John Naylor <john.nay...@2ndquadrant.com> writes:
> -As for the graph algorithm, I'd have to play with it to understand
> how it works.
I improved the comment about how come the hash table entry assignment
works. One thing I'm not clear about myself is
# A cycle-free graph is either empty or has some vertex of degree 1.
That sounds like a standard graph theory result, but I'm not familiar
with a proof for it.
regards, tom lane
#----------------------------------------------------------------------
#
# PerfectHash.pm
# Perl module that constructs minimal perfect hash functions
#
# This code constructs a minimal perfect hash function for the given
# set of keys, using an algorithm described in
# "An optimal algorithm for generating minimal perfect hash functions"
# by Czech, Havas and Majewski in Information Processing Letters,
# 43(5):256-264, October 1992.
# This implementation is loosely based on NetBSD's "nbperf",
# which was written by Joerg Sonnenberger.
#
# The resulting hash function is perfect in the sense that if the presented
# key is one of the original set, it will return the key's index in the set
# (in range 0..N-1). However, the caller must still verify the match,
# as false positives are possible. Also, the hash function may return
# values that are out of range (negative, or >= N). This indicates that
# the presented key is definitely not in the set.
#
#
# Portions Copyright (c) 1996-2019, PostgreSQL Global Development Group
# Portions Copyright (c) 1994, Regents of the University of California
#
# src/tools/PerfectHash.pm
#
#----------------------------------------------------------------------
package PerfectHash;
use strict;
use warnings;
# At runtime, we'll compute two simple hash functions of the input key,
# and use them to index into a mapping table. The hash functions are just
# multiply-and-add in uint32 arithmetic, with different multipliers but
# the same initial seed. All the complexity in this module is concerned
# with selecting hash parameters that will work and building the mapping
# table.
# We support making case-insensitive hash functions, though this only
# works for a strict-ASCII interpretation of case insensitivity,
# ie, A-Z maps onto a-z and nothing else.
my $case_insensitive = 0;
#
# Construct a C function implementing a perfect hash for the given keys.
# The C function definition is returned as a string.
#
# The keys can be any set of Perl strings; it is caller's responsibility
# that there not be any duplicates. (Note that the "strings" can be
# binary data, but endianness is the caller's problem.)
#
# The name to use for the function is caller-specified, but its signature
# is always "int f(const void *key, size_t keylen)". The caller may
# prepend "static " to the result string if it wants a static function.
#
# If $ci is true, the function is case-insensitive, for the limited idea
# of case-insensitivity explained above.
#
sub generate_hash_function
{
my ($keys_ref, $funcname, $ci) = @_;
# It's not worth passing this around as a parameter; just use a global.
$case_insensitive = $ci;
# Try different hash function parameters until we find a set that works
# for these keys. In principle we might need to change multipliers,
# but these two multipliers are chosen to be primes that are cheap to
# calculate via shift-and-add, so don't change them without care.
my $hash_mult1 = 31;
my $hash_mult2 = 2053;
# We just try successive hash seed values until we find one that works.
# (Commonly, random seeds are tried, but we want reproducible results
# from this program so we don't do that.)
my $hash_seed;
my @subresult;
for ($hash_seed = 0; $hash_seed < 1000; $hash_seed++)
{
@subresult =
_construct_hash_table($keys_ref, $hash_mult1, $hash_mult2,
$hash_seed);
last if @subresult;
}
# Choke if we didn't succeed in a reasonable number of tries.
die "failed to generate perfect hash" if !@subresult;
# Extract info from the function result array.
my $elemtype = $subresult[0];
my @hashtab = @{ $subresult[1] };
my $nhash = scalar(@hashtab);
