mincross.c

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/*************************************************************************
 * Copyright (c) 2011 AT&T Intellectual Property 
 * All rights reserved. This program and the accompanying materials
 * are made available under the terms of the Eclipse Public License v1.0
 * which accompanies this distribution, and is available at
 * https://www.eclipse.org/legal/epl-v10.html
 *
 * Contributors: Details at https://graphviz.org
 *************************************************************************/
 
 
/* 
 * dot_mincross(g) takes a ranked graphs, and finds an ordering
 * that avoids edge crossings.  clusters are expanded.
 * N.B. the rank structure is global (not allocated per cluster)
 * because mincross may compare nodes in different clusters.
 */
 
#include <assert.h>
#include <cgraph/cgraph.h>
#include <dotgen/dot.h>
#include <inttypes.h>
#include <limits.h>
#include <stdbool.h>
#include <stdint.h>
#include <stdlib.h>
#include <string.h>
#include <util/alloc.h>
#include <util/bitarray.h>
#include <util/exit.h>
#include <util/gv_math.h>
#include <util/itos.h>
#include <util/list.h>
#include <util/streq.h>
 
struct adjmatrix_t {
  size_t nrows;
  size_t ncols;
  uint8_t *data;    
  size_t allocated; 
};
 
static bool matrix_get(adjmatrix_t *me, size_t row, size_t col) {
  assert(me != NULL);
 
  // if this index is beyond anything set, infer it as unset
  const size_t index = row * me->ncols + col;
  const size_t byte_index = index / 8;
  const size_t bit_index = index % 8;
  if (byte_index >= me->allocated) {
    return false;
  }
 
  return (me->data[byte_index] >> bit_index) & 1;
}
 
static void matrix_set(adjmatrix_t *me, size_t row, size_t col) {
  assert(me != NULL);
 
  // if we are updating beyond allocated space, expand the backing store
  const size_t index = row * me->ncols + col;
  const size_t byte_index = index / 8;
  const size_t bit_index = index % 8;
  if (byte_index >= me->allocated) {
    me->data = gv_recalloc(me->data, me->allocated, byte_index + 1,
                           sizeof(me->data[0]));
    me->allocated = byte_index + 1;
  }
 
  me->data[byte_index] |= (uint8_t)(UINT8_C(1) << bit_index);
}
 
/* #define DEBUG */
#define MARK(v)         (ND_mark(v))
#define saveorder(v)    (ND_coord(v)).x
#define flatindex(v)    ((size_t)ND_low(v))
 
        /* forward declarations */
static bool medians(graph_t * g, int r0, int r1);
static int nodeposcmpf(const void *, const void *);
static int edgeidcmpf(const void *, const void *);
static void flat_breakcycles(graph_t * g);
static void flat_reorder(graph_t * g);
static void flat_search(graph_t * g, node_t * v);
static void init_mincross(graph_t * g);
static void merge2(graph_t * g);
static void init_mccomp(graph_t *g, size_t c);
static void cleanup2(graph_t *g, int64_t nc);
static int64_t mincross_clust(graph_t *g);
static int64_t mincross(graph_t *g, int startpass);
static void mincross_step(graph_t * g, int pass);
static void mincross_options(graph_t * g);
static void save_best(graph_t * g);
static void restore_best(graph_t * g);
static adjmatrix_t *new_matrix(size_t i, size_t j);
static void free_matrix(adjmatrix_t * p);
static int ordercmpf(const void *, const void *);
static int64_t ncross(void);
#ifdef DEBUG
void check_rs(graph_t * g, int null_ok);
void check_order(void);
void check_vlists(graph_t * g);
void node_in_root_vlist(node_t * n);
#endif
 
 
        /* mincross parameters */
static int MinQuit;
static const double Convergence = .995;
 
static graph_t *Root;
static int GlobalMinRank, GlobalMaxRank;
static edge_t **TE_list;
static int *TI_list;
static bool ReMincross;
 
#if defined(DEBUG) && DEBUG > 1
static void indent(graph_t* g)
{
  if (g->parent) {
    fprintf (stderr, "  ");
    indent(g->parent);
  }
}
 
static void nname(node_t *v, FILE *stream) {
        if (ND_node_type(v)) {
                if (ND_ranktype(v) == CLUSTER)
                        fprintf(stream, "v%s_%p", agnameof(ND_clust(v)), v);
                else
                        fprintf(stream, "v_%p", v);
        } else
                fputs(agnameof(v), stream);
}
static void dumpg (graph_t* g)
{
    int j, i, r;
    node_t* v;
    edge_t* e;
 
    fprintf (stderr, "digraph A {\n");
    for (r = GD_minrank(g); r <= GD_maxrank(g); r++) {
        fprintf (stderr, "  subgraph {rank=same  ");
        for (i = 0; i < GD_rank(g)[r].n; i++) {
          v = GD_rank(g)[r].v[i];
          if (i > 0)
            fputs(" -> ", stderr);
          nname(v, stderr);
        }
        if (i > 1) fprintf (stderr, " [style=invis]}\n");
        else fprintf (stderr, " }\n");
    }
    for (r = GD_minrank(g); r < GD_maxrank(g); r++) {
        for (i = 0; i < GD_rank(g)[r].n; i++) {
          v = GD_rank(g)[r].v[i];
          for (j = 0; (e = ND_out(v).list[j]); j++) {
             nname(v, stderr);
             fputs(" -> ", stderr);
             nname(aghead(e), stderr);
             fputc('\n', stderr);
          }
        }
    }
    fprintf (stderr, "}\n");
}
static void dumpr (graph_t* g, int edges)
{
    int j, i, r;
    node_t* v;
    edge_t* e;
 
    for (r = GD_minrank(g); r <= GD_maxrank(g); r++) {
        fprintf (stderr, "[%d] ", r);
        for (i = 0; i < GD_rank(g)[r].n; i++) {
          v = GD_rank(g)[r].v[i];
          nname(v, stderr);
          fprintf(stderr, "(%.02f,%d) ", saveorder(v),ND_order(v));
        }
        fprintf (stderr, "\n");
    }
    if (edges == 0) return;
    for (r = GD_minrank(g); r < GD_maxrank(g); r++) {
        for (i = 0; i < GD_rank(g)[r].n; i++) {
          v = GD_rank(g)[r].v[i];
          for (j = 0; (e = ND_out(v).list[j]); j++) {
             nname(v, stderr);
             fputs(" -> ", stderr);
             nname(aghead(e), stderr);
             fputc('\n', stderr);
          }
        }
    }
}
#endif
 
typedef struct {
    Agrec_t h;
    int x, lo, hi;
    Agnode_t* np;
} info_t;
 
#define ND_x(n) (((info_t*)AGDATA(n))->x)
#define ND_lo(n) (((info_t*)AGDATA(n))->lo)
#define ND_hi(n) (((info_t*)AGDATA(n))->hi)
#define ND_np(n) (((info_t*)AGDATA(n))->np)
#define ND_idx(n) (ND_order(ND_np(n)))
 
static void
emptyComp (graph_t* sg)
{
    Agnode_t* n;
    Agnode_t* nxt;
 
    for (n = agfstnode(sg); n; n = nxt) {
        nxt = agnxtnode (sg, n);
        agdelnode(sg,n);
    }
}
 
#define isBackedge(e) (ND_idx(aghead(e)) > ND_idx(agtail(e)))
 
static Agnode_t*
findSource (Agraph_t* g, Agraph_t* sg)
{
    Agnode_t* n;
 
    for (n = agfstnode(sg); n; n = agnxtnode(sg, n))
        if (agdegree(g,n,1,0) == 0) return n;
    return NULL;
}
 
static int
topsort (Agraph_t* g, Agraph_t* sg, Agnode_t** arr)
{
    Agnode_t* n;
    Agedge_t* e;
    Agedge_t* nxte;
    int cnt = 0;
 
    while ((n = findSource(g, sg))) {
        arr[cnt++] = ND_np(n); 
        agdelnode(sg, n);
        for (e = agfstout(g, n); e; e = nxte) {
            nxte = agnxtout(g, e); 
            agdeledge(g, e);
        }
    }
    return cnt;
}
 
static int
getComp (graph_t* g, node_t* n, graph_t* comp, int* indices)
{
    int backedge = 0;
    Agedge_t* e;
 
