AI-Pathfinding-SMAstar
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lib/AI/Pathfinding/SMAstar.pm view on Meta::CPAN
my $invocant = shift;
my $class = ref($invocant) || $invocant;
my $self = {
_priority_queue => AI::Pathfinding::SMAstar::PriorityQueue->new(),
_state_eval_func => undef,
_state_goal_p_func => undef,
_state_num_successors_func => undef,
_state_successors_iterator => undef,
_show_prog_func => undef,
_state_get_data_func => undef,
@_, # attribute override
};
return bless $self, $class;
}
sub state_eval_func {
my $self = shift;
if (@_) { $self->{_state_eval_func} = shift }
return $self->{_state_eval_func};
}
sub state_goal_p_func {
my $self = shift;
if (@_) { $self->{_state_goal_p_func} = shift }
return $self->{_state_goal_p_func};
}
sub state_num_successors_func {
my $self = shift;
if (@_) { $self->{_state_num_successors_func} = shift }
return $self->{_state_num_successors_func};
}
sub state_successors_iterator {
my $self = shift;
if (@_) { $self->{_state_successors_iterator} = shift }
return $self->{_state_successors_iterator};
}
sub state_get_data_func {
my $self = shift;
if (@_) { $self->{_state_get_data_func} = shift }
return $self->{_state_get_data_func};
}
sub show_prog_func {
my $self = shift;
if (@_) { $self->{_show_prog_func} = shift }
return $self->{_show_prog_func};
}
###################################################################
#
# Add a state from which to begin the search. There can
# be multiple start-states.
#
###################################################################
sub add_start_state
{
my ($self, $state) = @_;
my $state_eval_func = $self->{_state_eval_func};
my $state_goal_p_func = $self->{_state_goal_p_func};
my $state_num_successors_func = $self->{_state_num_successors_func},
my $state_successors_iterator = $self->{_state_successors_iterator},
my $state_get_data_func = $self->{_state_get_data_func};
# make sure required functions have been defined
if(!defined($state_eval_func)){
croak "SMAstar: evaluation function is not defined\n";
}
if(!defined($state_goal_p_func)){
croak "SMAstar: goal function is not defined\n";
}
if(!defined($state_num_successors_func)){
croak "SMAstar: num successors function is not defined\n";
}
if(!defined($state_successors_iterator)){
croak "SMAstar: successor iterator is not defined\n";
}
# create a path object from this state
my $state_obj = AI::Pathfinding::SMAstar::Path->new(
_state => $state,
_eval_func => $state_eval_func,
_goal_p_func => $state_goal_p_func,
_num_successors_func => $state_num_successors_func,
_successors_iterator => $state_successors_iterator,
_get_data_func => $state_get_data_func,
);
my $fcost = AI::Pathfinding::SMAstar::Path::fcost($state_obj);
# check if the fcost of this node looks OK (is numeric)
unless(Scalar::Util::looks_like_number($fcost)){
croak "Error: f-cost of state is not numeric. Cannot add state to queue.\n";
}
$state_obj->f_cost($fcost);
# check if the num_successors function returns a number
my $num_successors = $state_obj->get_num_successors();
unless(Scalar::Util::looks_like_number($num_successors)){
croak "Error: Number of state successors is not numeric. Cannot add state to queue.\n";
}
# test out the iterator function to make sure it returns
# an object of the correct type
my $classname = ref($state);
my $test_successor_iterator = $state_obj->{_successors_iterator}->($state);
my $test_successor = $test_successor_iterator->($state);
my $succ_classname = ref($test_successor);
unless($succ_classname eq $classname){
croak "Error: Successor iterator method of object $classname does " .
