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| [[Combinatorial game theory]] has several ways of measuring '''game complexity'''. This article describes five of them: state-space complexity, game tree size, decision complexity, game-tree complexity, and computational complexity.
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| | |
| ==Measures of game complexity==
| |
| === State-space complexity ===
| |
| The '''state-space complexity''' of a game is the number of legal game positions reachable from the initial position of the game.<ref name="Allis1994"/>
| |
| | |
| When this is too hard to calculate, an [[upper bound]] can often be computed by including illegal positions or positions that can never arise in the course of a game.
| |
| | |
| === Game tree size ===
| |
| The '''game tree size''' is the total number of possible games that can be played: the number of leaf nodes in the [[game tree]] rooted at the game's initial position.
| |
| | |
| The game tree is typically vastly larger than the state space because the same positions can occur in many games by making moves in a different order (for example, in a [[tic-tac-toe]] game with two X and one O on the board, this position could have been reached in two different ways depending on where the first X was placed). An upper bound for the size of the game tree can sometimes be computed by simplifying the game in a way that only increases the size of the game tree (for example, by allowing illegal moves) until it becomes tractable.
| |
| | |
| However, for games where the number of moves is not limited (for example by the size of the board, or by a rule about repetition of position) the game tree is infinite.
| |
| | |
| === Decision trees ===
| |
| The next two measures use the idea of a '''decision tree'''. A decision tree is a subtree of the game tree, with each position labelled with "player A wins", "player B wins" or "drawn", if that position can be proved to have that value (assuming best play by both sides) by examining only other positions in the graph. (Terminal positions can be labelled directly; a position with player A to move can be labelled "player A wins" if any successor position is a win for A, or labelled "player B wins" if all successor positions are wins for B, or labelled "draw" if all successor positions are either drawn or wins for B. And correspondingly for positions with B to move.)
| |
| | |
| ==== Decision complexity ====
| |
| '''Decision complexity''' of a game is the number of leaf nodes in the smallest decision tree that establishes the value of the initial position.
| |
| | |
| ==== Game-tree complexity ====
| |
| The '''game-tree complexity''' of a game is the number of leaf nodes in the smallest ''full-width'' decision tree that establishes the value of the initial position.<ref name="Allis1994"/> A full-width tree includes all nodes at each depth. | |
| | |
| This is an estimate of the number of positions we would have to evaluate in a [[minimax]] search to determine the value of the initial position.
| |
| | |
| It's hard even to estimate the game-tree complexity, but for some games a reasonable lower bound can be given by raising the game's average [[branching factor]] to the power of the number of [[Ply (chess)|plies]] in an average game, or:
| |
| | |
| <math>GTC \geq b^d</math> | |
| | |
| === Computational complexity ===
| |
| The '''[[computational complexity]]''' of a game describes the [[Asymptotic analysis|asymptotic]] difficulty of a game as it grows arbitrarily large, expressed in [[big O notation]] or as membership in a [[complexity class]]. This concept doesn't apply to particular games, but rather to games that have been [[generalized game|generalized]] so they can be made arbitrarily large, typically by playing them on an ''n''-by-''n'' board. (From the point of view of computational complexity a game on a fixed size of board is a finite problem that can be solved in O(1), for example by a look-up table from positions to the best move in each position.)
| |
| | |
| ==Example: tic-tac-toe==
| |
| For [[tic-tac-toe]], a simple upper bound for the size of the state space is 3<sup>9</sup> = 19,683. (There are three states for each cell and nine cells.) This count includes many illegal positions, such as a position with five crosses and no noughts, or a position in which both players have a row of three. A more careful count, removing these illegal positions, gives 5,478. And when rotations and reflections of positions are considered identical, there are only 765 essentially different positions.
| |
| | |
| A simple upper bound for the size of the game tree is 9! = 362,880. (There are nine positions for the first move, eight for the second, and so on.) This includes illegal games that continue after one side has won. A more careful count gives 255,168 possible games. When rotations and reflections of positions are considered the same, there are only 26,830 possible games.