    ND_x(n) = 1;
    indices[agnnodes(comp)] = ND_idx(n);
    agsubnode(comp, n, 1);
    for (e = agfstout(g,n); e; e = agnxtout(g,e)) {
        if (isBackedge(e)) backedge++;
        if (!ND_x(aghead(e)))
            backedge += getComp(g, aghead(e), comp, indices);
    }
    for (e = agfstin(g,n); e; e = agnxtin(g,e)) {
        if (isBackedge(e)) backedge++;
        if (!ND_x(agtail(e)))
            backedge += getComp(g, agtail(e), comp, indices);
    }
    return backedge;
}
 
static void
fixLabelOrder (graph_t* g, rank_t* rk)
{
    int cnt;
    bool haveBackedge = false;
    Agraph_t* sg;
    Agnode_t* n;
    Agnode_t* nxtp;
    Agnode_t* v;
 
    for (n = agfstnode(g); n; n = nxtp) {
        v = nxtp = agnxtnode(g, n);
        for (; v; v = agnxtnode(g, v)) {
            if (ND_hi(v) <= ND_lo(n)) { 
                haveBackedge = true;
                agedge(g, v, n, NULL, 1);
            }
            else if (ND_hi(n) <= ND_lo(v)) { 
                agedge(g, n, v, NULL, 1);
            }
        }
    }
    if (!haveBackedge) return;
    
    sg = agsubg(g, "comp", 1);
    Agnode_t **arr = gv_calloc(agnnodes(g), sizeof(Agnode_t*));
    int *indices = gv_calloc(agnnodes(g), sizeof(int));
 
    for (n = agfstnode(g); n; n = agnxtnode(g,n)) {
        if (ND_x(n) || agdegree(g,n,1,1) == 0) continue;
        if (getComp(g, n, sg, indices)) {
            int i, sz = agnnodes(sg);
            cnt = topsort (g, sg, arr);
            assert (cnt == sz);
            qsort(indices, cnt, sizeof(int), ordercmpf);
            for (i = 0; i < sz; i++) {
                ND_order(arr[i]) = indices[i];
                rk->v[indices[i]] = arr[i];
            }
       }
       emptyComp(sg);
    }
    free(indices);
    free (arr);
}
 
/* Check that the ordering of labels for flat edges is consistent. 
 * This is necessary because dot_position will attempt to force the label
 * to be between the edge's vertices. This can lead to an infeasible problem.
 *
 * We check each rank for any flat edge labels (as dummy nodes) and create a
 * graph with a node for each label. If the graph contains more than 1 node, we
 * call fixLabelOrder to see if there really is a problem and, if so, fix it.
 */ 
void
checkLabelOrder (graph_t* g)
{
    int j, r, lo, hi;
    graph_t* lg = NULL;
    rank_t* rk;
    Agnode_t* u;
    Agnode_t* n;
    Agedge_t* e;
 
    for (r = GD_minrank(g); r <= GD_maxrank(g); r++) {
        rk = GD_rank(g)+r;
        for (j = 0; j < rk->n; j++) {
            u = rk->v[j];
            if ((e = ND_alg(u))) {
                if (!lg) lg = agopen ("lg", Agstrictdirected, 0);
                n = agnode(lg, ITOS(j), 1);
                agbindrec(n, "info", sizeof(info_t), true);
                lo = ND_order(aghead(ND_out(u).list[0]));
                hi = ND_order(aghead(ND_out(u).list[1]));
                if (lo > hi) {
                    SWAP(&lo, &hi);
                }
                ND_lo(n) = lo;
                ND_hi(n) = hi;
                ND_np(n) = u;
            }
        }
        if (lg) {
            if (agnnodes(lg) > 1) fixLabelOrder (lg, rk);
            agclose(lg);
            lg = NULL;
        }
    }
}
 
/* Minimize edge crossings
 * Note that nodes are not placed into GD_rank(g) until mincross()
 * is called.
 */
int dot_mincross(graph_t *g) {
    int64_t nc;
    char *s;
 
    /* check whether malformed input has led to empty cluster that the crossing
     * functions will not anticipate
     */
    {
        size_t i;
        for (i = 1; i <= (size_t)GD_n_cluster(g); ) {
            if (agfstnode(GD_clust(g)[i]) == NULL) {
              agwarningf("removing empty cluster\n");
              memmove(&GD_clust(g)[i], &GD_clust(g)[i + 1],
                ((size_t)GD_n_cluster(g) - i) * sizeof(GD_clust(g)[0]));
              --GD_n_cluster(g);
            } else {
              ++i;
            }
        }
    }
 
    init_mincross(g);
 
    size_t comp;
    for (nc = 0, comp = 0; comp < GD_comp(g).size; comp++) {
        init_mccomp(g, comp);
        const int64_t mc = mincross(g, 0);
        if (mc < 0) {
            return -1;
        }
        nc += mc;
    }
 
    merge2(g);
 
    /* run mincross on contents of each cluster */
    for (int c = 1; c <= GD_n_cluster(g); c++) {
        const int64_t mc = mincross_clust(GD_clust(g)[c]);
        if (mc < 0) {
            return -1;
        }
        nc += mc;
#ifdef DEBUG
        check_vlists(GD_clust(g)[c]);
        check_order();
#endif
    }
 
    if (GD_n_cluster(g) > 0 && (!(s = agget(g, "remincross")) || mapbool(s))) {
        mark_lowclusters(g);
        ReMincross = true;
        const int64_t mc = mincross(g, 2);
        if (mc < 0) {
            return -1;
        }
        nc = mc;
#ifdef DEBUG
        for (int c = 1; c <= GD_n_cluster(g); c++)
            check_vlists(GD_clust(g)[c]);
#endif
    }
    cleanup2(g, nc);
    return 0;
}
 
static adjmatrix_t *new_matrix(size_t i, size_t j) {
    adjmatrix_t *rv = gv_alloc(sizeof(adjmatrix_t));
    rv->nrows = i;
    rv->ncols = j;
    return rv;
}
 
static void free_matrix(adjmatrix_t * p)
{
    if (p) {
        free(p->data);
        free(p);
    }
}
 
static void init_mccomp(graph_t *g, size_t c) {
    int r;
 
    GD_nlist(g) = GD_comp(g).list[c];
    if (c > 0) {
        for (r = GD_minrank(g); r <= GD_maxrank(g); r++) {
            GD_rank(g)[r].v = GD_rank(g)[r].v + GD_rank(g)[r].n;
            GD_rank(g)[r].n = 0;
        }
    }
}
 
static int betweenclust(edge_t * e)
{
    while (ED_to_orig(e))
        e = ED_to_orig(e);
    return (ND_clust(agtail(e)) != ND_clust(aghead(e)));
}
 
static void do_ordering_node(graph_t *g, node_t *n, bool outflag) {
    int i, ne;
    node_t *u, *v;
    edge_t *e, *f, *fe;
    edge_t **sortlist = TE_list;
 
    if (ND_clust(n))
        return;
    if (outflag) {
        for (i = ne = 0; (e = ND_out(n).list[i]); i++)
            if (!betweenclust(e))
                sortlist[ne++] = e;
    } else {
        for (i = ne = 0; (e = ND_in(n).list[i]); i++)
            if (!betweenclust(e))
                sortlist[ne++] = e;
    }
    if (ne <= 1)
        return;
    /* write null terminator at end of list.
       requires +1 in TE_list alloccation */
    sortlist[ne] = 0;
    qsort(sortlist, ne, sizeof(sortlist[0]), edgeidcmpf);
    for (ne = 1; (f = sortlist[ne]); ne++) {
        e = sortlist[ne - 1];
        if (outflag) {
            u = aghead(e);
            v = aghead(f);
        } else {
            u = agtail(e);
            v = agtail(f);
        }
        if (find_flat_edge(u, v))
            return;
        fe = new_virtual_edge(u, v, NULL);
        ED_edge_type(fe) = FLATORDER;
        flat_edge(g, fe);
    }
}
 
