partition.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
 *************************************************************************/
 
#include "config.h"
#include <assert.h>
#include <common/boxes.h>
#include <limits.h>
#include <ortho/partition.h>
#include <ortho/trap.h>
#include <math.h>
#include <stdbool.h>
#include <stdint.h>
#include <stdio.h>
#include <stdlib.h>
#include <util/alloc.h>
#include <util/bitarray.h>
#include <util/gv_math.h>
#include <util/prisize_t.h>
 
#ifndef DEBUG
  #define DEBUG 0
#endif
 
#define NPOINTS 4   /* only rectangles */
 
#define TR_FROM_UP 1        /* for traverse-direction */
#define TR_FROM_DN 2
 
#define SP_SIMPLE_LRUP 1    /* for splitting trapezoids */
#define SP_SIMPLE_LRDN 2
#define SP_2UP_2DN     3
#define SP_2UP_LEFT    4
#define SP_2UP_RIGHT   5
#define SP_2DN_LEFT    6
#define SP_2DN_RIGHT   7
#define SP_NOSPLIT    -1
 
#define DOT(v0, v1) ((v0).x * (v1).x + (v0).y * (v1).y)
#define CROSS_SINE(v0, v1) ((v0).x * (v1).y - (v1).x * (v0).y)
#define LENGTH(v0) hypot((v0).x, (v0).y)
 
#ifndef HAVE_SRAND48
#define srand48 srand
#endif
#ifdef _WIN32
extern double drand48(void);
#endif
 
typedef struct {
  int vnum;
  int next;         /* Circularly linked list  */
  int prev;         /* describing the monotone */
  int marked;           /* polygon */
} monchain_t;
 
typedef struct {
  pointf pt;
  int vnext[4];         /* next vertices for the 4 chains */
  int vpos[4];          /* position of v in the 4 chains */
  int nextfree;
} vertexchain_t;
 
static int chain_idx;
static size_t mon_idx;
        /* Table to hold all the monotone */
        /* polygons . Each monotone polygon */
        /* is a circularly linked list */
static monchain_t* mchain;
        /* chain init. information. This */
        /* is used to decide which */
        /* monotone polygon to split if */
        /* there are several other */
        /* polygons touching at the same */
        /* vertex  */
static vertexchain_t* vert;
        /* contains position of any vertex in */
        /* the monotone chain for the polygon */
static int* mon;
 
/* return a new mon structure from the table */
#define newmon() (++mon_idx)
/* return a new chain element from the table */
#define new_chain_element() (++chain_idx)
 
static void
convert (boxf bb, int flip, int ccw, pointf* pts)
{
    pts[0] = bb.LL;
    pts[2] = bb.UR;
    if (ccw) {
        pts[1].x = bb.UR.x;
        pts[1].y = bb.LL.y;
        pts[3].x = bb.LL.x;
        pts[3].y = bb.UR.y;
    }
    else {
        pts[1].x = bb.LL.x;
        pts[1].y = bb.UR.y;
        pts[3].x = bb.UR.x;
        pts[3].y = bb.LL.y;
    }
    if (flip) {
        int i;
        for (i = 0; i < NPOINTS; i++) {
            double tmp = pts[i].y;
            pts[i].y = pts[i].x;
            pts[i].x = -tmp;
        }
    }
}
 
static int
store (segment_t* seg, int first, pointf* pts)
{
    int i, last = first + NPOINTS - 1;
    int j = 0;
 
    for (i = first; i <= last; i++, j++) {
        if (i == first) {
            seg[i].next = first+1;
            seg[i].prev = last;
        }
        else if (i == last) {
            seg[i].next = first;
            seg[i].prev = last-1;
        }
        else {
            seg[i].next = i+1;
            seg[i].prev = i-1;
        }
        seg[i].is_inserted = false;
        seg[seg[i].prev].v1 = seg[i].v0 = pts[j];
    }
    return (last+1);
}
 
static void genSegments(cell *cells, size_t ncells, boxf bb, segment_t *seg,
                        int flip) {
    int i = 1;
    pointf pts[4];
 
    convert (bb, flip, 1, pts);
    i = store (seg, i, pts);
    for (size_t j = 0; j < ncells; j++) {
        convert (cells[j].bb, flip, 0, pts);
        i = store (seg, i, pts);
    }
}
 