| |
| | |
| The computational complexity of tic-tac-toe depends on how it is [[generalized game|generalized]]. A natural generalization is to [[m,n,k-game|''m'',''n'',''k''-games]]: played on an ''m'' by ''n'' board with winner being the first player to get ''k'' in a row. It is immediately clear that this game can be solved in [[DSPACE]](''mn'') by searching the entire game tree. This places it in the important complexity class [[PSPACE]]. With some more work it can be shown to be [[PSPACE-complete]].<ref name="Reisch1980">{{cite journal | author = Stefan Reisch | title = Gobang ist PSPACE-vollstandig (Gomoku is PSPACE-complete) | journal = Acta Informatica | volume = 13 | page = 5966 | year = 1980}}</ref>
| |
| | |
| ==Complexities of some well-known games==
| |
| Due to the large size of game complexities, this table gives the ceiling of their [[logarithm]] to base 10. All of the following numbers should be considered with caution: seemingly-minor changes to the rules of a game can change the numbers (which are often rough estimates anyway) by tremendous factors, which might easily be much greater than the numbers shown.
| |
| | |
| {| class="wikitable sortable" cellpadding="5"
| |
| |-
| |
| !Game
| |
| !Board size
| |
| (positions)
| |
| !State-space complexity
| |
| (as [[logarithm|log]] to base 10)
| |
| !Game-tree complexity
| |
| (as [[logarithm|log]] to base 10)
| |
| !Average game length
| |
| ([[Ply (game theory)|plies]])
| |
| !Branching factor
| |
| !Ref
| |
| ![[Complexity class]] of suitable [[generalized game]]
| |
| |-
| |
| |[[Tic-tac-toe]]
| |
| |align="right"|9
| |
| |align="right"|3
| |
| |align="right"|5
| |
| |align="right"|9
| |
| |align="right"|4
| |
| |align="right"|
| |
| |[[PSPACE-complete]]<ref name="Reisch1980"/>
| |
| |-
| |
| |[[Sim (pencil game)|Sim]]
| |
| |align="right"|15
| |
| |align="right"|3
| |
| |align="right"|8
| |
| |align="right"|14
| |
| |align="right"|3.7
| |
| |align="right"|
| |
| |[[PSPACE-complete]]<ref>[http://portal.acm.org/citation.cfm?id=728124 Wolfgang Slany: The Complexity of Graph Ramsey Games]</ref>
| |
| |-
| |
| |[[Pentomino]]es
| |
| |align="right"|64
| |
| |align="right"|12
| |
| |align="right"|18
| |
| |align="right"|10
| |
| |align="right"|75
| |
| |align="right"|<ref name="GamesSolved"/> <ref>Hilarie K. Orman: ''Pentominoes: A First Player Win'' in ''[http://www.msri.org/publications/books/Book29/contents.html Games of no chance]'', MSRI Publications – Volume 29, 1996, pages 339-344. Online: [http://www.msri.org/publications/books/Book29/files/orman.pdf pdf].</ref>
| |
| |?, but in [[PSPACE]]
| |
| |-
| |
| |[[Kalah]] <ref> See van den Herik et al for rules. </ref>
| |
| |align="right"|14
| |
| |align="right"|13
| |
| |align="right"|18
| |
| |align="right"|
| |
| |align="right"|
| |
| |align="right"|<ref name="GamesSolved"/>
| |
| |Generalization is unclear
| |
| |-
| |
| |[[Connect Four]]
| |
| |align="right"|42
| |
| |align="right"|13
| |
| |align="right"|21
| |
| |align="right"|36
| |
| |align="right"|4
| |
| |align="right"|<ref>{{cite web | title = John's Connect Four Playground | author = John Tromp | year = 2010 | url = http://www.cwi.nl/~tromp/c4/c4.html}}</ref> <ref name="Allis1994"/>
| |
| |?, but in [[PSPACE]]
| |
| |-
| |
| |[[Domineering]] (8 × 8)
| |
| |align="right"|64
| |
| |align="right"|15
| |
| |align="right"|27
| |
| |align="right"|30
| |
| |align="right"|8
| |
