Informed Search & Exploration (II) - ntnu.edu.tw

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Artificial Intelligence, Spring, 2009

Informed Search &Exploration (II)

Instructor: Tsung-Che [email protected]

Department of Computer Science and Information EngineeringNational Taiwan Normal University

29Artificial Intelligence, Spring, 2009

Recursive Best-first Search

Its structure is similar to that of arecursive DFS.

It keeps track of the f-value of the bestalternative path available from anyancestor of the current node.

If the current node exceeds this limit, therecursion unwinds back to the alternativepath.

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A first glance

Artificial Intelligence: A Modern Approach, 2nd ed., Figure 4.5



Artificial Intelligence: A Modern Approach,2nd ed., Figure 4.6(a)

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Artificial Intelligence: A Modern Approach,2nd ed., Figure 4.6(a)

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Artificial Intelligence: A Modern Approach, 2nd ed., Figure 4.6(a)

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Artificial Intelligence: A Modern Approach, 2nd ed., Figure 4.6(b)

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Artificial Intelligence: A Modern Approach, 2nd ed., Figure 4.6(b)

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Artificial Intelligence: A Modern Approach, 2nd ed., Figure 4.6(c)

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Artificial Intelligence: A Modern Approach, 2nd ed., Figure 4.6(c)

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RBFS uses only linear space, but it suffersfrom excessive node re-generation.



RBFS is optimal if h(n) is admissible.Its time complexity depends on

the accuracy of the heuristic function, and how often the best path changes as nodes are

expanded.

The problem is caused by using too littlememory. It cannot check for repeated states other than

those on the current path.They may explore the same state many times.

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Simplified Memory-bounded A*

SMA* proceeds just like A*, expanding thebest leaf until the memory is full.

It drops the worst leaf node –the one withthe highest f-value. Like RBFS, it backs up the value of the

forgotten node to its parent.When all leaf nodes have the same f-value, it

expands the newest and deletes the oldest.



It regenerates the subtree only when allother paths have been shown to look worsethan the path it has forgotten.

It is complete if there is any reachablesolution.

It is optimal if any optimal solution isreachable.

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It might be the best general-purposealgorithm for finding optimal solutions.

On very hard problems, however, SMA* isforced to switch back and forth continually.

“Memory limitations can make a problemintractable from the point of view ofcomputation time.”


Tea Time (Roomba)

Officialhttp://www.youtube.com/watch?v=HqhIMFQNGCg

Unofficialhttp://www.youtube.com/watch?v=D0DEPpFL9OY

Wiimba http://www.youtube.com/watch?v=NqbcfSqPnLA

Pacmba http://www.youtube.com/watch?v=2wsP_nmk_iw

Surfin’bahttp://www.youtube.com/watch?v=tLbprdjTX0w

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Heuristic Functions

Revisit of 8-puzzleThe average solution cost for a random 8-

puzzle problem is about 22 steps.The branching factor is about 3.An exhaustive search to depth 22 would look at

about 322 3.1 1010 states. But by keeping track of repeated states, there

are only 9!/2 = 181,440 distinct states, which ismanageable.

Artificial Intelligence: A ModernApproach, 2nd ed., Figure 4.7


Heuristic Functions

But for 15-puzzle, there is roughly 1013

distinct states. (Try: http://www.ababasoft.com/kids/15.html)

It means that we need a good heuristicfunction.

Two common heuristic functions: h1: the number of misplaced tiles (8) h2: Manhattan distance (18)

Both are admissible.


[3] [1] [2]

[2] [3]

[2] [2] [3]

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Heuristic Functions

One way to evaluate a heuristic is theeffective branching factor b*.N: total number of nodes generated by A* d: solution depth

N + 1 = 1 + b* + (b*)2 + (b*)3 + … + (b*)d

e.g. A* finds a solution at depth 5 using 52nodes, then b* is about 1.92.


Heuristic Functions

The effective branching factor is fairlyconstant for sufficiently hard problems.

A well-designed heuristic would have avalue of b* close to 1.

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Heuristic Functions

Performance comparison


Data are averaged over 100 randomly generated instances.

30,000 times


Heuristic Functions

Is h2 always better than h1? For any node n, h2(n) h1(n). (h2 dominates h1.) Every node with g(n) + h(n) = f(n) < C* will be

surely be expanded.

This is the same as saying that every node withh(n) < C* g(n) will be expanded.

Every node that is surely expanded by A* withh2 will also surely be expanded with h1.

Yes.

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Heuristic Functions

“It is always better to use a heuristicfunction with higher values,

provided it does not overestimate and doesnot take too long computation time.”


Inventing Heuristic Functions

The relation between h1(h2) and 8-puzzleThey are perfectly accurate path lengths for

the relaxed 8-puzzle problem.

h1: the number of misplaced tiles (8)h2: Manhattan distance (18)


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An example of generating relaxed problems

Original:A tile can move from square A to square Bif A is adjacent to B and B is blank.

Relaxed:(1) A tile can move from A to B if A is adjacent to B.(2) A tile can move from A to B if B is blank.(3) A tile can move from A to B.

h2

h1

Notice that the relaxed problem should be easy to solve,or calculation of h(n) will be too time-consuming.



The cost of an optimal solution to a relaxedproblem is an admissible heuristic for theoriginal problem.

In addition, it obeys the triangle inequalityand is therefore consistent. (because the derived heuristic is an exact cost

for the relaxed problem)

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How do we select from a set of heuristicsh1, h2, …, hm?

“We need not make a choice!”

Define h(n) = max{h1(n), h2(n), …, hm(n)},then

h is admissible, consistent, and dominates allits component heuristics.



Admissible heuristics can also be derivedfrom the solution cost of a subproblem.


The heuristic issubstantially moreaccurate than Manhattandistance in some cases.

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The idea behind pattern databases is tostore these exact solution costs for everypossible subproblem instances.

Then we compute an admissible heuristicduring the search by looking up in thedatabase.

The database itself is constructed bysearching backwards from the goal state. The search cost is amortized over many subproblem

instances.



We can construct databases for 1-2-3-4,5-6-7-8, and so on.

These heuristics are then combined bytaking the maximum value.

The number of nodes generated by thisheuristic can be reduced by a factor of1000, comparing with that by theManhattan distance.

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Can we add, rather than taking themaximum of, the heuristics from the 1-2-3-4 and the 5-6-7-8 databases?No, if we add the total number of moves

directly.Yes, if we just count the number of moves of

the relevant tiles.

Using this kind of disjoint patterndatabases, it is possible to solve random15-puzzles in a few milliseconds. (reduction by afactor of 10,000)



Disjoint pattern databases do not alwayswork.

It requires that the problem can bedivided up in such a way that each moveaffects only one subproblem.

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Another solution is to learn fromexperience, which means solving lots of 8-puzzles, for instance.

From these instances, learning algorithmslike neural networks can be used.


A 15-Puzzle Solver

http://www.ic-net.or.jp/home/takaken/e/15pz/index.html

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Another 15-Puzzle Solver

http://createuniverses.blogspot.com/2008/01/15-puzzle-solver.html

Informed Search & Exploration (II) - ntnu.edu.tw

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