# OK, construct the hash function definition including the hash table.
my $f = '';
$f .= sprintf "int\n";
$f .= sprintf "%s(const void *key, size_t keylen)\n{\n", $funcname;
$f .= sprintf "\tstatic const %s h[%d] = {\n", $elemtype, $nhash;
for (my $i = 0; $i < $nhash; $i++)
{
$f .= sprintf "%s%6d,%s",
($i % 8 == 0 ? "\t\t" : " "),
$hashtab[$i],
($i % 8 == 7 ? "\n" : "");
}
$f .= sprintf "\n" if ($nhash % 8 != 0);
$f .= sprintf "\t};\n\n";
$f .= sprintf "\tconst unsigned char *k = key;\n";
$f .= sprintf "\tuint32\t\ta = %d;\n", $hash_seed;
$f .= sprintf "\tuint32\t\tb = %d;\n\n", $hash_seed;
$f .= sprintf "\twhile (keylen--)\n\t{\n";
$f .= sprintf "\t\tunsigned char c = *k++";
$f .= sprintf " | 0x20" if $case_insensitive; # see comment below
$f .= sprintf ";\n\n";
$f .= sprintf "\t\ta = a * %d + c;\n", $hash_mult1;
$f .= sprintf "\t\tb = b * %d + c;\n", $hash_mult2;
$f .= sprintf "\t}\n";
$f .= sprintf "\treturn h[a %% %d] + h[b %% %d];\n", $nhash, $nhash;
$f .= sprintf "}\n";
return $f;
}
# Calculate a hash function as the run-time code will do.
#
# If we are making a case-insensitive hash function, we implement that
# by OR'ing 0x20 into each byte of the key. This correctly transforms
# upper-case ASCII into lower-case ASCII, while not changing digits or
# dollar signs. (It does change '_', else we could just skip adjusting
# $cn here at all, for typical keyword strings.)
sub _calc_hash
{
my ($key, $mult, $seed) = @_;
my $result = $seed;
for my $c (split //, $key)
{
my $cn = ord($c);
$cn |= 0x20 if $case_insensitive;
$result = ($result * $mult + $cn) % 4294967296;
}
return $result;
}
# Attempt to construct a mapping table for a minimal perfect hash function
# for the given keys, using the specified hash parameters.
#
# Returns an array containing the mapping table element type name as the
# first element, and a ref to an array of the table values as the second.
#
# Returns an empty array on failure; then caller should choose different
# hash parameter(s) and try again.
sub _construct_hash_table
{
my ($keys_ref, $hash_mult1, $hash_mult2, $hash_seed) = @_;
my @keys = @{$keys_ref};
# This algorithm is based on a graph whose edges correspond to the
# keys and whose vertices correspond to entries of the mapping table.
# A key's edge links the two vertices whose indexes are the outputs of
# the two hash functions for that key. For K keys, the mapping
# table must have at least 2*K+1 entries, guaranteeing that there's at
# least one unused entry. (In principle, larger mapping tables make it
# easier to find a workable hash and increase the number of inputs that
# can be rejected due to touching unused hashtable entries. In
practice,
# neither effect seems strong enough to justify using a larger table.)
my $nedges = scalar @keys; # number of edges
my $nverts = 2 * $nedges + 1; # number of vertices
# Initialize the array of edges.
my @E = ();
foreach my $kw (@keys)
{
# Calculate hashes for this key.
# The hashes are immediately reduced modulo the mapping table
size.
my $hash1 = _calc_hash($kw, $hash_mult1, $hash_seed) % $nverts;
my $hash2 = _calc_hash($kw, $hash_mult2, $hash_seed) % $nverts;
# If the two hashes are the same for any key, we have to fail
# since this edge would itself form a cycle in the graph.
return () if $hash1 == $hash2;
# Add the edge for this key.
push @E, { left => $hash1, right => $hash2 };
}
# Initialize the array of vertices, giving them all empty lists
# of associated edges. (The lists will be hashes of edge numbers.)
my @V = ();
for (my $v = 0; $v < $nverts; $v++)
{
push @V, { edges => {} };
}
# Insert each edge in the lists of edges using its vertices.
for (my $e = 0; $e < $nedges; $e++)
{
my $v = $E[$e]{left};
$V[$v]{edges}->{$e} = 1;
$v = $E[$e]{right};
$V[$v]{edges}->{$e} = 1;
