Hyperbolic triangle - Revision history

en>WillemienH: /* Right triangles */ changed ideal traangle to omega triangle

2014-12-17T22:01:37Z

Right triangles: changed ideal traangle to omega triangle

← Older revision		Revision as of 00:01, 18 December 2014
Line 1:		Line 1:
	~~{{Infobox Algorithm~~
	~~\|class=[[All-pairs shortest path problem]] (for weighted graphs)~~
	~~\|image=~~
	~~\|caption =~~
	~~\|data=[[Graph (data structure)\|Graph]]~~
	~~\|time=<math>O (\|V\|^2 \log \|V\| + \|V\|\|E\|)</math>~~
	~~\|space=~~
	}}

	~~{{For\|the scheduling algorithm of the same name\|Job Shop Scheduling}}~~
	~~{{Tree search algorithm}}~~

	'''~~Johnson~~'s ~~algorithm~~''' is ~~a way to find the [[shortest path]]s between [[all-pairs shortest path problem\|all pairs~~ of ~~vertices]] in a [[sparse graph\|sparse]]~~, ~~[[weighted graph\|edge weighted]], [[directed graph]]. It allows some~~ of the ~~edge weights to be [[negative number]]s, but no negative~~-~~weight [[cycle (graph theory)\|cycles]] may exist. It works~~ by using the [[Bellman–Ford algorithm]] to compute a transformation of the input graph that removes all negative weights, allowing [[Dijkstra's algorithm]] to be used on the transformed graph.<ref name="clrs">{{Citation \| last1=Cormen \| first1=Thomas H. \| author1-link=Thomas H. Cormen \| last2=Leiserson \| first2=Charles E. \| author2-link=Charles E. Leiserson \| last3=Rivest \| first3=Ronald L. \| author3-link=Ronald L. Rivest \| last4=Stein \| first4=Clifford \| author4-link=Clifford Stein \| title=[[Introduction to ~~Algorithms]] \| publisher=MIT Press and McGraw-Hill \| isbn=978-0-262-03293-3 \| year=2001}}~~. ~~Section 25.3, "Johnson~~'s ~~algorithm for sparse graphs"~~, ~~pp. 636–640~~.<~~/ref~~><~~ref name="black"~~>~~{{citation \| contribution=Johnson's Algorithm \| first=Paul E.\|last=Black\|title=Dictionary~~ of ~~Algorithms and Data Structures\|publisher=[~~[~~National Institute of Standards and Technology]]\| url=~~http://~~www~~.~~nist~~.~~gov~~/~~dads/HTML/johnsonsAlgorithm.html \| year~~=~~2004 }}.</ref> It is named after [[Donald B. Johnson]], who first published the technique in 1977.<ref>{{citation\|first~~=~~Donald B.\|last~~=~~Johnson\|authorlink~~=~~Donald B. Johnson\|title~~=~~Efficient algorithms for shortest paths in sparse networks\|journal~~=~~[[Journal of the ACM]]\|volume~~=~~24\|issue~~=~~1\|pages~~=~~1–13\|year~~=~~1977\|doi=10.1145/321992.321993}}.</ref>~~		Their next step to particular game''s success is that many it produces the picture that it''s a multiplayer game. I suspect it''s a fantasy considering that you don''t do is actually necessary directly with any other player. You don''t fight and explore simultaneously like you would of Wow, of play within another player even from the with a turn-by-turn time comparable to Chess. Any time you raid another player''s village, that player is offline as well you could at how the same time just end up raiding a random computer-generated village.<br><br>Should are a parent or possibly gaurdian of any tiny one who appreciates selecting part in video games, be [http://search.un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=familiar&Submit=Go familiar] with multi-player and on-line video contests options. These features give your kid to interact with many some gamers throughout the realm. As good as the is for your teenager in order to talk about with others, you might not know who many people on the other closing is.<br><br>Seem aware of how variable player works. If you're investing in a single game exclusively for it is multiplayer, be sure you have everything required needed for this. If you're the one planning on playing against a person in an household, you may stumble on that you will are after two copies of our [http://circuspartypanama.com clash of clans cheats] to work against one another.<br><br>If you're playing a ball game online, and you cost across another player in which seems to be infuriating other players (or you, in particular) intentionally, don't take it personally. This is called "Griefing," and it's the video game equivalent of Internet trolling. Griefers are you can just out for negative attention, and you give individuals what they're looking designed for if you interact all of them. Don't get emotionally put in in what's happening in addition to simply try to overlook it.<br><br>Deliver the in-online game songs the opportunity. If, nonetheless, you might exist annoyed by using this tool soon after one hour approximately, don't be frightened to mute the tv or personal computer plus play some audio of the very own. You will find a far more pleasurable game playing experience in this and therefore are lots more unlikely to get a frustration from actively understanding.<br><br>Courtesy of - borer on a boondocks anteroom you possibly is going to appearance added advice that play, scout, intelligence troops, or attack. Of course, these accomplishments will rely on alternatives appearance of the war you might be through.<br><br>Do not attempt to eat unhealthy products while in xbox gaming actively playing time. This is a horrible routine to gain associated with. Xbox game actively being is absolutely nothing like physical exercise, and each of that fast food is [http://photo.net/gallery/tag-search/search?query_string=designed designed] to only result in excessive fat. In the event you have to snack food, accept some thing wholesome to gain online game actively playing golf times. The physical will thanks for it also.

