text.mod/graphviz.mod/graphviz/lib/neatogen/constraint.c

/*************************************************************************
 * 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
 *************************************************************************/


#include "config.h"
#include <cgraph/list.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>

/* 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;
}

/* intersectX0:
 * 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;
}

/* intersectY0:
 * 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;
}

/* mapGraphs:
 */
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


/* mkNConstraintG:
 * 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;
}
/* mkConstraintG:
 */
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)
{
    node_t *n;
    for (n = agfstnode(cg); n; n = agnxtnode(cg, n)) {
	free_list(ND_in(n));
	free_list(ND_out(n));
    }
    agclose(cg);
}

/* constrainX:
 * 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, int nnodes, intersectfn ifn,
                       int ortho)
{
    Dt_t *list = dtopen(&constr, Dtobag);
    nitem *p = nlist;
    graph_t *cg;
    int i;

    for (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 (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);
}

/* constrainY:
 * See constrainX.
 */
static void constrainY(graph_t* g, nitem* nlist, int nnodes, intersectfn ifn,
                       int ortho)
{
    Dt_t *list = dtopen(&constr, Dtobag);
    nitem *p = nlist;
    graph_t *cg;
    int i;

    for (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(cg, AGEDGE, "minlen", "");
	Agsym_t *rksym = agattr(cg, AGNODE, "rank", "");
	char buf[100];
	node_t *n;
	edge_t *e;
	for (n = agfstnode(cg); n; n = agnxtnode(cg, n)) {
	    snprintf(buf, sizeof(buf), "%d", ND_rank(n));
	    agxset(n, rksym, buf);
	    for (e = agfstedge(cg, n); e; e = agnxtedge(cg, e, n)) {
		snprintf(buf, sizeof(buf), "%d", ED_minlen(e));
		agxset(e, mlsym, buf);
	    }
	}
    }
#endif

    p = nlist;
    for (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);
}

/* overlaps:
 */
static int overlaps(nitem * p, int cnt)
{
    int i, j;
    nitem *pi = p;
    nitem *pj;

    for (i = 0; i < cnt - 1; i++) {
	pj = pi + 1;
	for (j = i + 1; j < cnt; j++) {
	    if (OVERLAP(pi->bb, pj->bb))
		return 1;
	    pj++;
	}
	pi++;
    }
    return 0;
}

/* initItem:
 */
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 = SCALE * (POINTS(ND_width(n)/2.0) + margin.x);
	h2 = 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;
}

/* cAdjust:
 * 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, i, nnodes = agnnodes(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 (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, int nn)
{
    info *p = nl;
    info *q;
    int i, j;
    double s, sc = 0;
    pointf pt;

    for (i = 0; i < nn; i++) {
	q = p + 1;
	for (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;
}

DEFINE_LIST(points, pointf)

static pointf *mkOverlapSet(info *nl, size_t nn, size_t *cntp) {
    info *p = nl;
    info *q;
    points_t S = {0};

    points_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;
		}
		points_append(&S, pt);
	    }
	    q++;
	}
	p++;
    }

    points_shrink_to_fit(&S);
    *cntp = points_size(&S);
    return points_detach(&S);
}

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;
}

/* computeScale:
 * 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;
}

/* scAdjust:
 * 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)
{
    int nnodes = agnnodes(g);
    info *nlist = gv_calloc(nnodes, sizeof(info));
    info *p = nlist;
    node_t *n;
    pointf s;
    int i;
    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;
	assert(nnodes >= 0);
	aarr = mkOverlapSet(nlist, (size_t)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 (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;
}