static void do_ordering(graph_t *g, bool outflag) {
    /* Order all nodes in graph */
    node_t *n;
 
    for (n = agfstnode(g); n; n = agnxtnode(g, n)) {
        do_ordering_node (g, n, outflag);
    }
}
 
static void do_ordering_for_nodes(graph_t * g)
{
    /* Order nodes which have the "ordered" attribute */
    node_t *n;
    const char *ordering;
 
    for (n = agfstnode(g); n; n = agnxtnode(g, n)) {
        if ((ordering = late_string(n, N_ordering, NULL))) {
            if (streq(ordering, "out"))
                do_ordering_node(g, n, true);
            else if (streq(ordering, "in"))
                do_ordering_node(g, n, false);
            else if (ordering[0])
                agerrorf("ordering '%s' not recognized for node '%s'.\n", ordering, agnameof(n));
        }
    }
}
 
/* handle case where graph specifies edge ordering
 * If the graph does not have an ordering attribute, we then
 * check for nodes having the attribute.
 * Note that, in this implementation, the value of G_ordering
 * dominates the value of N_ordering.
 */
static void ordered_edges(graph_t * g)
{
    char *ordering;
 
    if (!G_ordering && !N_ordering)
        return;
    if ((ordering = late_string(g, G_ordering, NULL))) {
        if (streq(ordering, "out"))
            do_ordering(g, true);
        else if (streq(ordering, "in"))
            do_ordering(g, false);
        else if (ordering[0])
            agerrorf("ordering '%s' not recognized.\n", ordering);
    }
    else
    {
        graph_t *subg;
 
        for (subg = agfstsubg(g); subg; subg = agnxtsubg(subg)) {
            /* clusters are processed by separate calls to ordered_edges */
            if (!is_cluster(subg))
                ordered_edges(subg);
        }
        if (N_ordering) do_ordering_for_nodes (g);
    }
}
 
static int64_t mincross_clust(graph_t *g) {
    int c;
 
    if (expand_cluster(g) != 0) {
        return -1;
    }
    ordered_edges(g);
    flat_breakcycles(g);
    flat_reorder(g);
    int64_t nc = mincross(g, 2);
    if (nc < 0) {
        return nc;
    }
 
    for (c = 1; c <= GD_n_cluster(g); c++) {
        const int64_t mc = mincross_clust(GD_clust(g)[c]);
        if (mc < 0) {
            return mc;
        }
        nc += mc;
    }
 
    save_vlist(g);
    return nc;
}
 
static bool left2right(graph_t *g, node_t *v, node_t *w) {
    /* CLUSTER indicates orig nodes of clusters, and vnodes of skeletons */
    if (!ReMincross) {
        if (ND_clust(v) != ND_clust(w) && ND_clust(v) && ND_clust(w)) {
            /* the following allows cluster skeletons to be swapped */
            if (ND_ranktype(v) == CLUSTER && ND_node_type(v) == VIRTUAL)
                return false;
            if (ND_ranktype(w) == CLUSTER && ND_node_type(w) == VIRTUAL)
                return false;
            return true;
        }
    } else {
        if (ND_clust(v) != ND_clust(w))
            return true;
    }
    adjmatrix_t *const M = GD_rank(g)[ND_rank(v)].flat;
    if (M == NULL)
        return false;
    if (GD_flip(g)) {
        SWAP(&v, &w);
    }
    return matrix_get(M, (size_t)flatindex(v), (size_t)flatindex(w));
}
 
static int64_t in_cross(node_t *v, node_t *w) {
    edge_t **e1, **e2;
    int inv, t;
    int64_t cross = 0;
 
    for (e2 = ND_in(w).list; *e2; e2++) {
        int cnt = ED_xpenalty(*e2);             
                
        inv = ND_order(agtail(*e2));
 
        for (e1 = ND_in(v).list; *e1; e1++) {
            t = ND_order(agtail(*e1)) - inv;
            if (t > 0 || (t == 0 && ED_tail_port(*e1).p.x > ED_tail_port(*e2).p.x))
                cross += ED_xpenalty(*e1) * cnt;
        }
    }
    return cross;
}
 
static int out_cross(node_t * v, node_t * w)
{
    edge_t **e1, **e2;
    int inv, cross = 0, t;
 
    for (e2 = ND_out(w).list; *e2; e2++) {
        int cnt = ED_xpenalty(*e2);
        inv = ND_order(aghead(*e2));
 
        for (e1 = ND_out(v).list; *e1; e1++) {
            t = ND_order(aghead(*e1)) - inv;
            if (t > 0 || (t == 0 && (ED_head_port(*e1)).p.x > (ED_head_port(*e2)).p.x))
                cross += ED_xpenalty(*e1) * cnt;
        }
    }
    return cross;
 
}
 
static void exchange(node_t * v, node_t * w)
{
    int vi, wi, r;
 
    r = ND_rank(v);
    vi = ND_order(v);
    wi = ND_order(w);
    ND_order(v) = wi;
    GD_rank(Root)[r].v[wi] = v;
    ND_order(w) = vi;
    GD_rank(Root)[r].v[vi] = w;
}
 
static int64_t transpose_step(graph_t *g, int r, bool reverse) {
    int i;
    node_t *v, *w;
 
    int64_t rv = 0;
    GD_rank(g)[r].candidate = false;
    for (i = 0; i < GD_rank(g)[r].n - 1; i++) {
        v = GD_rank(g)[r].v[i];
        w = GD_rank(g)[r].v[i + 1];
        assert(ND_order(v) < ND_order(w));
        if (left2right(g, v, w))
            continue;
        int64_t c0 = 0;
        int64_t c1 = 0;
        if (r > 0) {
            c0 += in_cross(v, w);
            c1 += in_cross(w, v);
        }
        if (GD_rank(g)[r + 1].n > 0) {
            c0 += out_cross(v, w);
            c1 += out_cross(w, v);
        }
        if (c1 < c0 || (c0 > 0 && reverse && c1 == c0)) {
            exchange(v, w);
            rv += c0 - c1;
            GD_rank(Root)[r].valid = false;
            GD_rank(g)[r].candidate = true;
 
            if (r > GD_minrank(g)) {
                GD_rank(Root)[r - 1].valid = false;
                GD_rank(g)[r - 1].candidate = true;
            }
            if (r < GD_maxrank(g)) {
                GD_rank(Root)[r + 1].valid = false;
                GD_rank(g)[r + 1].candidate = true;
            }
        }
    }
    return rv;
}
 
static void transpose(graph_t * g, bool reverse)
{
    int r;
 
    for (r = GD_minrank(g); r <= GD_maxrank(g); r++)
        GD_rank(g)[r].candidate = true;
    int64_t delta;
    do {
        delta = 0;
        for (r = GD_minrank(g); r <= GD_maxrank(g); r++) {
            if (GD_rank(g)[r].candidate) {
                delta += transpose_step(g, r, reverse);
            }
        }
    } while (delta >= 1);
}
 
static int64_t mincross(graph_t *g, int startpass) {
    const int endpass = 2;
    int maxthispass = 0, iter, trying, pass;
    int64_t cur_cross, best_cross;
 
    if (startpass > 1) {
        cur_cross = best_cross = ncross();
        save_best(g);
    } else
        cur_cross = best_cross = INT64_MAX;
    for (pass = startpass; pass <= endpass; pass++) {
        if (pass <= 1) {
            maxthispass = MIN(4, MaxIter);
            if (g == dot_root(g))
                if (build_ranks(g, pass) != 0) {
                    return -1;
                }
            if (pass == 0)
                flat_breakcycles(g);
            flat_reorder(g);
 