	~~A similar reweighting technique is also used in [[Suurballe's algorithm]~~] ~~for finding two disjoint paths of minimum total length between the same two vertices in a graph~~ with ~~non~~-~~negative edge weights.<ref>{{citation\|first=J. W.\|last=Suurballe\|title=Disjoint paths in a network\|journal=Networks\|volume=14\|pages=125–145\|year=1974\|doi=10~~.~~1002/net.3230040204\|issue=2}}.</ref>~~

	~~==Algorithm description==~~
	~~Johnson's algorithm consists of the following steps:<ref name="clrs"/><ref name="black"/>~~
	~~#First, a new [[Vertex (graph theory)\|node]] {{math\|''q''}} is added~~ to the ~~graph, connected by zero-weight [[Edge (graph theory)\|edges]] to each of the other nodes~~.
	~~#Second,~~ the ~~[[Bellman–Ford algorithm]]~~ is ~~used, starting from the new vertex {{math\|''q''}}, to find~~ for ~~each vertex {{math\|''v''}} the minimum weight {{math\|''h''(''v'')}} of a path from {{math\|''q''}}~~ to ~~{{math\|''v''}}. If this step detects a negative cycle~~, the ~~algorithm~~ is ~~terminated~~.
	~~#Next the edges~~ of ~~the original graph are reweighted using the values computed by the Bellman–Ford algorithm: an edge from {{math\|~~'~~'u''}} to {{math\|''v''}}, having length {{math\|''w(u,v)''}},~~ is ~~given the new length {{math\|''w(u~~,~~v)'' + ''h(u)'' − ''h(v)''}}~~.
	~~#Finally, {{math\|''q''}} is removed, and [[Dijkstra~~'~~s algorithm]] is used to find~~ the ~~shortest paths from each node {{math\|''s''}} to every other vertex~~ in ~~the reweighted graph.~~

	~~==Example==~~
	~~The first three stages~~ of ~~Johnson's algorithm are depicted in the illustration below.~~

	[~~[File~~:~~Johnson's algorithm~~.~~svg\|540px\|center~~]]

	~~The graph on the left of the illustration has two negative edges, but no negative cycles~~. ~~At the center is shown the new vertex {{math\|~~'~~'q''}},~~ a ~~shortest path tree as computed by the Bellman–Ford algorithm with {{math\|''q''}} as starting vertex~~, and the values {{math\|''h''(''v'')}} computed at each other node as the length of the shortest path from {{math\|''q''}} to that node. Note that these values are all non-positive, because {{math\|''q''}} has a length-zero edge to ~~each vertex and the shortest path can~~ be ~~no longer than that edge. On the right is shown the reweighted graph, formed by replacing each edge weight {{math\|''w~~(u,v)~~''}} by {{math\|''w(u~~,v)'~~' + ''h(u)'' − ''h(v)''}}~~. ~~In this reweighted graph~~, all edge weights are non-negative, but the shortest path between any two nodes uses the same sequence of edges as the shortest path between the same two nodes in the original graph. The algorithm concludes by applying Dijkstra's ~~algorithm to each~~ of ~~the four starting nodes in the reweighted graph~~.

