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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. | |||
==Definition== | |||
<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>: | |||
:{| | |||
| '''Capacity constraints''': || <math>\ f(u,v) \le c(u,v)</math>. The flow along an edge cannot exceed its capacity. | |||
|- | |||
| '''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). | |||
|- | |||
| '''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. | |||
|} | |||
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>. | |||
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. | |||
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> | |||
==Example== | |||
[[File:network flow.png|right|frame|100px|A flow network showing flow and capacity]] | |||
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. | |||
[[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 | |||
<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. | |||
==Applications== | |||
{{see also|Pipe network analysis}} | |||
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. | |||
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]]. | |||
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. | |||
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. | |||
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. | |||
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. | |||
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. | |||
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. | |||
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> | |||
==See also== | |||
* [[Centrality]] | |||
* [[Constructal theory]] | |||
* [[Ford–Fulkerson algorithm]] | |||
* [[Dinic's algorithm]] | |||
* [[Flow (computer networking)]] | |||
* [[Max-flow min-cut theorem]] | |||
* [[Oriented matroid]] | |||
* [[Shortest path problem]] | |||
==References== | |||
{{Reflist}} | |||
== Further reading == | |||
* {{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 }} | |||
* {{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 }} | |||
* {{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}} | |||
* {{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}} | |||
* {{cite book | author=Even, Shimon | title=Graph Algorithms | location=Rockville, Maryland | publisher=Computer Science Press | year=1979 | isbn=0-914894-21-8}} | |||
* {{cite book | author=Gibbons, Alan | title=Algorithmic Graph Theory | location=Cambridge | publisher=Cambridge University Press | year=1985 | isbn=0-521-28881-9 }} | |||
* {{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}} | |||
==External links== | |||
* [http://www-b2.is.tokushima-u.ac.jp/~ikeda/suuri/maxflow/Maxflow.shtml Maximum Flow Problem] | |||
* [http://www.topcoder.com/tc?module=Static&d1=tutorials&d2=maxFlow Maximum Flow] | |||
* [http://www.dis.uniroma1.it/~challenge9/download.shtml Real graph instances] | |||
* [http://www.di.unipi.it/di/groups/optimize/ Software, papers, test graphs, etc.] | |||
* [http://www.avglab.com/andrew/soft.html Software and papers for network flow problems] | |||
* [http://lemon.cs.elte.hu/ Lemon C++ library with several maximum flow and minimum cost circulation algorithms] | |||
* [http://quickgraph.codeplex.com/ QuickGraph], graph data structures and algorithms for .Net | |||
[[Category:Network flow]] | |||
[[Category:Graph algorithms]] | |||
[[Category:Operations research]] | |||
[[Category:Directed graphs]] | |||
Revision as of 05:54, 26 February 2013
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.
Definition
is a finite directed graph in which every edge has a non-negative, real-valued capacity . If , we assume that . We distinguish two vertices: a source and a sink . A flow in a flow network is a real function with the following three properties for all nodes and :
Capacity constraints: . The flow along an edge cannot exceed its capacity. Skew symmetry: . The net flow from to must be the opposite of the net flow from to (see example). Flow conservation: , unless or . The net flow to a node is zero, except for the source, which "produces" flow, and the sink, which "consumes" flow.
Notice that is the net flow from to . If the graph represents a physical network, and if there is a real flow of, for example, 4 units from to , and a real flow of 3 units from to , we have and .
The residual capacity of an edge is . This defines a residual network denoted , giving the amount of available capacity. See that there can be a path from to in the residual network, even though there is no path from to in the original network. Since flows in opposite directions cancel out, decreasing the flow from to is the same as increasing the flow from to . An augmenting path is a path in the residual network, where , , and . A network is at maximum flow if and only if there is no augmenting path in the residual network.
Should one need to model a network with more than one source, a supersource is introduced to the graph.[1] 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.[2]
Example
To the right you see a flow network with source labeled , sink , and four additional nodes. The flow and capacity is denoted . Notice how the network upholds skew symmetry, capacity constraints and flow conservation. The total amount of flow from to is 5, which can be easily seen from the fact that the total outgoing flow from is 5, which is also the incoming flow to . We know that no flow appears or disappears in any of the other nodes.
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 . This flow is not a maximum flow. There is available capacity along the paths , and , which are then the augmenting paths. The residual capacity of the first path is . 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.
Applications
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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.
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.
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.
The simplest and most common problem using flow networks is to find what is called the 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 of minimum capacity that separates the source and the sink.
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.
In a minimum cost flow problem, each edge has a given cost , and the cost of sending the flow across the edge is . The objective is to send a given amount of flow from the source to the sink, at the lowest possible price.