"not return an object of type $classname.\n";
}
# add this node to the queue
$self->{_priority_queue}->insert($state_obj);
}
###################################################################
#
# start the SMAstar search process
#
###################################################################
sub start_search
{
my ($self,
$log_function,
$str_function,
$max_states_in_queue,
$max_cost,
) = @_;
if(!defined($str_function)){
croak "SMAstar start_search: str_function is not defined.\n";
}
sma_star_tree_search(\($self->{_priority_queue}),
\&AI::Pathfinding::SMAstar::Path::is_goal,
\&AI::Pathfinding::SMAstar::Path::get_descendants_iterator_smastar,
\&AI::Pathfinding::SMAstar::Path::fcost,
\&AI::Pathfinding::SMAstar::Path::backup_fvals,
$log_function,
$str_function,
\&AI::Pathfinding::SMAstar::Path::progress,
$self->{_show_prog_func},
$max_states_in_queue,
$max_cost,
);
}
#################################################################
#
# SMAstar search
# Memory-bounded A* search
#
#
#################################################################
sub sma_star_tree_search
{
my ($priority_queue,
$goal_p,
$successors_func,
$eval_func,
$backup_func,
$log_function, # debug string func; represent state object as a string.
$str_function,
$prog_function,
$show_prog_func,
$max_states_in_queue,
$max_cost,
) = @_;
my $iteration = 0;
my $num_states_in_queue = $$priority_queue->size();
my $max_extra_states_in_queue = $max_states_in_queue;
$max_states_in_queue = $num_states_in_queue + $max_extra_states_in_queue;
my $max_depth = ($max_states_in_queue - $num_states_in_queue);
my $best; # the best candidate for expansion
if($$priority_queue->is_empty() || !$$priority_queue){
return;
}
else{
my $num_successors = 0;
# loop over the elements in the priority queue
while(!$$priority_queue->is_empty()){
lib/AI/Pathfinding/SMAstar.pm view on Meta::CPAN
if($a_seq eq $b_seq){
return 0;
}
elsif($a_seq lt $b_seq){
return -1;
}
else{
return 1;
}
}
1;
__END__
# Below is stub documentation for your module. You'd better edit it!
=head1 NAME
AI::Pathfinding::SMAstar - Simplified Memory-bounded A* Search
=head1 SYNOPSIS
use AI::Pathfinding::SMAstar;
=head2 EXAMPLE
##################################################################
#
# This example uses a hypothetical object called FrontierObj, and
# shows the functions that the FrontierObj class must feature in
# order to perform a path-search in a solution space populated by
# FrontierObj objects.
#
##################################################################
my $smastar = AI::Pathfinding::SMAstar->new(
# evaluates f(n) = g(n) + h(n), returns a number
_state_eval_func => \&FrontierObj::evaluate,
# when called on a node, returns 1 if it is a goal
_state_goal_p_func => \&FrontierObj::goal_test,
# must return the number of successors of a node
_state_num_successors_func => \&FrontierObj::get_num_successors,
# must return *one* successor at a time
_state_successors_iterator => \&FrontierObj::get_successors_iterator,
# can be any suitable string representation
_state_get_data_func => \&FrontierObj::string_representation,
# gets called once per iteration, useful for showing algorithm progress
_show_prog_func => \&FrontierObj::progress_callback,
);
# You can start the search from multiple start-states.
# Add the initial states to the smastar object before starting the search.
foreach my $frontierObj (@start_states){
$smastar->add_start_state($frontierObj);
}
#
# Start the search. If successful, $frontierGoalPath will contain the
# goal path. The optimal path to the goal node will be encoded in the
# ancestry of the goal path. $frontierGoalPath->antecedent() contains
# the goal path's parent path, and so forth back to the start path, which
# contains only the start state.
#
# $frontierGoalPath->state() contains the goal FrontierObj itself.
#
my $frontierGoalPath = $smastar->start_search(
\&log_function, # returns a string used for logging progress
\&str_function, # returns a string used to *uniquely* identify a node
$max_states_in_queue, # indicate the maximum states allowed in memory
$MAX_COST, # indicate the maximum cost allowed in search
);
In the example above, a hypothetical object, C<FrontierObj>, is used to
represent a state, or I<node> in your search space. To use SMA* search to
find a shortest path from a starting node to a goal in your search space, you must
define what a I<node> is, in your search space (or I<point>, or I<state>).
