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| In [[graph theory]], a '''flow network''' (also known as a '''transportation network''') is a [[directed graph]] where each edge has a '''capacity''' and each edge receives a flow. The amount of flow on an edge cannot exceed the capacity of the edge. Often in [[Operations Research]], a directed graph is called a '''network''', the vertices are called '''nodes''' and the edges are called '''arcs'''. A flow must satisfy the restriction that the amount of flow into a node equals the amount of flow out of it, except when it is a '''source''', which has more outgoing flow, or '''sink''', which has more incoming flow. A network can be used to model traffic in a road system, fluids in pipes, currents in an electrical circuit, or anything similar in which something travels through a network of nodes. | | In front of enjoying a brand most recent clash of clans crack tool, see the trick book. Most flash games possess a book you can buy individually. You might want to think about doing specific and studying it anyone decide to play, or even you should playing. In it manner, you can obtain the most out of your game play.<br><br>Video gaming are fun to play with your kids. This helps you learn much more details your kid's interests. Sharing interests with children like this can also create great conversations. It also gives you an opportunity to monitor growth and development of their skills.<br><br>clash of clans is a ideal game, which usually requires one to build your personal village, discover warriors, raid tools and build your own clan and so 4th. there is a lot a lot additionally to this video game and for every one of these you require jewels returning to play, as you which include. If you have any sort of inquiries regarding where and the best ways to utilize [http://circuspartypanama.com clash of clans hack free download], you could call us at the website. Clash of Clans hack allows you to get as many jewels as you desire. There is an unlimited amount of gems you could generate with all the Battle of Clans cheats you can buy online, however you need to be specific about the hyperlink you are using for the reason some of them but waste materials your as well as also dont get the individual anything more.<br><br>Guilds and clans have been recently popular ever since the primary beginning of first-person supply shooter and MMORPG game playing. World of WarCraft develops to the concept with their very own World associated Warcraft guilds. A real guild can easily always be understood as a having to do with players that band to the floor for companionship. People in the guild travel together again for fun and adventure while [http://browse.deviantart.com/?qh=§ion=&global=1&q=improving improving] in challenge and gold.<br><br>Several ways you can variety of participants what [http://Www.Dict.cc/?s=people+perform people perform] Clash of Clans across the world which offers you the chance which can crew up with clans that have been in players from different countries around the world and can also sit competitive towards other clans. This will make the game considerably more captivating as you will find a great deal of a number of strategies that might be applied by participants and this fact boosts the unpredictability half. Getting the right strategy november 23 is where the performer's skills are tested, although the game is simple perform and understand.<br><br>You will notice for yourself that our Money Compromise of Clans i fairly effective, 100 % invisible by the supervisor of the game, above all absolutely no price!<br><br>Now that you have read this composition, you need to have an easier time locating then loving video games inside your life. Notwithstanding your favored platform, from your cellphone to your own computer, playing furthermore enjoying video gaming allows for you to take the advantage of the worries of your busy week get details. |
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| ==Definition==
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| <math>G(V,E)</math> is a finite [[directed graph]] in which every [[edge (graph theory)|edge]] <math>\ (u,v) \in E</math> has a non-negative, real-valued capacity <math>\ c(u,v)</math>. If <math>\ (u, v) \not \in E</math>, we assume that <math>\ c(u, v) = 0</math>. We distinguish two vertices: a source <math>\ s</math> and a sink <math>\ t</math>. A flow in a flow network is a [[real number|real]] [[function (mathematics)|function]] <math>\ f:V \times V \rightarrow \mathbb{R}</math> with the following three properties for all nodes <math>\ u</math> and <math>\ v</math>: | |
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| :{|
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| | '''Capacity constraints''': || <math>\ f(u,v) \le c(u,v)</math>. The flow along an edge cannot exceed its capacity.
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| | '''Skew symmetry''': || <math>\ f(u,v) = - f(v,u)</math>. The net flow from <math>\ u</math> to <math>\ v</math> must be the opposite of the net flow from <math>\ v</math> to <math>\ u</math> (see example).
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| | '''Flow conservation''': || <math>\ \sum_{w \in V} f(u,w) = 0</math>, unless <math>\ u=s</math> or <math>\ u=t</math>. The net flow to a node is zero, except for the source, which "produces" flow, and the sink, which "consumes" flow.
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| Notice that <math>\ f(u,v)</math> is the ''net'' flow from <math>\ u</math> to <math>\ v</math>. If the graph represents a physical network, and if there is a real flow of, for example, 4 units from <math>\ u</math> to <math>\ v</math>, and a real flow of 3 units from <math>\ v</math> to <math>\ u</math>, we have <math>\ f(u,v)=1</math> and <math>\ f(v,u)=-1</math>.
