/ 
Algorithm-Networksort-2.02
River stage one • 1 direct dependent • 1 total dependent
/ Algorithm::Networksort

NAME

Algorithm::Networksort - Create Sorting Networks.

Bose-Nelson Sort for N = 4

Bose-Nelson sorting network, inputs = 4, drawn as a Knuth diagram.

SYNOPSIS

use Algorithm::Networksort;

my $inputs = 4;
my $algorithm = "bosenelson";

#
# Generate the sorting network (a list of comparators).
#
# nwsrt() is a convenient short-hand
# for Algorithm::Networksort->new().
#
my $nw = nwsrt(inputs => $inputs, algorithm => $algorithm);

#
# Print the title, the comparator list using the default
# format, and a text-based Knuth diagram.
#
print $nw->title(), "\n";
    $nw, "\n\n",
    $nw->graph_text(), "\n";

#
# Create an SVG image of the Knuth diagram.
# Set the diagram sizes.
#
$nw->graphsettings(indent => 12,
    hz_margin => 12, hz_sep=>12,
    vt_margin => 21, vt_sep => 12,
    compradius => 2, compline => 2,
    inputradius => 0, inputline => 2);

print $nw->graph_svg();

DESCRIPTION

This module will create sorting networks, a sequence of comparisons that do not depend upon the results of prior comparisons.

Since the sequences and their order never change, they can be very useful if deployed in hardware, or if used in software with a compiler that can take advantage of parallelism. Unfortunately a sorting network cannot be used for generic run-time sorting like quicksort, since the arrangement of the comparisons is fixed according to the number of elements to be sorted.

This module's main purpose is to create compare-and-swap macros (or functions, or templates) that one may insert into source code. It may also be used to create images of the sorting networks in either encapsulated postscript (EPS), scalar vector graphics (SVG), or in "ascii art" format.

Exported Functions

For convenience's sake, there are three exported functions to save typing and inconvenient look-ups.

nwsrt()

Simple function to save the programmer from the agony of typing Algorithm::Networksort->new():

use Algorithm::Networksort;

my $nw = nwsrt(inputs => 13, algorithm => 'bitonic');

nwsrt_algorithms()

Return a sorted list algorithm choices. Each one is a valid key for the algorithm argument of Algorithm::Networksort->new(), or nwsrt().

use Algorithm::Networksort;

my @alg_keys = nwsrt_algorithms();

print "The available keys for the algorithm argument are (",
        join(", ", @alg_keys), ")\n";

Or, for an even less likely example:

my $inputs = 6;

for my $al (nwsrt_algorithms())
{
	my $nw = nwsrt(inputs => $inputs, algorithm => $al);
	print $nw->title(), "\n", $nw, "\n";
}

nwsrt_title

Return a descriptive title for an algorithm, given the algorithm's key name.

$title = nwsrt_title($key);

These are the titles for the available algorithms. By themselves, they provide a readable list of choices for an interactive program. They are not to be confused with a sorting network's title, which may be set by the programmer.

Methods

new()

$nw1 = Algorithm::Networksort->new(inputs => $inputs,
                                   algorithm => 'batcher');

$nw2 = Algorithm::Networksort->new(inputs => $inputs,
                                   algorithm => 'oddevenmerge');

Returns an object that contains, among other things, a list of comparators that can sort $inputs items. The algorithm for generating the list may be chosen, but by default the sorting network is generated by the Bose-Nelson algorithm.

Arguments to new()

'inputs'

The number of inputs of the sorting network.

'algorithm'

The choices for the algorithm key are

'bosenelson'

Use the Bose-Nelson algorithm to generate the network. This is the most commonly implemented algorithm, recursive and simple to code.

'hibbard'

Use Hibbard's algorithm. This iterative algorithm was developed after the Bose-Nelson algorithm was published, and produces a different network "... for generating the comparisons one by one in the order in which they are needed for sorting," according to his article (see below).

'batcher'

Use Batcher's Merge Exchange algorithm. Merge Exchange is a real sort, in that in its usual form (for example, as described in Knuth) it can handle a variety of inputs. But while sorting it always generates an identical set of comparison pairs per array size, which lends itself to sorting networks.

