constraint.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 v2.0
 * which accompanies this distribution, and is available at
 * https://www.eclipse.org/org/documents/epl-2.0/EPL-2.0.html
 *
 * Contributors: Details at https://graphviz.org
 *************************************************************************/
 
 
#include "config.h"
#include <cgraph/cgraph.h>
#include <common/geom.h>
#include <math.h>
#include <neatogen/neato.h>
#include <neatogen/adjust.h>
#include <stddef.h>
#include <stdbool.h>
#include <util/alloc.h>
#include <util/gv_math.h>
#include <util/itos.h>
#include <util/list.h>
 
/* For precision, scale up before algorithms, then scale down */
#define SCALE 10   
#define SCALE2 (SCALE/2)
 
typedef struct nitem {
    Dtlink_t link;
    int val;
    point pos;                  /* position for sorting */
    node_t *np;                 /* base node */
    node_t *cnode;              /* corresponding node in constraint graph */
    node_t *vnode;              /* corresponding node in neighbor graph */
    box bb;
} nitem;
 
typedef int (*distfn) (box *, box *);
typedef int (*intersectfn) (nitem *, nitem *);
 
static int cmpitem(void *item1, void *item2) {
    const int *p1 = item1;
    const int *p2 = item2;
    if (*p1 < *p2) {
        return -1;
    }
    if (*p1 > *p2) {
        return 1;
    }
    return 0;
}
 
static Dtdisc_t constr = {
    offsetof(nitem, val),
    sizeof(int),
    offsetof(nitem, link),
    NULL,
    NULL,
    cmpitem,
};
 
static int distY(box * b1, box * b2)
{
    return ((b1->UR.y - b1->LL.y) + (b2->UR.y - b2->LL.y)) / 2;
}
 
static int distX(box * b1, box * b2)
{
    return ((b1->UR.x - b1->LL.x) + (b2->UR.x - b2->LL.x)) / 2;
}
 
/* Return true if boxes could overlap if shifted in y but don't,
 * or if actually overlap and an y move is smallest to remove overlap.
 * Otherwise (no x overlap or a x move is smaller), return false.
 * Assume q pos to above of p pos.
 */
static int intersectX0(nitem * p, nitem * q)
{
    int xdelta, ydelta;
    int v = p->bb.LL.x <= q->bb.UR.x && q->bb.LL.x <= p->bb.UR.x;
    if (v == 0)  /* no x overlap */
        return 0;
    if (p->bb.UR.y < q->bb.LL.y) /* but boxes don't really overlap */
        return 1;
    ydelta = distY(&p->bb,&q->bb) - (q->pos.y - p->pos.y);
    if (q->pos.x >= p->pos.x) 
        xdelta = distX(&p->bb,&q->bb) - (q->pos.x - p->pos.x); 
    else
        xdelta = distX(&p->bb,&q->bb) - (p->pos.x - q->pos.x); 
    return ydelta <= xdelta;
}
 
/* Return true if boxes could overlap if shifted in x but don't,
 * or if actually overlap and an x move is smallest to remove overlap.
 * Otherwise (no y overlap or a y move is smaller), return false.
 * Assume q pos to right of p pos.
 */
static int intersectY0(nitem * p, nitem * q)
{
    int xdelta, ydelta;
    int v = p->bb.LL.y <= q->bb.UR.y && q->bb.LL.y <= p->bb.UR.y;
    if (v == 0)  /* no y overlap */
        return 0;
    if (p->bb.UR.x < q->bb.LL.x) /* but boxes don't really overlap */
        return 1;
    xdelta = distX(&p->bb,&q->bb) - (q->pos.x - p->pos.x);
    if (q->pos.y >= p->pos.y) 
        ydelta = distY(&p->bb,&q->bb) - (q->pos.y - p->pos.y); 
    else
        ydelta = distY(&p->bb,&q->bb) - (p->pos.y - q->pos.y); 
    return xdelta <= ydelta;
}
 
static int intersectY(nitem * p, nitem * q)
{
  return p->bb.LL.y <= q->bb.UR.y && q->bb.LL.y <= p->bb.UR.y;
}
 