            if ((cur_cross = ncross()) <= best_cross) {
                save_best(g);
                best_cross = cur_cross;
            }
        } else {
            maxthispass = MaxIter;
            if (cur_cross > best_cross)
                restore_best(g);
            cur_cross = best_cross;
        }
        trying = 0;
        for (iter = 0; iter < maxthispass; iter++) {
            if (Verbose)
                fprintf(stderr,
                        "mincross: pass %d iter %d trying %d cur_cross %" PRId64 " best_cross %"
                        PRId64 "\n",
                        pass, iter, trying, cur_cross, best_cross);
            if (trying++ >= MinQuit)
                break;
            if (cur_cross == 0)
                break;
            mincross_step(g, iter);
            if ((cur_cross = ncross()) <= best_cross) {
                save_best(g);
                if (cur_cross < Convergence * (double)best_cross)
                    trying = 0;
                best_cross = cur_cross;
            }
        }
        if (cur_cross == 0)
            break;
    }
    if (cur_cross > best_cross)
        restore_best(g);
    if (best_cross > 0) {
        transpose(g, false);
        best_cross = ncross();
    }
 
    return best_cross;
}
 
static void restore_best(graph_t * g)
{
    node_t *n;
    int i, r;
 
    for (r = GD_minrank(g); r <= GD_maxrank(g); r++) {
        for (i = 0; i < GD_rank(g)[r].n; i++) {
            n = GD_rank(g)[r].v[i];
            ND_order(n) = saveorder(n);
        }
    }
    for (r = GD_minrank(g); r <= GD_maxrank(g); r++) {
        GD_rank(Root)[r].valid = false;
        qsort(GD_rank(g)[r].v, GD_rank(g)[r].n, sizeof(GD_rank(g)[0].v[0]),
              nodeposcmpf);
    }
}
 
static void save_best(graph_t * g)
{
    node_t *n;
    int i, r;
    for (r = GD_minrank(g); r <= GD_maxrank(g); r++) {
        for (i = 0; i < GD_rank(g)[r].n; i++) {
            n = GD_rank(g)[r].v[i];
            saveorder(n) = ND_order(n);
        }
    }
}
 
/* merges the connected components of g */
static void merge_components(graph_t * g)
{
    node_t *u, *v;
 
    if (GD_comp(g).size <= 1)
        return;
    u = NULL;
    for (size_t c = 0; c < GD_comp(g).size; c++) {
        v = GD_comp(g).list[c];
        if (u)
            ND_next(u) = v;
        ND_prev(v) = u;
        while (ND_next(v)) {
            v = ND_next(v);
        }
        u = v;
    }
    GD_comp(g).size = 1;
    GD_nlist(g) = GD_comp(g).list[0];
    GD_minrank(g) = GlobalMinRank;
    GD_maxrank(g) = GlobalMaxRank;
}
 
/* merge connected components, create globally consistent rank lists */
static void merge2(graph_t * g)
{
    int i, r;
    node_t *v;
 
    /* merge the components and rank limits */
    merge_components(g);
 
    /* install complete ranks */
    for (r = GD_minrank(g); r <= GD_maxrank(g); r++) {
        GD_rank(g)[r].n = GD_rank(g)[r].an;
        GD_rank(g)[r].v = GD_rank(g)[r].av;
        for (i = 0; i < GD_rank(g)[r].n; i++) {
            v = GD_rank(g)[r].v[i];
            if (v == NULL) {
                if (Verbose)
                    fprintf(stderr,
                            "merge2: graph %s, rank %d has only %d < %d nodes\n",
                            agnameof(g), r, i, GD_rank(g)[r].n);
                GD_rank(g)[r].n = i;
                break;
            }
            ND_order(v) = i;
        }
    }
}
 
static void cleanup2(graph_t *g, int64_t nc) {
    int i, j, r, c;
    node_t *v;
    edge_t *e;
 
    if (TI_list) {
        free(TI_list);
        TI_list = NULL;
    }
    if (TE_list) {
        free(TE_list);
        TE_list = NULL;
    }
    /* fix vlists of clusters */
    for (c = 1; c <= GD_n_cluster(g); c++)
        rec_reset_vlists(GD_clust(g)[c]);
 
    /* remove node temporary edges for ordering nodes */
    for (r = GD_minrank(g); r <= GD_maxrank(g); r++) {
        for (i = 0; i < GD_rank(g)[r].n; i++) {
            v = GD_rank(g)[r].v[i];
            ND_order(v) = i;
            if (ND_flat_out(v).list) {
                for (j = 0; (e = ND_flat_out(v).list[j]); j++)
                    if (ED_edge_type(e) == FLATORDER) {
                        delete_flat_edge(e);
                        free(e->base.data);
                        free(e);
                        j--;
                    }
            }
        }
        free_matrix(GD_rank(g)[r].flat);
    }
    if (Verbose)
        fprintf(stderr, "mincross %s: %" PRId64 " crossings, %.2f secs.\n",
                agnameof(g), nc, elapsed_sec());
}
 
static node_t *neighbor(node_t * v, int dir)
{
    node_t *rv = NULL;
assert(v);
    if (dir < 0) {
        if (ND_order(v) > 0)
            rv = GD_rank(Root)[ND_rank(v)].v[ND_order(v) - 1];
    } else
        rv = GD_rank(Root)[ND_rank(v)].v[ND_order(v) + 1];
assert(rv == 0 || (ND_order(rv)-ND_order(v))*dir > 0);
    return rv;
}
 
static bool is_a_normal_node_of(graph_t *g, node_t *v) {
    return ND_node_type(v) == NORMAL && agcontains(g, v);
}
 
static bool is_a_vnode_of_an_edge_of(graph_t *g, node_t *v) {
    if (ND_node_type(v) == VIRTUAL
        && ND_in(v).size == 1 && ND_out(v).size == 1) {
        edge_t *e = ND_out(v).list[0];
        while (ED_edge_type(e) != NORMAL)
            e = ED_to_orig(e);
        if (agcontains(g, e))
            return true;
    }
    return false;
}
 
static bool inside_cluster(graph_t *g, node_t *v) {
  return is_a_normal_node_of(g, v) || is_a_vnode_of_an_edge_of(g, v);
}
 
static node_t *furthestnode(graph_t * g, node_t * v, int dir)
{
    node_t *rv = v;
    for (node_t *u = v; (u = neighbor(u, dir)); ) {
        if (is_a_normal_node_of(g, u))
            rv = u;
        else if (is_a_vnode_of_an_edge_of(g, u))
            rv = u;
    }
    return rv;
}
 
void save_vlist(graph_t * g)
{
    int r;
 
    if (GD_rankleader(g))
        for (r = GD_minrank(g); r <= GD_maxrank(g); r++) {
            GD_rankleader(g)[r] = GD_rank(g)[r].v[0];
        }
}
 
void rec_save_vlists(graph_t * g)
{
    int c;
 
    save_vlist(g);
    for (c = 1; c <= GD_n_cluster(g); c++)
        rec_save_vlists(GD_clust(g)[c]);
}
 
 
void rec_reset_vlists(graph_t * g)
{
    // fix vlists of sub-clusters
    for (int c = 1; c <= GD_n_cluster(g); c++)
        rec_reset_vlists(GD_clust(g)[c]);
 
    if (GD_rankleader(g))
        for (int r = GD_minrank(g); r <= GD_maxrank(g); r++) {
            node_t *const v = GD_rankleader(g)[r];
            if (v == NULL) {
                continue;
            }
#ifdef DEBUG
            node_in_root_vlist(v);
#endif
            node_t *const u = furthestnode(g, v, -1);
            node_t *const w = furthestnode(g, v, 1);
            GD_rankleader(g)[r] = u;
#ifdef DEBUG
            assert(GD_rank(dot_root(g))[r].v[ND_order(u)] == u);
#endif
            GD_rank(g)[r].v = GD_rank(dot_root(g))[r].v + ND_order(u);
            GD_rank(g)[r].n = ND_order(w) - ND_order(u) + 1;
        }
}
 