'bitonic'

Use Batcher's bitonic algorithm. A bitonic sequence is a sequence that monotonically increases and then monotonically decreases. The bitonic sort algorithm works by recursively splitting sequences into bitonic sequences using so-called "half-cleaners". These bitonic sequences are then merged into a fully sorted sequence. Bitonic sort is a very efficient sort and is especially suited for implementations that can exploit network parallelism.

'oddevenmerge'

Use Batcher's Odd-Even Merge algorithm. This sort works in a similar way to a regular merge sort, except that in the merge phase the sorted halves are merged by comparing even elements separately from odd elements. This algorithm creates very efficient networks in both comparators and stages.

'bubble'

Use a naive bubble-sort/insertion-sort algorithm. Since this algorithm produces more comparison pairs than the other algorithms, it is only useful for illustrative purposes.

'oddeventrans'

Use a naive odd-even transposition sort. This is a primitive sort closely related to bubble sort except that it is more parallel. Because other algorithms are more efficient, this sort is included for illustrative purposes.

'balanced'

This network is described in the 1983 paper "The Balanced Sorting Network" by M. Dowd, Y. Perl, M Saks, and L. Rudolph. It is not a particularly efficient sort but it has some interesting properties due to the fact that it is constructed as a series of successive identical sub-blocks, somewhat like with 'oddeventrans'.

'none'

Do not generate a set of comparators. Instead, take the set from an outside source, using the comparators option.

#
# Test our own 5-input network.
#
@cmptr = ([1,2], [0,2], [0,1], [3,4], [0,3], [1,4], [2,4], [1,3], [2,3]);

$nw = Algorithm::Networksort->new(inputs => 5,
            algorithm => 'none',
            comparators => [@cmptr]);

Internally, this is what nwsrt_best() of Algorithm::Networksort::Best uses.

'comparators'

The list of comparators provided by the user instead of by an algorithm.

comparators()

The comparators in their 'raw' form, as generated by its algorithm.

For the comparators re-ordered in such a way as to take advantage of parallelism, see network().

network()

Returns the comparators re-ordered from the 'raw' form, providing a parallelized version of the comparator list; e.g., the best order possible to prevent stalling in a CPU's pipeline.

This is the form used when printing the sorting network using formats().

algorithm_name()

Return the full text name of the algorithm, given its key name.

sort()

Sort an array using the network. This is meant for testing purposes only - looping around an array of comparators in order to sort an array in an interpreted language is not the most efficient mechanism for using a sorting network.

This function uses the <=> operator for comparisons.

my @digits = (1, 8, 3, 0, 4, 7, 2, 5, 9, 6);
my $nw = Algorithm::Networksort->new(
                    inputs => (scalar @digits),
                    algorithm => 'hibbard');
$nw->sort(\@digits);
print join(", ", @digits);

statistics()

Return statistics on the last sort() call. Currently only "swaps", a count of the number of exchanges, is returned.

my(@d, %nw_stats);
my @digits = (1, 8, 3, 0, 4, 7, 2, 5, 9, 6);
my $inputs = scalar @digits;
my $nw_batcher = Algorithm::Networksort->new(inputs => $inputs, algorithm => 'batcher');
my $nw_bn = Algorithm::Networksort->new(inputs => $inputs, algorithm => 'bosenelson');

#
# List will wind up sorted, so copy it for our first test run.
#
@d = @digits;
$nw_batcher->sort(\@d);
%nw_stats = $nw_batcher->statistics();
print "The Batcher Merge-Exchange network took ",
    $nw_stats{swaps}, " exchanges to sort the array."

@d = @digits;
$nw_bn->sort(\@d);
%nw_stats = $nw_bn->statistics();
print "The Bose-Nelson network took ",
    $nw_stats{swaps}, " exchanges to sort the array."