static int intersectX(nitem * p, nitem * q)
{
  return p->bb.LL.x <= q->bb.UR.x && q->bb.LL.x <= p->bb.UR.x;
}
 
static void mapGraphs(graph_t * g, graph_t * cg, distfn dist)
{
    node_t *n;
    edge_t *e;
    edge_t *ce;
    node_t *t;
    node_t *h;
    nitem *tp;
    nitem *hp;
    int delta;
 
    for (n = agfstnode(g); n; n = agnxtnode(g, n)) {
        tp = ND_alg(n);
        t = tp->cnode;
        for (e = agfstout(g, n); e; e = agnxtout(g, e)) {
            hp = ND_alg(aghead(e));
            delta = dist(&tp->bb, &hp->bb);
            h = hp->cnode;
            ce = agedge(cg, t, h, NULL, 1);
            agbindrec(ce, "Agedgeinfo_t", sizeof(Agedgeinfo_t), true);
            ED_weight(ce) = 1;
            if (ED_minlen(ce) < delta) {
                if (ED_minlen(ce) == 0.0) {
                    elist_append(ce, ND_out(t));
                    elist_append(ce, ND_in(h));
                }
                ED_minlen(ce) = delta;
            }
        }
    }
}
 
#if defined(DEBUG) && DEBUG > 1
static int
indegree (graph_t * g, node_t *n)
{
  edge_t *e;
  int cnt = 0;
  for (e = agfstin(g,n); e; e = agnxtin(g,e)) cnt++;
  return cnt; 
}
 
static int
outdegree (graph_t * g, node_t *n)
{
  edge_t *e;
  int cnt = 0;
  for (e = agfstout(g,n); e; e = agnxtout(g,e)) cnt++;
  return cnt; 
}
 
static void
validate(graph_t * g)
{
    node_t *n;
    edge_t *e;
    int    i, cnt;
  
    cnt = 0;
    for (n = GD_nlist(g);n; n = ND_next(n)) {
      assert(outdegree(g,n) == ND_out(n).size);
      for (i = 0; (e = ND_out(n).list[i]); i++) {
        assert(agtail(e) == n);
        assert( e == agfindedge(g, n, aghead(e))); 
      }
      assert(indegree(g,n) == ND_in(n).size);
      for (i = 0; (e = ND_in(n).list[i]); i++) {
        assert(aghead(e) == n);
        assert( e == agfindedge(g, agtail(e), n)); 
      }
      cnt++;
    }
 
    assert (agnnodes(g) == cnt); 
}
#endif
 
 
/* Similar to mkConstraintG, except it doesn't enforce orthogonal
 * ordering. If there is overlap, as defined by intersect, the
 * nodes will kept/pushed apart in the current order. If not, no
 * constraint is enforced. If a constraint edge is added, and it
 * corresponds to a real edge, we increase the weight in an attempt
 * to keep the resulting shift short. 
 */
static graph_t *mkNConstraintG(graph_t * g, Dt_t * list,
                              intersectfn intersect, distfn dist)
{
    nitem *p;
    nitem *nxp;
    node_t *n;
    edge_t *e;
    node_t *lastn = NULL;
    graph_t *cg = agopen("cg", Agstrictdirected, NULL);
    agbindrec(cg, "Agraphinfo_t", sizeof(Agraphinfo_t), true);  // graph custom data
 
    for (p = (nitem *)dtflatten(list); p; p = (nitem *)dtlink(list, p)) {
        n = agnode(cg, agnameof(p->np), 1);      /* FIX */
        agbindrec(n, "Agnodeinfo_t", sizeof(Agnodeinfo_t), true); //node custom data
        ND_alg(n) = p;
        p->cnode = n;
        alloc_elist(0, ND_in(n));
        alloc_elist(0, ND_out(n));
        if (lastn) {
            ND_next(lastn) = n;
            lastn = n;
        } else {
            lastn = GD_nlist(cg) = n;
        }
    }
    for (p = (nitem *)dtflatten(list); p; p = (nitem *)dtlink(list, p)) {
        for (nxp = (nitem *)dtlink(link, p); nxp; nxp = (nitem *)dtlink(list, nxp)) {
            e = NULL;
            if (intersect(p, nxp)) {
                double delta = dist(&p->bb, &nxp->bb);
                e = agedge(cg, p->cnode, nxp->cnode, NULL, 1);
                agbindrec(e, "Agedgeinfo_t", sizeof(Agedgeinfo_t), true);   // edge custom data
                assert (delta <= 0xFFFF);
                ED_minlen(e) = delta;
                ED_weight(e) = 1;
            }
            if (e && agfindedge(g,p->np, nxp->np)) {
                ED_weight(e) = 100;
            }
        }
    }
   
