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| {{Infobox Algorithm
| | All of the trophies from all within the members in your kin get added up and as well , divided by 2 to ascertain your clans overall medals. Playing many different kinds of games will leave your gaming time more enjoyable. and your league also determines any battle win bonus. 5 star rating and is known to be somewhat addictive as players frequently devote several hours enjoying the game. She makes a speciality of beauty salon business start-up and client fascination.<br><br> |
| |class=[[Single-source shortest path problem]] (for weighted directed graphs)
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| |image=
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| |caption =
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| |data=[[Graph (data structure)|Graph]]
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| |time=<math>O (|V| |E|)</math>
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| |space=<math>O (|V|)</math>
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| }}
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| {{Tree search algorithm}}
| | Progressing to the higher level: what's important when it comes to game, but when looking at Clash of Clans, you get a lot more [http://Www.Ehow.com/search.html?s=subtle+actions subtle actions]. Despite making use of clash of clans hack tools, you may acquire experience points as a result of matching on top linked to other players. Increased metabolism purpose of earning Player vs player is to enable any enhancements for your indigneous group. The augmentation consists of better strive equipment, properties, troops and thus tribe people.<br><br>Throne Rush has an equivalent for just about everything in Clash. Instead associated with an Town Hall, it incorporates a Castle. Instead from Clans, it has Brotherhoods. Instead of Trophies, it has Morale. Perhaps the one thing it takes to the next step is its Immortal Personalities. clash of clans has a Barbarian King and a substantial Archer Queen which are typical special units that can be reused in battle inside they just require times of time to stop back to full healthy. Throne Rush has similar heroes that could be hired, but they are more extreme and more packed. They play almost the same way, although i think players will indulge in using four or twelve Immortal Heroes instead among just two, as for an extended time as they dont ruin the balance of online game too severely.<br><br>A very good method to please your kids with a gaming system and ensure they be placed fit is to purchase a Wii. This video gaming system needs real task to play. Your children won't be working for hours on end playing clash of clans hack. They really need to be moving around as tips on how to play the games for this particular system.<br><br>The website not only provides complete tools, there is besides that Clash of Clans compromise no survey by any kind of. Strict anti ban system allow users to utilize shed pounds and play without much hindrance. If enthusiastic gamers are interested in qualifing for the program, they are definitely required to visit this wonderful site and obtain the hack tool trainer now. The name of the internet business is Amazing Cheats. A number of web-sites have different types from software by which men or women can get past tough stages in the video game.<br><br>Also, the association alcazar together with your war abject is generally altered versus one with your whole village, so the device charge end up presently abounding seaprately. Soldiers donated to a combat abject is going turn out to be acclimated to avert the piece adjoin all attacks in the course of action year. Unlike you rregular apple though, there is no impose to appeal troops on your war base; they are unquestionably automatically open. Actual troops can be requested in case you aspirations however.<br><br>Watch for game of the year versions of major title of the article. When you liked this article as well as you wish to be given details about Clash Of Clans Hack Android [[http://prometeu.net Prometeu.Net]] i implore you to pay a visit to our site. These often come out a spring or maybe more after the original title, but form lots of the down loadable and extra content which had been released in stages big event first title. These games offer a much bang for the dollars. |
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| The '''Bellman–Ford algorithm''' is an [[algorithm]] that computes [[shortest path]]s from a single source [[vertex (graph theory)|vertex]] to all of the other vertices in a [[weighted digraph]].<ref name=Bang>{{harvtxt|Bang-Jensen|Gutin|2000}}</ref>
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| It is slower than [[Dijkstra's algorithm]] for the same problem, but more versatile, as it is capable of handling graphs in which some of the edge weights are negative numbers.
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| The algorithm is usually named after two of its developers, [[Richard Bellman]] and [[L. R. Ford, Jr.|Lester Ford, Jr.]], who published it in 1958 and 1956, respectively; however, [[Edward F. Moore]] also published the same algorithm in 1957, and for this reason it is also sometimes called the '''Bellman–Ford–Moore algorithm'''.<ref name=Bang/>
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| Negative edge weights are found in various applications of graphs, hence the usefulness of this algorithm.{{sfnp|Sedgewick|2002}}
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| If a graph contains a "negative cycle" (i.e. a [[cycle (graph theory)|cycle]] whose edges sum to a negative value) that is reachable from the source, then there is no ''cheapest'' path: any path can be made cheaper by one more [[Walk (graph theory)|walk]] around the negative cycle. In such a case, the Bellman–Ford algorithm can detect negative cycles and report their existence. {{sfnp|Kleinberg|Tardos|2006}}<ref name=Bang/>
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| ==Algorithm==
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| [[File:Bellman-Ford worst-case example.svg|thumb|In this example graph, assuming that A is the source and edges are processed in the worst order, from right to left, it requires the full |V|−1 or 4 iterations for the distance estimates to converge. Conversely, if the edges are processed in the best order, from left to right, the algorithm converges in a single iteration.]]
