|
|
| Line 1: |
Line 1: |
| {{Distinguish|Dykstra's projection algorithm}}
| | == sagte Belash und bemerkte Nike Air Jordan Schweiz == |
|
| |
|
| {{more footnotes|date=September 2012}}
| | Das Leben war nicht einfach, auch in der größten Haus in der Stadt. Dezember bis Februar, es war der einzige Raum mit Wärme, sagte Belash und bemerkte, dass Brennholz knapp war. Also, ich an einen anderen Freund, der $ 50 hat zu gehen und leihen die $ 50 Dollar zu gehen, kaufen die Dope. Ich gehe kaufen die Dope für 50 $ zu wissen, dass ich bald verkaufen sie an meinen Freund für $ 100 und ich werde 50 $ in den Deal gemacht haben. <br><br>BeesNeez Erzählung Studio, Geometrie Erzählung Studio mit Jahren [http://www.balmer-bd.ch/demo/cache.html Nike Air Jordan Schweiz] der wiederkehr auf den Fernen Nordküste von New South Wales. Mathe-Lektion 5. Verzeichnis der Free Knitting Patterns Online. Anfänger Strickanleitung, Tipps, Tricks, Testimonials, Links und vieles mehr. <br><br>Wurden zunächst bewusst das Video von einem Tippgeber, der uns mit dem Eigentümer des Video angeschlossen, nachdem wir nach Toronto gemacht, schrieb Cook. In relativ konstante Kommunikation mit dem Tippgeber seitdem. Wenn du ihm den Kopf geben am gleichen Ort zur gleichen Zeit in der gleichen Weise, wenn irgendwann wird [http://www.latinmusicshop.ch/database/upload.php Longchamp Le Pliage] es ein bisschen langweilig. So sicher sein, von Zeit zu Zeit zu experimentieren und die Position. <br><br>Gestern machten wir uns entschlossen, unsere eigene Kleiderformen für die Etsy Labs machen und dachten, dass wir die Liebe mit euch zu teilen, wenn Sie zu einem machen wollen. 19. Die They Killed Kenny Wieder Trope als in der populären Kultur verwendet wird, mit einer Liste [http://www.htmi.ch/InfoBack/HaodeUpload/Filer.aspx Polo Ralph Lauren Schweiz] von Beispielen aus allen Medien. Gleiche alte Lied (informal) pflegte zu sagen, dass eine Situation oder jemand das Verhalten gleich bleibt, vor allem wenn es langweilig oder ärgerlich ist "Wie die Arbeit gehen? "Oh. <br><br>Unterlassen Nagellackentferner mit Aceton und nehmen Sie Nagellack höchstens einmal pro Woche. Sie können immer Ihre Nägel während der Woche, wenn Ihre Nagellack-Chips. Leider ist meine Begeisterung wurde bald durch meine Unfähigkeit, um E-Mail-Konten gemildert. Nach verbrachte ich eine gute Stunde oder so, daran zu arbeiten gab es auf. <br><br>Über Bremen Castings Inc.: Besitz dieser Familie Gießerei und Maschinenhalle wurde 1939 in Bremen, Indiana gegründet. Mit über 70 Jahren Erfahrung, ist BCI weltweit für seine Qualität Grau-und Sphäroguss bearbeitet Gussteile bekannt. Sprechen Sie mit Ihrem Kind nicht zu ihm auf. Beim Gespräch mit Ihrem Kind, einwerfen, einige seiner highrolling Emotionen in das Gespräch zwischen den beiden von Ihnen. <br><br>Diese liebenswerte Kuriositäten Dinge tun, normale Linsen nur aboutlike [http://www.latinmusicshop.ch/config/router.php Nike Air Max 90] Biege träumen. Tipp sie so oder so, und plötzlich das Bild hat eine seltsame, schöne, softfocus Look mit nur ein bisschen scharf. Ich bin 21. Ich habe immer Behandlung und derzeit 100 mg zoloft.<ul> |
| {{Infobox algorithm
| | |
| |class=[[Search algorithm]]
| | <li>[http://demon-inferno.net/forum/viewtopic.php?f=10&t=258303 http://demon-inferno.net/forum/viewtopic.php?f=10&t=258303]</li> |
| |image=[[Image:Dijkstra Animation.gif|Dijkstra's algorithm runtime]]
| | |
| |caption = Dijkstra's algorithm. It picks the unvisited vertex with the lowest-distance, calculates the distance through it to each unvisited neighbor, and updates the neighbor's distance if smaller. Mark visited (set to red) when done with neighbors.
| | <li>[http://showtime2.no-ip.info/forum-php/forum/thread.php?threadid=228763&sid= http://showtime2.no-ip.info/forum-php/forum/thread.php?threadid=228763&sid=]</li> |
| |data=[[Graph (data structure)|Graph]]
| | |
| |time= <math>O(|E| + |V| \log|V|)</math>
| | <li>[http://zhaigounet.com/news/html/?46029.html http://zhaigounet.com/news/html/?46029.html]</li> |
| |best-time=
| | |
| |average-time=
| | <li>[http://enseignement-lsf.com/spip.php?article64#forum18281168 http://enseignement-lsf.com/spip.php?article64#forum18281168]</li> |
| |space=
| | |
| |optimal=
| | </ul> |
| |complete=
| |
| }}
| |
| {{graph search algorithm}}
| |
|
| |
|
| '''Dijkstra's algorithm''', conceived by [[computer scientist]] [[Edsger W. Dijkstra|Edsger Dijkstra]] in 1956 and published in 1959,<ref name="Dijkstra Interview">{{cite journal|last=Dijkstra|first=Edsger|coauthors=Thomas J. Misa, Editor|title=An Interview with Edsger W. Dijkstra|journal=Communications of the ACM|date=August 2010|month=08|volume=53|issue=8|pages=41–47|accessdate=2010-08-12|quote=What is the shortest way to travel from Rotterdam to Groningen? It is the algorithm for the shortest path which I designed in about 20 minutes. One morning I was shopping with my young fiancée, and tired, we sat down on the café terrace to drink a cup of coffee and I was just thinking about whether I could do this, and I then designed the algorithm for the shortest path.|doi=10.1145/1787234.1787249}}</ref><ref>{{harvnb|Dijkstra|1959}}</ref> is a [[graph search algorithm]] that solves the single-source [[shortest path problem]] for a [[graph (mathematics)|graph]] with non-negative [[edge (graph theory)|edge]] path costs, producing a [[shortest path tree]]. This algorithm is often used in [[routing]] and as a [[subroutine]] in other graph algorithms.