| |align="right"|<ref name="GamesSolved"/>
| |
| |?, but in [[PSPACE]]; in [[P (complexity class)|P]] for certain dimensions<ref> {{cite paper| authors=Michael Lachmann | author2= Cristopher Moore | author3=Ivan Rapaport|title=Who wins domineering on rectangular boards? | volume=MSRI Combinatorial Game Theory Research Workshop | date=July 2000 }}</ref>
| |
| |-
| |
| |[[Congkak]]
| |
| |align="right"|14
| |
| |align="right"|15
| |
| |align="right"|33
| |
| |align="right"|
| |
| |align="right"|
| |
| |align="right"|<ref name="GamesSolved">{{cite journal | title= Games solved: Now and in the future | author = H. J. van den Herik | author2 = J. W. H. M. Uiterwijk | author3 = J. van Rijswijck | year = 2002 | journal = Artificial Intelligence | volume = 134 | issue=1–2 | pages=277–311 | doi= 10.1016/S0004-3702(01)00152-7}} </ref>
| |
| |
| |
| |-
| |
| |[[English draughts|English draughts (8x8) (checkers)]]
| |
| |align="right"|32
| |
| |align="right"|20 or 18
| |
| |align="right"|31
| |
| |align="right"|70
| |
| |align="right"|2.8
| |
| |align="right"|<ref>{{cite journal | author = Jonathan Schaeffer et al. | title = Checkers is Solved | journal = Science | date = July 6, 2007 | doi=10.1126/science.1144079 | volume=317 |issue= 5844 |pages= 1518–1522 | pmid=17641166 }}</ref> or <ref name="Allis1994"/>
| |
| |[[EXPTIME-complete]]<ref name="robson1984">{{cite journal | author = J. M. Robson | title = N by N checkers is Exptime complete | journal = [[SIAM Journal on Computing]], | volume = 13 | issue = 2 | pages = 252–267 | year = 1984 | doi = 10.1137/0213018}}</ref>
| |
| |-
| |
| |[[Oware|Awari]]<ref>See Allis 1994 for rules</ref>
| |
| |align="right"|12
| |
| |align="right"|12
| |
| |align="right"|32
| |
| |align="right"|60
| |
| |align="right"|3.5
| |
| |align="right"|<ref name="Allis1994"/>
| |
| |Generalization is unclear
| |
| |-
| |
| |[[Qubic]]
| |
| |align="right"|64
| |
| |align="right"|30
| |
| |align="right"|34
| |
| |align="right"|20
| |
| |align="right"|54.2
| |
| |align="right"|<ref name=Allis1994/>
| |
| |[[PSPACE-complete]]<ref name="Reisch1980"/>
| |
| |-
| |
| |[[Fanorona]]
| |
| |align="right"|45
| |
| |align="right"|21
| |
| |align="right"|46
| |
| |align="right"|44
| |
| |align="right"|11
| |
| |align="right"|<ref name="Schadd2008">{{cite journal| author = M.P.D. Schadd, M.H.M. Winands, J.W.H.M. Uiterwijk, H.J. van den Herik and M.H.J. Bergsma | year = 2008 | title = Best Play in Fanorona leads to Draw | journal = [[New Mathematics and Natural Computation]] | volume = 4 |issue = 3 | pages = 369–387| url = https://dke.maastrichtuniversity.nl/m.winands/documents/Fanorona.pdf| doi = 10.1142/S1793005708001124}}</ref>
| |
| |?, but in [[EXPTIME]]
| |
| |-
| |
| |[[Nine Men's Morris]]
| |
| |align="right"|24
| |
| |align="right"|10
| |
| |align="right"|50
| |
| |align="right"|50
| |
| |align="right"|10
| |
| |align="right"|<ref name="Allis1994"/>
| |
| |?, but in [[EXPTIME]]
| |
| |-
| |
| |[[International draughts|International draughts (10x10)]]
| |
| |align="right"|50
| |
| |align="right"|30
| |
| |align="right"|54
| |
| |align="right"|90
| |
| |align="right"|4
| |
| |align="right"|<ref name="Allis1994"/>
| |
| |[[EXPTIME-complete]]<ref name="robson1984"/>
| |
| |-
| |
| |[[Chinese checkers]] (2 sets)
| |
| |align="right"|121
| |
| |align="right"|23
| |
| |align="right"|
| |
| |align="right"|
| |
| |align="right"|
| |
