
        }
        printf("\n");
    }
    printf("\n");
}

static void gen(int order, random_state *rs, int debug)
{
    digit *sq;

    solver_show_working = debug;

    sq = latin_generate(order, rs);
    latin_print(sq, order);
    if (latin_check(sq, order)) {
        fprintf(stderr, "Square is not a latin square!");
        exit(1);
    }

    sfree(sq);
}

void test_soak(int order, random_state *rs)
{
    digit *sq;
    int n = 0;
    time_t tt_start, tt_now, tt_last;

    solver_show_working = 0;
    tt_now = tt_start = time(NULL);

    while(1) {
        sq = latin_generate(order, rs);
        sfree(sq);
        n++;

        tt_last = time(NULL);
        if (tt_last > tt_now) {
            tt_now = tt_last;
            printf("%d total, %3.1f/s\n", n,
                   (double)n / (double)(tt_now - tt_start));
        }
    }
}

void usage_exit(const char *msg)
{
    if (msg)
        fprintf(stderr, "%s: %s\n", quis, msg);
    fprintf(stderr, "Usage: %s [--seed SEED] --soak <params> | [game_id [game_id ...]]\n", quis);
    exit(1);
}

int main(int argc, char *argv[])
{
    int i, soak = 0;
    random_state *rs;
    time_t seed = time(NULL);

    quis = argv[0];
    while (--argc > 0) {
        const char *p = *++argv;
        if (!strcmp(p, "--soak"))
            soak = 1;
        else if (!strcmp(p, "--seed")) {
            if (argc == 0)
                usage_exit("--seed needs an argument");
            seed = (time_t)atoi(*++argv);
            argc--;
        } else if (*p == '-')
                usage_exit("unrecognised option");
        else
            break; /* finished options */
    }

    rs = random_new((void*)&seed, sizeof(time_t));

    if (soak == 1) {
        if (argc != 1) usage_exit("only one argument for --soak");
        test_soak(atoi(*argv), rs);
    } else {
        if (argc > 0) {
            for (i = 0; i < argc; i++) {
                gen(atoi(*argv++), rs, 1);
            }
        } else {
            while (1) {
                i = random_upto(rs, 20) + 1;
                gen(i, rs, 0);
            }
        }
    }
    random_free(rs);
    return 0;
}

#endif

/* vim: set shiftwidth=4 tabstop=8: */