| | == Klasse 68 Mbt Zürich == |
|
| |
|
| For a given source [[vertex (graph theory)|vertex]] (node) in the graph, the algorithm finds the path with lowest cost (i.e. the shortest path) between that vertex and every other vertex. It can also be used for finding costs of shortest paths from a single vertex to a single destination vertex by stopping the algorithm once the shortest path to the destination vertex has been determined. For example, if the vertices of the graph represent cities and edge path costs represent driving distances between pairs of cities connected by a direct road, Dijkstra's algorithm can be used to find the shortest route between one city and all other cities. As a result, the shortest path first is widely used in network [[routing protocol]]s, most notably [[IS-IS]] and [[OSPF]] (Open Shortest Path First).
| | Die Leute werden fragen, was er in seinen ersten sieben Monaten mit seinem Arbeitgeber geleistet hat. In meiner Erfahrung, die viele von [http://www.vpod-ngo.ch/database/file.asp Mbt Zürich] uns sagen und glauben an Dinge wie "Someday". Ich hoffe, häufiger werden Modding und unter einige Ausflüge in den Bergen (ich lebe in Salt Lake City). Ich hoffe, ein paar gute Bilder von meinem X an diesem Wochenende haben. <br><br>Trip Advisor ist ein Segen und eine Tyrannei, aber ich kann kein Verdienst in einem Hochzeits-Homepage zu finden. Fotos von der (jetzt) glückliche Paar dienen nur zu ärgern, wenn ich bedenke, wie verarmten Familienmitglieder sind eingeladen zu berappen Hunderte von Pfund, eine Zeremonie, die auf sehr [http://www.rossi-praxis.ch/bilder/links/public.asp Nike Air Force] engen Linien und mit wenig Raum für Heiterkeit ausgeführt werden, erscheint besuchen. <br><br>Hallo, alle zusammen. Ich dachte, wie schwer es war, jetzt in Japan zu leben, da die Doppel Tragödien der Erdbeben-und Tsunami am 11. Ehrlich gesagt, habe ich seit Jahren ein großer Fan von Windows. Aber seit Microsoft hat für eine "benutzerfreundlicher" und "Universal"-OS gedrängt fühle ich, dass es schwieriger ist, Windows 8 verwenden, weil es für den durchschnittlichen joe konzipiert. <br><br>Viele Menschen haben so viel in ihre sozialen Netzwerke investiert, wäre es verheerend sein, um die Informationen, die sie enthalten, wie Freundeslisten, Fotos, auch ein Archiv Ihre Kommentare und Status-Updates zu verlieren. Dies ist ein Backup-Dienst für Facebook, Twitter, Google Viadeo, mit LinkedIN auch bald kommen Dienste wie Instagram Flickr offenbar in der Pipeline. <br><br>Er hat eine Leidenschaft für gesundes und natürliches Wohnen beinhaltet dieses Wissen in einem Stil, der Behandlung von Krankheiten und Prävention, die Sie finden gewann in den meisten Arztpraxen. Durch die Beschränkung des Einsatzes von Antibiotika unter Verwendung sciencebased natürliche Behandlungsansätze, wann immer möglich, und die Konzentration auf gute Ernährung und Immunsystem Gesundheit, Dr. <br><br>Diese Sammlung von Informationen existieren zu präsentieren und Informationen über Internet-Katalog Kultur und Geschichte. Lustig oder sterben alle Internet-Meme Bilder und Worte am meisten angesehen in diesem Monat Lustige Videos, lustige Bilder, lustige Witze, Top-Ten-Listen, Beschriftung Wettbewerbe, und lustig. <br><br>Es ist ein Evolved teilgenommen Directory. Mouvement 3D-Formen (Klasse 68) Erfahren Sie mehr über threeUniversal Formen, ein Einblick in die jedoch hinter den Umfragen und die Nachrichten, die Sie Komponisten zu hören. Bob ist ein gefragter Referent bei La Leche Liga Konferenzen, einschließlich Colorado, Texas, Ohio, North Carolina, Kalifornien / Nevada und der Internationalen LLL-Konferenz 2007 sowie verschiedene Elternschaft Gesundheit Messen. Er [http://www.vpod-ngo.ch/database/file.asp Mbt Shop] hat für Zeitgenössische Kinderheilkunde, Newsweek Japan und Dutzend der regionalen [http://www.scheffler.ch/webscripts/frame.asp Vibram Five Fingers Schweiz] Elternzeitschriften in der ganzen Nation geschrieben.<ul> |
| | | |
| Dijkstra's original algorithm does not use a [[min-priority queue]] and runs in <math>O(|V|^2)</math> (where <math>|V|</math> is the number of vertices). The idea of this algorithm is also given in {{harv|Leyzorek|Gray|Johnson|Ladew|1957}}. The implementation based on a min-priority queue implemented by a [[Fibonacci heap]] and running in <math>O(|E|+|V|\log|V|)</math> (where <math>|E|</math> is the number of edges) is due to {{harv|Fredman|Tarjan|1984}}. This is [[Asymptotic computational complexity|asymptotically]] the fastest known single-source [[shortest path problem|shortest-path algorithm]] for arbitrary [[directed graph]]s with unbounded non-negative weights.