}
# Now we attempt to prove the graph acyclic.
# A cycle-free graph is either empty or has some vertex of degree 1.
# Removing the edge attached to that vertex doesn't change this
property,
# so doing that repeatedly will reduce the size of the graph.
# If the graph is empty at the end of the process, it was acyclic.
# We track the order of edge removal so that the next phase can process
# them in reverse order of removal.
my @output_order = ();
# Consider each vertex as a possible starting point for edge-removal.
for (my $startv = 0; $startv < $nverts; $startv++)
{
my $v = $startv;
# If vertex v is of degree 1 (i.e. exactly 1 edge connects to
it),
# remove that edge, and then consider the edge's other vertex
to see
# if it is now of degree 1. The inner loop repeats until
reaching a
# vertex not of degree 1.
while (scalar(keys(%{ $V[$v]{edges} })) == 1)
{
# Unlink its only edge.
my $e = (keys(%{ $V[$v]{edges} }))[0];
delete($V[$v]{edges}->{$e});
# Unlink the edge from its other vertex, too.
my $v2 = $E[$e]{left};
$v2 = $E[$e]{right} if ($v2 == $v);
delete($V[$v2]{edges}->{$e});
# Push e onto the front of the output-order list.
unshift @output_order, $e;
# Consider v2 on next iteration of inner loop.
$v = $v2;
}
}
# We succeeded only if all edges were removed from the graph.
return () if (scalar(@output_order) != $nedges);
# OK, build the hash table of size $nverts.
my @hashtab = (0) x $nverts;
# We need a "visited" flag array in this step, too.
my @visited = (0) x $nverts;
# The goal is that for any key, the sum of the hash table entries for
# its first and second hash values is the desired output (i.e., the key
# number). By assigning hash table values in the selected edge order,
# we can guarantee that that's true. This works because the edge first
# removed from the graph (and hence last to be visited here) must have
# at least one vertex it shared with no other edge; hence it will have
at
# least one vertex (hashtable entry) still unvisited when we reach it
here,
# and we can assign that unvisited entry a value that makes the sum come
# out as we wish. By induction, the same holds for all the other edges.
foreach my $e (@output_order)
{
my $l = $E[$e]{left};
my $r = $E[$e]{right};
if (!$visited[$l])
{
# $hashtab[$r] might be zero, or some previously
assigned value.
$hashtab[$l] = $e - $hashtab[$r];
}
else
{
die "oops, doubly used hashtab entry" if $visited[$r];
# $hashtab[$l] might be zero, or some previously
assigned value.
$hashtab[$r] = $e - $hashtab[$l];
}
# Now freeze both of these hashtab entries.
$visited[$l] = 1;
$visited[$r] = 1;
}
# Detect range of values needed in hash table.
my $hmin = $nedges;
my $hmax = 0;
for (my $v = 0; $v < $nverts; $v++)
{
$hmin = $hashtab[$v] if $hashtab[$v] < $hmin;
$hmax = $hashtab[$v] if $hashtab[$v] > $hmax;
}
# Choose width of hashtable entries. In addition to the actual values,
# we need to be able to store a flag for unused entries, and we wish to
# have the property that adding any other entry value to the flag gives
# an out-of-range result (>= $nedges).
my $elemtype;
my $unused_flag;
if ( $hmin >= -0x7F
&& $hmax <= 0x7F
&& $hmin + 0x7F >= $nedges)
{
# int8 will work
$elemtype = 'int8';
$unused_flag = 0x7F;
}
elsif ($hmin >= -0x7FFF
&& $hmax <= 0x7FFF
&& $hmin + 0x7FFF >= $nedges)
{
# int16 will work
$elemtype = 'int16';
$unused_flag = 0x7FFF;
}
elsif ($hmin >= -0x7FFFFFFF
&& $hmax <= 0x7FFFFFFF
&& $hmin + 0x3FFFFFFF >= $nedges)
{
# int32 will work
$elemtype = 'int32';
$unused_flag = 0x3FFFFFFF;
}
else
{
die "hash table values too wide";
}
# Set any unvisited hashtable entries to $unused_flag.
for (my $v = 0; $v < $nverts; $v++)
{
$hashtab[$v] = $unused_flag if !$visited[$v];
}
return ($elemtype, \@hashtab);
}
1;