    for (p = (nitem *)dtflatten(list); p; p = (nitem *)dtlink(list, p)) {
        n = p->cnode;
        for (e = agfstout(cg,n); e; e = agnxtout(cg,e)) {
            elist_append(e, ND_out(n));
            elist_append(e, ND_in(aghead(e)));
        }
    }
 
    /* We could remove redundant constraints here. However, the cost of doing 
     * this may be a good deal more than the time saved in network simplex. 
     * Also, if the graph is changed, the ND_in and ND_out data has to be 
     * updated.
     */
    return cg;
}
 
static graph_t *mkConstraintG(Dt_t * list, intersectfn intersect, distfn dist) {
    nitem *p;
    nitem *nxt = NULL;
    nitem *nxp;
    graph_t *vg;
    node_t *prev = NULL;
    node_t *root = NULL;
    node_t *n = NULL;
    edge_t *e;
    int lcnt, cnt;
    int oldval = -INT_MAX;
    node_t *lastn = NULL;
    graph_t *cg = agopen("cg", Agstrictdirected, NULL);
    agbindrec(cg, "Agraphinfo_t", sizeof(Agraphinfo_t), true);  // graph custom data
 
    /* count distinct nodes */
    cnt = 0;
    for (p = (nitem *)dtflatten(list); p; p = (nitem *)dtlink(list, p)) {
        if (oldval != p->val) {
            oldval = p->val;
            cnt++;
        }
    }
 
    /* construct basic chain to enforce left to right order */
    oldval = -INT_MAX;
    lcnt = 0;
    for (p = (nitem *)dtflatten(list); p; p = (nitem *)dtlink(list, p)) {
        if (oldval != p->val) {
            oldval = p->val;
            /* n = newNode (cg); */
            n = agnode(cg, agnameof(p->np), 1); /* FIX */
            agbindrec(n, "Agnodeinfo_t", sizeof(Agnodeinfo_t), true); //node custom data
            ND_alg(n) = p;
            if (root) {
                ND_next(lastn) = n;
                lastn = n;
            } else {
                root = n;
                lastn = GD_nlist(cg) = n;
            }
            alloc_elist(lcnt, ND_in(n));
            if (prev) {
                if (prev == root)
                    alloc_elist(2 * (cnt - 1), ND_out(prev));
                else
                    alloc_elist(cnt - lcnt - 1, ND_out(prev));
                e = agedge(cg, prev, n, NULL, 1);
                agbindrec(e, "Agedgeinfo_t", sizeof(Agedgeinfo_t), true);   // edge custom data
                ED_minlen(e) = SCALE;
                ED_weight(e) = 1;
                elist_append(e, ND_out(prev));
                elist_append(e, ND_in(n));
            }
            lcnt++;
            prev = n;
        }
        p->cnode = n;
    }
    alloc_elist(0, ND_out(prev));
 