| | <li>[http://team-impressive.de/index.php?site=news_comments&newsID=148 http://team-impressive.de/index.php?site=news_comments&newsID=148]</li> |
| | | |
| == Algorithm ==
| | <li>[http://annuncianimali.altervista.org/index.php?page=item&id=133810 http://annuncianimali.altervista.org/index.php?page=item&id=133810]</li> |
| [[Image:Dijkstras progress animation.gif|thumb|Illustration of Dijkstra's algorithm search for finding path from a start node (lower left, red) to a goal node (upper right, green) in a [[robotics|robot]] [[motion planning]] problem. Open nodes represent the "tentative" set. Filled nodes are visited ones, with color representing the distance: the greener, the farther. Nodes in all the different directions are explored uniformly, appearing as a more-or-less circular [[wavefront]] as Dijkstra's algorithm uses a [[consistent heuristic|heuristic]] identically equal to 0.]]
| | |
| | | <li>[http://www.cert-centers.com/forum.php?mod=viewthread&tid=1255846 http://www.cert-centers.com/forum.php?mod=viewthread&tid=1255846]</li> |
| Let the node at which we are starting be called the '''initial node'''. Let the '''distance of node ''Y''''' be the distance from the '''initial node''' to ''Y''. Dijkstra's algorithm will assign some initial distance values and will try to improve them step by step.
| | |
| | | <li>[http://www.tianmizhidu.com/forum.php?mod=viewthread&tid=624168 http://www.tianmizhidu.com/forum.php?mod=viewthread&tid=624168]</li> |
| # Assign to every node a tentative distance value: set it to zero for our initial node and to infinity for all other nodes.
| | |
| # Mark all nodes unvisited. Set the initial node as current. Create a set of the unvisited nodes called the ''unvisited set'' consisting of all the nodes.
| | </ul> |
| # For the current node, consider all of its unvisited neighbors and calculate their ''tentative'' distances. For example, if the current node ''A'' is marked with a <!--tentative ((commented out because the distance is NOT tentative.. it cannot change while visiting the neighbors.. so calling it tentative implies that it can still change, which is confusing))--> distance of 6, and the edge connecting it with a neighbor ''B'' has length 2, then the distance to ''B'' (through ''A'') will be 6 + 2 = 8.
| |
| # When we are done considering all of the neighbors of the current node, mark the current node as visited and remove it from the ''unvisited set''. A visited node will never be checked again. <!-- its distance recorded now is final and minimal. ((commented out because the distance is final for the current-node at the point it is taken from the queue.. the distance cannot be dropped while visiting the neighbors.. yet, the current-node remains unvisited until this step. This is inconsistent and confusing.)) -->
| |
| # If the destination node has been marked visited (when planning a route between two specific nodes) or if the smallest tentative distance among the nodes in the ''unvisited set'' is infinity (when planning a complete traversal; occurs when there is no connection between the initial node and remaining unvisited nodes), then stop. The algorithm has finished.
| |
| # Select the unvisited node that is marked with the smallest tentative distance, and set it as the new "current node" then go back to step 3.
| |
| | |
| == Description ==
| |
| :'''''Note:''' For ease of understanding, this discussion uses the terms '''intersection''', '''road''' and '''map''' — however, formally these terms are '''vertex''', '''edge''' and '''graph''', respectively.''
| |
| | |
| Suppose you would like to find the shortest path between two [[Intersection (road)|intersections]] on a city map, a starting point and a destination. The order is conceptually simple: to start, mark the distance to every intersection on the map with infinity. This is done not to imply there is an infinite distance, but to note that intersection has not yet been ''visited''; some variants of this method simply leave the intersection unlabeled. Now, at each iteration, select a ''current'' intersection. For the first iteration, the current intersection will be the starting point and the distance to it (the intersection's label) will be zero. For subsequent iterations (after the first), the current intersection will be the closest unvisited intersection to the starting point—this will be easy to find.
| |
| | |
| From the current intersection, update the distance to every unvisited intersection that is directly connected to it. This is done by determining the sum of the distance between an unvisited intersection and the value of the current intersection, and [[Graph labeling|relabeling]] the unvisited intersection with this value if it is less than its current value. In effect, the intersection is relabeled if the path to it through the current intersection is shorter than the previously known paths. To facilitate shortest path identification, in pencil, mark the road with an arrow pointing to the relabeled intersection if you label/relabel it, and erase all others pointing to it. After you have updated the distances to each [[Neighbourhood (graph theory)|neighboring intersection]], mark the current intersection as ''visited'' and select the unvisited intersection with lowest distance (from the starting point) – or the lowest label—as the current intersection. Nodes marked as visited are labeled with the shortest path from the starting point to it and will not be revisited or returned to.
| |
| | |
| Continue this process of updating the neighboring intersections with the shortest distances, then marking the current intersection as visited and moving onto the closest unvisited intersection until you have marked the destination as visited. Once you have marked the destination as visited (as is the case with any visited intersection) you have determined the shortest path to it, from the starting point, and can trace your way back, following the arrows in reverse.
| |
| | |
| Of note is the fact that this algorithm makes no attempt to direct "exploration" towards the destination as one might expect. Rather, the sole consideration in determining the next "current" intersection is its distance from the starting point. This algorithm, therefore "expands outward" from the starting point, interactively considering every node that is closer in terms of shortest path distance until it reaches the destination. When understood in this way, it is clear how the algorithm necessarily finds the shortest path, however, it may also reveal one of the algorithm's weaknesses: its relative slowness in some topologies.