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| Like [[Dijkstra's Algorithm]], Bellman–Ford is based on the principle of [[Relaxation (approximation)|relaxation]], in which an approximation to the correct distance is gradually replaced by more accurate values until eventually reaching the optimum solution. In both algorithms, the approximate distance to each vertex is always an overestimate of the true distance, and is replaced by the minimum of its old value with the length of a newly found path.
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| However, Dijkstra's algorithm [[Greedy algorithm|greedily]] selects the minimum-weight node that has not yet been processed, and performs this relaxation process on all of its outgoing edges; in contrast, the Bellman–Ford algorithm simply relaxes ''all'' the edges, and does this |''V'' | − 1 times, where |''V'' | is the number of vertices in the graph. In each of these repetitions, the number of vertices with correctly calculated distances grows, from which it follows that eventually all vertices will have their correct distances. This method allows the Bellman–Ford algorithm to be applied to a wider class of inputs than Dijkstra.
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| Bellman–Ford runs in {{math| [[Big O notation|O]]({{!}} V {{!}} {{dot}} {{!}} E {{!}}) }} time, where {{math|{{!}} V {{!}} }} and {{math|{{!}} E {{!}} }} are the number of vertices and edges respectively.
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| '''procedure''' BellmanFord(''list'' vertices, ''list'' edges, ''vertex'' source)
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| ''// This implementation takes in a graph, represented as lists of vertices and edges,''
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| ''// and fills two arrays (distance and predecessor) with shortest-path information''
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|
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| ''// Step 1: initialize graph''
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| '''for each''' vertex v '''in''' vertices:
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| '''if''' v '''is''' source '''then''' distance[v] := 0
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| '''else''' distance[v] := '''infinity'''
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| predecessor[v] := '''null'''
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|
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| ''// Step 2: relax edges repeatedly''
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| <!-- The outer loop iterates |V|-1 times. See, e.g, CLRS Ch. 24.1. The value of i is unused. -->
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| '''for''' i '''from''' 1 '''to''' size(vertices)-1:
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| '''for each''' edge (u, v) '''with''' weight w '''in''' edges:
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| '''if''' distance[u] + w < distance[v]:
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| distance[v] := distance[u] + w
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| predecessor[v] := u
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|
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| ''// Step 3: check for negative-weight cycles''
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| '''for each''' edge (u, v) '''with''' weight w '''in''' edges:
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| '''if''' distance[u] + w < distance[v]:
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| '''error''' "Graph contains a negative-weight cycle"
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| ==Proof of correctness==
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| The correctness of the algorithm can be shown by [[mathematical induction|induction]]. The precise statement shown by induction is:
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| '''Lemma'''. After ''i'' repetitions of ''for'' cycle:
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| * If Distance(''u'') is not infinity, it is equal to the length of some path from ''s'' to ''u'';
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| * If there is a path from ''s'' to ''u'' with at most ''i'' edges, then Distance(u) is at most the length of the shortest path from ''s'' to ''u'' with at most ''i'' edges.
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| '''Proof'''. For the base case of induction, consider <code>i=0</code> and the moment before ''for'' cycle is executed for the first time. Then, for the source vertex, <code>source.distance = 0</code>, which is correct. For other vertices ''u'', <code>u.distance = '''infinity'''</code>, which is also correct because there is no path from ''source'' to ''u'' with 0 edges.
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| For the inductive case, we first prove the first part. Consider a moment when a vertex's distance is updated by
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| <code>v.distance := u.distance + uv.weight</code>. By inductive assumption, <code>u.distance</code> is the length of some path from ''source'' to ''u''. Then <code>u.distance + uv.weight</code> is the length of the path from ''source'' to ''v'' that follows the path from ''source'' to ''u'' and then goes to ''v''.
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| For the second part, consider the shortest path from ''source'' to ''u'' with at most ''i'' edges. Let ''v'' be the last vertex before ''u'' on this path. Then, the part of the path from ''source'' to ''v'' is the shortest path from ''source'' to ''v'' with at most ''i-1'' edges. By inductive assumption, <code>v.distance</code> after ''i''−1 cycles is at most the length of this path. Therefore, <code>uv.weight + v.distance</code> is at most the length of the path from ''s'' to ''u''. In the ''i<sup>th</sup>'' cycle, <code>u.distance</code> gets compared with <code>uv.weight + v.distance</code>, and is set equal to it if <code>uv.weight + v.distance</code> was smaller. Therefore, after ''i'' cycles, <code>u.distance</code> is at most the length of the shortest path from ''source'' to ''u'' that uses at most ''i'' edges.