| |align="right"|<ref name=Bell_Halma>{{cite journal|author=G.I. Bell|title=The Shortest Game of Chinese Checkers and Related Problems |journal=Integers|year=2009|arxiv=0803.1245|url = http://emis.ams.org}}</ref>
| |
| |[[EXPTIME]]-complete <ref name=pebble>{{cite journal|title=Classes of Pebble Games and Complete Problems|journal= SIAM Journal on Computing| volume = 8| year = 1979 |pages= 574–586|author=Takumi Kasai, Akeo Adachi, and Shigeki Iwata|doi=10.1137/0208046|issue=4}} Proves completeness of the generalization to arbitrary graphs.</ref>
| |
| |-
| |
| |[[Chinese checkers]] (6 sets)
| |
| |align="right"|121
| |
| |align="right"|78
| |
| |align="right"|
| |
| |align="right"|
| |
| |align="right"|
| |
| |align="right"|<ref name=Bell_Halma/>
| |
| |[[EXPTIME]]-complete <ref name=pebble/>
| |
| |-
| |
| |[[Lines of Action]]
| |
| |align="right"|64
| |
| |align="right"|23
| |
| |align="right"|64
| |
| |align="right"|44
| |
| |align="right"|29
| |
| |align="right"|<ref name="Winands2004">{{cite thesis | author = Mark H.M. Winands | year = 2004 | title = Informed Search in Complex Games | degree = Ph.D. |publisher= Maastricht University, Maastricht, The Netherlands | isbn = 90-5278-429-9 | url = https://dke.maastrichtuniversity.nl/m.winands/documents/informed_search.pdf}}</ref>
| |
| |?, but in [[EXPTIME]]
| |
| |-
| |
| |[[Reversi]] (Othello)
| |
| |align="right"|64
| |
| |align="right"|28
| |
| |align="right"|58
| |
| |align="right"|58
| |
| |align="right"|10
| |
| |align="right"|<ref name="Allis1994"/>
| |
| |[[PSPACE-complete]]<ref>{{cite journal | author = S. Iwata and T. Kasai | title = The Othello game on an n*n board is PSPACE-complete | journal = Theor. Comp. Sci. | issue = 2 | year = 1994 | pages = 329–340 | volume = 123 | doi = 10.1016/0304-3975(94)90131-7}}</ref>
| |
| |-
| |
| |[[OnTop (board game)|OnTop]] (2p base game)
| |
| |align="right"|72
| |
| |align="right"|88
| |
| |align="right"|62
| |
| |align="right"|31
| |
| |align="right"|23.77
| |
| |align="right"|<ref name="OnTopComputer">{{cite thesis | title=Analysis and Implementation of the Game OnTop |
| |
| url = https://project.dke.maastrichtuniversity.nl/games/files/msc/Briesemeister_Thesis.pdf |
| |
| author = Robert Briesemeister | year=2009 | publisher = Maastricht University, Dept of Knowledge Engineering }} </ref>
| |
| |
| |
| |-
| |
| |[[Hex (board game)|Hex (11x11)]]
| |
| |align="right"|121
| |
| |align="right"|57
| |
| |align="right"|98
| |
| |align="right"|40
| |
| |align="right"|280
| |
| |align="right"|<ref name="GamesSolved"/>
| |
| |[[PSPACE-complete]]<ref>{{cite journal | author = Stefan Reisch | title = Hex ist PSPACE-vollständig (Hex is PSPACE-complete) | journal = Acta Inf. | issue = 15 | year = 1981 | pages = 167–191}}</ref>
| |
| |-
| |
| |[[Gomoku]] (15x15, freestyle)
| |
| |align="right"|225
| |
| |align="right"|105
| |
| |align="right"|70
| |
| |align="right"|30
| |
| |align="right"|210
| |
| |align="right"|<ref name="Allis1994"/>
| |
| |[[PSPACE-complete]]<ref name="Reisch1980"/>
| |
| |-
| |
| |[[Go (game)|Go (9x9)]]
| |
| |align="right"|81
| |
| |align="right"|38
| |
| |align="right"|
| |
| |align="right"|45
| |
| |align="right"|
| |
| |align="right"|<ref name="cwi">{{cite web | title = Combinatorics of Go | author = John Tromp and Gunnar Farnebäck | year = 2007 | url = http://www.cwi.nl/~tromp/go/gostate.ps}} This paper derives the bounds 48<log(log(''N''))<171 on the number of possible games ''N''.</ref> <ref name="Allis1994"/>
