path: root/support/graphbase/gb_rand.w

% This file is part of the Stanford GraphBase (c) Stanford University 1992
\def\title{GB\_\thinspace RAND}
@i boilerplate.w %<< legal stuff: PLEASE READ IT BEFORE MAKING ANY CHANGES!

\prerequisite{GB\_\thinspace GRAPH}
@*Random graphs. This GraphBase module provides two external
subroutines called |random_graph| and |random_bigraph|, which generate
graphs in which the arcs or edges have been selected ``at random.''  A
third subroutine, |random_lengths|, randomizes the lengths of the arcs
of a given graph.  The performance of algorithms on such graphs can
fruitfully be compared to their performance on the nonrandom graphs
generated by other GraphBase routines.

Before reading this code, the reader should be familiar with the
basic data structures and conventions described in |gb_graph|. The
routines in |gb_graph| are loaded together with all GraphBase applications,
and the programs below are typical illustrations of how to use them.

@f Graph int /* module |gb_graph| defines several types including these */
@f Vertex int
@f Arc int
@f Area int

@(gb_rand.h@>=
extern Graph *random_graph();
 /* users of |gb_rand| should include this header info */
extern Graph *random_bigraph();
extern int random_lengths();

@ Here is an overview of the file \.{gb\_rand.c}, the \Cee\ code from which
users can obtain the routines |random_graph| and |random_bigraph|:

@p
#include "gb_graph.h" /* this header file teaches \Cee\ about GraphBase */
#include "gb_flip.h" /* we will use the |gb_flip| routines for random numbers */
@<Private declarations@>@;
@<Internal functions@>@;
@<External functions@>

@ The procedure |random_graph(n,m,multi,self,directed,dist_from,dist_to,min_len,max_len,seed)|
is designed to produce a pseudo-random graph with |n| vertices and |m| arcs or
edges, using pseudo-random numbers that depend on |seed| in a system-independent
fashion. The remaining parameters specify a variety of options:
$$\vcenter{\halign{#\hfil\cr
|multi!=0| permits duplicate arcs;\cr
|self!=0| permits self-loops (arcs from a vertex to itself);\cr
|directed!=0| makes the graph directed; otherwise each arc becomes an undirected
  edge;\cr
|dist_from| and |dist_to| specify probability distributions on the arcs;\cr
|min_len| and |max_len| bound the arc lengths, which will be uniformly
distributed between these limits.\cr
}}$$
If |dist_from| or |dist_to| are |NULL|, the probability distribution is
uniform over vertices; otherwise the \\{dist} parameter points to an array of
|n| nonnegative integers that sum to $2^{30}$, specifying the respective
probabilities (times $2^{30}$) that each given vertex will appear as the
source or destination of the random arcs.

A special option |multi=-1| is provided. This acts exactly like |multi=1|, except
that arcs are not physically duplicated in computer memory---they are replaced
by a single arc whose length is the minimum of all arcs having a common source
and destination.

The vertices are named simply |"0"|, |"1"|, |"2"|, and so on.

@ Examples: |random_graph(1000,5000,0,0,0,NULL,NULL,1,1,0)| creates a random
undirected graph with 1000 vertices and 5000 edges (hence 10000 arcs) of
length~1, having
no duplicate edges or self-loops. There are ${1000\choose2}=499500$ possible
undirected edges on 1000 vertices, hence there are exactly $499500\choose5000$
possible graphs meeting these specifications; every such graph would be
equally likely, if |random_graph| had access to an ideal source of
random numbers. The GraphBase programs are designed to be
system-independent, so that identical graphs will be obtained by
everybody who asks for |random_graph(1000,5000,0,0,0,NULL,NULL,1,1,0)|.
Equivalent experiments on algorithms for graph manipulation can therefore
be performed by researchers in different parts of the world.

The subroutine call |random_graph(1000,5000,0,0,0,NULL,NULL,1,1,s)|
will produce different graphs when the random seed |s| varies;
however, the graph for any particular value of~|s| will be the same on
all computers. The seed value can be any integer in the range $0\le s<2^{31}$.

