Maximum Flow Network

7/26/2019 Maximum Flow Network

http://slidepdf.com/reader/full/maximum-flow-network 1/30

Maximum Flow Network

Submitted By:-



Contents:

Maximum Flow Network

Cuts and Flows

Residual Network

Augmenting Path Ford-Fulkerson Method

Matching



Network Flows



Types of Networks Internet

Telephone

Cell

Highways

Rail

Electrical Power

Water

Sewer

Gas

…



Network Flow• Instance:

• A Network is a directed graph G

• Edges represent pipes that carry flow

• Each edge (u,v) has a maximum capacity c(u,v)



I. If (u,v)E, we assume that c(u,v)=0.

II. Two distinct vertices :a source s and a sink t.

Typical example:

Here IEI>=IVI-1; E represents edges and V represents vertices.

ae

f

s

a

t

3/3 2/2

1/4

2/3

2/2

0/2

3/3

2/3

Notation:

Flow/Capacity



G=(V,E): a flow network with capacity function c.

s-- the source and t-- the sink.

A flow in G: a real-valued function f:V*V Rsatisfying the following two properties:

Capacity constraint: For all u,v V,

we require f(u,v) c( u,v).

Flow conservation: For all u V-{s,t}, we require

vine vout e

e f e f .. ..

)()(

Flow:

When (u,v) E, there can be no flow from u to v and f(e)=0



Max Flow of a Network

Max flow is the flow from s-t that maximizes net flow out of the

source.

f

s

a

t

3/3 2/2

1/4

2/3

2/2

0/2

3/3

2/3

g j

Here |f|=f(s,g) + f(s,f) =3+2=5



uts of Flow Networks

Cuts: A cut (S,T)of flow network G=(V,E) is a partition of Vinto S and T=V-S such that sS and tT

11/16

12/1215/20

7/7

4/411/14

4/91/4

8/13



Residual Networks

The residual capacity of an edge (u, v) in a network with a flow f is

given by:

),(),(),( vu f vucvuc f

The residual network of a graph G induced by a flow f is the graph

including only the edges with positive residual capacity, i.e.,

( , ), wheref f

G V E {( , ) : ( , ) 0}f f

E u v V V c u v

The network given by the undirected arcs andresidual capacities is called residual network.



Example of Residual Network

Flow Network:

Residual Network:



Augmenting Paths

Augmenting path=path in the residual network

Given a flow network G=(V,E) and a flow f, an augmenting path P

is a simple path from s to t in residual network Gf .

An augmenting path is a directed path from the source to the sink in

the residual network such that every arc on this path has positive residual

capacity.

The minimum of these residual capacities is called the residual

capacity of the augmenting path.



Example: G =(V,E) be the given graph as shown below find, residual

graph and augmenting path.

Solution:

Original graph G=(V,E)

(a)Flow f (e)

(b)Arc e=(v, w) ϵ E

Flow Network:



S V111/16

Flow

capacity

Residual Graph: Gf =(V, Ef )

(a) Residual arcs e= (V, w) and eR =(W,V)

(b) “Undo” flow sent.

S V15

Residual Capacity

11

Residual Capacity



Therefore, we can represent the residual graph as follows

3

Residual Network:

Augmenting Path

The residual capacity of this augmenting path is 4



Ford-Fulkerson Method

Depends on two ideas-

1. Residual networks

2. Augmented path



Augmenting path

Original Network

Example:

Flow Network

Resulting Flow =4



CSE 310119

Residual Network Flow Network

Resulting Flow =11




Resulting Flow =19




Resulting Flow =23



Residual Network

No augmenting pathMaxflow=23



MAXIMUM BIPARTITE MATCHING



BIPARTITE GRAPH

A Bipartite graph G(V,E) is a graph in which all vertices

are divided into disjoint subsets say L and R, such that

every edge in E, is between a vertex in L and a vertex in R



Suppose we have a set of

people L and set of jobs R.

Each person can do onlysome of the jobs

Can model this as a bipartite

graph

X

u

People Tasks

L R



MATCHING

A matching in a graph is a subset of edges in which no two

edges are adjacent. It may also be an entire graph consisting

of edges without common vertices.



A matching in a graph is a subset M of E , such that for all vertices v in V , at

most one edge of M is incident on v .

BIPARTITE MATCHING



MAXIMUM MATCHING

A maximum matching is a matching with the largest possible

number of edges; it is globally optimal.



So, want a maximum matching: one that contains as many

edges as

possible.



Advantages

Cost is low

Time is redeuced

Disadvantages

• It is an unidirectional network



THANK YOU

Maximum Flow Network

Documents

Transcript of Maximum Flow Network