/* Generate a random permutation of the segments 1..n */
static void generateRandomOrdering(size_t n, int *permute) {
    for (size_t i = 0; i < n; i++) {
        assert(i < INT_MAX);
        permute[i] = (int)i + 1;
    }
 
    for (size_t i = 0; i < n; i++) {
        const size_t j = (size_t)((double)i + drand48() * (double)(n - i));
        if (j != i) {
            SWAP(&permute[i], &permute[j]);
        }
    }
}
 
/* Function returns true if the trapezoid lies inside the polygon */
static bool
inside_polygon (trap_t *t, segment_t* seg)
{
  int rseg = t->rseg;
 
  if (!t->is_valid)
    return false;
 
  if (t->lseg <= 0 || t->rseg <= 0)
    return false;
  
  if ((!is_valid_trap(t->u0) && !is_valid_trap(t->u1)) || (!is_valid_trap(t->d0) && !is_valid_trap(t->d1))) // triangle
    return greater_than(seg[rseg].v1, seg[rseg].v0);
  
  return false;
}
 
static double
get_angle (pointf *vp0, pointf *vpnext, pointf *vp1)
{
  pointf v0, v1;
  
  v0.x = vpnext->x - vp0->x;
  v0.y = vpnext->y - vp0->y;
 
  v1.x = vp1->x - vp0->x;
  v1.y = vp1->y - vp0->y;
 
  if (CROSS_SINE(v0, v1) >= 0)  /* sine is positive */
    return DOT(v0, v1)/LENGTH(v0)/LENGTH(v1);
  else
    return -1.0 * DOT(v0, v1)/LENGTH(v0)/LENGTH(v1) - 2;
}
 
/* (v0, v1) is the new diagonal to be added to the polygon. Find which */
/* chain to use and return the positions of v0 and v1 in p and q */ 
static void
get_vertex_positions (int v0, int v1, int *ip, int *iq)
{
  vertexchain_t *vp0, *vp1;
  int i;
  double angle, temp;
  int tp = 0, tq = 0;
 
  vp0 = &vert[v0];
  vp1 = &vert[v1];
  
  /* p is identified as follows. Scan from (v0, v1) rightwards till */
  /* you hit the first segment starting from v0. That chain is the */
  /* chain of our interest */
  
  angle = -4.0;
  for (i = 0; i < 4; i++)
    {
      if (vp0->vnext[i] <= 0)
        continue;
      if ((temp = get_angle(&vp0->pt, &(vert[vp0->vnext[i]].pt), 
                            &vp1->pt)) > angle)
        {
          angle = temp;
          tp = i;
        }
    }
 
  *ip = tp;
 
  /* Do similar actions for q */
 
  angle = -4.0;
  for (i = 0; i < 4; i++)
    {
      if (vp1->vnext[i] <= 0)
        continue;      
      if ((temp = get_angle(&vp1->pt, &(vert[vp1->vnext[i]].pt), 
                            &vp0->pt)) > angle)
        {
          angle = temp;
          tq = i;
        }
    }
 
  *iq = tq;
}
 
/* v0 and v1 are specified in anti-clockwise order with respect to 
 * the current monotone polygon mcur. Split the current polygon into 
 * two polygons using the diagonal (v0, v1) 
 */
static size_t make_new_monotone_poly(size_t mcur, int v0, int v1) {
  int p, q, ip, iq;
  const size_t mnew = newmon();
  int i, j, nf0, nf1;
  vertexchain_t *vp0, *vp1;
  
  vp0 = &vert[v0];
  vp1 = &vert[v1];
 
  get_vertex_positions(v0, v1, &ip, &iq);
 
  p = vp0->vpos[ip];
  q = vp1->vpos[iq];
 