    /* add immediate right neighbor constraints
     * Construct visibility graph, then perform transitive reduction.
     * Remaining outedges are immediate right neighbors.
     * FIX: Incremental algorithm to construct trans. reduction?
     */
    vg = agopen("vg", Agstrictdirected, NULL);
    for (p = (nitem *)dtflatten(list); p; p = (nitem *)dtlink(list, p)) {
        n = agnode(vg, agnameof(p->np), 1);  /* FIX */
        agbindrec(n, "Agnodeinfo_t", sizeof(Agnodeinfo_t), true);  //node custom data
        p->vnode = n;
        ND_alg(n) = p;
    }
    oldval = -INT_MAX;
    for (p = (nitem *)dtflatten(list); p; p = (nitem *)dtlink(list, p)) {
        if (oldval != p->val) { /* new pos: reset nxt */
            oldval = p->val;
            for (nxt = (nitem *)dtlink(link, p); nxt;
                 nxt = (nitem *)dtlink(list, nxt)) {
                if (nxt->val != oldval)
                    break;
            }
            if (!nxt)
                break;
        }
        for (nxp = nxt; nxp; nxp = (nitem *)dtlink(list, nxp)) {
            if (intersect(p, nxp))
                agedge(vg, p->vnode, nxp->vnode, NULL, 1);
        }
    }
 
    /* Remove redundant constraints here. However, the cost of doing this
     * may be a good deal more than the time saved in network simplex. Also,
     * if the graph is changed, the ND_in and ND_out data has to be updated.
     */
    mapGraphs(vg, cg, dist);
    agclose(vg);
 
    return cg;
}
 
static void closeGraph(graph_t * cg)
{
    for (node_t *n = agfstnode(cg); n; n = agnxtnode(cg, n)) {
        free_list(ND_in(n));
        free_list(ND_out(n));
    }
    agclose(cg);
}
 
/* Create the X constrains and solve. We use a linear objective function
 * (absolute values rather than squares), so we can reuse network simplex.
 * The constraints are encoded as a dag with edges having a minimum length.
 */
static void constrainX(graph_t *g, nitem *nlist, size_t nnodes, intersectfn ifn,
                       int ortho)
{
    Dt_t *list = dtopen(&constr, Dtobag);
    nitem *p = nlist;
    graph_t *cg;
 
    for (size_t i = 0; i < nnodes; i++) {
        p->val = p->pos.x;
        dtinsert(list, p);
        p++;
    }
    if (ortho)
        cg = mkConstraintG(list, ifn, distX);
    else
        cg = mkNConstraintG(g, list, ifn, distX);
    rank(cg, 2, INT_MAX);
 
    p = nlist;
    for (size_t i = 0; i < nnodes; i++) {
        int newpos, oldpos, delta;
        oldpos = p->pos.x;
        newpos = ND_rank(p->cnode);
        delta = newpos - oldpos;
        p->pos.x = newpos;
        p->bb.LL.x += delta;
        p->bb.UR.x += delta;
        p++;
    }
 
    closeGraph(cg);
    dtclose(list);
}
 
static void constrainY(graph_t *g, nitem *nlist, size_t nnodes, intersectfn ifn,
                       int ortho)
{
    Dt_t *list = dtopen(&constr, Dtobag);
    nitem *p = nlist;
    graph_t *cg;
 
    for (size_t i = 0; i < nnodes; i++) {
        p->val = p->pos.y;
        dtinsert(list, p);
        p++;
    }
    if (ortho)
        cg = mkConstraintG(list, ifn, distY);
    else
        cg = mkNConstraintG(g, list, ifn, distY);
    rank(cg, 2, INT_MAX);
#ifdef DEBUG
    {
        Agsym_t *mlsym = agattr_text(cg, AGEDGE, "minlen", "");
        Agsym_t *rksym = agattr_text(cg, AGNODE, "rank", "");
        node_t *n;
        edge_t *e;
        for (n = agfstnode(cg); n; n = agnxtnode(cg, n)) {
            agxset(n, rksym, ITOS(ND_rank(n)));
            for (e = agfstedge(cg, n); e; e = agnxtedge(cg, e, n)) {
                agxset(e, mlsym, ITOS(ED_minlen(e)));
            }
        }
    }
#endif
 
    p = nlist;
    for (size_t i = 0; i < nnodes; i++) {
        int newpos, oldpos, delta;
        oldpos = p->pos.y;
        newpos = ND_rank(p->cnode);
        delta = newpos - oldpos;
        p->pos.y = newpos;
        p->bb.LL.y += delta;
        p->bb.UR.y += delta;
        p++;
    }
 