/* The structures in crossing minimization and positioning require
 * that clusters have some node on each rank. This function recursively
 * guarantees this property. It takes into account nodes and edges in
 * a cluster, the latter causing dummy nodes for intervening ranks.
 * For any rank without node, we create a real node of small size. This
 * is stored in the subgraph sg, for easy removal later.
 *
 * I believe it is not necessary to do this for the root graph, as these
 * are laid out one component at a time and these will necessarily have a
 * node on each rank from source to sink levels.
 */
static Agraph_t *realFillRanks(Agraph_t *g, bitarray_t *ranks, Agraph_t* sg) {
    int i, c;
    Agedge_t* e;
    Agnode_t* n;
 
    for (c = 1; c <= GD_n_cluster(g); c++)
        sg = realFillRanks(GD_clust(g)[c], ranks, sg);
 
    if (dot_root(g) == g)
        return sg;
    bitarray_clear(ranks);
    for (n = agfstnode(g); n; n = agnxtnode(g,n)) {
        bitarray_set(ranks, ND_rank(n), true);
        for (e = agfstout(g,n); e; e = agnxtout(g,e)) {
            for (i = ND_rank(n)+1; i <= ND_rank(aghead(e)); i++) 
                bitarray_set(ranks, i, true);
        }
    }
    for (i = GD_minrank(g); i <= GD_maxrank(g); i++) {
        if (!bitarray_get(*ranks, i)) {
            if (!sg) {
                sg = agsubg (dot_root(g), "_new_rank", 1);
            }
            n = agnode (sg, NULL, 1);
            agbindrec(n, "Agnodeinfo_t", sizeof(Agnodeinfo_t), true);
            ND_rank(n) = i;
            ND_lw(n) = ND_rw(n) = 0.5;
            ND_ht(n) = 1;
            ND_UF_size(n) = 1;
            alloc_elist(4, ND_in(n));
            alloc_elist(4, ND_out(n));
            agsubnode (g, n, 1);
        }
    }
    return sg;
}
 
static void
fillRanks (Agraph_t* g)
{
    int rnks_sz = GD_maxrank(g) + 2;
    bitarray_t rnks = bitarray_new(rnks_sz);
    realFillRanks(g, &rnks, NULL);
    bitarray_reset(&rnks);
}
 
static void init_mincross(graph_t * g)
{
    int size;
 
    if (Verbose)
        start_timer();
 
    ReMincross = false;
    Root = g;
    /* alloc +1 for the null terminator usage in do_ordering() */
    size = agnedges(dot_root(g)) + 1;
    TE_list = gv_calloc(size, sizeof(edge_t*));
    TI_list = gv_calloc(size, sizeof(int));
    mincross_options(g);
    if (GD_flags(g) & NEW_RANK)
        fillRanks (g);
    class2(g);
    decompose(g, 1);
    allocate_ranks(g);
    ordered_edges(g);
    GlobalMinRank = GD_minrank(g);
    GlobalMaxRank = GD_maxrank(g);
}
 
static void flat_rev(Agraph_t * g, Agedge_t * e)
{
    int j;
    Agedge_t *rev;
 
    if (!ND_flat_out(aghead(e)).list)
        rev = NULL;
    else
        for (j = 0; (rev = ND_flat_out(aghead(e)).list[j]); j++)
            if (aghead(rev) == agtail(e))
                break;
    if (rev) {
        merge_oneway(e, rev);
        if (ED_edge_type(rev) == FLATORDER && ED_to_orig(rev) == 0)
            ED_to_orig(rev) = e;
        elist_append(e, ND_other(agtail(e)));
    } else {
        rev = new_virtual_edge(aghead(e), agtail(e), e);
        if (ED_edge_type(e) == FLATORDER)
            ED_edge_type(rev) = FLATORDER;
        else
            ED_edge_type(rev) = REVERSED;
        ED_label(rev) = ED_label(e);
        flat_edge(g, rev);
    }
}
 
static void flat_search(graph_t * g, node_t * v)
{
    int i;
    bool hascl;
    edge_t *e;
    adjmatrix_t *M = GD_rank(g)[ND_rank(v)].flat;
 
    ND_mark(v) = true;
    ND_onstack(v) = true;
    hascl = GD_n_cluster(dot_root(g)) > 0;
    if (ND_flat_out(v).list)
        for (i = 0; (e = ND_flat_out(v).list[i]); i++) {
            if (hascl && !(agcontains(g, agtail(e)) && agcontains(g, aghead(e))))
                continue;
            if (ED_weight(e) == 0)
                continue;
            if (ND_onstack(aghead(e))) {
                assert(flatindex(aghead(e)) < M->nrows);
                assert(flatindex(agtail(e)) < M->ncols);
                matrix_set(M, (size_t)flatindex(aghead(e)), (size_t)flatindex(agtail(e)));
                delete_flat_edge(e);
                i--;
                if (ED_edge_type(e) == FLATORDER)
                    continue;
                flat_rev(g, e);
            } else {
                assert(flatindex(aghead(e)) < M->nrows);
                assert(flatindex(agtail(e)) < M->ncols);
                matrix_set(M, (size_t)flatindex(agtail(e)), (size_t)flatindex(aghead(e)));
                if (!ND_mark(aghead(e)))
                    flat_search(g, aghead(e));
            }
        }
    ND_onstack(v) = false;
}
 
static void flat_breakcycles(graph_t * g)
{
    int i, r;
    node_t *v;
 
    for (r = GD_minrank(g); r <= GD_maxrank(g); r++) {
        bool flat = false;
        for (i = 0; i < GD_rank(g)[r].n; i++) {
            v = GD_rank(g)[r].v[i];
            ND_mark(v) = false;
            ND_onstack(v) = false;
            ND_low(v) = i;
            if (ND_flat_out(v).size > 0 && !flat) {
                GD_rank(g)[r].flat =
                    new_matrix((size_t)GD_rank(g)[r].n, (size_t)GD_rank(g)[r].n);
                flat = true;
            }
        }
        if (flat) {
            for (i = 0; i < GD_rank(g)[r].n; i++) {
                v = GD_rank(g)[r].v[i];
                if (!ND_mark(v))
                    flat_search(g, v);
            }
        }
    }
}
 
/* Allocate rank structure, determining number of nodes per rank.
 * Note that no nodes are put into the structure yet.
 */
void allocate_ranks(graph_t * g)
{
    int r, low, high;
    node_t *n;
    edge_t *e;
 
    int *cn = gv_calloc(GD_maxrank(g) + 2, sizeof(int)); // must be 0 based, not GD_minrank
    for (n = agfstnode(g); n; n = agnxtnode(g, n)) {
        cn[ND_rank(n)]++;
        for (e = agfstout(g, n); e; e = agnxtout(g, e)) {
            low = ND_rank(agtail(e));
            high = ND_rank(aghead(e));
            if (low > high) {
                SWAP(&low, &high);
            }
            for (r = low + 1; r < high; r++)
                cn[r]++;
        }
    }
    GD_rank(g) = gv_calloc(GD_maxrank(g) + 2, sizeof(rank_t));
    for (r = GD_minrank(g); r <= GD_maxrank(g); r++) {
        GD_rank(g)[r].an = GD_rank(g)[r].n = cn[r] + 1;
        GD_rank(g)[r].av = GD_rank(g)[r].v = gv_calloc(cn[r] + 1, sizeof(node_t*));
    }
    free(cn);
}
 