In a circulation problem, you have a lower bound on the edges, in addition to the upper bound . 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 and . The flow circulates through the network, hence the name of the problem.
In a network with gains or generalized network each edge has a 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.
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.[3]
See also
- Centrality
- Constructal theory
- Ford–Fulkerson algorithm
- Dinic's algorithm
- Flow (computer networking)
- Max-flow min-cut theorem
- Oriented matroid
- Shortest path problem
References
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
External links
- Maximum Flow Problem
- Maximum Flow
- Real graph instances
- Software, papers, test graphs, etc.
- Software and papers for network flow problems
- Lemon C++ library with several maximum flow and minimum cost circulation algorithms
- QuickGraph, graph data structures and algorithms for .Net
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The Institute of Property Brokers (IEA) has revealed really helpful commissions/fees for real estate transactions. With the steam of an overheated property market dying down, the excitement has now shifted to developers enjoying discount games to push new initiatives or clear previous stock. With so many ‘great offers', patrons are spoiled for choices. LONDON London mayor Boris Johnson mentioned excessive property costs are "the appropriate drawback to have" and that technology startups are interested in the town regardless of its "creaking" infrastructure. More detail SEOUL A prolonged property market stoop and low rates of interest are threatening the way forward for a uniquely South Korean residence lease system that traces its roots back to the 19th century. More detail Typical Sequence of a Venture Preview
Since 2009 until August 2013, there was eight property cooling measures and 1 automotive mortgage regulation. Here's a abstract of the measures and it's impact until date. Checklist of Property cooling measures by URA/MAS. b. Foreigners and non-individuals (corporate entities) shopping for any residential property pays an ABSD of 10%; c. Everlasting Residents (PRs) proudly owning one and buying the second and subsequent residential property will pay an ABSD of three%; and Anyway, the challenge won't be accomplished till 2019. Who knows what the property market will probably be like five years from now? Amongst others, new rules search to curb non-public property house owners from investing and cashing in on HDB resale flats All residential property loans will now only enable a maximum loan tenor of 35 years.
which runs alongside Holland Road and boasts essentially the most exclusive and costly properties in Singapore, such nearly as good class bungalows and high-finish condos and flats; and Here at New Zealand Property Solutions we purpose to search out the most suitable property investments in New Zealand for our abroad clients based in Singapore, Malaysia and lots of other developed nations. Many Singaporean and worldwide traders may not have the local market understanding to maximise the complete potential of capital gains and constructive cash flow that comes from New Zealand property funding. New Zealand Property Solutions gives the required information to ensure a sound funding, as well the difficult process of discovering the professional contacts required. condo prices in singapore Amenities
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Resolve Whether To Rent Or Buy Shopping for a house isn't only an investment, however a permanent tie to a location. More importantly, it may restrict job alternatives by making you location dependent. In the event you're unsure about whether or not you may be in the same metropolis in 5-8 years, it's best to hire. Watch In Sweden, one needs to get a license from the government so as to paint his own home. In New Zealand, it's unlawful if a cat leaves the home with out having three bells around its neck. Borgnine owned the home for nearly 60 years earlier than he died on the age of ninety five. It hit the real estate market Sept. 2012 for $3.395 million, a couple of months after Borgnine handed away. Posted by Edison Foo December 13, 2013 Open for booking NOW !!! Posted by Edison Foo October 22, 2013
Earlier than you decide to buy a House or Flat it's best to ensure that you have sufficientfunds obtainable to complete the acquisition. If you are financing your buy withyour CPF funds and/or a Mortgage you must contact the CPF Board and the Lender financial institution/financecompany upfront to signal all relevant application kinds and furnish all relevantinformation and paperwork to ensure that the CPF funds and/or Loan would be approvedand the funds will probably be obtainable. Along with the acquisition value you'll haveto make provision for the stamp fees and Lawyer's charges. Residential Property Act.
Do contact us if you want an opinion. We've good contacts and networks with local and foreign banks in Singapore. And we are in a position to provide you data and market talk from the bottom up. Singaporeans have traditionally been averse to wealth redistribution, partly due to the idea that focussing on equalising life opportunities is sufficient. When property taxes have been abolished in 2008, Singapore turned one of the few international locations that does not have capital gains (including property) or estate taxes. Some fear that property duties may result in capital flight. But rich individuals choose Singapore for many reasons, including business convenience and household security. Belgravia Villas Cluster Home @ Ang Mo Kio Open for Sale 19September Types of Residential Property - ↑ http://www.pedropinto.org.s3.amazonaws.com/publications/locating_source_diffusion_networks.pdf