| |
| |[[EXPTIME-complete]]<ref name="Robson1983">{{Cite book | author = J. M. Robson | chapter = The complexity of Go | title = Information Processing; Proceedings of IFIP Congress | year = 1983 | pages = 413–417}}</ref>
| |
| |-
| |
| |[[Chess]]
| |
| |align="right"|64
| |
| |align="right"|47
| |
| |align="right"|123
| |
| |align="right"|80
| |
| |align="right"|35
| |
| |align="right"|<ref name="Shannon1950">The size of the state space and game tree for chess were first estimated in {{cite journal | author = [[Claude Shannon]] | title = Programming a Computer for Playing Chess | journal = Philosophical Magazine | volume = 41 | issue = 314 | year = 1950 | url = http://archive.computerhistory.org/projects/chess/related_materials/text/2-0%20and%202-1.Programming_a_computer_for_playing_chess.shannon/2-0%20and%202-1.Programming_a_computer_for_playing_chess.shannon.062303002.pdf}} Shannon gave estimates of 10<sup>43</sup> and 10<sup>120</sup> respectively, smaller than the upper bound in the table,
| |
| which is detailed in [[Shannon number]].</ref>
| |
| |[[EXPTIME-complete ]] (without 50-move drawing rule) <ref name="Fraenkel1981">{{cite journal | author = [[Aviezri Fraenkel]] and D. Lichtenstein | title = Computing a perfect strategy for n×n chess requires time exponential in n | journal = J. Comb. Th. A | issue = 31 | year = 1981 | pages = 199–214}}</ref>
| |
| |-
| |
| |[[Connect6]]
| |
| |align="right"|361
| |
| |align="right"|172
| |
| |align="right"|140
| |
| |align="right"|30
| |
| |align="right"|46000
| |
| |align="right"|<ref name=EnhancePNConn6> {{cite journal | title=2009 Chinese Control and Decision Conference | doi=10.1109/CCDC.2009.5191963 | chapter=Enhancements of proof number search in connect6 | year=2009 | last1=Chang-Ming Xu | last2=Ma | first2=Z.M. | last3=Jun-Jie Tao | last4=Xin-He Xu | isbn=978-1-4244-2722-2 | pages=4525 }} </ref>
| |
| |[[PSPACE|PSPACE-complete]]<ref>[http://portal.acm.org/citation.cfm?id=1290250 On the fairness and complexity of generalized k-in-a-row games<!-- Bot generated title -->]</ref>
| |
| |-
| |
| |[[Backgammon]]
| |
| |align="right"|28
| |
| |align="right"|20
| |
| |align="right"|144
| |
| |align="right"|50-60
| |
| |align="right"|250
| |
| |align="right"|<ref>http://books.nips.cc/papers/txt/nips04/0259.txt</ref>
| |
| |Generalization is unclear
| |
| |-
| |
| |[[Xiangqi]]
| |
| |align="right"|90
| |
| |align="right"|48
| |
| |align="right"|150
| |
| |align="right"|95
| |
| |align="right"|38
| |
| |align="right"|<ref name="Allis1994">{{cite thesis | author = [[Victor Allis]] | year = 1994 | title = Searching for Solutions in Games and Artificial Intelligence | degree = Ph.D. |publisher= University of Limburg, Maastricht, The Netherlands | isbn = 90-900748-8-0 | url = https://project.dke.maastrichtuniversity.nl/games/files/phd/SearchingForSolutions.pdf}}</ref> <ref name="Hsu2004">{{cite journal | author = Shi-Jim Yen, Jr-Chang Chen, Tai-Ning Yang, and Shun-Chin Hsu | title = Computer Chinese Chess |date=March 2004 | journal = International Computer Games Association Journal | volume = 27 | issue = 1 | pages = 3–18 | url = http://www.csie.ndhu.edu.tw/~sjyen/Papers/2004CCC.pdf}}</ref>
| |
| |?, believed to be [[EXPTIME-complete]]
| |
| |-
| |
| |[[Abalone (board game)|Abalone]]
| |
| |align="right"|61
| |
| |align="right"|25
| |
| |align="right"|154
| |
| |align="right"|87
| |
| |align="right"|60