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| If there are no negative-weight cycles, then every shortest path visits each vertex at most once, so at step 3 no further improvements can be made. Conversely, suppose no improvement can be made. Then for any cycle with vertices ''v''[0], ..., ''v''[''k''−1],
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| <code>v[i].distance <= v[(i-1) mod k].distance + v[(i-1) mod k]v[i].weight</code>
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| Summing around the cycle, the ''v''[''i''].distance terms and the ''v''[''i''−1 (mod ''k'')] distance terms cancel, leaving
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| <code>0 <= sum from 1 to k of v[i-1 (mod k)]v[i].weight</code> | |
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| I.e., every cycle has nonnegative weight.
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| ==Finding negative cycles==
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| When the algorithm is used to find shortest paths, the existence of negative cycles is a problem, preventing the algorithm from finding a correct answer. However, since it terminates upon finding a negative cycle, the Bellman–Ford algorithm can be used for applications in which this is the target to be sought - for example in [[cycle-cancelling]] techniques in [[Flow network|network flow]] analysis.<ref name="Bang"/>
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| == Applications in routing ==
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| A distributed variant of the Bellman–Ford algorithm is used in [[distance-vector routing protocol]]s, for example the [[Routing Information Protocol]] (RIP). The algorithm is distributed because it involves a number of nodes (routers) within an [[autonomous system (Internet)|Autonomous system]], a collection of IP networks typically owned by an ISP.
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| It consists of the following steps:
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| # Each node calculates the distances between itself and all other nodes within the AS and stores this information as a table.
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| # Each node sends its table to all neighboring nodes.
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| # When a node receives distance tables from its neighbors, it calculates the shortest routes to all other nodes and updates its own table to reflect any changes.
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| The main disadvantages of the Bellman–Ford algorithm in this setting are as follows:
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| * It does not scale well.
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| * Changes in [[network topology]] are not reflected quickly since updates are spread node-by-node.
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| * [[Count to infinity#Count-to-infinity problem|Count to infinity]] (if link or node failures render a node unreachable from some set of other nodes, those nodes may spend forever gradually increasing their estimates of the distance to it, and in the meantime there may be routing loops).
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| ==Improvements==
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| The Bellman–Ford algorithm may be improved in practice (although not in the worst case) by the observation that, if an iteration of the main loop of the algorithm terminates without making any changes, the algorithm can be immediately terminated, as subsequent iterations will not make any more changes. With this early termination condition, the main loop may in some cases use many fewer than |''V''| − 1 iterations, even though the worst case of the algorithm remains unchanged.
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| {{harvtxt|Yen|1970}} described two more improvements to the Bellman–Ford algorithm for a graph without negative-weight cycles; again, while making the algorithm faster in practice, they do not change its O(|V|*|E|) worst case time bound. His first improvement reduces the number of relaxation steps that need to be performed within each iteration of the algorithm. If a vertex ''v'' has a distance value that has not changed since the last time the edges out of ''v'' were relaxed, then there is no need to relax the edges out of ''v'' a second time. In this way, as the number of vertices with correct distance values grows, the number whose outgoing edges need to be relaxed in each iteration shrinks, leading to a constant-factor savings in time for [[dense graph]]s.
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| Yen's second improvement first assigns some arbitrary linear order on all vertices and then partitions the set of all edges into two subsets. The first subset, ''E<sub>f</sub>'', contains all edges (''v<sub>i</sub>'', ''v<sub>j</sub>'') such that ''i'' < ''j''; the second, ''E<sub>b</sub>'', contains edges (''v<sub>i</sub>'', ''v<sub>j</sub>'') such that ''i'' > ''j''. Each vertex is visited in the order ''v<sub>1</sub>'', ''v<sub>2</sub>'', ..., ''v''<sub>|''V''|</sub>, relaxing each outgoing edge from that vertex in ''E<sub>f</sub>''. Each vertex is then visited in the order ''v''<sub>|''V''|</sub>, ''v''<sub>|''V''|−1</sub>, ..., ''v''<sub>1</sub>, relaxing each outgoing edge from that vertex in ''E<sub>b</sub>''. Each iteration of the main loop of the algorithm, after the first one, adds at least two edges to the set of edges whose relaxed distances match the correct shortest path distances: one from ''E<sub>f</sub>'' and one from ''E<sub>b</sub>''. This modification reduces the worst-case number of iterations of the main loop of the algorithm from |''V''| − 1 to |''V''|/2.<ref>Cormen et al., 2nd ed., Problem 24-1, pp. 614–615.</ref><ref name=Sedweb/>
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| Another improvement, by {{harvtxt|Bannister|Eppstein|2012}}, replaces the arbitrary linear order of the vertices used in Yen's second improvement by a [[random permutation]]. This change makes the worst case for Yen's improvement (in which the edges of a shortest path strictly alternate between the two subsets ''E<sub>f</sub>'' and ''E<sub>b</sub>'') very unlikely to happen. With a randomly permuted vertex ordering, the [[expected value|expected]] number of iterations needed in the main loop is at most |''V''|/3.<ref name=Sedweb>See Sedgewick's [http://algs4.cs.princeton.edu/44sp/ web exercises] for ''Algorithms'', 4th ed., exercises 5 and 11 (retrieved 2013-01-30).</ref>
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| ===Original sources===
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| *{{cite journal
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| | last = Bellman | first = Richard | authorlink = Richard Bellman
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| | mr = 0102435
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| | journal = Quarterly of Applied Mathematics
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| | pages = 87–90
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| | title = On a routing problem
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| | volume = 16
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| | year = 1958
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| | ref = harv}}
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| *{{cite book
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| |authorlink=L. R. Ford, Jr. | last=Ford Jr. | first=Lester R.
