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| :''This article is about the game; for the graph theory property, see [[Planar graph]].''
| | Hi there, I am Alyson Pomerleau and I believe it seems fairly good when you say it. Ohio is exactly where my house is but my spouse wants us to move. Distributing manufacturing has been his occupation for some time. The favorite hobby for him and his children is style and he'll be starting some thing else alongside with it.<br><br>Feel free to visit my blog post :: psychics online ([http://kjhkkb.net/xe/notice/374835 http://kjhkkb.net/]) |
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| '''Planarity''' is a [[puzzle computer game]] by John Tantalo, based on a concept by Mary Radcliffe at [[Western Michigan University]].<ref>{{citation|contribution=Cat's Cradle on Steroids|title=Today @ PC World|magazine=[[PCWorld]]|date=August 1, 2005|first=Yardena|last=Arar|url=http://blogs.pcworld.com/staffblog/archives/000802.html|archiveurl=http://web.archive.org/web/20090604060302/http://blogs.pcworld.com/staffblog/archives/000802.html|archivedate=2009-06-04}}</ref>
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| The name comes from the concept of [[planar graph]]s in graph theory; these are graphs that can be embedded in the [[Euclidean plane]] so that no edges intersect. By [[Fáry's theorem]], if a graph is planar, it can be drawn without crossings so that all of its edges are straight line segments. In the planarity game, the player is presented with a [[circular layout]] of a planar graph, with all the vertices placed on a single circle and with many crossings. The goal for the player is to eliminate all of the crossings and construct a straight-line embedding of the graph by moving the vertices one by one into better positions.
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| ==History and versions==
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| The game was written in [[Adobe Flash|Flash]] by John Tantalo at [[Case Western Reserve University]].<ref name="Case news">{{cite web | last =Massie | first =Laura | title =Case student develops booming on-line game | publisher = Case Western Reserve University News Center | date =2005-06-20 | url =http://www.case.edu/news/2005/7-05/planarity.htm | accessdate = 2007-09-30}}</ref> Online popularity and the local notoriety he gained placed Tantalo as one of Cleveland's most interesting people for 2006.<ref name = Observer>{{cite news |last=Castro|first=Laura|title=Case student one of Cleveland's "Most Interesting People"|publisher=The Observer|date=2005-11-18|url=http://observer.case.edu/Archives/Volume_38/Issue_11/Story_379/|accessdate= 2007-09-30}} {{Dead link|date=October 2010|bot=H3llBot}}</ref><ref name = "Cleveland Magazine">{{cite press release | title =Most Interesting People 2006 | publisher =Cleveland Magazine | date =January, 2006 | url =http://www.clevelandmagazine.com/ME2/dirmod.asp?sid=E73ABD6180B44874871A91F6BA5C249C&nm=Arts+%26+Entertainemnt&type=Publishing&mod=Publications%3A%3AArticle&mid=1578600D80804596A222593669321019&tier=4&id=5BE7EE32336E46ACA55FAE03A95E4E35 | accessdate = 2007-09-30 }}</ref> It in turn has inspired the creation of a [[GTK+]] version by [[Xiph.org]]'s [[Chris Montgomery]], which possesses additional level generation algorithms and the ability to manipulate multiple nodes at once.<ref>{{cite web | url=http://web.mit.edu/xiphmont/Public/gPlanarity.html | title=gPlanarity home }}</ref>
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| ==Puzzle generation algorithm==
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| The definition of the planarity puzzle does not depend on how the planar graphs in the puzzle are generated, but the original implementation uses the following algorithm:
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| # Generate a set of random lines in a plane such that no two lines are parallel and no three lines meet in a single point.
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| # Calculate the intersections of every line pair.
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| # Create a graph with a vertex for each intersection and an edge for each line segment connecting two intersections (the [[Arrangement of lines|arrangement]] of the lines).