A common example used for informed search methods, and one that is used in Russell's
original paper, is optimal puzzle solving, such as solving an 8 or 15-tile puzzle
in the least number of moves. If trying to solve such a puzzle, a I<node> in the
search space could be defined as a configuration of that puzzle (a paricular
ordering of the tiles).
There is an example provided in the /t directory of this module's distribution,
where SMA* is applied to the problem of finding the shortest palindrome that
contains a minimum number of letters specified, over a given list of words.
Once you have a definition and representation of a node in your search space, SMA*
search requires the following functions to work:
=over
=item *
B<State evaluation function> (C<_state_eval_func above>)
This function must return the cost of this node in the search space. In all
forms of A* search, this means the cost paid to arrive at this node along a
path, plus the estimated cost of going from this node to a goal state:
I<f(x) = g(n) + h(n)>
This function must be I<positive> and I<monotonic>, meaning that the path to a
successor node must be at least as expensive overall when compared to the path
to that node's antecedent. So if the nodes along a particular path are
labeled: 1 -> 2 -> 3, it must be at least as expensive to arrive at node 3 as
it is to arrive at node 2. This amounts to the evaluation of the following
assignment B<[1]> when calculating the cost of a successor of node I<x>:
I<f(successor) = max(f(x), g(successor) + h(successor))>
NOTE: Monotonicity is ensured in this implementation of SMA*, so even if your
function is not monotonic (which is possible, even given an admissible
heuristic), SMA* will assign the antecedent node's cost to a successor if
that successor's I<g+h> amounts to less than the antecedent's f-cost.
=item *
B<State goal function> (C<_state_goal_p_func> above)
Goal predicate function. This function must return 1 if the object argument is a
goal node, or 0 otherwise.
=item *
B<State number of successors function> (C<_state_num_successors_func> above)
This function must return the number of successors of the argument object/node,
i.e. all nodes that are reachable from this node via a single operation.
lib/AI/Pathfinding/SMAstar.pm view on Meta::CPAN
=item *
B<log string function> (C<log_function> above)
This is an arbitrary string used for logging. It also gets passed to
the show-progress function above.
=item *
B<str_function> (C<str_function> above)
This function returns a *unique* string representation of this node.
Uniqueness is required for SMA* to work properly.
=item *
B<max states allowed in memory> (C<max_states_in_queue> above)
An integer indicating the maximum number of expanded nodes to hold in
memory at any given time.
=item *
B<maximum cost> (C<MAX_COST> above)
An integer indicating the maximum cost, beyond which nodes will not
be expanded.
=back
=head1 DESCRIPTION
=head2 Overview
Simplified Memory-bounded A* search (or SMA* search) addresses some of the
limitations of conventional A* search, by bounding the amount of space required
to perform a shortest-path search. This module is an implementation of
SMA*, which was first introduced by Stuart Russell in 1992. SMA* is a simpler,
more efficient variation of the original MA* search introduced by P. Chakrabarti
et al. in 1989 (see references below).
=head2 Motivation and Comparison to A* Search
=head3 A* search
A* Search is an I<optimal> and I<complete> algorithm for computing a sequence of
operations leading from a system's start-state (node) to a specified goal.
In this context, I<optimal> means that A* search will return the shortest
(or cheapest) possible sequence of operations (path) leading to the goal,
and I<complete> means that A* will always find a path to
the goal if such a path exists.
In general, A* search works using a calculated cost function on each node along a
path, in addition to an I<admissible> heuristic estimating the distance from
that node to the goal. The cost is calculated as:
I<f(n) = g(n) + h(n)>
Where:
=over
=item *
I<n> is a state (node) along a path
=item *
I<g(n)> is the total cost of the path leading up to I<n>
=item *
I<h(n)> is the heuristic function, or estimated cost of the path from I<n>
to the goal node.
=back
For a given admissible heuristic function, it can be shown that A* search
is I<optimally efficient>, meaning that, in its calculation of the shortest
path, it expands fewer nodes in the search space than any other algorithm.