| |
| | |
| == Pseudocode ==
| |
| | |
| In the following algorithm, the code <code>u := vertex in ''Q'' with smallest dist[]</code>, searches for the vertex <code><var>u</var></code> in the vertex set <code><var>Q</var></code> that has the least <code>dist[<var>u</var>]</code> value. That vertex is removed from the priority queue <code><var>Q</var></code> and returned to the user. <code>dist_between(<var>u</var>, <var>v</var>)</code> calculates the length between the two neighbor-nodes <code><var>u</var></code> and <code><var>v</var></code>. The variable <code><var>alt</var></code> on lines 20 & 22 is the length of the path from the root node to the neighbor node <code><var>v</var></code> if it were to go through <code><var>u</var></code>. If this path is shorter than the current shortest path recorded for <code><var>v</var></code>, that current path is replaced with this <code><var>alt</var></code> path. The <code>previous</code> array is populated with a pointer to the "next-hop" node on the source graph to get the shortest route to the source.
| |
| | |
| 1 '''function''' Dijkstra(''Graph'', ''source''):
| |
| 2 '''for each''' vertex ''v'' in ''Graph'': ''// Initializations''
| |
| 3 dist[''v''] := infinity ; ''// Unknown distance function from''
| |
| 4 ''// source to v''
| |
| 5 previous[''v''] := undefined ; ''// Previous node in optimal path
| |
| 6 '''end for''' ''// from source''
| |
| 7
| |
| 8 dist[''source''] := 0 ; ''// Distance from source to source''
| |
| 9 ''Q'' := the set of all nodes in ''Graph'' ; ''// All nodes in the graph are''
| |
| 10 ''// unoptimized – thus are in Q''
| |
| 11 '''while''' ''Q'' '''is not''' empty: ''// The main loop''
| |
| 12 ''u'' := vertex in ''Q'' with smallest distance in dist[] ; ''// Source node in first case''
| |
| 13 remove ''u'' from ''Q'' ;
| |
| 14 '''if''' dist[''u''] = infinity:
| |
| 15 '''break''' ; ''// all remaining vertices are''
| |
| 16 '''end if''' ''// inaccessible from source''
| |
| 17
| |
| 18 '''for each''' neighbor ''v'' of ''u'': ''// where v has not yet been''
| |
| 19 ''// removed from Q.''
| |
| 20 ''alt'' := dist[''u''] + dist_between(''u'', ''v'') ;
| |
| 21 '''if''' ''alt'' < dist[''v'']: ''// Relax (u,v,a)''
| |
| 22 dist[''v''] := ''alt'' ;
| |
| 23 previous[''v''] := ''u'' ;
| |
| 24 decrease-key ''v'' in ''Q''; ''// Reorder v in the Queue''
| |
| 25 '''end if'''
| |
| 26 '''end for'''
| |
| 27 '''end while'''
| |
| 28 '''return''' dist;
| |
| 29 '''endfunction'''
| |
| | |
| If we are only interested in a shortest path between vertices <code><var>source</var></code> and <code><var>target</var></code>, we can terminate the search at line 13 if <code><var>u</var> = <var>target</var></code>.
| |
| Now we can read the shortest path from <code><var>source</var></code> to <code><var>target</var></code> by reverse iteration:
| |
| | |
| 1 ''S'' := empty sequence
| |
| 2 ''u'' := ''target''
| |
| 3 '''while''' previous[''u''] is defined: ''// Construct the shortest path with a stack S
| |
| 4 insert ''u'' at the beginning of ''S'' ''// Push the vertex into the stack
| |
| 5 ''u'' := previous[''u''] ''// Traverse from target to source
| |
| 6 '''end while''' ;
| |
| | |
| Now sequence <code><var>S</var></code> is the list of vertices constituting one of the shortest paths from <code><var>source</var></code> to <code><var>target</var></code>, or the empty sequence if no path exists.
| |
| | |
| A more general problem would be to find all the shortest paths between <code><var>source</var></code> and <code><var>target</var></code> (there might be several different ones of the same length). Then instead of storing only a single node in each entry of <code>previous[]</code> we would store all nodes satisfying the relaxation condition. For example, if both <code><var>r</var></code> and <code><var>source</var></code> connect to <code><var>target</var></code> and both of them lie on different shortest paths through <code><var>target</var></code> (because the edge cost is the same in both cases), then we would add both <code><var>r</var></code> and <code><var>source</var></code> to <code>previous[<var>target</var>]</code>. When the algorithm completes, <code>previous[]</code> data structure will actually describe a graph that is a subset of the original graph with some edges removed. Its key property will be that if the algorithm was run with some starting node, then every path from that node to any other node in the new graph will be the shortest path between those nodes in the original graph, and all paths of that length from the original graph will be present in the new graph. Then to actually find all these shortest paths between two given nodes we would use a path finding algorithm on the new graph, such as [[depth-first search]].
| |
| | |
| ===Using a priority queue===
| |
| | |
| A [[min-priority queue]] is an abstract data structure that provides 3 basic operations : <code>add_with_priority()</code>, <code>decrease_priority()</code> and <code>extract_min()</code>. As mentioned earlier, using such a data structure can lead to faster computing times than using a basic queue. Notably, [[Fibonacci heap]] {{harv|Fredman|Tarjan|1984}} or [[Brodal queue]] offer optimal implementations for those 3 operations. As the algorithm is slightly different, we mention it here, in pseudo-code as well :
| |
| | |
| 1 '''function''' Dijkstra(''Graph'', ''source''):
| |
| 2 dist[''source''] := 0 ''// Initializations
| |
| 3 '''for each''' vertex ''v'' in ''Graph'':
| |
| 4 '''if''' ''v'' ≠ ''source''
| |
| 5 dist[''v''] := infinity ''// Unknown distance from source to v
| |
| 6 previous[''v''] := undefined ''// Predecessor of v
| |
| 7 '''end if'''
| |
| 8 ''PQ''.add_with_priority(''v'',dist[''v''])
| |
| 9 '''end for'''
| |
| 10
| |
| 11
| |
| 12 '''while''' ''PQ'' is not empty: ''// The main loop
| |
| 13 ''u'' := ''PQ''.extract_min() ''// Remove and return best vertex
| |
| 14 '''for each''' neighbor v of u: ''// where v has not yet been removed from PQ.