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| If a graph is generated from <math>L</math> lines, then the graph will have exactly <math>\tbinom{L}{2} = \tfrac{L(L-1)}{2}</math> vertices (each line has <math>L-1</math> vertices, and each vertex is shared with one other line) and <math>L(L-2)</math> edges (each line contains <math>L-2</math> edges). The first level of Planarity is built with <math>L=4</math> lines, so it has <math>L(L-1)/2=6</math> vertices and <math>L(L-2)=8</math> edges. Each level after is generated by one more line than the last. If a level was generated with <math>L</math> lines, then the next level has <math>L</math> more vertices and <math>2L-1</math> more edges.
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| The best known algorithms from [[computational geometry]] for constructing the graphs of line arrangements solve the problem in <math>O(L^2)</math> time,<ref>{{citation | |
| | last1 = Chazelle | first1 = B. | author1-link = Bernard Chazelle
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| | last2 = Guibas | first2 = L. J. | author2-link = Leonidas J. Guibas
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| | last3 = Lee | first3 = D. T. | author3-link = Der-Tsai Lee
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| | doi = 10.1007/BF01934990
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| | issue = 1
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| | journal = BIT
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| | pages = 76–90
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| | title = The power of geometric duality
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| | volume = 25
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| | year = 1985}}</ref> linear in the size of the graph to be constructed, but they are somewhat complex. Alternatively and more simply, it is possible to index each crossing point by the pair of lines that cross at that point, sort the crossings along each line by their <math>x</math>-coordinates, and use this sorted ordering to generate the edges of the planar graph, in near-optimal <math>O(L^2\log L)</math> time. Once the vertices and edges of the graph have been generated, they may be placed evenly around a circle using a [[random permutation]].
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| ==Related theoretical research==
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| The problem of [[Planarity testing|determining whether a graph is planar]] can be solved in [[linear time]],<ref>{{citation
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| | last1 = Mehlhorn | first1 = K. | author1-link = Kurt Mehlhorn
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| | last2 = Mutzel | first2 = P.
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| | doi = 10.1007/s004539900046
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| | issue = 2
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| | journal = Algorithmica
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| | mr = 1394503
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| | pages = 233–242
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| | title = On the embedding phase of the Hopcroft and Tarjan planarity testing algorithm
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| | volume = 16
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| | year = 1996}}</ref> and any such graph is guaranteed to have a straight-line embedding by [[Fáry's theorem]], that can also be found from the planar embedding in linear time.<ref>{{Citation
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| | last1 = de Fraysseix | first1 = Hubert
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| | last2 = Pach | first2 = János | author2-link = János Pach
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| | last3 = Pollack | first3 = Richard
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| | title = How to draw a planar graph on a grid
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| | journal = Combinatorica
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| | year = 1990
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| | volume = 10
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| | pages = 41–51
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| | mr = 1075065
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| | doi = 10.1007/BF02122694}}</ref> Therefore, any puzzle could be solved in linear time by a computer. However, these puzzles are not as straightforward for human players to solve.
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| In the field of [[computational geometry]], the process of moving a subset of the vertices in a graph embedding to eliminate edge crossings has been studied by Pach and Tardos (2002),<ref>{{citation
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| | last1 = Pach | first1 = János | author1-link = János Pach
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| | last2 = Tardos | first2 = Gábor | author2-link = Gábor Tardos
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| | doi = 10.1007/s00454-002-2889-y
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| | issue = 4
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| | journal = Discrete & Computational Geometry
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| | pages = 585–592
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| | title = Untangling a polygon
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| | volume = 28
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| | year = 2002}}</ref> and others, inspired by the planarity puzzle.<ref>{{citation
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| | last1 = Bose | first1 = Prosenjit
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| | last2 = Dujmovic | first2 = Vida
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| | last3 = Hurtado | first3 = Ferran
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| | last4 = Langerman | first4 = Stefan
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| | last5 = Morin | first5 = Pat
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| | last6 = Wood | first6 = David R.