  /* At this stage, we have got the positions of v0 and v1 in the */
  /* desired chain. Now modify the linked lists */
 
  i = new_chain_element();      /* for the new list */
  j = new_chain_element();
 
  mchain[i].vnum = v0;
  mchain[j].vnum = v1;
 
  mchain[i].next = mchain[p].next;
  mchain[mchain[p].next].prev = i;
  mchain[i].prev = j;
  mchain[j].next = i;
  mchain[j].prev = mchain[q].prev;
  mchain[mchain[q].prev].next = j;
 
  mchain[p].next = q;
  mchain[q].prev = p;
 
  nf0 = vp0->nextfree;
  nf1 = vp1->nextfree;
 
  vp0->vnext[ip] = v1;
 
  vp0->vpos[nf0] = i;
  vp0->vnext[nf0] = mchain[mchain[i].next].vnum;
  vp1->vpos[nf1] = j;
  vp1->vnext[nf1] = v0;
 
  vp0->nextfree++;
  vp1->nextfree++;
 
#if DEBUG > 0
  fprintf(stderr, "make_poly: mcur = %" PRISIZE_T ", (v0, v1) = (%d, %d)\n",
          mcur, v0, v1);
  fprintf(stderr, "next posns = (p, q) = (%d, %d)\n", p, q);
#endif
 
  mon[mcur] = p;
  mon[mnew] = i;
  return mnew;
}
 
/* recursively visit all the trapezoids */
static void traverse_polygon(bitarray_t *visited, boxes_t *decomp,
                             segment_t *seg, traps_t *tr, size_t mcur,
                             size_t trnum, size_t from, int flip, int dir) {
  size_t mnew;
  int v0, v1;
 
  if (!is_valid_trap(trnum) || bitarray_get(*visited, trnum))
    return;
 
  trap_t *t = traps_at(tr, trnum);
 
  bitarray_set(visited, trnum, true);
  
  if (t->hi.y > t->lo.y + C_EPS && fp_equal(seg[t->lseg].v0.x, seg[t->lseg].v1.x) &&
      fp_equal(seg[t->rseg].v0.x, seg[t->rseg].v1.x)) {
      boxf newbox = {0};
      if (flip) {
          newbox.LL.x = t->lo.y;
          newbox.LL.y = -seg[t->rseg].v0.x;
          newbox.UR.x = t->hi.y;
          newbox.UR.y = -seg[t->lseg].v0.x;
      } else {
          newbox.LL.x = seg[t->lseg].v0.x;
          newbox.LL.y = t->lo.y;
          newbox.UR.x = seg[t->rseg].v0.x;
          newbox.UR.y = t->hi.y;
      }
      boxes_append(decomp, newbox);
  }
  
  /* We have much more information available here. */
  /* rseg: goes upwards   */
  /* lseg: goes downwards */
 
  /* Initially assume that dir = TR_FROM_DN (from the left) */
  /* Switch v0 and v1 if necessary afterwards */
 
 
  /* special cases for triangles with cusps at the opposite ends. */
  /* take care of this first */
  if (!is_valid_trap(t->u0) && !is_valid_trap(t->u1)) {
      if (is_valid_trap(t->d0) && is_valid_trap(t->d1)) { // downward opening triangle
          v0 = traps_get(tr, t->d1).lseg;
          v1 = t->lseg;
          if (from == t->d1)
            {
              mnew = make_new_monotone_poly(mcur, v1, v0);
              traverse_polygon(visited, decomp, seg, tr, mcur, t->d1, trnum, flip, TR_FROM_UP);
              traverse_polygon(visited, decomp, seg, tr, mnew, t->d0, trnum, flip, TR_FROM_UP);
            }
          else
            {
              mnew = make_new_monotone_poly(mcur, v0, v1);
              traverse_polygon (visited, decomp, seg, tr, mcur, t->d0, trnum, flip, TR_FROM_UP);
              traverse_polygon (visited, decomp, seg, tr, mnew, t->d1, trnum, flip, TR_FROM_UP);
            }
        }
      else
        {
          /* Just traverse all neighbours */
          traverse_polygon(visited, decomp, seg, tr, mcur, t->u0, trnum, flip, TR_FROM_DN);
          traverse_polygon(visited, decomp, seg, tr, mcur, t->u1, trnum, flip, TR_FROM_DN);
          traverse_polygon(visited, decomp, seg, tr, mcur, t->d0, trnum, flip, TR_FROM_UP);
          traverse_polygon(visited, decomp, seg, tr, mcur, t->d1, trnum, flip, TR_FROM_UP);
        }
    }
  
