orbital.algorithm.template
Class IterativeDeepeningAStar

java.lang.Object
  orbital.algorithm.template.GeneralSearch
      orbital.algorithm.template.GeneralBoundingSearch
          orbital.algorithm.template.IterativeDeepeningAStar

All Implemented Interfaces:

java.io.Serializable, AlgorithmicTemplate, EvaluativeAlgorithm, HeuristicAlgorithm

public class IterativeDeepeningAStar
extends GeneralBoundingSearch
implements HeuristicAlgorithm

Iterative Deepening A^* (IDA^*). A heuristic search algorithm.

Apart from the evaluation function, IDA^* has not much in common with A^* but resembles ID, instead. It is not an instance of best-first search, but of bounding search. Which also means that it has less administrative overhead than maintaining priority queues.

IDA^* is complete, optimal if the heuristic function h is admissible. It has a time complexity of O(b^d) and a space complexity of O(b*d).

The effective time complexity of IDA^* depends upon the number of different values the heuristic returns. If |h(S)| is small, IDA^* only has to go through very little iterations. Of course, if the heuristic has too few values, then it does not guide search at all.

Author:

André Platzer

See Also:

"Korf, R.E. (1985) Depth-first iterative deepening. An optimal admissible tree search. AIJ, 27(1), 97-109", Serialized Form

Nested Class Summary

Nested classes/interfaces inherited from class orbital.algorithm.template.GeneralSearch

GeneralSearch.OptionIterator

Nested classes/interfaces inherited from interface orbital.algorithm.template.HeuristicAlgorithm

HeuristicAlgorithm.Configuration, HeuristicAlgorithm.PatternDatabaseHeuristic

Nested classes/interfaces inherited from interface orbital.algorithm.template.EvaluativeAlgorithm

EvaluativeAlgorithm.EvaluationComparator

Constructor Summary

IterativeDeepeningAStar(Function heuristic)
Create a new instance of IDA^* search.

Method Summary

Function	complexity() O(bd) where b is the branching factor and d the solution depth.
protected java.util.Iterator	createTraversal(GeneralSearchProblem problem) Define a traversal order by creating an iterator for the problem's state space.
Function	getEvaluation() f(n) = g(n) + h(n).
Function	getHeuristic() Get the heuristic function used.
boolean	isOptimal() Optimal if heuristic is admissible.
protected boolean	isOutOfBounds(java.lang.Object node) Compare f(n) with bound, normally and in case update cheapest pruned bound.
void	setHeuristic(Function heuristic) Set the heuristic function to use.
protected java.lang.Object	solveImpl(GeneralSearchProblem problem) Solve with bounds f(n0), f(n1), f(n2), ...
Function	spaceComplexity() O(b*d) where b is the branching factor and d the solution depth.

Methods inherited from class orbital.algorithm.template.GeneralBoundingSearch

getBound, isContinuedWhenFound, processSolution, search, setBound, setBound, setContinuedWhenFound

Methods inherited from class orbital.algorithm.template.GeneralSearch

getProblem, solve

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Methods inherited from interface orbital.algorithm.template.AlgorithmicTemplate

solve, spaceComplexity

Constructor Detail

IterativeDeepeningAStar

public IterativeDeepeningAStar(Function heuristic)

Create a new instance of IDA^* search. Which is a bounding search using the evaluation function f(n) = g(n) + h(n).

Parameters:

heuristic - the heuristic cost function h:S->R embedded in the evaluation function f.

See Also:

getEvaluation()

Method Detail

getHeuristic

public Function getHeuristic()

Description copied from interface: HeuristicAlgorithm

Get the heuristic function used.

Specified by:

getHeuristic in interface HeuristicAlgorithm

Returns:

the heuristic cost function h:S->R estimating h^*.

setHeuristic

public void setHeuristic(Function heuristic)

Description copied from interface: HeuristicAlgorithm

Set the heuristic function to use.