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| The '''residual capacity''' of an edge is <math>\ c_f(u,v) = c(u,v) - f(u,v)</math>. This defines a '''residual network''' denoted <math>\ G_f(V,E_f)</math>, giving the amount of ''available'' capacity. See that there can be a path from <math>\ u</math> to <math>\ v</math> in the residual network, even though there is no path from <math>\ u</math> to <math>\ v</math> in the original network. Since flows in opposite directions cancel out, ''decreasing'' the flow from <math>\ v</math> to <math>\ u</math> is the same as ''increasing'' the flow from <math>\ u</math> to <math>\ v</math>. An '''augmenting path''' is a path <math>\ (u_1,u_2,\dots,u_k)</math> in the residual network, where <math>\ u_1=s</math>, <math>\ u_k=t</math>, and <math>\ c_f(u_i, u_{i+1}) > 0</math>. A network is at maximum flow if and only if there is no augmenting path in the residual network.
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| Should one need to model a network with more than one source, a '''supersource''' is introduced to the graph.<ref>{{DADS|Supersource|supersource}}</ref> This consists of a vertex connected to each of the sources with edges of infinite capacity, so as to act as a global source. A similar construct for sinks is called a '''supersink'''.<ref>{{DADS|Supersink|supersink}}</ref>
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| ==Example==
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| [[File:network flow.png|right|frame|100px|A flow network showing flow and capacity]]
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| To the right you see a flow network with source labeled <math>s</math>, sink <math>t</math>, and four additional nodes. The flow and capacity is denoted <math>f/c</math>. Notice how the network upholds skew symmetry, capacity constraints and flow conservation. The total amount of flow from <math>s</math> to <math>t</math> is 5, which can be easily seen from the fact that the total outgoing flow from <math>s</math> is 5, which is also the incoming flow to <math>t</math>. We know that no flow appears or disappears in any of the other nodes.
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| [[File:Network flow residual SVG.svg|right|thumb|332px|Residual network for the above flow network, showing residual capacities]] | |
| Below you see the residual network for the given flow. Notice how there is positive residual capacity on some edges where the original capacity is zero, for example for the edge <math>(d,c)</math>. This flow is not a [[max flow|maximum flow]]. There is available capacity along the paths <math>(s,a,c,t)</math>, <math>(s,a,b,d,t)</math> and <math>(s,a,b,d,c,t)</math>, which are then the augmenting paths. The residual capacity of the first path is
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| <math>\min(c(s,a)-f(s,a), c(a,c)-f(a,c), c(c,t)-f(c,t))</math> <math>= \min(5-3, 3-2, 2-1) = \min(2, 1, 1) = 1</math>. Notice that as long as there exists some path with a positive residual capacity, the flow will not be maximum. The residual capacity for some path is the minimum residual capacity of all edges in that path.
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| ==Applications== | |
| {{see also|Pipe network analysis}}
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| Picture a series of water pipes, fitting into a network. Each pipe is of a certain diameter, so it can only maintain a flow of a certain amount of water. Anywhere that pipes meet, the total amount of water coming into that junction must be equal to the amount going out, otherwise we would quickly run out of water, or we would have a build up of water. We have a water inlet, which is the source, and an outlet, the sink. A flow would then be one possible way for water to get from source to sink so that the total amount of water coming out of the outlet is consistent. Intuitively, the total flow of a network is the rate at which water comes out of the outlet.
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| Flows can pertain to people or material over transportation networks, or to electricity over [[electrical distribution]] systems. For any such physical network, the flow coming into any intermediate node needs to equal the flow going out of that node. This conservation constraint was formalized as [[Kirchhoff's current law]].
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| Flow networks also find applications in [[ecology]]: flow networks arise naturally when considering the flow of nutrients and energy between different organizations in a [[food web]]. The mathematical problems associated with such networks are quite different from those that arise in networks of fluid or traffic flow. The field of ecosystem network analysis, developed by [[Robert Ulanowicz]] and others, involves using concepts from [[information theory]] and [[thermodynamics]] to study the evolution of these networks over time.
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| The simplest and most common problem using flow networks is to find what is called the [[maximum flow problem|maximum flow]], which provides the largest possible total flow from the source to the sink in a given graph. There are many other problems which can be solved using max flow algorithms, if they are appropriately modeled as flow networks, such as [[bipartite matching]], the [[assignment problem]] and the [[transportation problem]]. Maximum flow problems can be solved efficiently with the [[Relabel-to-front_algorithm]]. The [[max-flow min-cut theorem]] states that finding a maximal network flow is equivalent to finding a [[Cut (graph theory)|cut]] of minimum capacity that separates the source and the sink.
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| In a [[multi-commodity flow problem]], you have multiple sources and sinks, and various "commodities" which are to flow from a given source to a given sink. This could be for example various goods that are produced at various factories, and are to be delivered to various given customers through the ''same'' transportation network.