| |
| 15 ''alt'' = dist[''u''] + length(''u'', ''v'')
| |
| 16 '''if''' ''alt'' < dist[''v''] ''// Relax the edge (u,v)
| |
| 17 dist[''v''] := ''alt''
| |
| 18 previous[''v''] := ''u''
| |
| 19 ''PQ''.decrease_priority(''v'',''alt'')
| |
| 20 '''end if'''
| |
| 21 '''end for'''
| |
| 22 '''end while'''
| |
| 23 '''return''' previous[]
| |
| | |
| It should be noted that other data structures can be used to achieve even faster computing times in practice.<ref name=chen_07>{{cite book|first1=M.|last1=Chen|first2=R. A.|last2=Chowdhury|first3=V.|last3=Ramachandran|first4=D. L.|last4=Roche|first5=L.|last5=Tong|title=Priority Queues and Dijkstra’s Algorithm — UTCS Technical Report TR-07-54 — 12 October 2007|publisher=The University of Texas at Austin, Department of Computer Sciences|location=Austin, Texas|year=2007|url=http://www.cs.sunysb.edu/~rezaul/papers/TR-07-54.pdf|ref=chen}}</ref>
| |
| | |
| == Running time ==
| |
| | |
| An upper bound of the running time of Dijkstra's algorithm on a graph with edges <math>E</math> and vertices <math>V</math> can be expressed as a function of <math>|E|</math> and <math>|V|</math> using [[Big O notation#Graph theory|big-O notation]].
| |
| | |
| For any implementation of vertex set <math>Q</math> the running time is in <math>O(|E| \cdot dk_Q + |V| \cdot em_Q)</math>, where <math>dk_Q</math> and <math>em_Q</math> are times needed to perform decrease key and extract minimum operations in set <math>Q</math>, respectively.
| |
| | |
| The simplest implementation of the Dijkstra's algorithm stores vertices of set <math>Q</math> in an ordinary linked list or array, and extract minimum from <math>Q</math> is simply a linear search through all vertices in <math>Q</math>. In this case, the running time is <math>O(|E| + |V|^2) = O(|V|^2)</math>.
| |
| | |
| For [[sparse graph]]s, that is, graphs with far fewer than <math>O(|V|^2)</math> edges, Dijkstra's algorithm can be implemented more efficiently by storing the graph in the form of [[adjacency list]]s and using a [[self-balancing binary search tree]], [[binary heap]], [[pairing heap]], or [[Fibonacci heap]] as a [[priority queue]] to implement extracting minimum efficiently. With a self-balancing binary search tree or binary heap, the algorithm requires <math>\Theta((|E| + |V|) \log |V|)</math> time (which is dominated by <math>\Theta( | E | \log | V | )</math>, assuming the graph is connected). To avoid <math>O(|V|)</math> look-up in decrease-key step on a vanilla binary heap, it is necessary to maintain a supplementary index mapping each vertex to the heap's index (and keep it up to date as priority queue <math>Q</math> changes), making it take only <math>O(\log|V|)</math> time instead. The [[Fibonacci heap]] improves this to <math>O(|E| + |V| \log|V|)</math>.
| |
| | |
| Note that for [[directed acyclic graph]]s, it is possible to find shortest paths from a given starting vertex in linear time, by processing the vertices in a topological order, and calculating the path length for each vertex to be the minimum length obtained via any of its incoming edges.<ref>http://www.boost.org/doc/libs/1_44_0/libs/graph/doc/dag_shortest_paths.html</ref>
| |
| | |
| == Related problems and algorithms ==
| |
| | |
| The functionality of Dijkstra's original algorithm can be extended with a variety of modifications. For example, sometimes it is desirable to present solutions which are less than mathematically optimal. To obtain a ranked list of less-than-optimal solutions, the optimal solution is first calculated. A single edge appearing in the optimal solution is removed from the graph, and the optimum solution to this new graph is calculated. Each edge of the original solution is suppressed in turn and a new shortest-path calculated. The secondary solutions are then ranked and presented after the first optimal solution.
| |
| | |
| Dijkstra's algorithm is usually the working principle behind [[link-state routing protocol]]s, [[OSPF]] and [[IS-IS]] being the most common ones.
| |
| | |
| Unlike Dijkstra's algorithm, the [[Bellman–Ford algorithm]] can be used on graphs with negative edge weights, as long as the graph contains no [[negative cycle]] reachable from the source vertex ''s''. The presence of such cycles means there is no shortest path, since the total weight becomes lower each time the cycle is traversed. It is possible to adapt Dijkstra's algorithm to handle negative weight edges by combining it with the Bellman-Ford algorithm (to remove negative edges and detect negative cycles), such an algorithm is called [[Johnson's algorithm]].
| |
| | |
| The [[A-star algorithm|A* algorithm]] is a generalization of Dijkstra's algorithm that cuts down on the size of the subgraph that must be explored, if additional information is available that provides a lower bound on the "distance" to the target. This approach can be viewed from the perspective of [[linear programming]]: there is a natural [[Shortest path problem#Linear programming formulation|linear program for computing shortest paths]], and solutions to its [[dual linear program]] are feasible if and only if they form a [[consistent heuristic]] (speaking roughly, since the sign conventions differ from place to place in the literature). This feasible dual / consistent heuristic defines a non-negative [[reduced cost]] and A* is essentially running Dijkstra's algorithm with these reduced costs. If the dual satisfies the weaker condition of [[Admissible heuristic|admissibility]], then A* is instead more akin to the Bellman–Ford algorithm.
