flow
On a digraph, define a sink to be a vertex without-degree zero and a source to be a vertex within-degree zero.Let be a digraph with non-negative weights and with exactlyone sink and exactly one source.A flow on is an assignmentof values to each edge of satisfying certain rules:
- 1.
For any edge , we must have (where is the weight of ).
- 2.
For any vertex , excluding the source and the sink, let bethe set of edges incident
to and let be the set of edgesincident from .Then we must have
Let be the edges incident from the source, and let bethe set of edges incident to the sink.If is a flow, then
We will refer to this quantity as the amount of flow.
Note that a flow given by trivially satisfies these conditions.We are typically more interested in maximum flows, where the amountof flow is maximized for a particular graph.
We may interpret a flow as a means of transmitting something througha network.Suppose we think of the edges in a graph as pipes, with the weightscorresponding with the capacities of the pipes; we are pouring waterinto the system through the source and draining it through the sink.Then the first rule requires that we do not pump more water througha pipe than is possible, and the second rule requires that any waterentering a junction of pipes must leave.Under this interpretation, the maximum amount of flow corresponds tothe maximum amount of water we could pump through this network.
Instead of water in pipes, one may think of electric charge in a network of conductors.Rule (2) above is one of Kirchoff’s two laws for such networks; the othersays that the sum of the voltage drops around any circuit is zero.
Title | flow |
Canonical name | Flow |
Date of creation | 2013-03-22 13:00:50 |
Last modified on | 2013-03-22 13:00:50 |
Owner | bgins (4516) |
Last modified by | bgins (4516) |
Numerical id | 6 |
Author | bgins (4516) |
Entry type | Definition |
Classification | msc 05C20 |
Classification | msc 94C15 |
Synonym | network flow |
Related topic | MaximumFlowMinimumCutTheorem |
Related topic | MaximumFlowminimumCutTheorem |
Defines | maximum flow |
Defines | source |
Defines | sink |
Defines | Kirchoff’s law |