    closeGraph(cg);
    dtclose(list);
}
 
static int overlaps(nitem *p, size_t cnt) {
    nitem *pi = p;
    nitem *pj;
 
    for (size_t i = 0; i + 1 < cnt; i++) {
        pj = pi + 1;
        for (size_t j = i + 1; j < cnt; j++) {
            if (OVERLAP(pi->bb, pj->bb))
                return 1;
            pj++;
        }
        pi++;
    }
    return 0;
}
 
static void initItem(node_t * n, nitem * p, expand_t margin)
{
    int x = POINTS(SCALE * ND_pos(n)[0]);
    int y = POINTS(SCALE * ND_pos(n)[1]);
    int w2, h2;
    box b;
 
    if (margin.doAdd) {
        w2 = d2i(SCALE * (POINTS(ND_width(n) / 2.0) + margin.x));
        h2 = d2i(SCALE * (POINTS(ND_height(n) / 2.0) + margin.y));
    }
    else {
        w2 = POINTS(margin.x * SCALE2 * ND_width(n));
        h2 = POINTS(margin.y * SCALE2 * ND_height(n));
    }
 
    b.LL.x = x - w2;
    b.LL.y = y - h2;
    b.UR.x = x + w2;
    b.UR.y = y + h2;
 
    p->pos.x = x;
    p->pos.y = y;
    p->np = n;
    p->bb = b;
}
 
/* Use optimization to remove overlaps.
 * Modifications;
 *  - do y;x then x;y and use the better one
 *  - for all overlaps (or if overlap with leftmost nodes), add a constraint;
 *     constraint could move both x and y away, or the smallest, or some
 *     mixture.
 *  - follow by a scale down using actual shapes
 * We use an optimization based on Marriott, Stuckey, Tam and He,
 * "Removing Node Overlapping in Graph Layout Using Constrained Optimization",
 * Constraints,8(2):143--172, 2003.
 * We solve 2 constraint problem, one in X, one in Y. In each dimension,
 * we require relative positions to remain the same. That is, if two nodes
 * have the same x originally, they have the same x at the end, and if one
 * node is to the left of another, it remains to the left. In addition, if
 * two nodes could overlap by moving their X coordinates, we insert a constraint
 * to keep the two nodes sufficiently apart. Similarly, for Y.
 * 
 * mode = AM_ORTHOXY => first X, then Y
 * mode = AM_ORTHOYX => first Y, then X
 * mode = AM_ORTHO   => first X, then Y
 * mode = AM_ORTHO_YX   => first Y, then X
 * In the last 2 cases, relax the constraints as follows: during the X pass,
 * if two nodes actually intersect and a smaller move in the Y direction
 * will remove the overlap, we don't force the nodes apart in the X direction,
 * but leave it for the Y pass to remove any remaining overlaps. Without this,
 * the X pass will remove all overlaps, and the Y pass only compresses in the
 * Y direction, causing a skewing of the aspect ratio.
 */
int cAdjust(graph_t * g, int mode)
{
    expand_t margin;
    int ret;
    const size_t nnodes = agnnodes_z(g);
    nitem *nlist = gv_calloc(nnodes, sizeof(nitem));
    nitem *p = nlist;
    node_t *n;
 
    margin = sepFactor (g);
 
    for (n = agfstnode(g); n; n = agnxtnode(g, n)) {
        initItem(n, p, margin);
        p++;
    }
 
    if (overlaps(nlist, nnodes)) {
        point pt;
 