  else if (!is_valid_trap(t->d0) && !is_valid_trap(t->d1)) {
      if (is_valid_trap(t->u0) && is_valid_trap(t->u1)) { // upward opening triangle
          v0 = t->rseg;
          v1 = traps_get(tr, t->u0).rseg;
          if (from == t->u1)
            {
              mnew = make_new_monotone_poly(mcur, v1, v0);
              traverse_polygon(visited, decomp, seg, tr, mcur, t->u1, trnum, flip, TR_FROM_DN);
              traverse_polygon(visited, decomp, seg, tr, mnew, t->u0, trnum, flip, TR_FROM_DN);
            }
          else
            {
              mnew = make_new_monotone_poly(mcur, v0, v1);
              traverse_polygon(visited, decomp, seg, tr, mcur, t->u0, trnum, flip, TR_FROM_DN);
              traverse_polygon(visited, decomp, seg, tr, mnew, t->u1, trnum, flip, TR_FROM_DN);
            }
        }
      else
        {
          /* Just traverse all neighbours */
          traverse_polygon(visited, decomp, seg, tr, mcur, t->u0, trnum, flip, TR_FROM_DN);
          traverse_polygon(visited, decomp, seg, tr, mcur, t->u1, trnum, flip, TR_FROM_DN);
          traverse_polygon(visited, decomp, seg, tr, mcur, t->d0, trnum, flip, TR_FROM_UP);
          traverse_polygon(visited, decomp, seg, tr, mcur, t->d1, trnum, flip, TR_FROM_UP);
        }
    }
  
  else if (is_valid_trap(t->u0) && is_valid_trap(t->u1)) {
      if (is_valid_trap(t->d0) && is_valid_trap(t->d1)) { // downward + upward cusps
          v0 = traps_get(tr, t->d1).lseg;
          v1 = traps_get(tr, t->u0).rseg;
          if ((dir == TR_FROM_DN && t->d1 == from) ||
              (dir == TR_FROM_UP && t->u1 == from))
            {
              mnew = make_new_monotone_poly(mcur, v1, v0);
              traverse_polygon(visited, decomp, seg, tr, mcur, t->u1, trnum, flip, TR_FROM_DN);
              traverse_polygon(visited, decomp, seg, tr, mcur, t->d1, trnum, flip, TR_FROM_UP);
              traverse_polygon(visited, decomp, seg, tr, mnew, t->u0, trnum, flip, TR_FROM_DN);
              traverse_polygon(visited, decomp, seg, tr, mnew, t->d0, trnum, flip, TR_FROM_UP);
            }
          else
            {
              mnew = make_new_monotone_poly(mcur, v0, v1);
              traverse_polygon(visited, decomp, seg, tr, mcur, t->u0, trnum, flip, TR_FROM_DN);
              traverse_polygon(visited, decomp, seg, tr, mcur, t->d0, trnum, flip, TR_FROM_UP);
              traverse_polygon(visited, decomp, seg, tr, mnew, t->u1, trnum, flip, TR_FROM_DN);
              traverse_polygon(visited, decomp, seg, tr, mnew, t->d1, trnum, flip, TR_FROM_UP);
            }
        }
      else                      /* only downward cusp */
        {
          if (equal_to(t->lo, seg[t->lseg].v1)) {
              v0 = traps_get(tr, t->u0).rseg;
              v1 = seg[t->lseg].next;
 