| |
| | |
| The process that underlies Dijkstra's algorithm is similar to the greedy process used in [[Prim's algorithm]]. Prim's purpose is to find a [[minimum spanning tree]] that connects all nodes in the graph; Dijkstra is concerned with only two nodes. Prim's does not evaluate the total weight of the path from the starting node, only the individual path.
| |
| | |
| [[Breadth-first search]] can be viewed as a special-case of Dijkstra's algorithm on unweighted graphs, where the priority queue degenerates into a FIFO queue.
| |
| | |
| == Dynamic programming perspective ==
| |
| | |
| From a [[dynamic programming]] point of view, Dijkstra's algorithm is a successive approximation scheme that solves the dynamic programming functional equation for the shortest path problem by the '''Reaching''' method.<ref name=sniedovich_06>{{cite journal | last = Sniedovich | first = M. | title = Dijkstra’s algorithm revisited: the dynamic programming connexion | journal = Journal of Control and Cybernetics | volume = 35 | issue = 3 | pages = 599–620 | year = 2006 | url = http://matwbn.icm.edu.pl/ksiazki/cc/cc35/cc3536.pdf | format = [[PDF]]}} [http://www.ifors.ms.unimelb.edu.au/tutorial/dijkstra_new/index.html Online version of the paper with interactive computational modules.]</ref><ref name=denardo_03>{{cite book | last = Denardo | first = E.V. | title = Dynamic Programming: Models and Applications | publisher = [[Dover Publications]] | location = Mineola, NY | year = 2003 | isbn = 978-0-486-42810-9}}</ref><ref name=sniedovich_10>{{cite book | last = Sniedovich | first = M. | title = Dynamic Programming: Foundations and Principles | publisher = [[Francis & Taylor]] | year = 2010 | isbn = 978-0-8247-4099-3 }}</ref>
| |
| | |
| In fact, Dijkstra's explanation of the logic behind the algorithm,<ref>{{harvnb|Dijkstra|1959|p=270}}</ref> namely
| |
| {{quote|
| |
| '''Problem 2.''' Find the path of minimum total length between two given nodes <math>P</math> and <math>Q</math>.
| |
| | |
| We use the fact that, if <math>R</math> is a node on the minimal path from <math>P</math> to <math>Q</math>, knowledge of the latter implies the knowledge of the minimal path from <math>P</math> to <math>R</math>.
| |
| }}
| |
| | |
| is a paraphrasing of [[Richard Bellman|Bellman's]] famous [[Principle of Optimality]] in the context of the shortest path problem.
| |
| | |
| == See also == | |
| {{portal|Computer science}}
| |
| * [[A* search algorithm]]
| |
| * [[Euclidean shortest path]]
| |
| * [[Flood fill]]
| |
| * [[Longest path problem]]
| |
| * [[Johnson's algorithm]]
| |
| * [[Bellman–Ford algorithm]]
| |
| * [[Floyd–Warshall algorithm]]
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| == References ==
| |
| * {{cite journal | authorlink = Edsger W. Dijkstra | first1 = E. W. | last1 = Dijkstra | url= http://www-m3.ma.tum.de/twiki/pub/MN0506/WebHome/dijkstra.pdf | title = A note on two problems in connexion with graphs | journal = Numerische Mathematik | volume = 1 | year = 1959 | pages = 269–271 | ref = harv | doi = 10.1007/BF01386390}}
| |
| * {{cite book | author1-link = Thomas H. Cormen | first1 = Thomas H. | last1 = Cormen | author2-link = Charles E. Leiserson | first2 = Charles E. | last2 = Leiserson | author3-link = Ronald L. Rivest | first3 = Ronald L. | last3 = Rivest | author4-link = Clifford Stein | first4 = Clifford | last4 = Stein | title = [[Introduction to Algorithms]] | edition = Second | publisher = [[MIT Press]] and [[McGraw–Hill]] | year = 2001 | isbn = 0-262-03293-7 | chapter = Section 24.3: Dijkstra's algorithm | pages = 595–601}}
| |
| * {{cite conference|first1=Michael Lawrence|last1=Fredman|authorlink1=Michael Fredman|first2=Robert E.|last2=Tarjan|authorlink2=Robert Tarjan|title=Fibonacci heaps and their uses in improved network optimization algorithms|conference=25th Annual Symposium on Foundations of Computer Science|year=1984|publisher=[[IEEE]]|pages=338–346|url=http://www.computer.org/portal/web/csdl/doi/10.1109/SFCS.1984.715934|ref=harv|doi=10.1109/SFCS.1984.715934}}
| |
| * {{cite journal|first1=Michael Lawrence|last1=Fredman|authorlink1=Michael Fredman|first2=Robert E.|last2=Tarjan|authorlink2=Robert Tarjan|title=Fibonacci heaps and their uses in improved network optimization algorithms|journal=Journal of the Association for Computing Machinery|volume=34|year=1987|pages=596–615|url=http://portal.acm.org/citation.cfm?id=28874|ref=harv|doi=10.1145/28869.28874|issue=3}}
| |