/* install a node at the current right end of its rank */
int install_in_rank(graph_t *g, node_t *n) {
    int i, r;
 
    r = ND_rank(n);
    i = GD_rank(g)[r].n;
    if (GD_rank(g)[r].an <= 0) {
        agerrorf("install_in_rank, line %d: %s %s rank %d i = %d an = 0\n",
              __LINE__, agnameof(g), agnameof(n), r, i);
        return -1;
    }
 
    GD_rank(g)[r].v[i] = n;
    ND_order(n) = i;
    GD_rank(g)[r].n++;
    assert(GD_rank(g)[r].n <= GD_rank(g)[r].an);
#ifdef DEBUG
    {
        node_t *v;
 
        for (v = GD_nlist(g); v; v = ND_next(v))
            if (v == n)
                break;
        assert(v != NULL);
    }
#endif
    if (ND_order(n) > GD_rank(Root)[r].an) {
        agerrorf("install_in_rank, line %d: ND_order(%s) [%d] > GD_rank(Root)[%d].an [%d]\n",
              __LINE__, agnameof(n), ND_order(n), r, GD_rank(Root)[r].an);
        return -1;
    }
    if (r < GD_minrank(g) || r > GD_maxrank(g)) {
        agerrorf("install_in_rank, line %d: rank %d not in rank range [%d,%d]\n",
              __LINE__, r, GD_minrank(g), GD_maxrank(g));
        return -1;
    }
    if (GD_rank(g)[r].v + ND_order(n) >
        GD_rank(g)[r].av + GD_rank(Root)[r].an) {
        agerrorf("install_in_rank, line %d: GD_rank(g)[%d].v + ND_order(%s) [%d] > GD_rank(g)[%d].av + GD_rank(Root)[%d].an [%d]\n",
              __LINE__, r, agnameof(n),ND_order(n), r, r, GD_rank(Root)[r].an);
        return -1;
    }
    return 0;
}
 
/*      install nodes in ranks. the initial ordering ensure that series-parallel
 *      graphs such as trees are drawn with no crossings.  it tries searching
 *      in- and out-edges and takes the better of the two initial orderings.
 */
int build_ranks(graph_t *g, int pass) {
    int i, j;
    node_t *n, *ns;
    edge_t **otheredges;
    node_queue_t q = {0};
    for (n = GD_nlist(g); n; n = ND_next(n))
        MARK(n) = false;
 
#ifdef DEBUG
    {
        edge_t *e;
        for (n = GD_nlist(g); n; n = ND_next(n)) {
            for (i = 0; (e = ND_out(n).list[i]); i++)
                assert(!MARK(aghead(e)));
            for (i = 0; (e = ND_in(n).list[i]); i++)
                assert(!MARK(agtail(e)));
        }
    }
#endif
 
    for (i = GD_minrank(g); i <= GD_maxrank(g); i++)
        GD_rank(g)[i].n = 0;
 
    const bool walkbackwards = g != agroot(g); // if this is a cluster, need to
                                               // walk GD_nlist backward to
                                               // preserve input node order
    if (walkbackwards) {
        for (ns = GD_nlist(g); ND_next(ns); ns = ND_next(ns)) {
            ;
        }
    } else {
        ns = GD_nlist(g);
    }
    for (n = ns; n; n = walkbackwards ? ND_prev(n) : ND_next(n)) {
        otheredges = pass == 0 ? ND_in(n).list : ND_out(n).list;
        if (otheredges[0] != NULL)
            continue;
        if (!MARK(n)) {
            MARK(n) = true;
            node_queue_push_back(&q, n);
            while (!node_queue_is_empty(&q)) {
                node_t *n0 = node_queue_pop_front(&q);
                if (ND_ranktype(n0) != CLUSTER) {
                    if (install_in_rank(g, n0) != 0) {
                        node_queue_free(&q);
                        return -1;
                    }
                    enqueue_neighbors(&q, n0, pass);
                } else {
                    const int rc = install_cluster(g, n0, pass, &q);
                    if (rc != 0) {
                        node_queue_free(&q);
                        return rc;
                    }
                }
            }
        }
    }
    assert(node_queue_is_empty(&q));
    for (i = GD_minrank(g); i <= GD_maxrank(g); i++) {
        GD_rank(Root)[i].valid = false;
        if (GD_flip(g) && GD_rank(g)[i].n > 0) {
            node_t **vlist = GD_rank(g)[i].v;
            int num_nodes_1 = GD_rank(g)[i].n - 1;
            int half_num_nodes_1 = num_nodes_1 / 2;
            for (j = 0; j <= half_num_nodes_1; j++)
                exchange(vlist[j], vlist[num_nodes_1 - j]);
        }
    }
 
    if (g == dot_root(g) && ncross() > 0)
        transpose(g, false);
    node_queue_free(&q);
    return 0;
}
 
void enqueue_neighbors(node_queue_t *q, node_t *n0, int pass) {
    edge_t *e;
 
    if (pass == 0) {
        for (size_t i = 0; i < ND_out(n0).size; i++) {
            e = ND_out(n0).list[i];
            if (!MARK(aghead(e))) {
                MARK(aghead(e)) = true;
                node_queue_push_back(q, aghead(e));
            }
        }
    } else {
        for (size_t i = 0; i < ND_in(n0).size; i++) {
            e = ND_in(n0).list[i];
            if (!MARK(agtail(e))) {
                MARK(agtail(e)) = true;
                node_queue_push_back(q, agtail(e));
            }
        }
    }
}
 
static bool constraining_flat_edge(Agraph_t *g, Agedge_t *e) {
  if (ED_weight(e) == 0)
    return false;
  if (!inside_cluster(g, agtail(e)))
    return false;
  if (!inside_cluster(g, aghead(e)))
    return false;
  return true;
}
 
DEFINE_LIST(nodes, node_t *)
 
/* construct nodes reachable from 'here' in post-order.
* This is the same as doing a topological sort in reverse order.
*/
static void postorder(graph_t *g, node_t *v, nodes_t *list, int r) {
    edge_t *e;
    int i;
 
    MARK(v) = true;
    if (ND_flat_out(v).size > 0) {
        for (i = 0; (e = ND_flat_out(v).list[i]); i++) {
            if (!constraining_flat_edge(g, e)) continue;
            if (!MARK(aghead(e)))
                postorder(g, aghead(e), list, r);
        }
    }
    assert(ND_rank(v) == r);
    nodes_append(list, v);
}
 
static void flat_reorder(graph_t * g)
{
    int i, r, local_in_cnt, local_out_cnt, base_order;
    node_t *v;
    nodes_t temprank = {0};
    edge_t *flat_e, *e;
 
    if (!GD_has_flat_edges(g))
        return;
    for (r = GD_minrank(g); r <= GD_maxrank(g); r++) {
        if (GD_rank(g)[r].n == 0) continue;
        base_order = ND_order(GD_rank(g)[r].v[0]);
        for (i = 0; i < GD_rank(g)[r].n; i++)
            MARK(GD_rank(g)[r].v[i]) = false;
        nodes_clear(&temprank);
 
        /* construct reverse topological sort order in temprank */
        for (i = 0; i < GD_rank(g)[r].n; i++) {
            if (GD_flip(g)) v = GD_rank(g)[r].v[i];
            else v = GD_rank(g)[r].v[GD_rank(g)[r].n - i - 1];
 
            local_in_cnt = local_out_cnt = 0;
            for (size_t j = 0; j < ND_flat_in(v).size; j++) {
                flat_e = ND_flat_in(v).list[j];
                if (constraining_flat_edge(g, flat_e)) local_in_cnt++;
            }
            for (size_t j = 0; j < ND_flat_out(v).size; j++) {
                flat_e = ND_flat_out(v).list[j];
                if (constraining_flat_edge(g, flat_e)) local_out_cnt++;
            }
            if (local_in_cnt == 0 && local_out_cnt == 0)
                nodes_append(&temprank, v);
            else {
                if (!MARK(v) && local_in_cnt == 0) {
                    postorder(g, v, &temprank, r);
                }
            }
        }
 
        if (nodes_size(&temprank) > 0) {
            if (!GD_flip(g)) {
                nodes_reverse(&temprank);
            }
            for (i = 0; i < GD_rank(g)[r].n; i++) {
                v = GD_rank(g)[r].v[i] = nodes_get(&temprank, (size_t)i);
                ND_order(v) = i + base_order;
            }
 