To get a random directed graph, allowing self-loops and repeated arcs,
and with a uniform distribution on vertices, ask for
$$\hbox{|random_graph(n,m,1,1,1,NULL,NULL,1,1,s)|}.$$
Each of the $m$ arcs of that digraph has probability $1/n^2$ of being from
$u$ to $v$, for all $u$ and~$v$. If self-loops are disallowed (by
changing `|1,1,1|' to `|1,0,1|'), each arc has probability
$1/(n^2-n)$ of being from $u$ to $v$, for all $u\ne v$.

To get a random directed graph in which vertex $k$ is twice as likely
as vertex $k+1$ to be the source of an arc but only half as likely to
be the destination of an arc, for all~$k$, try
$$\hbox{|random_graph(25,m,1,1,1,d0,d1,0,255,s)|}$$
where the arrays |d0| and |d1| have the static declarations
$$\vbox{
\hbox{|long d0[31]={0x20000000,0x10000000,@t\dots@>,4,2,1,1};|}
\hbox{|long d1[31]={1,1,2,4,@t\dots@>,0x10000000,0x20000000};|}}$$
then about 1/4 of the arcs will run from 0 to 30, while arcs
from 30 to 0 will be extremely rare (occurring with probability $2^{-60}$).
Incidentally, the arc lengths in this example will be random bytes,
uniformly distributed between 0 and 255, because the |min_len=0| and
|max_len=255|.

If we forbid repeated arcs in this example, by setting |multi=0|, the
effect is to discard all arcs having the same source and destination
as a previous arc, regardless of length. In such a case |m| had better not
be too large, because the algorithm will keep going until it has found
|m| distinct arcs, and many arcs are quite rare indeed; they will
probably not be found until hundreds of centuries have elapsed.

A random bipartite graph can also be obtained as a special case of
|random_graph|; this case is explained below.

Semantics:
If |multi=directed=0| and |self!=0|, we have an undirected graph without duplicate
edges but with self-loops permitted. A self-loop then consists of
two identical self-arcs, in spite of the fact that |multi=0|.

@ If the |random_graph| routine encounters a problem, it returns
|NULL|, after putting a code number into the external variable
|panic_code|. This code number identifies the type of failure.
Otherwise |random_graph| returns a pointer to the newly created graph
and leaves |panic_code| unchanged. The |gb_alloc_trouble| will be
cleared to zero after |random_graph| has acted.

@d panic(c) @+{@+panic_code=c;@+gb_alloc_trouble=0;@+return NULL;@+}

@<External f...@>=
Graph *random_graph(n,m,multi,self,directed,dist_from,dist_to,min_len,max_len,seed)
  unsigned n; /* number of vertices desired */
  unsigned long m; /* number of arcs or edges desired */
  int multi; /* allow duplicate arcs? */
  int self; /* allow self loops? */
  int directed; /* directed graph? */
  long *dist_from; /* distribution of arc sources */
  long *dist_to; /* distribution of of arc destinations */
  long min_len,max_len; /* bounds on random lengths */
  long seed; /* random number seed */
{@+@<Local variables@>@;
@#
  if (n==0) panic(bad_specs); /* we gotta have a vertex */
  if (min_len>max_len) panic(very_bad_specs); /* what are you trying to do? */
  if (((unsigned long)(max_len))-((unsigned long)(min_len))>=
      ((unsigned long)0x80000000)) panic(bad_specs+1); /* too much range */
  @<Check the distribution parameters@>;
  gb_init_rand(seed);
  @<Create a graph with |n| vertices and no arcs@>;
  @<Build tables for nonuniform distributions, if needed@>;
  for (mm=m; mm; mm--)
    @<Add a random arc or a random edge@>;
trouble: if (gb_alloc_trouble) {
    gb_recycle(new_graph);
    panic(alloc_fault); /* oops, we ran out of memory somewhere back there */
  }
  gb_free(new_graph->aux_data);
  return new_graph;
}

@ @<Local var...@>=
Graph *new_graph; /* the graph constructed by |random_graph| */
long mm; /* the number of arcs or edges we still need to generate */
register int k; /* vertex being processed */

@ @d dist_code(x) (x? "dist": "0")

@<Create a graph with |n| vertices and no arcs@>=
new_graph=gb_new_graph(n);
if (new_graph==NULL)
  panic(no_room); /* out of memory before we're even started */
for (k=0; k<n; k++) {
  sprintf(name_buffer,"%d",k);
  (new_graph->vertices+k)->name=gb_save_string(name_buffer);
}
sprintf(new_graph->id,"random_graph(%u,%lu,%d,%d,%d,%s,%s,%ld,%ld,%ld)",@|
 n,m,multi>0?1:multi<0?-1:0,self?1:0,directed?1:0,@|
 dist_code(dist_from),dist_code(dist_to),min_len,max_len,seed);