| * {{cite journal | first1 = F. Benjamin | last1 = Zhan | first2 = Charles E. | last2 = Noon |date=February 1998 | title = Shortest Path Algorithms: An Evaluation Using Real Road Networks | journal = [[Transportation Science]] | volume = 32 | issue = 1 | pages = 65–73 | doi = 10.1287/trsc.32.1.65}}
| |
| * {{cite book|first1=M.|last1=Leyzorek|first2=R. S.|last2=Gray|first3=A. A.|last3=Johnson|first4=W. C.|last4=Ladew|first5=S. R.|last5=Meaker, Jr.|first6=R. M.|last6=Petry|first7=R. N.|last7=Seitz|title=Investigation of Model Techniques — First Annual Report — 6 June 1956 — 1 July 1957 — A Study of Model Techniques for Communication Systems|publisher=Case Institute of Technology|location=Cleveland, Ohio|year=1957|ref=harv}}
| |
| * {{cite journal|first1=D.E.|last1=[[Donald Knuth|Knuth]]|title=A Generalization of Dijkstra's Algorithm|journal=[[Information Processing Letters]]|volume=6|number=1|pages=1–5|year=1977}}
| |
| | |
| == External links ==
| |
| {{external links|date=September 2012}}
| |
| {{Commons category|Dijkstra's algorithm}}
| |
| * C/C++
| |
| ** [https://github.com/xtaci/algorithms/blob/master/include/dijkstra.h Dijkstra's Algorithm in C++]
| |
| ** [http://www.boost.org/doc/libs/1_43_0/libs/graph/doc/dijkstra_shortest_paths.html Implementation in Boost C++ library]
| |
| ** [http://www.rawbytes.com/dijkstras-algorithm-in-c/ Dijkstra's Algorithm in C Programming Language]
| |
| * Java
| |
| ** [http://www.dgp.toronto.edu/people/JamesStewart/270/9798s/Laffra/DijkstraApplet.html Applet by Carla Laffra of Pace University]
| |
| ** [http://students.ceid.upatras.gr/~papagel/english/java_docs/minDijk.htm Visualization of Dijkstra's Algorithm]
| |
| ** [http://www-b2.is.tokushima-u.ac.jp/~ikeda/suuri/dijkstra/Dijkstra.shtml Shortest Path Problem: Dijkstra's Algorithm]
| |
| ** [http://www.unf.edu/~wkloster/foundations/DijkstraApplet/DijkstraApplet.htm Dijkstra's Algorithm Applet]
| |
| ** [http://code.google.com/p/annas/ Open Source Java Graph package with implementation of Dijkstra's Algorithm]
| |
| ** [http://www.stackframe.com/software/PathFinder A Java library for path finding with Dijkstra's Algorithm and example Applet]
| |
| ** [https://github.com/graphhopper/graphhopper/tree/90879ad05c4dfedf0390d44525065f727b043357/core/src/main/java/com/graphhopper/routing Dijkstra's algorithm as bidirectional version in Java]
| |
| * C#/.Net
| |
| ** [http://www.codeproject.com/KB/recipes/ShortestPathCalculation.aspx Dijkstra's Algorithm in C#]
| |
| ** [http://www.codeproject.com/KB/recipes/FastHeapDijkstra.aspx Fast Priority Queue Implementation of Dijkstra's Algorithm in C#]
| |
| ** [http://quickgraph.codeplex.com/ QuickGraph, Graph Data Structures and Algorithms for .NET]
| |
| * [http://optlab-server.sce.carleton.ca/POAnimations2007/DijkstrasAlgo.html Dijkstra's Algorithm Simulation]
| |
| * [http://purl.umn.edu/107247 Oral history interview with Edsger W. Dijkstra], [[Charles Babbage Institute]] University of Minnesota, Minneapolis.
| |
| * [http://www.cs.sunysb.edu/~skiena/combinatorica/animations/dijkstra.html Animation of Dijkstra's algorithm]
| |
| * [http://bonsaicode.wordpress.com/2011/01/04/programming-praxis-dijkstra’s-algorithm/ Haskell implementation of Dijkstra's Algorithm] on Bonsai code
| |
| * [http://hansolav.net/sql/graphs.html Implementation in T-SQL]
| |
| * [http://www.mathworks.com/matlabcentral/fileexchange/20025-advanced-dijkstras-minimum-path-algorithm A MATLAB program for Dijkstra's algorithm]
| |
| * [http://www.turb0js.com/a/Dijkstra%27s_Algorithm Step through Dijkstra's Algorithm in an online JavaScript Debugger]
| |
| | |
| {{Use dmy dates|date=February 2011}}
| |
| | |
| [[Category:1959 in computer science]]
| |
| [[Category:Graph algorithms]]
| |
| [[Category:Search algorithms]]
| |
| [[Category:Routing algorithms]]
| |
| [[Category:Combinatorial optimization]]
| |
| [[Category:Articles with example pseudocode]]
| |
| [[Category:Dutch inventions]]
| |
sagte Belash und bemerkte Nike Air Jordan Schweiz
Das Leben war nicht einfach, auch in der größten Haus in der Stadt. Dezember bis Februar, es war der einzige Raum mit Wärme, sagte Belash und bemerkte, dass Brennholz knapp war. Also, ich an einen anderen Freund, der $ 50 hat zu gehen und leihen die $ 50 Dollar zu gehen, kaufen die Dope. Ich gehe kaufen die Dope für 50 $ zu wissen, dass ich bald verkaufen sie an meinen Freund für $ 100 und ich werde 50 $ in den Deal gemacht haben.
BeesNeez Erzählung Studio, Geometrie Erzählung Studio mit Jahren Nike Air Jordan Schweiz der wiederkehr auf den Fernen Nordküste von New South Wales. Mathe-Lektion 5. Verzeichnis der Free Knitting Patterns Online. Anfänger Strickanleitung, Tipps, Tricks, Testimonials, Links und vieles mehr.
