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| {{Refimprove|date=February 2010}}
| | Andera is what you can contact her but she by no means truly liked that name. Mississippi is exactly where her home is but her spouse desires them to move. It's not a common factor but what I like doing is to climb but I don't have the time lately. Credit authorising is how she tends to make a living.<br><br>Also visit my web page ... psychic readers ([http://jplusfn.gaplus.kr/xe/qna/78647 jplusfn.gaplus.kr]) |
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| '''SSS*''' is a [[search algorithm]], introduced by George Stockman in 1979, that conducts a [[state space search]] traversing a [[game tree]] in a [[Best-first search|best-first]] fashion similar to that of the [[A* search algorithm]].
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| SSS* is based on the notion of [[solution tree]]s. Informally, a solution tree can be formed from any arbitrary game tree by pruning the number of branches at each [[minimax|MAX]] node to one. Such a tree represents a complete strategy for MAX, since it specifies exactly one MAX action for every possible sequence of moves might be made by the opponent. Given a game tree, SSS* searches through the space of partial solution trees, gradually analyzing larger and larger subtrees, eventually producing a single solution tree with the same root and Minimax value as the original game tree. SSS* never examines a node that [[alpha-beta pruning]] would prune, and may prune some branches that alpha-beta would not. Stockman speculated that SSS* may therefore be a better general algorithm than alpha-beta. However, [[Judea Pearl|Igor Roizen]] and [[Judea Pearl]] have shown<ref>{{cite journal|last=Roizen|first=Igor|coauthors=Judea Pearl|title=A minimax algorithm better than alpha-beta?: Yes and No|journal=Artificial Intelligence|year=1983|month=March|volume=1-2}}</ref> that the savings in the number of positions that SSS* evaluates relative to alpha/beta is limited and generally not enough to compensate for the increase in other resources (e.g., the storing and sorting of a list of nodes made necessary by the best-first nature of the algorithm). However, [[Aske Plaat]], [[Jonathan Schaeffer]], Wim Pijls and Arie de Bruin have shown that a sequence of null-window alpha-beta calls is equivalent to SSS* (i.e., it expands the same nodes in the same order) when alpha-beta is used with a [[transposition table]], as is the case in all game-playing programs for chess, checkers, etc. Now the storing and sorting of the OPEN list were no longer necessary. This allowed the implementation of (an algorithm equivalent to) SSS* in tournament quality game-playing programs. Experiments showed that it did indeed perform better than [[Alpha-Beta]] in practice, but that it did not beat [[NegaScout]].<ref>{{cite journal|last=Plaat|first=Aske|coauthors=Jonathan Schaeffer, Wim Pijls, and Arie de Bruin|title=Best-first Fixed-depth Minimax Algorithms|journal=Artificial Intelligence|year=1996|month=november|volume=87|issue=1-2}}</ref>
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| The reformulation of a best-first algorithm as a sequence of depth-first calls prompted the formulation of a class of null-window alpha-beta algorithms, of which [[MTD-f]] is the best known example.
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| ==Algorithm==
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| There is a [[priority queue]] OPEN that stores states <math>(J, s, h)</math> or the nodes, where <math>J</math> - node identificator ([[Dewey's notation]] is used to identify nodes, <math>\epsilon</math> is a root), <math>s\in\{L,S\}</math> - state of the node <math>J</math> (L - the node is live, which means it's not solved yet and S - the node is solved), <math>h\in(-\infty, \infty)</math> - value of the solved node. Items in OPEN queue are sorted descending by their <math>h</math> value. If more than one node has the same value of <math>h</math>, a node left-most in the tree is chosen.
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| OPEN := { (e,L,inf) }
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| while (true) // repeat until stopped
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| pop an element p=(J,s,h) from the head of the OPEN queue
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| if J == e and s == S
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| STOP the algorithm and return h as a result
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| else
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| apply Gamma operator for p
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| <math>\Gamma</math> operator for <math>p=(J,s,h)</math> is defined in the following way:
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| if s == L
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| if J is a terminal node
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| (1.) add (J,S,min(h,value(J))) to OPEN
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| else if J is a MIN node
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| (2.) add (J.1,L,h) to OPEN
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| else
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| (3.) for j=1..number_of_children(J) add (J.j,L,h) to OPEN
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| else
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| if J is a MIN node
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| (4.) add (parent(J),S,h) to OPEN
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| remove from OPEN all the states that are associated with the children of parent(J)
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| else if is_last_child(J) // if J is the last child of parent(J)
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| (5.) add (parent(J),S,h) to OPEN
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| else
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| (6.) add (parent(J).(k+1),L,h) to OPEN // add state associated with the next child of parent(J) to OPEN
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| ==External links==
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| *[http://chessprogramming.wikispaces.com/SSS*+and+Dual* Chess Programming Wiki]
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| *[http://www.cse.msu.edu/~stockman/ George Stockman's website]
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| *[http://plaat.nl/mtdf.html introduction to the MTD-f algorithm]
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| ==References==
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| {{Reflist}}
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| <references />
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| [[Category:Search algorithms]]
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