@ @<Private d...@>=
static char name_buffer[]="9999999999";

@ @d rand_len (min_len==max_len?min_len:min_len+gb_unif_rand(max_len-min_len))

@<Add a random arc or a random edge@>=
{@+register Vertex *u,*v;
repeat:
  if (dist_from)
    @<Generate a random vertex |u| according to |dist_from|@>@;
  else u=new_graph->vertices+gb_unif_rand(n);
  if (dist_to)
    @<Generate a random vertex |v| according to |dist_to|@>@;
  else v=new_graph->vertices+gb_unif_rand(n);
  if (u==v && !self) goto repeat;
  if (multi<=0)
    @<Search for duplicate arcs or edges; |goto repeat| or |done| if found@>;
  if (directed) gb_new_arc(u,v,rand_len);
  else gb_new_edge(u,v,rand_len);
done:;
}

@ When we decrease the length of an existing edge, we use the fact that
its two arcs are adjacent in memory. If |u==v| in this case, we encounter
the first of two mated arcs before seeing the second; hence the mate of
the arc we find is in location |a+1| when |u<=v|, and in location
|a-1| when |u>v|.

We must exit to location |trouble| if memory has been exhausted;
otherwise there is a danger of an infinite loop, with |dummy_arc->next
=dummy_arc|.

@<Search for duplicate arcs or edges; |goto repeat| or |done| if found@>=
if (gb_alloc_trouble) goto trouble;
else {@+register Arc *a;
  long len; /* length of new arc or edge being combined with previous */
  for (a=u->arcs; a; a=a->next)
    if (a->tip==v)
      if (multi==0) goto repeat; /* reject a duplicate arc */
      else { /* |multi<0| */
        len=rand_len;
        if (len<a->len) {
          a->len=len;
          if (!directed) {
            if (u<=v) (a+1)->len=len;
            else (a-1)->len=len;
          }
        }
        goto done;
      }
}

@* Nonuniform random number generation. The |random_graph| procedure is
complete except for the parts that handle general distributions |dist_from|
and |dist_to|. First, we had better check the input to make sure that
it is well formed; otherwise disaster can ensue later. This part
of the program is easy:

 @<Check the distribution parameters@>=
{@+register long acc; /* sum of probabilities */
  register long *p; /* pointer to current probability of interest */
  if (dist_from) {
    for (acc=0,@,p=dist_from; p<dist_from+n; p++) {
      if (*p<0) panic(invalid_operand);
        /* |dist_from| contains a negative entry */
      if (*p>0x40000000-acc) panic(invalid_operand+1);
        /* probability too high */
      acc+=*p;
    }
    if (acc!=0x40000000)
      panic(invalid_operand+2); /* |dist_from| table doesn't sum to $2^{30}$ */
  }
  if (dist_to) {
    for (acc=0,@,p=dist_to; p<dist_to+n; p++) {
      if (*p<0) panic(invalid_operand+5);
         /* |dist_to| contains a negative entry */
      if (*p>0x40000000-acc) panic(invalid_operand+6);
         /* probability too high */ 
     acc+=*p;
    }
    if (acc!=0x40000000)
      panic(invalid_operand+7); /* |dist_to| table doesn't sum to $2^{30}$ */
  }
}

@ We generate nonuniform distributions by using Alistair J. Walker's alias
method (see, for example, {\sl Seminumerical Algorithms}, second edition,
exercise 3.4.1--7). This involves setting up ``magic'' tables
of length |nn|, where |nn| is the smallest power of~2 that is |>=n|.