Methods For Printing

The network object by default prints itself in a grouped format; that is

my $nw = nwsrt(inputs => 8, algorithm => 'bosenelson');
print $nw . "\n";

Will result in the output

[[0,1], [2,3], [4,5], [6,7],
[0,2], [1,3], [4,6], [5,7],
[1,2], [5,6], [0,4], [3,7],
[1,5], [2,6],
[1,4], [3,6],
[2,4], [3,5],
[3,4]]

If you had shifted the array index by one using index_base():

my $nw = nwsrt(inputs => 8, algorithm => 'bosenelson');
$nw->index_base([1 .. 8]);
print $nw . "\n";

This would have resulted in the output

[[1,2], [3,4], [5,6], [7,8],
[1,3], [2,4], [5,7], [6,8],
[2,3], [6,7], [1,5], [4,8],
[2,6], [3,7],
[2,5], [4,7],
[3,5], [4,6],
[4,5]]

For a wider variety of outputs, use formats() and index_base() as described below.

formats()

An array reference of format strings, for use in formatted printing (see formatted()). You may use as many sprintf-style formats as you like to form your output.

$nw->formats([ "swap(%d, %d) ", "if ($card[%d] < $card[%d]);\n" ]);

index_base()

The values to use to reference array indices in formatted printing (see formatted()). By default, array indices are zero-based. To use a different index base (most commonly, one-based array indexing), use this method.

$nw->index_base([1 .. $inputs]);

formatted()

$string = $nw->formatted();

Returns a formatted string that represents the list of comparators.

If no formats have been provided via the formats() method, the default format will be used: an array of arrays as represented in Perl.

Likewise, the network sorting pairs are zero-based. If you want the pairs written out for some sequence other than 0, 1, 2, ... then you can provide that using inputs_base().

Example 0: you want a string in the default format.

print $nw->formatted();

Example 1: you want the output to look like the default format, but one-based instead of zero-based.

$nw->index_base([1..$inputs]);
print $nw->formatted();

Example 2: you want a simple list of SWAP macros.

$nw->formats([ "SWAP(%d, %d);\n" ]);
print $nw->formatted();

Example 3: as with example 2, but the SWAP values need to be one-based instead of zero-based.

$nw->index_base([1..$inputs]);
$nw->formats([ "SWAP(%d, %d);\n" ]);
print $nw->formatted();

Example 4: you want a series of comparison and swap statements.

$nw->formats([ "if (v[%d] < v[%d]) then\n",
            "    exchange(v, %d, %d)\nend if\n" ]);
print $nw->formatted();

Example 5: you want the default format to use letters, not numbers.

$nw->index_base( [('a'..'z')[0..$inputs]] );
$nw->formats([ "[%s,%s]," ]);      # Note that we're using the string flag.

my $string = '[' . $nw->formatted();
substr($string, -1, 1) = ']';    # Overwrite the trailing comma.

print $string, "\n";

group()

Takes the comparator list and returns a list of comparator lists, each sub-list representing a group of comparators that can be operate without interfering with each other, depending on what is needed for interference-free grouping.

There is one option available, 'grouping', that will produce a grouping that represents parallel operations of comparators. Its values may be:

'graph'

Group the comparators as parallel as possible for graphing.

'parallel'

Arrange the sequence in parallel so that it has a minimum depth. This, after flattening the lists into a single list again, is what is used to produce the sequence in network().

number

Arrange the comparators every number count. This is for simple printing of the comparators, without any grouping for graphing or maximum parallelization.

The chances that you will need to use this function are slim, but the following code snippet may represent an example:

my $nw = Algorithm::Networksort->new(inputs => 8, algorithm => 'batcher');

print "There are ", $nw->length(),
    " comparators in this network, grouped into\n",
    $nw->depth(), " parallel operations.\n\n";

print $nw, "\n";

my @grouped_network = $nw->group(grouping=>'graph');
print "\nThis will be graphed in ", scalar @grouped_network,
    " columns.\n";

This will produce:

There are 19 comparators in this network, grouped into 6 parallel operations.

[[0,4], [1,5], [2,6], [3,7]]
[[0,2], [1,3], [4,6], [5,7]]
[[2,4], [3,5], [0,1], [6,7]]
[[2,3], [4,5]]
[[1,4], [3,6]]
[[1,2], [3,4], [5,6]]

This will be graphed in 11 columns.