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| In a [[minimum cost flow problem]], each edge <math>u,v</math> has a given cost <math>k(u,v)</math>, and the cost of sending the flow <math>f(u,v)</math> across the edge is <math>f(u,v) \cdot k(u,v)</math>. The objective is to send a given amount of flow from the source to the sink, at the lowest possible price.
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| In a [[circulation problem]], you have a lower bound <math>l(u,v)</math> on the edges, in addition to the upper bound <math>c(u,v)</math>. Each edge also has a cost. Often, flow conservation holds for ''all'' nodes in a circulation problem, and there is a connection from the sink back to the source. In this way, you can dictate the total flow with <math>l(t,s)</math> and <math>c(t,s)</math>. The flow ''circulates'' through the network, hence the name of the problem.
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| In a '''network with gains''' or '''generalized network''' each edge has a '''[[gain graph|gain]]''', a real number (not zero) such that, if the edge has gain ''g'', and an amount ''x'' flows into the edge at its tail, then an amount ''gx'' flows out at the head.
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| In a '''source localization problem''', an algorithm tries to identify the most likely source node of information diffusion through a partially observed network. This can be done in linear time for trees and cubic time for arbitrary networks and has applications ranging from tracking mobile phone users to identifying the originating village of disease outbreaks.<ref>http://www.pedropinto.org.s3.amazonaws.com/publications/locating_source_diffusion_networks.pdf</ref>
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| ==See also==
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| * [[Centrality]]
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| * [[Constructal theory]]
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| * [[Ford–Fulkerson algorithm]]
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| * [[Dinic's algorithm]]
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| * [[Flow (computer networking)]]
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| * [[Max-flow min-cut theorem]]
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| * [[Oriented matroid]]
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| * [[Shortest path problem]]
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| ==References==
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| {{Reflist}}
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| == Further reading ==
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| * {{cite book | author=George T. Heineman, Gary Pollice, and Stanley Selkow | title= Algorithms in a Nutshell | publisher=[[Oreilly Media]] | year=2008 | chapter=Chapter 8:Network Flow Algorithms | pages = 226–250 | isbn=978-0-596-51624-6 }}
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| * {{cite book | author=[[Ravindra K. Ahuja]], [[Thomas L. Magnanti]], and [[James B. Orlin]] | title= Network Flows: Theory, Algorithms and Applications | publisher=Prentice Hall | year=1993 | isbn=0-13-617549-X }}
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| * {{cite book | author=[[Béla Bollobás|Bollobás, Béla]] | title= Graph Theory: An Introductory Course | location=Heidelberg | publisher=Springer-Verlag | year=1979 | isbn=3-540-90399-2}}
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| * {{cite book | author=[[Gary Theodore Chartrand|Chartrand, Gary]] & Oellermann, Ortrud R. | title=Applied and Algorithmic Graph Theory | location=New York | publisher=McGraw-Hill | year=1993 | isbn=0-07-557101-3}}
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| * {{cite book | author=Even, Shimon | title=Graph Algorithms | location=Rockville, Maryland | publisher=Computer Science Press | year=1979 | isbn=0-914894-21-8}}
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| * {{cite book | author=Gibbons, Alan | title=Algorithmic Graph Theory | location=Cambridge | publisher=Cambridge University Press | year=1985 | isbn=0-521-28881-9 }}
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| * {{cite book | author = [[Thomas H. Cormen]], [[Charles E. Leiserson]], [[Ronald L. Rivest]], and [[Clifford Stein]] | title = [[Introduction to Algorithms]] | origyear = 1990 | edition = 2nd | year = 2001 | publisher = MIT Press and McGraw-Hill | pages = 696–697 | chapter = 26 | isbn = 0-262-03293-7}}
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| ==External links==
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| * [http://www-b2.is.tokushima-u.ac.jp/~ikeda/suuri/maxflow/Maxflow.shtml Maximum Flow Problem]
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| * [http://www.topcoder.com/tc?module=Static&d1=tutorials&d2=maxFlow Maximum Flow]
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| * [http://www.dis.uniroma1.it/~challenge9/download.shtml Real graph instances]
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| * [http://www.di.unipi.it/di/groups/optimize/ Software, papers, test graphs, etc.]
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| * [http://www.avglab.com/andrew/soft.html Software and papers for network flow problems]
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| * [http://lemon.cs.elte.hu/ Lemon C++ library with several maximum flow and minimum cost circulation algorithms]
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| * [http://quickgraph.codeplex.com/ QuickGraph], graph data structures and algorithms for .Net
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| [[Category:Network flow]]
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| [[Category:Graph algorithms]]
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| [[Category:Operations research]]
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| [[Category:Directed graphs]]
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