@f magic_entry int

@<Local v...@>=
long nn=1; /* this will be increased to $2^{\lceil\mskip1mu\lg n\rceil}$ */
int kk=31; /* this will be decreased to $31-\lceil\mskip1mu\lg n\rceil$ */
magic_entry *dist_from_table, *dist_to_table; /* alias tables */

@ @<Build...@>=
{
  if (dist_from) {
    while (nn<n) nn+=nn, kk--;
    dist_from_table=walker(n,nn,dist_from,new_graph);
  }
  if (dist_to) {
    while (nn<n) nn+=nn, kk--;
    dist_to_table=walker(n,nn,dist_to,new_graph);
  }
  if (gb_alloc_trouble) {
    gb_recycle(new_graph);
    panic(alloc_fault); /* oops, we ran out of memory somewhere back there */
  }
}

@ @<Private...@>=
typedef struct {
  long prob; /* a probability, multiplied by $2^{31}$ and translated */
  long inx; /* index that might be selected */
} magic_entry;

@ Once the magic tables have been set up, we will be able to generate
nonuniform vertices by using the following code:

@<Generate a random vertex |u|...@>=
{@+register magic_entry *magic;
  register long uu=gb_next_rand(); /* uniform random number */
  k=uu>>kk;
  magic=dist_from_table+k;
  if (uu<=magic->prob) u=new_graph->vertices+k;
  else u=new_graph->vertices+magic->inx;
}

@ @<Generate a random vertex |v|...@>=
{@+register magic_entry *magic;
  register long uu=gb_next_rand(); /* uniform random number */
  k=uu>>kk;
  magic=dist_to_table+k;
  if (uu<=magic->prob) v=new_graph->vertices+k;
  else v=new_graph->vertices+magic->inx;
}

@ So all we have to do is set up those magic tables. If |uu| is a uniform
random integer between 0 and $2^{31}-1$, the index |k=uu>>kk| will be a
uniform random integer between 0
and |nn-1|, because of the relation between |nn| and |kk|. Once |k| is
computed in the code above, we will select vertex~|k| with probability
|(p+1-(k<<kk))|/$2^{31}$, where |p=magic->prob| and |magic| is the $k$th
element of the magic table; otherwise we will select
vertex |magic->inx|. The trick is to set things up so that each vertex
is selected with the proper overall probability.

Let's imagine that the given distribution vector has length |nn|,
instead of~|n|, by extending it if necessary with zeroes. Then the
average entry among these |nn| integers is exactly $t=2^{30}/|nn|$.
If some entry, say entry~|i|, exceeds |t|, there must be another entry
that's less than |t|, say entry~|j|. We can set the $j$th entry
of the magic table so that its |prob| field selects vertex~$j$ with the
correct probability, and so that its |vert| field equals~|i|. Then
we are selecting vertex~|i| with a certain residual probability, so we
subtract that residual from |i|'s present probability, and repeat the
process with vertex~|j| eliminated. The average of the remaining entries
is still~|t|, so we can repeat this procedure until all remaining entries
are exactly equal to~|t|. The rest is easy.

During the calculation, we will maintain two linked lists of
|(prob,vert)| pairs; the |hi| list will contain entries with |prob>t|,
and the |lo| list will contain the rest. We'll call these list
elements `nodes'; and we'll use the field names |key| and~|j| instead
of |prob| and |vert| during this part of the computation.

@<Private...@>=
typedef struct node_struct {
  long key; /* a numeric quantity */
  struct node_struct *link; /* the next node on the list */
  int j; /* a vertex number to be selected with probability $|key|/2^{30}$ */
} node;
static Area temp_nodes; /* nodes will be allocated in this area */
static node *base_node; /* beginning of a block of nodes */

@ @<Internal...@>=
static magic_entry *walker(n,nn,dist,g)
  int n; /* length of |dist| vector */
  long nn; /* $2^{\lceil\mskip1mu\lg n\rceil}$ */
  register long *dist; /* start of distribution table, which sums to $2^{30}$ */
  Graph *g; /* tables will be allocated for this graph's vertices */
{@+magic_entry *table; /* this will be the magic table we compute */
  long t; /* average |key| value */
  node *hi=NULL, *lo=NULL; /* nodes not yet included in magic table */
  register node *p, *q; /* pointer variables for list manipulation */
  register int *r; /* pointer variable to traverse the |dist| table */
  base_node=gb_alloc_type(nn,@[node@],temp_nodes);
  table=gb_alloc_type(nn,@[magic_entry@],g->aux_data);
  if (!gb_alloc_trouble) {
    @<Initialize the |hi| and |lo| lists@>;
    while (hi) @<Remove a |lo| element and match it with a |hi| element;
        deduct the residual probability from that |hi|~element@>;
    while (lo) @<Remove a |lo| element of |key| value |t|@>;
  }
  gb_free(temp_nodes);
  return table; /* if |gb_alloc_trouble| is nonzero, the table is empty */
}