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| | doi = 10.1007/s00454-008-9125-3
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| | issue = 4
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| | journal = Discrete & Computational Geometry
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| | pages = 570–585
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| | title = A polynomial bound for untangling geometric planar graphs
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| | volume = 42
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| | year = 2008}}</ref><ref>{{citation
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| | last = Cibulka | first = Josef
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| | doi = 10.1007/s00454-009-9150-x
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| | issue = 2
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| | journal = Discrete & Computational Geometry
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| | pages = 402–411
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| | title = Untangling Polygons and Graphs
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| | volume = 43
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| | year = 2009}}</ref><ref>{{citation
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| | last1 = Goaoc | first1 = Xavier
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| | last2 = Kratochvíl | first2 = Jan
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| | last3 = Okamoto | first3 = Yoshio
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| | last4 = Shin | first4 = Chan-Su
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| | last5 = Spillner | first5 = Andreas
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| | last6 = Wolff | first6 = Alexander
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| | doi = 10.1007/s00454-008-9130-6
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| | issue = 4
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| | journal = Discrete & Computational Geometry
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| | pages = 542–569
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| | title = Untangling a Planar Graph
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| | volume = 42
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| | year = 2009}}</ref> The results of these researchers shows that (in theory, assuming that the field of play is the infinite plane rather than a bounded rectangle) it is always possible to solve the puzzle while leaving <math>n^\epsilon</math> of the <math>n</math> input vertices fixed in place at their original positions, for a constant <math>\epsilon</math> that has not been determined precisely but lies between 1/4 and 1/2. When the planar graph to be untangled is a [[cycle graph]], a larger number of vertices may be fixed in place. However, determining the largest number of vertices that may be left in place for a particular input puzzle (or equivalently, the smallest number of moves needed to solve the puzzle) is [[NP-complete]].
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| {{harvtxt|Verbitsky|2008}} has shown that the randomized [[circular layout]] used for the initial state of Planarity is nearly the worst possible in terms of its [[crossing number (graph theory)|number of crossings]]: regardless of what planar graph is to be tangled, the [[expected value]] of the number of crossings for this layout is within a factor of three of the largest number of crossings among all layouts.<ref>{{citation
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| | last = Verbitsky | first = Oleg
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| | arxiv = 0705.3748
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| | doi = 10.1016/j.tcs.2008.02.032
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| | issue = 1-3
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| | journal = Theoretical Computer Science
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| | mr = 2412266
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| | pages = 294–300
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| | title = On the obfuscation complexity of planar graphs
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| | volume = 396
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| | year = 2008}}</ref>
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| ==References==
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| {{reflist}}
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| ==External links==
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| *www.planarity.net — the original Flash game (Warning: site triggers virus alerts, 10 Sept, 2013)
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| *[http://www.arcadegameland.com/game/1206/Planarity.html HTML5 Planarity] — Planarity implemented in HTML5
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| *[http://ccl.northwestern.edu/netlogo/ NetLogo System] — Included as sample program (game) in NetLogo system
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| *[http://www.jasondavies.com/planarity/ Planarity] — Version using [[SVG]] and the [http://mbostock.github.com/d3/ d3] [[JavaScript]] library
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| *[http://www.libavg.de/site/projects/libavg/wiki/Planarity Multitouch Planarity] — Multiplayer- and multitouch-enabled version written in [[Python (programming language)|Python]] using libavg.
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| [[Category:Puzzle video games]]
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| [[Category:Mathematical games]]
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| [[Category:Planar graphs]]
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Hi there, I am Alyson Pomerleau and I believe it seems fairly good when you say it. Ohio is exactly where my house is but my spouse wants us to move. Distributing manufacturing has been his occupation for some time. The favorite hobby for him and his children is style and he'll be starting some thing else alongside with it.
Feel free to visit my blog post :: psychics online (http://kjhkkb.net/)