To be admissible, the heuristic I<h(n)> can never over-estimate the distance
from the node to the goal. Note that if the heuristic I<h(n)> is set to
zero, A* search reduces to I<Branch and Bound> search. If the cost-so-far
I<g(n)> is set to zero, A* reduces to I<Greedy Best-first> search (which is
neither complete nor optimal). If both I<g(n)> and I<h(n)> are set to zero,
the search becomes I<Breadth-first>, which is complete and optimal, but not
optimally efficient.
The space complexity of A* search is bounded by an exponential of the
branching factor of the search-space, by the length of the longest path
examined during the search. This is can be a problem particularly if the
branching factor is large, because the algorithm may run out of memory.
=head3 SMA* Search
Like A* search, SMA* search is an optimal and complete algorithm for finding
a least-cost path. Unlike A*, SMA* will not run out of memory, I<unless the size
of the shortest path exceeds the amount of space in available memory>.
SMA* addresses the possibility of running out of memory
by pruning the portion of the search-space that is being examined. It relies on
the I<pathmax>, or I<monotonicity> constraint on I<f(n)> to remove the shallowest
of the highest-cost nodes from the search queue when there is no memory left to
expand new nodes. It records the best costs of the pruned nodes within their
antecedent nodes to ensure that crucial information about the search space is
not lost. To facilitate this mechanism, the search queue is best maintained
as a search-tree of search-trees ordered by cost and depth, respectively.
=head4 Nothing is for free
The pruning of the search queue allows SMA* search to utilize all available
memory for search without any danger of overflow. It can, however, make
SMA* search significantly slower than a theoretical unbounded-memory search,
due to the extra bookkeeping it must do, and because nodes may need to be
re-expanded (the overall number of node expansions may increase).
In this way there is a trade-off between time and space.
It can be shown that of the memory-bounded variations of A* search, such MA*, IDA*,
Iterative Expansion, etc., SMA* search expands the least number of nodes on average.
However, for certain classes of problems, guaranteeing optimality can be costly.
This is particularly true in solution spaces where:
=over
=item *
the branching factor of the search space is large
=item *
there are many equivalent optimal solutions (or shortest paths)
=back
For solution spaces with these characteristics, stochastic methods or
approximation algorithms such as I<Simulated Annealing> can provide a
massive reduction in time and space requirements, while introducing a
tunable probability of producing a sub-optimal solution.
=head1 METHODS
=head2 new()
my $smastar = AI::Pathfinding::SMAstar->new();
Creates a new SMA* search object.
=head2 start_search()
my $frontierGoalObj = $smastar->start_search(
\&log_function, # returns a string used for logging progress
\&str_function, # returns a string used to *uniquely* identify a node
$max_states_in_queue, # indicate the maximum states allowed in memory
$MAX_COST, # indicate the maximum cost allowed in search
);
Initiates a memory-bounded search. When calling this function, pass a handle to
a function for recording current status( C<log_function> above- this can be
an empty subroutine if you don't care), a function that returns a *unique* string
representing a node in the search-space (this *cannot* be an empty subroutine), a
maximum number of expanded states to store in the queue, and a maximum cost
value (beyond which the search will cease).
=head2 state_eval_func()
$smastar->state_eval_func(\&FrontierObj::evaluate);
Set or get the handle to the function that returns the cost of the object
argument (node) in the search space.
=head2 state_goal_p_func()
$smastar->state_goal_p_func(\&FrontierObj::goal_test);
Set/get the handle to the goal predicate function. This is a function
that returns 1 if the argument object is a goal node, or 0 otherwise.
=head2 state_num_successors_func()
$smastar->state_num_successors_func(\&FrontierObj::get_num_successors);
Set/get the handle to the function that returns the number of successors
of this the object argument (node).
=head2 state_successors_iterator()
$smastar->state_successors_iterator(\&FrontierObj::get_successors_iterator);
Set/get the handle to the function that returns iterator that produces the
next successor of this node.
=head2 state_get_data_func()
$smastar->state_get_data_func(\&FrontierObj::string_representation);
Set/get the handle to the function that returns a string
representation of this node.
=head2 show_prog_func()
$smatar->show_prog_func(\&FrontierObj::progress_callback);
Sets/gets the callback function for displaying the progress of the search.
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