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| |title=Network Flow Theory
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| |date=August 14, 1956
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| |series=Paper P-923
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| |publisher=RAND Corporation
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| |location=Santa Monica, California
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| |url=http://www.rand.org/pubs/papers/P923.html}}
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| *{{cite conference
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| | last = Moore | first = Edward F. | authorlink = Edward F. Moore
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| | title = The shortest path through a maze
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| | location = Cambridge, Mass.
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| | mr = 0114710
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| | pages = 285–292
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| | publisher = Harvard Univ. Press
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| | booktitle = Proc. Internat. Sympos. Switching Theory 1957, Part II
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| | year = 1959
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| | ref = harv}}
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| *{{cite journal
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| | last = Yen | first = Jin Y.
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| | mr = 0253822
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| | journal = Quarterly of Applied Mathematics
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| | pages = 526–530
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| | title = An algorithm for finding shortest routes from all source nodes to a given destination in general networks
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| | volume = 27
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| | year = 1970
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| | ref = harv}}
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| *{{cite conference|title=Randomized speedup of the Bellman–Ford algorithm|first1=M. J.|last1=Bannister|first2=D.|last2=Eppstein|author2-link=David Eppstein|arxiv=1111.5414|booktitle=Analytic Algorithmics and Combinatorics (ANALCO12), Kyoto, Japan|year=2012|pages=41–47|url=http://siam.omnibooksonline.com/2012ANALCO/data/papers/005.pdf|ref=harv}}
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| ===Secondary sources===
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| *{{Cite book|first1=Jørgen |last1=Bang-Jensen|first2=Gregory|last2=Gutin|year=2000|title=Digraphs: Theory, Algorithms and Applications|edition=First |isbn=978-1-84800-997-4|chapter=Section 2.3.4: The Bellman-Ford-Moore algorithm|url=http://www.cs.rhul.ac.uk/books/dbook/|ref=harv}}
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| *{{Introduction to Algorithms}}, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 24.1: The Bellman–Ford algorithm, pp. 588–592. Problem 24-1, pp. 614–615. Third Edition. MIT Press, 2009. ISBN 978-0-262-53305-8. Section 24.1: The Bellman–Ford algorithm, pp. 651–655.
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| *{{cite book | first1 = George T. | last1 = Heineman | first2 = Gary | last2 = Pollice | first3 = Stanley | last3 = Selkow | title= Algorithms in a Nutshell | publisher=[[O'Reilly Media]] | year=2008 | chapter=Chapter 6: Graph Algorithms | pages = 160–164 | isbn=978-0-596-51624-6 | ref = harv }}
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| *{{cite book|last1=Kleinberg|first1=Jon|author1-link=Jon Kleinberg|last2=Tardos|first2=Éva|author2-link=Éva Tardos|year=2006|title=Algorithm Design|location=New York|publisher=Pearson Education, Inc.|ref=harv}}
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| *{{Cite book|first=Robert |last=Sedgewick|authorlink=Robert Sedgewick (computer scientist)|year=2002|title= Algorithms in Java|edition=3rd |isbn= 0-201-36121-3|chapter=Section 21.7: Negative Edge Weights|url=http://safari.oreilly.com/0201361213/ch21lev1sec7|ref=harv}}
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| ==External links==
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| * [https://github.com/xtaci/algorithms/blob/master/include/bellman_ford.h C++ code example]
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| * [http://code.google.com/p/annas/ Open Source Java Graph package with Bellman-Ford Algorithms]
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| {{DEFAULTSORT:Bellman-Ford algorithm}}
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| [[Category:Graph algorithms]]
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| [[Category:Polynomial-time problems]]
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| [[Category:Articles with example C code]]
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| [[Category:Articles with example pseudocode]]
| |
| [[Category:Dynamic programming]]
| |
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