@ @<Initialize the |hi| and |lo| lists@>=
t=0x40000000/nn; /* this division is exact */
p=base_node;
while (nn>n) {
  p->key=0;
  p->link=lo;
  p->j=--nn;
  lo=p++;
}
for (dist=dist+n-1; n>0; dist--,p++) {
  p->key=*dist;
  p->j=--n;
  if (*dist>t)
    p->link=hi,@, hi=p;
  else p->link=lo,@, lo=p;
}

@ When we change the scale factor from $2^{30}$ to $2^{31}$, we need to
be careful lest integer overflow occur. The introduction of register |x| into
this code removes the risk.

@<Remove a |lo| element and match it with a |hi| element...@>=
{register magic_entry *r; register long x;
  p=hi,@, hi=p->link;
  q=lo,@, lo=q->link;
  r=table+q->j;
  x=t*q->j+q->key-1;
  r->prob=x+x+1;
  r->inx=p->j;
  /* we have just given |q->key| units of probability to vertex |q->j|,
     and |t-q->key| units to vertex |p->j| */
  if ((p->key-=t-q->key)>t)
    p->link=hi,@, hi=p;
  else p->link=lo,@, lo=p;
}

@ When all remaining entries have the average probability, the
|vert| component need not be set, because it will never be used.

@<Remove a |lo| element of |key| value |t|@>=
{register magic_entry *r; register long x;
  q=lo, lo=q->link;
  r=table+q->j;
  x=t*q->j+t-1;
  r->prob=x+x+1;
  /* that's |t| units of probability for vertex |q->j| */
}

@*Random bipartite graphs. The procedure call
$$\hbox{|random_bigraph(n1,n2,m,multi,dist1,dist2,min_len,max_len,seed)|}$$
is designed to produce a pseudo-random bipartite graph
with |n1| vertices in one part and |n2| in the other, having |m| edges.
The remaining parameters |multi|, |dist1|, |dist2|, |min_len|, |max_len|,
and |seed| have the same meaning as the analogous parameters of |random_graph|.

In fact, |random_bigraph| does its work by reducing its parameters
to a special case of |random_graph|. Almost all that needs to be done is
to pad |dist1| with |n2| trailing zeroes and |dist2| with |n1| leading
zeroes. The only slightly tricky part occurs when |dist1| and/or |dist2| are
null, since non-null distribution vectors summing exactly to $2^{30}$ must then
be fabricated.

@<External f...@>=
Graph *random_bigraph(n1,n2,m,multi,dist1,dist2,min_len,max_len,seed)
  unsigned n1,n2; /* number of vertices desired in each part */
  unsigned long m; /* number of edges desired */
  int multi; /* allow duplicate edges? */
  long *dist1, *dist2; /* distribution of edge endpoints */
  long min_len,max_len; /* bounds on random lengths */
  long seed; /* random number seed */
{@+int n=n1+n2; /* total number of vertices */
  Area new_dists;
  long *dist_from, *dist_to;
  Graph *new_graph;
  init_area(new_dists);
  if (n1==0 || n2==0) panic(bad_specs); /* illegal options */
  if (min_len>max_len) panic(very_bad_specs); /* what are you trying to do? */
  if (((unsigned long)(max_len))-((unsigned long)(min_len))>=
      ((unsigned long)0x80000000)) panic(bad_specs+1); /* too much range */
  dist_from=gb_alloc_type(n,@[long@],new_dists);
  dist_to=gb_alloc_type(n,@[long@],new_dists);
  if (gb_alloc_trouble) {
    gb_free(new_dists);
    panic(no_room+2); /* no room for auxiliary distribution tables */
  }
  @<Compute the entries of |dist_from| and |dist_to|@>;
  new_graph=random_graph(n,m,multi,0,0,dist_from,dist_to,min_len,max_len,seed);
  sprintf(new_graph->id,"random_bigraph(%u,%u,%lu,%d,%s,%s,%ld,%ld,%ld)",@|
    n1,n2,m,multi>0?1:multi<0?-1:0,dist_code(dist1),dist_code(dist2),@|
    min_len,max_len,seed);
  mark_bipartite(new_graph,n1);
  gb_free(new_dists);
  return new_graph;
}