        switch ((adjust_mode)mode) {
        case AM_ORTHOXY:
            constrainX(g, nlist, nnodes, intersectY, 1);
            constrainY(g, nlist, nnodes, intersectX, 1);
            break;
        case AM_ORTHOYX:
            constrainY(g, nlist, nnodes, intersectX, 1);
            constrainX(g, nlist, nnodes, intersectY, 1);
            break;
        case AM_ORTHO :
            constrainX(g, nlist, nnodes, intersectY0, 1);
            constrainY(g, nlist, nnodes, intersectX, 1);
            break;
        case AM_ORTHO_YX :
            constrainY(g, nlist, nnodes, intersectX0, 1);
            constrainX(g, nlist, nnodes, intersectY, 1);
            break;
        case AM_PORTHOXY:
            constrainX(g, nlist, nnodes, intersectY, 0);
            constrainY(g, nlist, nnodes, intersectX, 0);
            break;
        case AM_PORTHOYX:
            constrainY(g, nlist, nnodes, intersectX, 0);
            constrainX(g, nlist, nnodes, intersectY, 0);
            break;
        case AM_PORTHO_YX :
            constrainY(g, nlist, nnodes, intersectX0, 0);
            constrainX(g, nlist, nnodes, intersectY, 0);
            break;
        case AM_PORTHO :
        default :
            constrainX(g, nlist, nnodes, intersectY0, 0);
            constrainY(g, nlist, nnodes, intersectX, 0);
            break;
        }
        p = nlist;
        for (size_t i = 0; i < nnodes; i++) {
            n = p->np;
            pt = p->pos;
            ND_pos(n)[0] = PS2INCH(pt.x) / SCALE;
            ND_pos(n)[1] = PS2INCH(pt.y) / SCALE;
            p++;
        }
        ret = 1;
    }
    else ret = 0;
    free(nlist);
    return ret;
}
 
typedef struct {
    pointf pos;                 /* position for sorting */
    boxf bb;
    double wd2;
    double ht2;
    node_t *np;
} info;
 
static int sortf(const void *x, const void *y) {
    const pointf *p = x;
    const pointf *q = y;
    if (p->x < q->x)
        return -1;
    else if (p->x > q->x)
        return 1;
    else if (p->y < q->y)
        return -1;
    else if (p->y > q->y)
        return 1;
    else
        return 0;
}
 
static double compress(info *nl, size_t nn) {
    info *p = nl;
    info *q;
    double s, sc = 0;
    pointf pt;
 
    for (size_t i = 0; i < nn; i++) {
        q = p + 1;
        for (size_t j = i + 1; j < nn; j++) {
            if (OVERLAP(p->bb, q->bb))
                return 0;
            if (p->pos.x == q->pos.x)
                pt.x = HUGE_VAL;
            else {
                pt.x = (p->wd2 + q->wd2) / fabs(p->pos.x - q->pos.x);
            }
            if (p->pos.y == q->pos.y)
                pt.y = HUGE_VAL;
            else {
                pt.y = (p->ht2 + q->ht2) / fabs(p->pos.y - q->pos.y);
            }
            if (pt.y < pt.x)
                s = pt.y;
            else
                s = pt.x;
            if (s > sc)
                sc = s;
            q++;
        }
        p++;
    }
    return sc;
}
 
static pointf *mkOverlapSet(info *nl, size_t nn, size_t *cntp) {
    info *p = nl;
    info *q;
    LIST(pointf) S = {0};
 
    LIST_APPEND(&S, (pointf){0});
 
    for (size_t i = 0; i < nn; i++) {
        q = p + 1;
        for (size_t j = i + 1; j < nn; j++) {
            if (OVERLAP(p->bb, q->bb)) {
                pointf pt;
                if (p->pos.x == q->pos.x)
                    pt.x = HUGE_VAL;
                else {
                    pt.x = (p->wd2 + q->wd2) / fabs(p->pos.x - q->pos.x);
                    if (pt.x < 1)
                        pt.x = 1;
                }
                if (p->pos.y == q->pos.y)
                    pt.y = HUGE_VAL;
                else {
                    pt.y = (p->ht2 + q->ht2) / fabs(p->pos.y - q->pos.y);
                    if (pt.y < 1)
                        pt.y = 1;
                }
                LIST_APPEND(&S, pt);
            }
            q++;
        }
        p++;
    }
 