            /* nonconstraint flat edges must be made LR */
            for (i = 0; i < GD_rank(g)[r].n; i++) {
                v = GD_rank(g)[r].v[i];
                if (ND_flat_out(v).list) {
                    for (size_t j = 0; (e = ND_flat_out(v).list[j]); j++) {
                        if ((!GD_flip(g) && ND_order(aghead(e)) < ND_order(agtail(e))) ||
                                  (GD_flip(g) && ND_order(aghead(e)) > ND_order(agtail(e)))) {
                            assert(!constraining_flat_edge(g, e));
                            delete_flat_edge(e);
                            j--;
                            flat_rev(g, e);
                        }
                    }
                }
            }
            /* postprocess to restore intended order */
        }
        /* else do no harm! */
        GD_rank(Root)[r].valid = false;
    }
    nodes_free(&temprank);
}
 
static void reorder(graph_t * g, int r, bool reverse, bool hasfixed)
{
    int changed = 0, nelt;
    node_t **vlist = GD_rank(g)[r].v;
    node_t **lp, **rp, **ep = vlist + GD_rank(g)[r].n;
 
    for (nelt = GD_rank(g)[r].n - 1; nelt >= 0; nelt--) {
        lp = vlist;
        while (lp < ep) {
            /* find leftmost node that can be compared */
            while (lp < ep && ND_mval(*lp) < 0)
                lp++;
            if (lp >= ep)
                break;
            /* find the node that can be compared */
            bool sawclust = false;
            bool muststay = false;
            for (rp = lp + 1; rp < ep; rp++) {
                if (sawclust && ND_clust(*rp))
                    continue;   /* ### */
                if (left2right(g, *lp, *rp)) {
                    muststay = true;
                    break;
                }
                if (ND_mval(*rp) >= 0)
                    break;
                if (ND_clust(*rp))
                    sawclust = true;    /* ### */
            }
            if (rp >= ep)
                break;
            if (!muststay) {
                const double p1 = ND_mval(*lp);
                const double p2 = ND_mval(*rp);
                if (p1 > p2 || (p1 >= p2 && reverse)) {
                    exchange(*lp, *rp);
                    changed++;
                }
            }
            lp = rp;
        }
        if (!hasfixed && !reverse)
            ep--;
    }
 
    if (changed) {
        GD_rank(Root)[r].valid = false;
        if (r > 0)
            GD_rank(Root)[r - 1].valid = false;
    }
}
 
static void mincross_step(graph_t * g, int pass)
{
    int r, other, first, last, dir;
 
    bool reverse = pass % 4 < 2;
 
    if (pass % 2 == 0) {        /* down pass */
        first = GD_minrank(g) + 1;
        if (GD_minrank(g) > GD_minrank(Root))
            first--;
        last = GD_maxrank(g);
        dir = 1;
    } else {                    /* up pass */
        first = GD_maxrank(g) - 1;
        last = GD_minrank(g);
        if (GD_maxrank(g) < GD_maxrank(Root))
            first++;
        dir = -1;
    }
 
    for (r = first; r != last + dir; r += dir) {
        other = r - dir;
        bool hasfixed = medians(g, r, other);
        reorder(g, r, reverse, hasfixed);
    }
    transpose(g, !reverse);
}
 
static int local_cross(elist l, int dir)
{
    int i, j;
    int cross = 0;
    edge_t *e, *f;
    bool is_out = dir > 0;
    for (i = 0; (e = l.list[i]); i++) {
        if (is_out)
            for (j = i + 1; (f = l.list[j]); j++) {
                if ((ND_order(aghead(f)) - ND_order(aghead(e)))
                         * (ED_tail_port(f).p.x - ED_tail_port(e).p.x) < 0)
                    cross += ED_xpenalty(e) * ED_xpenalty(f);
        } else
            for (j = i + 1; (f = l.list[j]); j++) {
                if ((ND_order(agtail(f)) - ND_order(agtail(e)))
                        * (ED_head_port(f).p.x - ED_head_port(e).p.x) < 0)
                    cross += ED_xpenalty(e) * ED_xpenalty(f);
            }
    }
    return cross;
}
 
static int64_t rcross(graph_t *g, int r) {
    int top, bot, max, i, k;
    node_t **rtop, *v;
 
    int64_t cross = 0;
    max = 0;
    rtop = GD_rank(g)[r].v;
 
    int *Count = gv_calloc(GD_rank(Root)[r + 1].n + 1, sizeof(int));
 
    for (top = 0; top < GD_rank(g)[r].n; top++) {
        edge_t *e;
        if (max > 0) {
            for (i = 0; (e = ND_out(rtop[top]).list[i]); i++) {
                for (k = ND_order(aghead(e)) + 1; k <= max; k++)
                    cross += Count[k] * ED_xpenalty(e);
            }
        }
        for (i = 0; (e = ND_out(rtop[top]).list[i]); i++) {
            int inv = ND_order(aghead(e));
            if (inv > max)
                max = inv;
            Count[inv] += ED_xpenalty(e);
        }
    }
    for (top = 0; top < GD_rank(g)[r].n; top++) {
        v = GD_rank(g)[r].v[top];
        if (ND_has_port(v))
            cross += local_cross(ND_out(v), 1);
    }
    for (bot = 0; bot < GD_rank(g)[r + 1].n; bot++) {
        v = GD_rank(g)[r + 1].v[bot];
        if (ND_has_port(v))
            cross += local_cross(ND_in(v), -1);
    }
    free(Count);
    return cross;
}
 
static int64_t ncross(void) {
    int r;
 
    graph_t *g = Root;
    int64_t count = 0;
    for (r = GD_minrank(g); r < GD_maxrank(g); r++) {
        if (GD_rank(g)[r].valid)
            count += GD_rank(g)[r].cache_nc;
        else {
            const int64_t nc = GD_rank(g)[r].cache_nc = rcross(g, r);
            count += nc;
            GD_rank(g)[r].valid = true;
        }
    }
    return count;
}
 
static int ordercmpf(const void *x, const void *y) {
  const int *i0 = x;
  const int *i1 = y;
  if (*i0 < *i1) {
    return -1;
  }
  if (*i0 > *i1) {
    return 1;
  }
  return 0;
}
 
/* Calculate a mval for nodes with no in or out non-flat edges.
 * Assume (ND_out(n).size == 0) && (ND_in(n).size == 0)
 * Find flat edge a->n where a has the largest order and set
 * n.mval = a.mval+1, assuming a.mval is defined (>=0).
 * If there are no flat in edges, find flat edge n->a where a 
 * has the smallest order and set * n.mval = a.mval-1, assuming 
 * a.mval is > 0.
 * Return true if n.mval is left -1, indicating a fixed node for sorting.
 */
static bool flat_mval(node_t * n)
{
    int i;
    edge_t *e, **fl;
    node_t *nn;
 
    if (ND_flat_in(n).size > 0) {
        fl = ND_flat_in(n).list;
        nn = agtail(fl[0]);
        for (i = 1; (e = fl[i]); i++)
            if (ND_order(agtail(e)) > ND_order(nn))
                nn = agtail(e);
        if (ND_mval(nn) >= 0) {
            ND_mval(n) = ND_mval(nn) + 1;
            return false;
        }
    } else if (ND_flat_out(n).size > 0) {
        fl = ND_flat_out(n).list;
        nn = aghead(fl[0]);
        for (i = 1; (e = fl[i]); i++)
            if (ND_order(aghead(e)) < ND_order(nn))
                nn = aghead(e);
        if (ND_mval(nn) > 0) {
            ND_mval(n) = ND_mval(nn) - 1;
            return false;
        }
    }
    return true;
}
 