@ The relevant identity we need here is the replicative law for the
floor function:
$$\left\lfloor x\over n\right\rfloor+\left\lfloor x+1\over n\right\rfloor
+ \cdots + \left\lfloor x+n-1\over n\right\rfloor = \lfloor x\rfloor\,.$$

@<Compute the entries...@>=
{@+register long *p, *q; /* traversers of the dists */
  register int k; /* vertex count */
  p=dist1; q=dist_from;
  if (p)
    while (p<dist1+n1) *q++=*p++;
  else for (k=0; k<n1; k++) *q++=(0x40000000+k)/n1;
  p=dist2; q=dist_to+n1;
  if (p)
    while (p<dist2+n2) *q++=*p++;
  else for (k=0; k<n2; k++) *q++=(0x40000000+k)/n2;
}

@* Random lengths. The subroutine call
$$\hbox{|random_lengths(g,directed,min_len,max_len,dist,seed)|}$$
takes an existing graph and assigns new lengths to
each of its arcs. The lengths will be uniformly distributed between
|min_len| and |max_len| inclusive, if |dist=NULL|; otherwise |dist|
should be a probability distribution vector of length |max_len-min_len+1|,
like those in |random_graph|.

If |directed=0|, pairs of arcs $u\to v$ and $v\to u$ will be regarded as
a single edge, both arcs receiving the same length.

The procedure returns a nonzero value if something goes wrong; in that
case, graph |g| will not have been changed.

Alias tables for generating nonuniform random lengths will survive
in |g->aux_data|.

@<External f...@>=
int random_lengths(g,directed,min_len,max_len,dist,seed)
  Graph *g; /* graph whose lengths will be randomized */
  int directed; /* is it directed? */
  long min_len,max_len; /* bounds on random lengths */
  long *dist; /* distribution of lengths */
  long seed; /* random number seed */
{@+register Vertex *u,*v; /* current vertices of interest */
  register Arc *a; /* current arc of interest */
  long nn=1, kk=31; /* variables for nonuniform generation */
  magic_entry *dist_table; /* alias table for nonuniform generation */
  if (g==NULL) panic(missing_operand); /* where is |g|? */
  gb_init_rand(seed);
  if (min_len>max_len) return 102; /* what are you trying to do? */
  if (((unsigned long)(max_len))-((unsigned long)(min_len))>=
      ((unsigned long)0x80000000)) return 103; /* too much range */
  @<Check |dist| for validity, and set up the |dist_table|@>;
  sprintf(buffer,",%d,%ld,%ld,%s,%ld)",directed?1:0,@|
     min_len,max_len,dist_code(dist),seed);
  make_compound_id(g,"random_lengths(",g,buffer);
  @<Run through all arcs and assign new lengths@>;
  return 0;
}

@ @<Private dec...@>=
static char buffer[]="1,-1000000001,-1000000000,dist,1000000000)";

@ @<Check |dist| for validity...@>=
if (dist) {@+register long acc; /* sum of probabilities */
  register long *p; /* pointer to current probability of interest */
  register n=max_len-min_len+1;
  for (acc=0,p=dist; p<dist+n; p++) {
    if (*p<0) return -1; /* negative probability */
    if (*p>0x40000000-acc) return 1; /* probability too high */
    acc+=*p;
  }
  if (acc!=0x40000000) return 2; /* probabilities don't sum to 1 */
  while (nn<n) nn+=nn,kk--;
  dist_table=walker(n,nn,dist,g);
  if (gb_alloc_trouble) {
    gb_alloc_trouble=0;
    return 66; /* not enough room to generate the magic tables */
  }
}

@ @<Run through all arcs and assign new lengths@>=
for (u=g->vertices;u<g->vertices+g->n;u++)
  for (a=u->arcs;a;a=a->next) {
    v=a->tip;
    if (directed==0 && u>v) a->len=(a-1)->len;
    else {@+register long len; /* a random length */
      if (dist==0) len=rand_len;
      else {@+long uu=gb_next_rand();
        long k=uu>>kk;
        magic_entry *magic=dist_table+k;
        if (uu<=magic->prob) len=min_len+k;
        else len=min_len+magic->inx;
      }
      a->len=len;
      if (directed==0 && u==v && a->next==a+1) (++a)->len=len;
    }
  }

@* Index. Here is a list that shows where the identifiers of this program are
defined and used.