gb_rand.w

模拟器提供了一个简单易用的平台

% This file is part of the Stanford GraphBase (c) Stanford University 1993
@i boilerplate.w %<< legal stuff: PLEASE READ IT BEFORE MAKING ANY CHANGES!
@i gb_types.w

\def\title{GB\_\,RAND}

\prerequisite{GB\_\,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 {\sc GB\_\,GRAPH}. The
routines in {\sc GB\_\,GRAPH} are loaded together with all GraphBase
applications, and the programs below are typical illustrations of how
to use them.

@d random_graph r_graph /* abbreviations for Procrustean external linkage */
@d random_bigraph r_bigraph
@d random_lengths r_lengths

@(gb_rand.h@>=
#define random_graph r_graph
 /* users of {\sc GB\_\,RAND} should include this header info */
#define random_bigraph r_bigraph
#define random_lengths r_lengths
extern Graph *random_graph();
extern Graph *random_bigraph();
extern long random_lengths();

@ Here is an overview of the file \.{gb\_rand.c}, the \CEE/ code for the
routines in question.

@p
#include "gb_graph.h" /* this header file teaches \CEE/ about GraphBase */
#include "gb_flip.h"
 /* we will use the {\sc GB\_\,FLIP} routines for random numbers */
@h@#
@<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.

Warning: Users need to initialize the random number generator before
calling |random_graph|, or any other GraphBase procedure that
consumes random numbers. (See {\sc GB\_\,FLIP}.)

@ 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
$$\hbox{|random_graph(31,m,1,1,1,d0,d1,0,255,s)|}$$
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 |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_trouble_code| will be
cleared to zero after |random_graph| has acted.

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

@<External f...@>=
Graph *random_graph(n,m,multi,self,directed,dist_from,dist_to,min_len,max_len,
                       seed)
  unsigned long n; /* number of vertices desired */
  unsigned long m; /* number of arcs or edges desired */
  long multi; /* allow duplicate arcs? */
  long self; /* allow self loops? */
  long directed; /* directed graph? */
  long *dist_from; /* distribution of arc sources */
  long *dist_to; /* distribution 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_trouble_code) {
    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 long 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,"%ld",k);
  (new_graph->vertices+k)->name=gb_save_string(name_buffer);
}
sprintf(new_graph->id,"random_graph(%lu,%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+1))

@<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_trouble_code) 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|. Before attempting to generate those distributions, we had better
check them to make sure that the specifications are well formed;
otherwise disaster might 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 Walker's alias
@^Walker, Alistair J.@>
method (see, for example, {\sl Seminumerical Algorithms}, second edition,
exercise 3.4.1--7). Walker's method 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 $2^{\lceil\mskip1mu\lg n\rceil}$ */
long kk=31; /* this will be decreased to $31-\lceil\mskip1mu\lg n\rceil$ */
magic_entry *from_table, *to_table; /* alias tables */

@ @<Build...@>=
{
  if (dist_from) {
    while (nn<n) nn+=nn, kk--;
    from_table=walker(n,nn,dist_from,new_graph);
  }