              if (dir == TR_FROM_UP && t->u0 == from)
                {
                  mnew = make_new_monotone_poly(mcur, v1, v0);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->u0, trnum, flip, TR_FROM_DN);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->d0, trnum, flip, TR_FROM_UP);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->u1, trnum, flip, TR_FROM_DN);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->d1, trnum, flip, TR_FROM_UP);
                }
              else
                {
                  mnew = make_new_monotone_poly(mcur, v0, v1);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->u1, trnum, flip, TR_FROM_DN);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->d0, trnum, flip, TR_FROM_UP);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->d1, trnum, flip, TR_FROM_UP);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->u0, trnum, flip, TR_FROM_DN);
                }
            }
          else
            {
              v0 = t->rseg;
              v1 = traps_get(tr, t->u0).rseg;   
              if (dir == TR_FROM_UP && t->u1 == from)
                {
                  mnew = make_new_monotone_poly(mcur, v1, v0);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->u1, trnum, flip, TR_FROM_DN);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->d1, trnum, flip, TR_FROM_UP);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->d0, trnum, flip, TR_FROM_UP);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->u0, trnum, flip, TR_FROM_DN);
                }
              else
                {
                  mnew = make_new_monotone_poly(mcur, v0, v1);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->u0, trnum, flip, TR_FROM_DN);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->d0, trnum, flip, TR_FROM_UP);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->d1, trnum, flip, TR_FROM_UP);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->u1, trnum, flip, TR_FROM_DN);
                }
            }
        }
    }
  else if (is_valid_trap(t->u0) || is_valid_trap(t->u1)) { // no downward cusp
      if (is_valid_trap(t->d0) && is_valid_trap(t->d1)) { // only upward cusp
          if (equal_to(t->hi, seg[t->lseg].v0)) {
              v0 = traps_get(tr, t->d1).lseg;
              v1 = t->lseg;
              if (!(dir == TR_FROM_DN && t->d0 == from))
                {
                  mnew = make_new_monotone_poly(mcur, v1, v0);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->u1, trnum, flip, TR_FROM_DN);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->d1, trnum, flip, TR_FROM_UP);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->u0, trnum, flip, TR_FROM_DN);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->d0, trnum, flip, TR_FROM_UP);
                }
              else
                {
                  mnew = make_new_monotone_poly(mcur, v0, v1);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->d0, trnum, flip, TR_FROM_UP);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->u0, trnum, flip, TR_FROM_DN);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->u1, trnum, flip, TR_FROM_DN);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->d1, trnum, flip, TR_FROM_UP);
                }
            }
          else
            {
              v0 = traps_get(tr, t->d1).lseg;
              v1 = seg[t->rseg].next;
 
              if (dir == TR_FROM_DN && t->d1 == from)
                {
                  mnew = make_new_monotone_poly(mcur, v1, v0);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->d1, trnum, flip, TR_FROM_UP);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->u1, trnum, flip, TR_FROM_DN);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->u0, trnum, flip, TR_FROM_DN);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->d0, trnum, flip, TR_FROM_UP);
                }
              else
                {
                  mnew = make_new_monotone_poly(mcur, v0, v1);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->u0, trnum, flip, TR_FROM_DN);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->d0, trnum, flip, TR_FROM_UP);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->u1, trnum, flip, TR_FROM_DN);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->d1, trnum, flip, TR_FROM_UP);
                }
            }
        }
      else                      /* no cusp */
        {
          if (equal_to(t->hi, seg[t->lseg].v0) && equal_to(t->lo, seg[t->rseg].v0)) {
              v0 = t->rseg;
              v1 = t->lseg;
              if (dir == TR_FROM_UP)
                {
                  mnew = make_new_monotone_poly(mcur, v1, v0);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->u0, trnum, flip, TR_FROM_DN);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->u1, trnum, flip, TR_FROM_DN);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->d1, trnum, flip, TR_FROM_UP);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->d0, trnum, flip, TR_FROM_UP);
                }
              else
                {
                  mnew = make_new_monotone_poly(mcur, v0, v1);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->d1, trnum, flip, TR_FROM_UP);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->d0, trnum, flip, TR_FROM_UP);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->u0, trnum, flip, TR_FROM_DN);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->u1, trnum, flip, TR_FROM_DN);
                }
            }
          else if (equal_to(t->hi, seg[t->rseg].v1) &&
                   equal_to(t->lo, seg[t->lseg].v1)) {
              v0 = seg[t->rseg].next;
              v1 = seg[t->lseg].next;
 
