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Algorithm::SetCovering 0.05
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NAME
Algorithm::SetCovering - Algorithms to solve the "set covering problem"
SYNOPSIS
use Algorithm::SetCovering;
my $alg = Algorithm::SetCovering->new(
columns => 4,
mode => "greedy");
$alg->add_row(1, 0, 1, 0);
$alg->add_row(1, 1, 0, 0);
$alg->add_row(1, 1, 1, 0);
$alg->add_row(0, 1, 0, 1);
$alg->add_row(0, 0, 1, 1);
my @to_be_opened = (@ARGV || (1, 1, 1, 1));
my @set = $alg->min_row_set(@to_be_opened);
print "To open (@to_be_opened), we need ",
scalar @set, " keys:\n";
for(@set) {
print "$_: ", join('-', $alg->row($_)), "\n";
}
DESCRIPTION
Consider having M keys and N locks. Every key opens one or more locks:
| lock1 lock2 lock3 lock4
-----+------------------------
key1 | x x
key2 | x x
key3 | x x x
key4 | x x
key5 | x x
Given an arbitrary set of locks you have to open (e.g. 2,3,4), the task
is to find a minimal set of keys to accomplish this. In the example
above, the set [key4, key5] fulfils that condition.
The underlying problem is called "set covering problem" and the
corresponding decision problem is NP-complete.
Methods
$alg = Algorithm::SetCovering->new(columns => $cols, [mode => $mode]);
Create a new Algorithm::SetCovering object. The mandatory parameter
"columns" needs to be set to the number of columns in the matrix
(the number of locks in the introductory example).
"mode" is optional and selects an algorithm for finding the
solution. The following values for "mode" are implemented:
"brute_force"
Will iterate over all permutations of keys. Only recommended for
very small numbers of keys.
"greedy"
Greedy algorithm. Scales O(mn^2). Can't do much better for a
NP-hard problem.
The default for "mode" is set to "greedy".
$alg->add_row(@columns)
Add a new row to the matrix. In the example above, this adds one key
and specifies which locks it is able to open.
$alg->add_row(1,0,0,1);
specifies that the new key can open locks #1 and #4.
The number of elements in @columns needs to match the previously
defined number of columns.
$alg->min_row_set(@columns_to_cover)
Determines a minimal set of keys to cover a given set of locks and
returns an array of index numbers for those keys.
Defines which columns have to be covered by passing in an array with
true values on element positions that need to be covered. For
example,
my @idx_set = $alg->min_row_set(1,1,0,1);
specifies that all but the third column have to be covered and
returns an array of index numbers into an array, defined previously
(and implicitely) via successive add_row() commands.
If no set of keys can be found that satisfies the given requirement,
an empty list is returned.
If you've forgotten which locks the key referred to by a certain
index number can open, use the "rows()" method to find out:
my(@opens_locks) = $alg->rows($idx_set[0]);
will give back an array of 0's and 1's, basically returning the very
parameters we've passed on to the add_row() command previously.
Strategies
Currently, the module implements the Greedy algorithm and also (just for
scientific purposes) a dumb brute force method, creating all possible
combinations of keys, sorting them by the number of keys used
(combinations with fewer keys have priority) and trying for each of them
if it fits the requirement of opening a given number of locks.
This obviously won't scale beyond a really small number of keys (N),
because the number of permutations will be 2**N-2.
The Greedy Algorithm, on the other hand scales with O(mn^2), with m
being the number of keys and n being the number of locks.
Limitations
Julien Gervais-Bird points out: The greedy
algorithm does not always return the minimal set of keys. Consider this
example:
| lock1 lock2 lock3 lock4 lock5 lock6
-----+------------------------------------
key1 | x x x
key2 | x x
key3 | x x
key4 | x x
The minimal set of keys to open all the locks is (key2, key3, key4),
however the greedy algorithm will return (key1,key2,key3,key4) because
key1 opens more locks than any other key.
AUTHOR
Mike Schilli, 2003,
Thanks to the friendly guys on rec.puzzles, who provided me with
valuable input to analyze the problem and explained the algorithm:
Craig
Robert Israel
Patrick Hamlyn
COPYRIGHT AND LICENSE
Copyright 2003 by Mike Schilli
This library is free software; you can redistribute it and/or modify it
under the same terms as Perl itself.