    LIST_SHRINK_TO_FIT(&S);
    pointf *ret;
    LIST_DETACH(&S, &ret, cntp);
    return ret;
}
 
static pointf computeScaleXY(pointf *aarr, size_t m) {
    double cost, bestcost;
    pointf scale;
 
    aarr[0].x = 1;
    aarr[0].y = HUGE_VAL;
    qsort(aarr + 1, m - 1, sizeof(pointf), sortf);
 
    pointf *barr = gv_calloc(m, sizeof(pointf));
    barr[m - 1].x = aarr[m - 1].x;
    barr[m - 1].y = 1;
    for (size_t k = m - 2; m > 1; k--) {
        barr[k].x = aarr[k].x;
        barr[k].y = fmax(aarr[k + 1].y, barr[k + 1].y);
        if (k == 0) {
            break;
        }
    }
 
    size_t best = 0;
    bestcost = HUGE_VAL;
    for (size_t k = 0; k < m; k++) {
        cost = barr[k].x * barr[k].y;
        if (cost < bestcost) {
            bestcost = cost;
            best = k;
        }
    }
    assert(bestcost < HUGE_VAL);
    scale.x = barr[best].x;
    scale.y = barr[best].y;
 
    free(barr);
    return scale;
}
 
/* For each (x,y) in aarr, scale has to be bigger than the smallest one.
 * So, the scale is the max min.
 */
static double computeScale(pointf *aarr, size_t m) {
    double sc = 0;
    double v;
    pointf p;
 
    aarr++;
    for (size_t i = 1; i < m; i++) {
        p = *aarr++;
        v = fmin(p.x, p.y);
        if (v > sc)
            sc = v;
    }
    return sc;
}
 
/* Scale the layout.
 * equal > 0  => scale uniformly in x and y to remove overlaps
 * equal = 0  => scale separately in x and y to remove overlaps
 * equal < 0  => scale down uniformly in x and y to remove excess space
 * The last assumes there are no overlaps at present.
 * Based on Marriott, Stuckey, Tam and He,
 * "Removing Node Overlapping in Graph Layout Using Constrained Optimization",
 * Constraints,8(2):143--172, 2003.
 */
int scAdjust(graph_t * g, int equal)
{
    const size_t nnodes = agnnodes_z(g);
    info *nlist = gv_calloc(nnodes, sizeof(info));
    info *p = nlist;
    node_t *n;
    pointf s;
    expand_t margin;
    pointf *aarr;
 
    margin = sepFactor (g);
    if (margin.doAdd) {
        /* we use inches below */
        margin.x = PS2INCH(margin.x);
        margin.y = PS2INCH(margin.y);
    }
 
    for (n = agfstnode(g); n; n = agnxtnode(g, n)) {
        double w2, h2;
        if (margin.doAdd) {
            w2 = ND_width(n) / 2.0 + margin.x;
            h2 = ND_height(n) / 2.0 + margin.y;
        }
        else {
            w2 = margin.x * ND_width(n) / 2.0;
            h2 = margin.y * ND_height(n) / 2.0;
        }
        p->pos.x = ND_pos(n)[0];
        p->pos.y = ND_pos(n)[1];
        p->bb.LL.x = p->pos.x - w2;
        p->bb.LL.y = p->pos.y - h2;
        p->bb.UR.x = p->pos.x + w2;
        p->bb.UR.y = p->pos.y + h2;
        p->wd2 = w2;
        p->ht2 = h2;
        p->np = n;
        p++;
    }
 
    if (equal < 0) {
        s.x = s.y = compress(nlist, nnodes);
        if (s.x == 0) {         /* overlaps exist */
            free(nlist);
            return 0;
        }
        if (Verbose) fprintf(stderr, "compress %g \n", s.x);
    } else {
        size_t m;
        aarr = mkOverlapSet(nlist, nnodes, &m);
 
        if (m == 1) { // no overlaps
            free(aarr);
            free(nlist);
            return 0;
        }
 
        if (equal) {
            s.x = s.y = computeScale(aarr, m);
        } else {
            s = computeScaleXY(aarr, m);
        }
        free(aarr);
        if (Verbose) fprintf(stderr, "scale by %g,%g \n", s.x, s.y);
    }
 
    p = nlist;
    for (size_t i = 0; i < nnodes; i++) {
        ND_pos(p->np)[0] = s.x * p->pos.x;
        ND_pos(p->np)[1] = s.y * p->pos.y;
        p++;
    }
 
    free(nlist);
    return 1;
}