              if (dir == TR_FROM_UP)
                {
                  mnew = make_new_monotone_poly(mcur, v1, v0);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->u0, trnum, flip, TR_FROM_DN);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->u1, trnum, flip, TR_FROM_DN);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->d1, trnum, flip, TR_FROM_UP);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->d0, trnum, flip, TR_FROM_UP);
                }
              else
                {
                  mnew = make_new_monotone_poly(mcur, v0, v1);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->d1, trnum, flip, TR_FROM_UP);
                  traverse_polygon(visited, decomp, seg, tr, mcur, t->d0, trnum, flip, TR_FROM_UP);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->u0, trnum, flip, TR_FROM_DN);
                  traverse_polygon(visited, decomp, seg, tr, mnew, t->u1, trnum, flip, TR_FROM_DN);
                }
            }
          else                  /* no split possible */
            {
              traverse_polygon(visited, decomp, seg, tr, mcur, t->u0, trnum, flip, TR_FROM_DN);
              traverse_polygon(visited, decomp, seg, tr, mcur, t->d0, trnum, flip, TR_FROM_UP);
              traverse_polygon(visited, decomp, seg, tr, mcur, t->u1, trnum, flip, TR_FROM_DN);
              traverse_polygon(visited, decomp, seg, tr, mcur, t->d1, trnum, flip, TR_FROM_UP);
            }
        }
    }
}
 
static void
monotonate_trapezoids(int nsegs, segment_t *seg, traps_t *tr, 
    int flip, boxes_t *decomp) {
    int i;
    bitarray_t visited = bitarray_new(traps_size(tr));
 
    mchain = gv_calloc(traps_size(tr), sizeof(monchain_t));
    vert = gv_calloc(nsegs + 1, sizeof(vertexchain_t));
    mon = gv_calloc(nsegs, sizeof(int));
 
  /* First locate a trapezoid which lies inside the polygon */
  /* and which is triangular */
    size_t j;
    for (j = 0; j < traps_size(tr); j++)
        if (inside_polygon(traps_at(tr, j), seg)) break;
    const size_t tr_start = j;
  
  /* Initialise the mon data-structure and start spanning all the */
  /* trapezoids within the polygon */
 
    for (i = 1; i <= nsegs; i++) {
        mchain[i].prev = seg[i].prev;
        mchain[i].next = seg[i].next;
        mchain[i].vnum = i;
        vert[i].pt = seg[i].v0;
        vert[i].vnext[0] = seg[i].next; /* next vertex */
        vert[i].vpos[0] = i;    /* locn. of next vertex */
        vert[i].nextfree = 1;
    }
 
    chain_idx = nsegs;
    mon_idx = 0;
    mon[0] = 1;                 /* position of any vertex in the first */
                                /* chain  */
  
  /* traverse the polygon */
    if (is_valid_trap(traps_get(tr, tr_start).u0))
        traverse_polygon(&visited, decomp, seg, tr, 0, tr_start,
                         traps_get(tr, tr_start).u0, flip, TR_FROM_UP);
    else if (is_valid_trap(traps_get(tr, tr_start).d0))
        traverse_polygon(&visited, decomp, seg, tr, 0, tr_start,
                         traps_get(tr, tr_start).d0, flip, TR_FROM_DN);
  
    bitarray_reset(&visited);
    free (mchain);
    free (vert);
    free (mon);
}
 
static bool rectIntersect(boxf *d, const boxf r0, const boxf r1) {
    double t = fmax(r0.LL.x, r1.LL.x);
    d->UR.x = fmin(r0.UR.x, r1.UR.x);
    d->LL.x = t;
    
    t = fmax(r0.LL.y, r1.LL.y);
    d->UR.y = fmin(r0.UR.y, r1.UR.y);
    d->LL.y = t;
 
