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title="Wikipedia:When to cite">introducing</a> more precise citations.</span> <span class="date-container"><i>(<span class="date">January 2013</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-One_source plainlinks metadata ambox ambox-content ambox-one_source" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>relies largely or entirely on a <a href="/wiki/Wikipedia:Articles_with_a_single_source" title="Wikipedia:Articles with a single source">single source</a></b>.<span class="hide-when-compact"> Relevant discussion may be found on the <a href="/wiki/Talk:Combinatorial_search##" title="Talk:Combinatorial search">talk page</a>. Please help <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Combinatorial_search&action=edit">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">introducing citations to additional sources</a>.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Combinatorial+search%22">"Combinatorial search"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Combinatorial+search%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Combinatorial+search%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Combinatorial+search%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Combinatorial+search%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Combinatorial+search%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">May 2024</span>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Computer_science" title="Computer science">computer science</a> and <a href="/wiki/Artificial_intelligence" title="Artificial intelligence">artificial intelligence</a>, <b>combinatorial search</b> studies <a href="/wiki/Search_algorithms" class="mw-redirect" title="Search algorithms">search algorithms</a> for solving instances of problems that are believed to be hard in general, by efficiently exploring the usually large solution space of these instances. Combinatorial search algorithms achieve this efficiency by reducing the effective size of the search space or employing heuristics. Some algorithms are guaranteed to find the optimal solution, while others may only return the best solution found in the part of the state space that was explored. </p><p>Classic combinatorial search problems include solving the <a href="/wiki/Eight_queens_puzzle" title="Eight queens puzzle">eight queens puzzle</a> or evaluating moves in games with a large <a href="/wiki/Game_tree" title="Game tree">game tree</a>, such as <a href="/wiki/Reversi" title="Reversi">reversi</a> or <a href="/wiki/Chess" title="Chess">chess</a>. </p><p>A study of <a href="/wiki/Computational_complexity_theory" title="Computational complexity theory">computational complexity theory</a> helps to motivate combinatorial search. Combinatorial search algorithms are typically concerned with problems that are <a href="/wiki/NP-hard" class="mw-redirect" title="NP-hard">NP-hard</a>. Such problems are not believed to be efficiently solvable in general. However, the various approximations of complexity theory suggest that some instances (e.g. "small" instances) of these problems could be efficiently solved. This is indeed the case, and such instances often have important practical ramifications. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Combinatorial_search&action=edit&section=1" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Common algorithms for solving combinatorial search problems include: </p> <ul><li><a href="/wiki/A*_search_algorithm" title="A* search algorithm">A* search algorithm</a></li> <li><a href="/wiki/Alpha%E2%80%93beta_pruning" title="Alpha–beta pruning">Alpha–beta pruning</a></li> <li><a href="/wiki/Branch-and-bound" class="mw-redirect" title="Branch-and-bound">Branch-and-bound</a></li> <li><a href="/wiki/Minimax" title="Minimax">Minimax</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Lookahead">Lookahead</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Combinatorial_search&action=edit&section=2" title="Edit section: Lookahead"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Lookahead is an important component of combinatorial search, which specifies, roughly, how deeply the <a href="/wiki/Graph_(data_structure)" class="mw-redirect" title="Graph (data structure)">graph</a> representing the problem is explored. The need for a specific limit on lookahead comes from the large problem graphs in many applications, such as <a href="/wiki/Computer_chess" title="Computer chess">computer chess</a> and <a href="/wiki/Computer_Go" title="Computer Go">computer Go</a>. A naive <a href="/wiki/Breadth-first_search" title="Breadth-first search">breadth-first search</a> of these graphs would quickly consume all the memory of any modern computer. By setting a specific lookahead limit, the algorithm's time can be carefully controlled; its time <a href="/wiki/Exponential_growth" title="Exponential growth">increases exponentially</a> as the lookahead limit increases. </p><p>More sophisticated search techniques such as <a href="/wiki/Alpha%E2%80%93beta_pruning" title="Alpha–beta pruning">alpha–beta pruning</a> are able to eliminate entire subtrees of the search tree from consideration. When these techniques are used, lookahead is not a precisely defined quantity, but instead either the maximum depth searched or some type of average. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Combinatorial_search&action=edit&section=3" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Brute-force_search" title="Brute-force search">Brute-force search</a></li> <li><a href="/wiki/Combinatorial_explosion" title="Combinatorial explosion">Combinatorial explosion</a></li> <li><a href="/wiki/Combinatorial_optimization" title="Combinatorial optimization">Combinatorial optimization</a></li> <li><a href="/wiki/Search_algorithm" title="Search algorithm">Search algorithm</a></li> <li><a href="/wiki/State_space_search" title="State space search">State space search</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Combinatorial_search&action=edit&section=4" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Russell and Norvig. <i><a href="/wiki/Artificial_Intelligence:_A_Modern_Approach" title="Artificial Intelligence: A Modern Approach">Artificial Intelligence: A Modern Approach</a></i>.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Combinatorial_search&oldid=1235533628">https://en.wikipedia.org/w/index.php?title=Combinatorial_search&oldid=1235533628</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Analysis_of_algorithms" title="Category:Analysis of algorithms">Analysis of algorithms</a></li><li><a href="/wiki/Category:Combinatorial_optimization" title="Category:Combinatorial optimization">Combinatorial optimization</a></li><li><a href="/wiki/Category:Computational_complexity_theory" title="Category:Computational complexity theory">Computational complexity theory</a></li><li><a href="/wiki/Category:Game_artificial_intelligence" title="Category:Game artificial intelligence">Game artificial intelligence</a></li><li><a href="/wiki/Category:Search_algorithms" title="Category:Search algorithms">Search algorithms</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_lacking_in-text_citations_from_January_2013" title="Category:Articles lacking in-text citations from January 2013">Articles lacking in-text citations from January 2013</a></li><li><a href="/wiki/Category:All_articles_lacking_in-text_citations" title="Category:All articles lacking in-text citations">All articles lacking in-text citations</a></li><li><a href="/wiki/Category:Articles_needing_additional_references_from_May_2024" title="Category:Articles needing additional references from May 2024">Articles needing additional references from May 2024</a></li><li><a href="/wiki/Category:All_articles_needing_additional_references" title="Category:All articles needing additional references">All articles needing additional references</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 19 July 2024, at 19:42<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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