An heuristic cost function h:S->R is estimating the cost to get from a node n to a goal G. For several heuristic algorithms this heuristic function needs to be admissible

h =< h^* i.e. h is a lower bound for the effective cost function h^*. Then the pathmax heuristic f(s') := max {f(s), g(s') + h(s')} (for transitions s->s'=t(s,a) with a in A(s)) is monotonic.

A heuristic cost function h is monotonic :<=> the f-costs (with h) do not decrease in any path from the initial state <=> h obeys the triangular inequality

for all s in S,a in A(s) h(s) =< h(s') + c(s,a) with s' = t(s,a) i.e. the sum of the costs from A to C and from C to B must not be less than the cost from A to B.

A simple improvement for heuristic functions is using pathmax.

Specified by:

setHeuristic in interface HeuristicAlgorithm

Parameters:

heuristic - the heuristic cost function h:S->R estimating h^*. h will be embedded in the evaluation function f.

See Also:

"Pearl, J. Heuristics: Intelligent Search Strategies for Computer Problem Solving. Addison-Wesley, Reading, Massachusetts. 1984."

getEvaluation

public Function getEvaluation()

f(n) = g(n) + h(n).

Specified by:

getEvaluation in interface EvaluativeAlgorithm

Specified by:

getEvaluation in interface HeuristicAlgorithm

Returns:

the evaluation function f:S->R used to evaluate (either utility or cost) value of states.

complexity

public Function complexity()

O(b^d) where b is the branching factor and d the solution depth.

Specified by:

complexity in interface AlgorithmicTemplate

Returns:

the function f for which the solve() method of this algorithm runs in O(f(n)) assuming the algorithmic problem hook to run in O(1).

See Also:

AlgorithmicTemplate.solve(AlgorithmicProblem)

isOptimal

public boolean isOptimal()

Optimal if heuristic is admissible.

Specified by:

isOptimal in class GeneralSearch

Returns:

whether this search algorithm is optimal, i.e. whether solutions found are guaranteed to be optimal.

isOutOfBounds

protected boolean isOutOfBounds(java.lang.Object node)

Compare f(n) with bound, normally and in case update cheapest pruned bound.

Overrides:

isOutOfBounds in class GeneralBoundingSearch

Parameters:

node - the node to check.

Returns:

whether the node is out of current bounds.

See Also:

nextBound

solveImpl

protected java.lang.Object solveImpl(GeneralSearchProblem problem)

Solve with bounds f(n₀), f(n₁), f(n₂), ... until a solution is found. Where n₀=root, and n_i is the cheapest node pruned in iteration i-1.

Overrides:

solveImpl in class GeneralSearch

Returns:

the solution found by GeneralSearch.search(Iterator).

See Also:

GeneralSearch.search(Iterator)

createTraversal

protected java.util.Iterator createTraversal(GeneralSearchProblem problem)

Description copied from class: GeneralSearch

Define a traversal order by creating an iterator for the problem's state space.

Lays a linear order through the state space which the search can simply follow sequentially. Thus a traversal policy effectively reduces a search problem through a graph to a search problem through a linear sequence of states. Of course, the mere notion of a traversal policy does not yet solve the task of finding a good order of states, but only encapsulate it. Complete search algorithms result from traversal policies that have a linear sequence through the whole state space.

Parameters:

problem - the problem whose state space to create a traversal iterator for.

Returns:

an iterator over the options of the problem's state space thereby encapsulating and hiding the traversal order.

See Also:

Factory Method, GeneralSearch.OptionIterator

spaceComplexity

public Function spaceComplexity()

O(b*d) where b is the branching factor and d the solution depth.

Returns:

the function f for which the solve() method of this algorithm consumes memory with an amount in O(f(n)) assuming the algorithmic problem hook uses space in O(1).

See Also:

AlgorithmicTemplate.solve(AlgorithmicProblem)