    return !(d->LL.x >= d->UR.x || d->LL.y >= d->UR.y);
}
 
#if DEBUG > 1
static void
dumpTrap (trap_t* tr, int n)
{
    int i;
    for (i = 1; i <= n; i++) {
      tr++;
      fprintf(stderr, "%d : %d %d (%f,%f) (%f,%f) %" PRISIZE_T " %" PRISIZE_T
              " %" PRISIZE_T " %" PRISIZE_T "\n", i, tr->lseg, tr->rseg,
              tr->hi.x, tr->hi.y, tr->lo.x, tr->lo.y, tr->u0, tr->u1, tr->d0,
              tr->d1);
      fprintf(stderr, "    %" PRISIZE_T " %" PRISIZE_T " %d %s\n", tr->sink,
              tr->usave, tr->uside, tr->is_valid ? "valid" : "invalid");
    }
    fprintf (stderr, "====\n");
}
 
static void
dumpSegs (segment_t* sg, int n)
{
    int i;
    for (i = 1; i <= n; i++) {
      sg++;
      fprintf(stderr, "%d : (%f,%f) (%f,%f) %d %" PRISIZE_T " %" PRISIZE_T
              " %d %d\n", i, sg->v0.x, sg->v0.y, sg->v1.x, sg->v1.y,
              (int)sg->is_inserted, sg->root0,  sg->root1, sg->next, sg->prev);
    }
    fprintf (stderr, "====\n");
}
#endif
 
boxf *partition(cell *cells, size_t ncells, size_t *nrects, boxf bb) {
    const size_t nsegs = 4 * (ncells + 1);
    segment_t* segs = gv_calloc(nsegs + 1, sizeof(segment_t));
    int* permute = gv_calloc(nsegs, sizeof(int));
 
    if (DEBUG) {
        fprintf(stderr, "cells = %" PRISIZE_T " segs = %" PRISIZE_T
                " traps = dynamic\n", ncells, nsegs);
    }
    genSegments(cells, ncells, bb, segs, 0);
    if (DEBUG) {
        fprintf(stderr, "%" PRISIZE_T "\n\n", ncells + 1);
        for (size_t i = 1; i <= nsegs; i++) {
            if (i%4 == 1) fprintf(stderr, "4\n");
            fprintf (stderr, "%f %f\n", segs[i].v0.x, segs[i].v0.y);
            if (i%4 == 0) fprintf(stderr, "\n");
        }
    }
    srand48(173);
    generateRandomOrdering(nsegs, permute);
    assert(nsegs <= INT_MAX);
    traps_t hor_traps = construct_trapezoids((int)nsegs, segs, permute);
    if (DEBUG) {
        fprintf(stderr, "hor traps = %" PRISIZE_T "\n", traps_size(&hor_traps));
    }
    boxes_t hor_decomp = {0};
    monotonate_trapezoids((int)nsegs, segs, &hor_traps, 0, &hor_decomp);
    traps_free(&hor_traps);
 
    genSegments(cells, ncells, bb, segs, 1);
    generateRandomOrdering(nsegs, permute);
    traps_t ver_traps = construct_trapezoids((int)nsegs, segs, permute);
    if (DEBUG) {
        fprintf(stderr, "ver traps = %" PRISIZE_T "\n", traps_size(&ver_traps));
    }
    boxes_t vert_decomp = {0};
    monotonate_trapezoids((int)nsegs, segs, &ver_traps, 1, &vert_decomp);
    traps_free(&ver_traps);
 
    boxes_t rs = {0};
    for (size_t i = 0; i < boxes_size(&vert_decomp); ++i)
        for (size_t j = 0; j < boxes_size(&hor_decomp); ++j) {
            boxf newbox = {0};
            if (rectIntersect(&newbox, boxes_get(&vert_decomp, i),
                              boxes_get(&hor_decomp, j)))
                boxes_append(&rs, newbox);
        }
 
    free (segs);
    free (permute);
    boxes_free(&hor_decomp);
    boxes_free(&vert_decomp);
    *nrects = boxes_size(&rs);
    return boxes_detach(&rs);
}