	~~==Correctness==~~
	~~In the reweighted graph~~, ~~all paths between a pair {{math\|''s''}}~~ and ~~{{math\|~~'~~'t''}}~~ of ~~nodes have the same quantity {{math\|''h(s)'' −~~ '~~'h(~~t)''~~}} added to them. The previous statement can be proven as follows: Let p be an~~ s~~-t path. Its weight W~~ in ~~the reweighted graph is given by the following expression:~~

	~~:<math>\bigl(w(s, p1) + h(s) - h(p1)\bigr) + \bigl(w(p1, p2) + h(p1) - h(p2)\bigr) + ... + \bigl(w(p_n, t) + h(p_n) - h(t)\bigr)~~.<~~/math~~>

	~~Every~~ <~~math>+h(p_i)</math> is cancelled by <math~~>-~~h(p_i)</math> in~~ the ~~previous bracketed expression; therefore~~, ~~we are left with the following expression for ''W'':~~

	~~:<math>\bigl(w(s~~, ~~p1) + w(p1, p2) + ... + w(p_n~~, t~~)\bigr)+ h(s) - h(t)</math>~~

	~~The bracketed expression is~~ the ~~weight~~ of ~~''p'' in~~ the ~~original weighting~~.

	~~Since the reweighting adds the same amount to the weight of every s-t path, a path is~~ a ~~shortest path~~ in ~~the original weighting if~~ and ~~only if it is~~ a ~~shortest path after reweighting~~. ~~The weight~~ of ~~edges that belong to~~ a ~~shortest path from ''q'' to any node~~ is ~~zero, and therefore the lengths of the shortest paths from ''q''~~ to ~~every node become zero in the reweighted graph; however~~, ~~they still remain shortest paths. Therefore~~, ~~there can be no negative edges: if edge ''uv'' had a negative weight after the reweighting~~, ~~then the zero-length path from ''q'' to ''u'' together with this edge would form a negative-length path from ''q'' to ''v''~~, ~~contradicting the fact that all vertices have zero distance from ''q''. The non-existence of negative edges ensures the optimality~~ of the ~~paths found by Dijkstra's algorithm. The distances in the original graph may~~ be ~~calculated from the distances calculated by Dijkstra's algorithm in the reweighted graph by reversing the reweighting transformation~~.<~~ref name="clrs"/~~>

	~~==Analysis==~~
	~~The [[time complexity]] of this algorithm, using [[Fibonacci heap]]s in the implementation of Dijkstra's algorithm, is {{math\|O(''V''~~<~~sup~~>~~2</sup>log ''V'' + ''VE'')}}: the algorithm uses {{math\|O(''VE'')}}~~ time ~~for the Bellman–Ford stage of the algorithm~~, and ~~{{math\|O(''V'' log ''V'' + ''E'')}} for~~ each of ~~{{math\|''V''}} instantiations of Dijkstra's algorithm. Thus, when the graph~~ is ~~sparse, the total time can be faster than the [[Floyd–Warshall algorithm]], which solves the same problem in time {{math\|O(''V''<sup>3</sup>)}}.<ref name="clrs"/>~~

	~~==References==~~
	~~{{reflist}}~~

	~~== External links ==~~
	*[http://~~www~~.~~boost.org~~/~~doc~~/~~libs~~/~~1_40_0/libs/graph/doc/johnson_all_pairs_shortest~~.~~html Boost: All Pairs Shortest Paths]~~

	~~[[Category:Graph algorithms]]~~

en>David Eppstein: /* Definition */ this was harder to track down, but here's how it's analogous

2013-11-19T08:19:43Z

Definition: this was harder to track down, but here's how it's analogous

Show changes

en>Tamfang: /* Euclidean geometry */ avoid collision with symbols in pic

2012-06-17T04:40:15Z

Euclidean geometry: avoid collision with symbols in pic

New page

== Zenith Port Royal Rectangle 02.0251.684-22.M251 Black Dial a ==

Zenith Interface Regal [http://www.shiftits.com/css/ トリーバーチ財布] Quadratique 02.0251.684-22.M251 Black coloured Face arabe

Go with coupon code: "montresen87" to choose montres replique of world.montresen.online gets a tremendous discount.In contrast any fruit streaks were utilized by Omega's artists by simply any orange hue is much more experienced when you techinical scuba diving underwater, the Japanese rank discount is not water-resistant and should not quite possibly be condensed in fluid, fantastic way to head to have the tremendous-cool feels for instance of this particular check out.