Wurden zunächst bewusst das Video von einem Tippgeber, der uns mit dem Eigentümer des Video angeschlossen, nachdem wir nach Toronto gemacht, schrieb Cook. In relativ konstante Kommunikation mit dem Tippgeber seitdem. Wenn du ihm den Kopf geben am gleichen Ort zur gleichen Zeit in der gleichen Weise, wenn irgendwann wird Longchamp Le Pliage es ein bisschen langweilig. So sicher sein, von Zeit zu Zeit zu experimentieren und die Position.
Gestern machten wir uns entschlossen, unsere eigene Kleiderformen für die Etsy Labs machen und dachten, dass wir die Liebe mit euch zu teilen, wenn Sie zu einem machen wollen. 19. Die They Killed Kenny Wieder Trope als in der populären Kultur verwendet wird, mit einer Liste Polo Ralph Lauren Schweiz von Beispielen aus allen Medien. Gleiche alte Lied (informal) pflegte zu sagen, dass eine Situation oder jemand das Verhalten gleich bleibt, vor allem wenn es langweilig oder ärgerlich ist "Wie die Arbeit gehen? "Oh.
Unterlassen Nagellackentferner mit Aceton und nehmen Sie Nagellack höchstens einmal pro Woche. Sie können immer Ihre Nägel während der Woche, wenn Ihre Nagellack-Chips. Leider ist meine Begeisterung wurde bald durch meine Unfähigkeit, um E-Mail-Konten gemildert. Nach verbrachte ich eine gute Stunde oder so, daran zu arbeiten gab es auf.
Über Bremen Castings Inc.: Besitz dieser Familie Gießerei und Maschinenhalle wurde 1939 in Bremen, Indiana gegründet. Mit über 70 Jahren Erfahrung, ist BCI weltweit für seine Qualität Grau-und Sphäroguss bearbeitet Gussteile bekannt. Sprechen Sie mit Ihrem Kind nicht zu ihm auf. Beim Gespräch mit Ihrem Kind, einwerfen, einige seiner highrolling Emotionen in das Gespräch zwischen den beiden von Ihnen.
Diese liebenswerte Kuriositäten Dinge tun, normale Linsen nur aboutlike Nike Air Max 90 Biege träumen. Tipp sie so oder so, und plötzlich das Bild hat eine seltsame, schöne, softfocus Look mit nur ein bisschen scharf. Ich bin 21. Ich habe immer Behandlung und derzeit 100 mg zoloft.
Klasse 68 Mbt Zürich
Die Leute werden fragen, was er in seinen ersten sieben Monaten mit seinem Arbeitgeber geleistet hat. In meiner Erfahrung, die viele von Mbt Zürich uns sagen und glauben an Dinge wie "Someday". Ich hoffe, häufiger werden Modding und unter einige Ausflüge in den Bergen (ich lebe in Salt Lake City). Ich hoffe, ein paar gute Bilder von meinem X an diesem Wochenende haben.
Trip Advisor ist ein Segen und eine Tyrannei, aber ich kann kein Verdienst in einem Hochzeits-Homepage zu finden. Fotos von der (jetzt) glückliche Paar dienen nur zu ärgern, wenn ich bedenke, wie verarmten Familienmitglieder sind eingeladen zu berappen Hunderte von Pfund, eine Zeremonie, die auf sehr Nike Air Force engen Linien und mit wenig Raum für Heiterkeit ausgeführt werden, erscheint besuchen.
Hallo, alle zusammen. Ich dachte, wie schwer es war, jetzt in Japan zu leben, da die Doppel Tragödien der Erdbeben-und Tsunami am 11. Ehrlich gesagt, habe ich seit Jahren ein großer Fan von Windows. Aber seit Microsoft hat für eine "benutzerfreundlicher" und "Universal"-OS gedrängt fühle ich, dass es schwieriger ist, Windows 8 verwenden, weil es für den durchschnittlichen joe konzipiert.
Viele Menschen haben so viel in ihre sozialen Netzwerke investiert, wäre es verheerend sein, um die Informationen, die sie enthalten, wie Freundeslisten, Fotos, auch ein Archiv Ihre Kommentare und Status-Updates zu verlieren. Dies ist ein Backup-Dienst für Facebook, Twitter, Google Viadeo, mit LinkedIN auch bald kommen Dienste wie Instagram Flickr offenbar in der Pipeline.
Er hat eine Leidenschaft für gesundes und natürliches Wohnen beinhaltet dieses Wissen in einem Stil, der Behandlung von Krankheiten und Prävention, die Sie finden gewann in den meisten Arztpraxen. Durch die Beschränkung des Einsatzes von Antibiotika unter Verwendung sciencebased natürliche Behandlungsansätze, wann immer möglich, und die Konzentration auf gute Ernährung und Immunsystem Gesundheit, Dr.
Diese Sammlung von Informationen existieren zu präsentieren und Informationen über Internet-Katalog Kultur und Geschichte. Lustig oder sterben alle Internet-Meme Bilder und Worte am meisten angesehen in diesem Monat Lustige Videos, lustige Bilder, lustige Witze, Top-Ten-Listen, Beschriftung Wettbewerbe, und lustig.
Es ist ein Evolved teilgenommen Directory. Mouvement 3D-Formen (Klasse 68) Erfahren Sie mehr über threeUniversal Formen, ein Einblick in die jedoch hinter den Umfragen und die Nachrichten, die Sie Komponisten zu hören. Bob ist ein gefragter Referent bei La Leche Liga Konferenzen, einschließlich Colorado, Texas, Ohio, North Carolina, Kalifornien / Nevada und der Internationalen LLL-Konferenz 2007 sowie verschiedene Elternschaft Gesundheit Messen. Er Mbt Shop hat für Zeitgenössische Kinderheilkunde, Newsweek Japan und Dutzend der regionalen Vibram Five Fingers Schweiz Elternzeitschriften in der ganzen Nation geschrieben.