| |
| |align="right"|<ref name=PasChor>{{cite web|last=Chorus|first=Pascal|title=Implementing a Computer Player for Abalone Using Alpha-Beta and Monte-Carlo Search|url=https://project.dke.maastrichtuniversity.nl/games/files/msc/pcreport.pdf|publisher=Dept of Knowledge Engineering, Maastricht University|accessdate=29 March 2012}}</ref>
| |
| |?, but in [[EXPTIME]]
| |
| |-
| |
| |[[Havannah]]
| |
| |align="right"|271
| |
| |align="right"|127
| |
| |align="right"|157
| |
| |align="right"|66
| |
| |align="right"|240
| |
| |align="right"|<ref name="GamesSolved"/> <ref>{{cite web|last=Joosten|first=B|title=Creating a Havannah Playing Agent|url=https://project.dke.maastrichtuniversity.nl/games/files/bsc/bscHavannah.pdf|accessdate=29 March 2012}}</ref>
| |
| |?, but in [[PSPACE]]
| |
| |-
| |
| |[[Quoridor]]
| |
| |align="right"|81
| |
| |align="right"|42
| |
| |align="right"|162
| |
| |align="right"|91
| |
| |align="right"|60
| |
| |align="right"|<ref name=MasterQuor> {{cite thesis | author = Lisa Glendenning | title = Mastering Quoridor | year = 2005 | department = Computer Science | degree = B.Sc. | publisher = [[University of New Mexico]] | month = May | url = http://hyperion.cs.washington.edu/attachments/15/glendenning_ugrad_thesis.pdf }}</ref>
| |
| |?, but in [[PSPACE]]
| |
| |-
| |
| |[[Carcassonne (board game)|Carcassonne]] (2p base game)
| |
| |align="right"|72
| |
| |align="right"|>40
| |
| |align="right"|195
| |
| |align="right"|71
| |
| |align="right"|55
| |
| |align="right"|<ref name="CarcassoneComputer">{{cite thesis | title=Implementing a Computer Player for Carcassonne |
| |
| url = https://project.dke.maastrichtuniversity.nl/games/files/msc/MasterThesisCarcassonne.pdf |
| |
| author = Cathleen Heyden | year=2009 | publisher = Maastricht University, Dept of Knowledge Engineering }} </ref>
| |
| |Generalization is unclear
| |
| |-
| |
| |[[Game of the Amazons|Amazons (10x10)]]
| |
| |align="right"|100
| |
| |align="right"|40
| |
| |align="right"|212
| |
| |align="right"|84
| |
| |align="right"|374 or 299<ref>The lower branching factor is for the second player.</ref>
| |
| |align="right"|<ref name="Kloetzer_themonte-carlo">{{cite journal | author = Julien Kloetzer | author2 = Hiroyuki Iida | author3 = Bruno Bouzy | title = The Monte-Carlo Approach in Amazons | year = 2007 | url = http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.79.7640}} </ref> <ref name="Hengens_thesis">{{cite paper | author = P. P. L. M. Hensgens | title = A Knowledge-Based Approach of the Game of Amazons | year = 2001 | publisher = Universiteit Maastricht, Institute for Knowledge and Agent Technology | url = https://project.dke.maastrichtuniversity.nl/games/files/msc/Hensgens_thesis.pdf}} </ref>
| |
| |[[PSPACE-complete]]<ref>{{cite arxiv | author = R. A. Hearn | title = Amazons is PSPACE-complete | date = 2005-02-02 | eprint = cs.CC/0502013 | class = cs.CC}}</ref>
| |
| |-
| |
| |[[Go (game)|Go (13x13)]]
| |
| |align="right"|169
| |
| |align="right"|79
| |
| |align="right"|
| |
| |align="right"|90
| |
| |align="right"|
| |
| |align="right"|<ref name="Allis1994"/> <ref name="cwi" />
| |
| |[[EXPTIME-complete]]<ref name="Robson1983" />
| |
| |-
| |
| |[[Shogi]]
| |
| |align="right"|81
| |
| |align="right"|71
| |
| |align="right"|226
| |
| |align="right"|115
| |
| |align="right"|92
| |