Methods For Graphing

graph_eps()

Returns a string that graphs out the network's comparators. The format will be encapsulated postscript.

my $nw = nwsrt(inputs = 4, algorithm => 'bitonic');

print $nw->graph_eps();

graph_svg()

Returns a string that creates a Knuth diagram of a network.

$nw = nwsrt(inputs => 4, algorithm => 'bitonic');
$diagram = $nw->graph_svg();    # Using default attributes.

The string will consist of SVG drawing tags enclosed by <svg> and </svg> tags.

The attributes of the diagram can be changed via colorsettings() and graphsettings().

$nw = nwsrt_best(name => 'floyd09');   # See Algorithm::Networksort::Best

$nw->colorsettings(compbegin => '#04c', compend => '#00c');
$diagram = $nw->graph_svg();

Embedded in a web page, this will produce

9-input Network by Robert W. Floyd

graph_text()

Returns a string that graphs out the network's comparators in plain text.

my $nw = Algorithm::Networksort(inputs = 4, algorithm => 'bitonic');

print $nw->graph_text();

This will produce

o--^-----^--^--o
   |     |  |   
o--v--^--|--v--o
      |  |      
o--^--v--|--^--o
   |     |  |   
o--v-----v--v--o

colorsettings()

Sets the colors of the graph parts, currently for Postscript and SVG output only.

The parts are named.

my %old_colors = $nw->colorsettings(compbegin => "#cc0044", compend => "#cc4400");

You may use web colors names in place of the hex triplet if creating an SVG document (Postscript will not understand it, however). The shorthand hexadecimal form (using only three digits) is also valid, as it will be converted into the standard six digit form before writing out the diagram.

'inputbegin'

Opening of input line.

'inputline'

The input line.

'inputend'

Closing of the input line.

'compbegin'

Opening of the comparator.

'compline'

The comparator line.

'compend'

Closing of the comparator line.

'foreground'

Default color for the graph as a whole.

'background'

Color of the background.

All parts not named are reset to 'undef' (which means calling colorsettings() with no arguments resets everything), and will be colored with the 'foreground' color. The foreground color itself has a default value of '#000000' (black). The one exception is 'background', which has no default color at all.

graphsettings()

Alter diagram properties such as line width or margin size.

#
# Set hz_margin, saving its old value for later.
#
my %old_gset = $nw->graphsettings(hz_margin => 12);

Options

SVG measurements are in pixels.

'hz_margin'

Default value: 18. The horizontal spacing between the edges of the graphic and the sorting network.

'hz_sep'

Default value: 12. The spacing separating the horizontal lines (the input lines).

'vt_margin'

Default value: 21. The vertical spacing between the edges of the graphic and the sorting network.

'vt_sep'

Default value: 12. The spacing separating the vertical lines (the comparators).

'indent'

Default value: 9. The indention between the start of the input lines and the placement of the first comparator. The same value spaces the placement of the final comparator and the end of the input lines.

'compline'

Default value: 2. Width of the lines used to draw comparators.

'compradius'

Default value: 2. Radii of the circles used on either end of the comparator lines.

'inputline'

Default value: 2. Width of the lines used to draw input lines.

'inputradius'

Default value: 2. Radii of the circles used on either end of the input lines.

Options for graph_text():

'inputbegin'

Default value: "o-". The starting characters for the input line.

'inputline'

Default value: "---". The characters that make up an input line.

'inputcompline'

Default value: "-|-". The characters that make up an input line that has a comparator crossing over it.

'inputend'

Default value: "-o\n". The characters that make up the end of an input line.

'compbegin'

Default value: "-^-". The characters that make up an input line with the starting point of a comparator.

'compend'

Default value: "-v-". The characters that make up an input line with the end point of a comparator.

'gapbegin'

Default value: " " (two spaces). The characters that start the gap between the input lines.

'gapcompline'

Default value: " | " (space vertical bar space). The characters that make up the gap with a comparator passing through.

'gapnone'

Default value: " " (three spaces). The characters that make up the space between the input lines.

'gapend'

Default value: " \n" (two spaces and a newline). The characters that end the gap between the input lines.

textsettings()

Alter the text settings for the ascii-art characters.

