% This file is part of the Stanford GraphBase (c) Stanford University 1990
\nocon

@* Introduction. This is a hastily written implementation of hull insertion.

@f Graph int /* |gb_graph| defines the |Graph| type and a few others */
@f Vertex int
@f Arc int
@f Area int

@p
#include "gb_graph.h"
#include "gb_miles.h"
int n=128;
@<Global variables@>@;
@<Procedures@>@;
@#
main()
{
  @<Local variables@>@;
  Graph *g=miles(128,0,0,0,0,0,0);
@#
  mems=ccs=0;
  @<Find convex hull of |g|@>;
  printf("Total of %d mems and %d calls on ccw.\n",mems,ccs);
}

@ I'm instrumenting this in a simple way.

@d o mems++
@d oo mems+=2

@<Glob...@>=
int mems; /* memory accesses */
int ccs; /* calls on |ccw| */
int serial_no=1; /* used to disambiguate entries with equal coordinates */

@*Data structures.
For now, each vertex is represented by two coordinates stored in the
utility fields |x.I| and |y.I|. I'm also putting a serial number into
|z.I|, so that I can check whether different algorithms generate
identical hulls.

A vertex |v| in the convex hull also has a successor |v->succ| and
and predecessor |v->pred|, stored in utility fields |u| and |v|.

This implementation is the simplest one I know; it simply walks around the
current convex hull each time, therefore not really bad if the current
hull never gets big.

@d succ u.V
@d pred v.V

@ @<Glob...@>=
Vertex *rover; /* one of the vertices in the convex hull */

@ We assume that the vertices have been given to us in a GraphBase-type
graph. The algorithm begins with a trivial hull that contains
only the first two vertices.

@<Initialize the data structures@>=
o,u=g->vertices;
v=u+1;
u->z.I=0; v->z.I=1;
oo,u->succ=u->pred=v;
oo,v->succ=v->pred=u;
rover=u;
if (n<150) printf("Beginning with (%s; %s)\n",u->name,v->name);

@ We'll probably need a bunch of local variables to do elementary operations on
data structures.

@<Local...@>=
Vertex *u,*v,*vv,*w;

@*Hull updating.
The main loop of the algorithm updates the data structure incrementally
by adding one new vertex at a time. If the new vertex lies outside the
current convex hull, we put it into the cycle and possibly delete some
vertices that were previously part of the hull.

@<Find convex hull of |g|@>=
@<Initialize the data structures@>;
for (oo,vv=g->vertices+2;vv<g->vertices+g->n;vv++) {
  vv->z.I=++serial_no;
  @<Go around the current hull; |continue| if |vv| is inside it@>;
  @<Update the convex hull, knowing that |vv| lies outside the consecutive
     hull vertices |u| and |v|@>;
}
@<Print the convex hull@>;

@ Let me do the easy part first, since it's bedtime and I can worry about
the rest tomorrow.

@<Print the convex hull@>=
u=rover;
printf("The convex hull is:\n");
do {
  printf("  %s\n",u->name);
  u=u->succ;
} while (u!=rover);

@ @<Go around...@>=
u=rover;
do {
  o,v=u->succ;
  if (ccw(u,vv,v)) goto found;
  u=v;
} while (u!=rover);
continue;
found:;

@ @<Update the convex hull, knowing that |vv| lies outside the consecutive
     hull vertices |u| and |v|@>=
if (u==rover) {
  while (1) {
    o,w=u->pred;
    if (w==v) break;
    if (ccw(vv,w,u)) break;
    u=w;
  }
  rover=w;
}
while (1) {
  if (v==rover) break;
  o,w=v->succ;
  if (ccw(w,vv,v)) break;
  v=w;
}
oo,u->succ=v->pred=vv;
oo,vv->pred=u;@+vv->succ=v;
if (n<150) printf("New hull sequence (%s; %s; %s)\n",u->name,vv->name,v->name);
 
@*Determinants. I need code for the primitive function |ccw|.
Floating-point arithmetic suffices for my purposes.

We want to evaluate the determinant
$$ccw(u,v,w)=\left\vert\matrix{u(x)&u(y)&1\cr v(x)&v(y)&1\cr w(x)&w(y)&1\cr}
 \right\vert=\left\vert\matrix{u(x)-w(x)&u(y)-w(y)\cr v(x)-w(x)&v(y)-w(y)\cr}
 \right\vert\,.$$

@<Proc...@>=
int ccw(u,v,w)
  Vertex *u,*v,*w;
{@+register double wx=(double)w->x.I, wy=(double)w->y.I;
  register double det=((double)u->x.I-wx)*((double)v->y.I-wy)
         -((double)u->y.I-wy)*((double)v->x.I-wx);
  Vertex *uu=u,*vv=v,*ww=w,*t;
  if (det==0) {
    det=1;
    if (u->x.I>v->x.I || (u->x.I==v->x.I && (u->y.I>v->y.I ||
         (u->y.I==v->y.I && u->z.I>v->z.I)))) {
           t=u;@+u=v;@+v=t;@+det=-det;
    }
    if (v->x.I>w->x.I || (v->x.I==w->x.I && (v->y.I>w->y.I ||
         (v->y.I==w->y.I && v->z.I>w->z.I)))) {
           t=v;@+v=w;@+w=t;@+det=-det;
    }
    if (u->x.I>v->x.I || (u->x.I==v->x.I && (u->y.I>v->y.I ||
         (u->y.I==v->y.I && u->z.I<v->z.I)))) {
           det=-det;
    }
  }
  if (n<150) printf("cc(%s; %s; %s) is %s\n",uu->name,vv->name,ww->name,
    det>0? "true": "false");
  ccs++;
  return (det>0);
}