| |align="right"|<ref name="Hsu2004"/> <ref> {{cite journal | title = Computer shogi | doi = 10.1016/S0004-3702(01)00157-6 | journal = Artificial Intelligence | volume=134 | issue=1–2 |date=january 2002 | pages=121–144 | author= Hiroyuki Iida, Makoto Sakuta, [[Jeff Rollason]] }} </ref>
| |
| |[[EXPTIME-complete]]<ref>{{cite journal | author = H. Adachi, H. Kamekawa, and S. Iwata | title = Shogi on n × n board is complete in exponential time | journal = Trans. IEICE | volume= J70-D | pages = 1843–1852 | year = 1987}}</ref>
| |
| |-
| |
| |[[Arimaa]]
| |
| |align="right"|64
| |
| |align="right"|43
| |
| |align="right"|402
| |
| |align="right"|92
| |
| |align="right"|17281
| |
| |align="right"|<ref name="Cox2006">{{cite web | title = Analysis and Implementation of the Game Arimaa | author = Christ-Jan Cox | year = 2006 | url = https://project.dke.maastrichtuniversity.nl/games/files/msc/Cox_thesis1.pdf}}</ref> <ref name="Wu2011">{{cite web | title = Move Ranking and Evaluation in the Game of Arimaa | author = David Jian Wu | year = 2011 | url = http://icosahedral.net/downloads/djwuthesis.pdf}}</ref> <ref name="Haskin2006">{{cite web | title = A Look at the Arimaa Branching Factor | author = Brian Haskin | year = 2006 | url = http://arimaa.janzert.com/bf_study/}}</ref>
| |
| |?, but in [[EXPTIME]]
| |
| |-
| |
| |[[Go (game)|Go (19x19)]]
| |
| |align="right"|361
| |
| |align="right"|171
| |
| |align="right"|360
| |
| |align="right"|150
| |
| |align="right"|250
| |
| |align="right"|<ref name="cwi" /> <ref name="Allis1994"/>
| |
| |[[EXPTIME-complete]]<ref name="Robson1983" />
| |
| |-
| |
| |[[Stratego]]
| |
| |align="right"|92
| |
| |align="right"|115
| |
| |align="right"|535
| |
| |align="right"|381
| |
| |align="right"|21.739
| |
| |align="right"|<ref name="ArtsStratego"> {{cite thesis | author = A.F.C. Arts | title = Competitive Play in Stratego | year = 2010 | url =https://project.dke.maastrichtuniversity.nl/games/files/msc/Arts_thesis.pdf | university = Maastricht }}</ref>
| |
| |
| |
| |-
| |
| |[[Computer bridge|Double dummy bridge]]<ref>Double dummy bridge (i.e. double dummy problems in the context of [[contract bridge]]) is not a proper board game but has a similar game tree, and is studied in [[computer bridge]], which motivates including it in the list. The bridge table can be regarded as having one slot for each player and trick to play a card in, which corresponds to board size 52. Game-tree complexity is a very weak upper bound: 13! to the power of 4 players regardless of legality. State-space complexity is for one given deal; likewise regardless of legality but with many transpositions eliminated. Note that the last 4 plies are always forced moves with branching factor 1.</ref>
| |
| |align="right"|(52)
| |
| |align="right"|<17
| |
| |align="right"|<40
| |
| |align="right"|52
| |
| |align="right"|5.6
| |
| |align="right"|
| |
| |
| |
| |}
| |
| | |
| ==See also==
| |
| *[[Go and mathematics ]]
| |
| *[[Solved game]]
| |
| *[[Shannon number]]
| |
| *[[List of NP-complete problems#Games and puzzles|list of NP-complete games and puzzles]]
| |
| *[[List of PSPACE-complete problems#Games and puzzles|list of PSPACE-complete games and puzzles]]
| |
| | |
| ==Notes and references==
| |
| <references/>
| |
| | |
| ==External links==
| |
| * [[David Eppstein]]'s [http://www.ics.uci.edu/~eppstein/cgt/hard.html Computational Complexity of Games and Puzzles]
| |
| | |
| [[Category:Combinatorial game theory]]
| |
| [[Category:Game theory]]
| |