#
# Set different beginning and ending characters.
#
my %old_gset = $nw->textsettings(
        inputbegin => '---', gapbegin => '   ',
        inputend => '---\n', gapend => '   \n'
    );

Options

'inputbegin'

Default value: "o-". The starting characters for the input line.

'inputline'

Default value: "---". The characters that make up an input line.

'inputcompline'

Default value: "-|-". The characters that make up an input line that has a comparator crossing over it.

'inputend'

Default value: "-o\n". The characters that make up the end of an input line.

'compbegin'

Default value: "-^-". The characters that make up an input line with the starting point of a comparator.

'compend'

Default value: "-v-". The characters that make up an input line with the end point of a comparator.

'gapbegin'

Default value: " " (two spaces). The characters that start the gap between the input lines.

'gapcompline'

Default value: " | " (space vertical bar space). The characters that make up the gap with a comparator passing through.

'gapnone'

Default value: " " (three spaces). The characters that make up the space between the input lines.

'gapend'

Default value: " \n" (two spaces and a newline). The characters that end the gap between the input lines.

ACKNOWLEDGMENTS

Doug Hoyte provided the code for the bitonic, odd-even merge, odd-even transposition, balanced, and bubble sort algorithms, and the idea for what became the statistics() method.

Morwenn found documentation errors and networks that went into Algorithm::Networksort::Best.

SEE ALSO

Bose and Nelson's algorithm.

  • Bose and Nelson, "A Sorting Problem", Journal of the ACM, Vol. 9, 1962, pp. 282-296.

  • Joseph Celko, "Bose-Nelson Sort", Doctor Dobb's Journal, September 1985.

  • Frederick Hegeman, "Sorting Networks", The C/C++ User's Journal, February 1993.

  • Joe Celko, Joe Celko's SQL For Smarties (third edition). Implementing Bose-Nelson sorting network in SQL.

    This material isn't in either the second or fourth edition of the book.

  • Joe Celko, Joe Celko's Thinking in Sets: Auxiliary, Temporal, and Virtual Tables in SQL.

    The sorting network material removed from the third edition of SQL For Smarties seems to have been moved to this book.

Hibbard's algorithm.

  • T. N. Hibbard, "A Simple Sorting Algorithm", Journal of the ACM Vol. 10, 1963, pp. 142-50.

Batcher's Merge Exchange algorithm.

  • Code for Kenneth Batcher's Merge Exchange algorithm was derived from Knuth's The Art of Computer Programming, Vol. 3, section 5.2.2.

Batcher's Bitonic algorithm

  • Kenneth Batcher, "Sorting Networks and their Applications", Proc. of the AFIPS Spring Joint Computing Conf., Vol. 32, 1968, pp. 307-3114. A PDF of this article may be found at http://www.cs.kent.edu/~batcher/sort.pdf.

    The paper discusses both the Odd-Even Merge algorithm and the Bitonic algorithm.

  • Dr. Hans Werner Lang has written a detailed discussion of the bitonic sort algorithm here: http://www.iti.fh-flensburg.de/lang/algorithmen/sortieren/bitonic/bitonicen.htm

  • T. H. Cormen, E. E. Leiserson, R. L. Rivest, Introduction to Algorithms, first edition, McGraw-Hill, 1990, section 28.3.

  • T. H. Cormen, E. E. Leiserson, R. L. Rivest, C. Stein, Introduction to Algorithms, 2nd edition, McGraw-Hill, 2001, section 27.3.

Algorithm discussion

  • Donald E. Knuth, The Art of Computer Programming, Vol. 3: Sorting and Searching (2nd ed.), Addison Wesley Longman Publishing Co., Inc., Redwood City, CA, 1998.

  • Sherenaz W. Al-Haj Baddar and Kenneth E. Batcher, Designing Sorting Networks: A New Paradigm, Springer-Verlag, 2011

  • Kenneth Batcher's web site (http://www.cs.kent.edu/~batcher/) lists his publications, including his paper listed above.

AUTHOR

John M. Gamble may be found at jgamble@cpan.org