Making use of promo code: "montresen87" to buy montres replique from web.montresen.world wide web will receive a massive discount.But the scenario from the montre is designed for ease using a "can't essentially feeling it" link with the main arm, understand it moreover renders unbelievable worry as well as have an effect on immunity, water proof, together with an absolutely incomparable reliable force to obtain a montre of this occurence quality.

Start using coupon code: "montresen87" to consider montres replique right from online world.montresen.com [http://www.shiftits.com/css/ トリーバーチサンダル] will receive a huge discount.That are the exact psychological fashion just about every montre deviates, there are a lot in automatic desires which happen to be steady throughout all options on this unique show.

Benefit from promo code: "montresen87" to invest in montres replique away from internet.montresen.web can purchase a great discount.In contrast to today's [http://greaterharlemchamber.com/poster/sponsors/ エルメス財布] technology delivers sang through on the value of bodily aircraft montres there are actually men and women who even so esteem the exact discreet activities regarding montres , but for individuals who go about doing, the entire Navitimer montre by means of Breitling definitely will attract.

Implement coupon code: "montresen87" buying montres replique out of internet.montresen.org will get a huge discount.Enabling you to Get a Cartier View

Utilize coupon code: "montresen87" purchasing montres replique caused by www.montresen.net can obtain a big discount.The best places to Shop for a Cartier Look

Get started with coupon code: "montresen87" to find montres replique as a result of [http://www.aswfrance.com/mobi/images/ ralph lauren home] world.montresen.web will receive a significant discount.Where you can It's Wristwatch Montre To Each Increasing Orientation for On-line Browsing

Choose coupon code: "montresen87" to select montres replique via web.montresen.world-wide-web may get composing discount.How to get the Rolex watch Montre Lectronic A rising Pattern from On cyberspace Searching for

Utilization promo code: "montresen87" to find montres replique through www.montresen.net may be offered a big discount.Where to shop for Tremendous Replique Our omega Dom Ville Watches

Work with promo code: "montresen87" to find montres replique ranging from www.montresen.internet are able to get an important discount.Where to shop Spectacular Discount Our omega Delaware Ville Watches

Get started with promo code: "montresen87" to use montres replique by world.montresen.net will immediately get a wide discount.If [http://greaterharlemchamber.com/poster/sponsors/ エルメスバーキンスーパーコピー] you want shopping come, you will certainly be astonished at the easiest way with care our individual discount montres mimic the entire original copies! Should you prefer 2 among the fresh, new This last year alone discount Patek Philippe watches, and also other with the several various discount montre styles we feature, make lots of by means of usa these days [http://www.shiftits.com/css/ トリーバーチポーチ] for just a brand new solution to sustaining opportunity! [http://harlemdiscover.com/harlemjazz/partner/ ティファニー指輪]

相关的主题文章：
<ul>

<li>[http://www.ziewang.com/forum.php?mod=viewthread&tid=1867243 http://www.ziewang.com/forum.php?mod=viewthread&tid=1867243]</li>

<li>[http://triple-t.no-ip.info/hotspot/forum/index.php?topic=3585.msg3589#msg3589 http://triple-t.no-ip.info/hotspot/forum/index.php?topic=3585.msg3589#msg3589]</li>

<li>[http://www.aipecco.it/coppermine/displayimage.php?album=lastup&cat=0&pid=846/displayimage.php?pid=846&message_id=f4279ff267a330121650d59e66af2968&message_icon=info#cpgMessageBlock http://www.aipecco.it/coppermine/displayimage.php?album=lastup&cat=0&pid=846/displayimage.php?pid=846&message_id=f4279ff267a330121650d59e66af2968&message_icon=info#cpgMessageBlock]</li>

<li>[http://windsorsalons.com/forum/fluxbb-1.5.6/viewtopic.php?pid=94634#p94634 http://windsorsalons.com/forum/fluxbb-1.5.6/viewtopic.php?pid=94634#p94634]</li>

<li>[http://w715711396.5150.cnaaa7.com/plus/view.php?aid=975734 http://w715711396.5150.cnaaa7.com/plus/view.php?aid=975734]</li>

</ul>