#define VAL(node,port) (MC_SCALE * ND_order(node) + (port).order)
 
static bool medians(graph_t * g, int r0, int r1)
{
    int i, j0, lspan, rspan, *list;
    node_t *n, **v;
    edge_t *e;
    bool hasfixed = false;
 
    list = TI_list;
    v = GD_rank(g)[r0].v;
    for (i = 0; i < GD_rank(g)[r0].n; i++) {
        n = v[i];
        size_t j = 0;
        if (r1 > r0)
            for (j0 = 0; (e = ND_out(n).list[j0]); j0++) {
                if (ED_xpenalty(e) > 0)
                    list[j++] = VAL(aghead(e), ED_head_port(e));
        } else
            for (j0 = 0; (e = ND_in(n).list[j0]); j0++) {
                if (ED_xpenalty(e) > 0)
                    list[j++] = VAL(agtail(e), ED_tail_port(e));
            }
        switch (j) {
        case 0:
            ND_mval(n) = -1;
            break;
        case 1:
            ND_mval(n) = list[0];
            break;
        case 2:
            ND_mval(n) = (list[0] + list[1]) / 2;
            break;
        default:
            qsort(list, j, sizeof(int), ordercmpf);
            if (j % 2)
                ND_mval(n) = list[j / 2];
            else {
                /* weighted median */
                size_t rm = j / 2;
                size_t lm = rm - 1;
                rspan = list[j - 1] - list[rm];
                lspan = list[lm] - list[0];
                if (lspan == rspan)
                    ND_mval(n) = (list[lm] + list[rm]) / 2;
                else {
                    double w = list[lm] * (double)rspan + list[rm] * (double)lspan;
                    ND_mval(n) = w / (lspan + rspan);
                }
            }
        }
    }
    for (i = 0; i < GD_rank(g)[r0].n; i++) {
        n = v[i];
        if (ND_out(n).size == 0 && ND_in(n).size == 0)
            hasfixed |= flat_mval(n);
    }
    return hasfixed;
}
 
static int nodeposcmpf(const void *x, const void *y) {
// Suppress Clang/GCC -Wcast-qual warning. Casting away const here is acceptable
// as the later usage is const. We need the cast because the macros use
// non-const pointers for genericity.
#ifdef __GNUC__
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wcast-qual"
#endif
  node_t **n0 = (node_t **)x;
  node_t **n1 = (node_t **)y;
#ifdef __GNUC__
#pragma GCC diagnostic pop
#endif
  if (ND_order(*n0) < ND_order(*n1)) {
    return -1;
  }
  if (ND_order(*n0) > ND_order(*n1)) {
    return 1;
  }
  return 0;
}
 
static int edgeidcmpf(const void *x, const void *y) {
// Suppress Clang/GCC -Wcast-qual warning. Casting away const here is acceptable
// as the later usage is const. We need the cast because the macros use
// non-const pointers for genericity.
#ifdef __GNUC__
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wcast-qual"
#endif
  edge_t **e0 = (edge_t **)x;
  edge_t **e1 = (edge_t **)y;
#ifdef __GNUC__
#pragma GCC diagnostic pop
#endif
  if (AGSEQ(*e0) < AGSEQ(*e1)) {
    return -1;
  }
  if (AGSEQ(*e0) > AGSEQ(*e1)) {
    return 1;
  }
  return 0;
}
 
/* following code deals with weights of edges of "virtual" nodes */
#define ORDINARY        0
#define SINGLETON       1
#define VIRTUALNODE     2
#define NTYPES          3
 
#define C_EE            1
#define C_VS            2
#define C_SS            2
#define C_VV            4
 
static int table[NTYPES][NTYPES] = {
    /* ordinary */ {C_EE, C_EE, C_EE},
    /* singleton */ {C_EE, C_SS, C_VS},
    /* virtual */ {C_EE, C_VS, C_VV}
};
 
static int endpoint_class(node_t * n)
{
    if (ND_node_type(n) == VIRTUAL)
        return VIRTUALNODE;
    if (ND_weight_class(n) <= 1)
        return SINGLETON;
    return ORDINARY;
}
 
void virtual_weight(edge_t * e)
{
    int t;
    t = table[endpoint_class(agtail(e))][endpoint_class(aghead(e))];
 
    /* check whether the upcoming computation will overflow */
    assert(t >= 0);
    if (INT_MAX / t < ED_weight(e)) {
        agerrorf("overflow when calculating virtual weight of edge\n");
        graphviz_exit(EXIT_FAILURE);
    }
 
    ED_weight(e) *= t;
}
 
#ifdef DEBUG
void check_rs(graph_t * g, int null_ok)
{
    int i, r;
    node_t *v, *prev;
 
    fprintf(stderr, "\n\n%s:\n", agnameof(g));
    for (r = GD_minrank(g); r <= GD_maxrank(g); r++) {
        fprintf(stderr, "%d: ", r);
        prev = NULL;
        for (i = 0; i < GD_rank(g)[r].n; i++) {
            v = GD_rank(g)[r].v[i];
            if (v == NULL) {
                fprintf(stderr, "NULL\t");
                if (!null_ok)
                    abort();
            } else {
                fprintf(stderr, "%s(%f)\t", agnameof(v), ND_mval(v));
                assert(ND_rank(v) == r);
                assert(v != prev);
                prev = v;
            }
        }
        fprintf(stderr, "\n");
    }
}
 
void check_order(void)
{
    int i, r;
    node_t *v;
    graph_t *g = Root;
 
    for (r = GD_minrank(g); r <= GD_maxrank(g); r++) {
        assert(GD_rank(g)[r].v[GD_rank(g)[r].n] == NULL);
        for (i = 0; (v = GD_rank(g)[r].v[i]); i++) {
            assert(ND_rank(v) == r);
            assert(ND_order(v) == i);
        }
    }
}
#endif
 
static void mincross_options(graph_t * g)
{
    char *p;
    double f;
 
    /* set default values */
    MinQuit = 8;
    MaxIter = 24;
 
    p = agget(g, "mclimit");
    if (p && (f = atof(p)) > 0.0) {
        MinQuit = MAX(1, scale_clamp(MinQuit, f));
        MaxIter = MAX(1, scale_clamp(MaxIter, f));
    }
}
 
#ifdef DEBUG
void check_exchange(node_t * v, node_t * w)
{
    int i, r;
    node_t *u;
 
    if (ND_clust(v) == NULL && ND_clust(w) == NULL)
        return;
    assert(ND_clust(v) == NULL || ND_clust(w) == NULL);
    assert(ND_rank(v) == ND_rank(w));
    assert(ND_order(v) < ND_order(w));
    r = ND_rank(v);
 
    for (i = ND_order(v) + 1; i < ND_order(w); i++) {
        u = GD_rank(dot_root(v))[r].v[i];
        if (ND_clust(u))
            abort();
    }
}
 
void check_vlists(graph_t * g)
{
    int c, i, j, r;
    node_t *u;
 
    for (r = GD_minrank(g); r <= GD_maxrank(g); r++) {
        for (i = 0; i < GD_rank(g)[r].n; i++) {
            u = GD_rank(g)[r].v[i];
            j = ND_order(u);
            assert(GD_rank(Root)[r].v[j] == u);
        }
        if (GD_rankleader(g)) {
            u = GD_rankleader(g)[r];
            j = ND_order(u);
            assert(GD_rank(Root)[r].v[j] == u);
        }
    }
    for (c = 1; c <= GD_n_cluster(g); c++)
        check_vlists(GD_clust(g)[c]);
}
 
void node_in_root_vlist(node_t * n)
{
    node_t **vptr;
 
    for (vptr = GD_rank(Root)[ND_rank(n)].v; *vptr; vptr++)
        if (*vptr == n)
            break;
    if (*vptr == 0)
        abort();
}
#endif                          /* DEBUG code */