# netflow

Document Sample

```					                                                                                                                                 JN 2006.05
Flow in transport networks

3                   2                                                              3/3               2/2
s: source, start
3                                t: sink, terminate
s                                             t                                           s               1/ 3                   t
weights >0: arc capacity
f / w: flow and capacity
1                   3                                                              1/1               2/3

Df. transport network:
G = ( V, E, w, s, t ), E ! V x V , e = u-v with u ! v (no loops), w: E " R+ (positive reals), s, t # V , s ! t

Df. flow f : E " R+ =[ 0 .. \$ )
a) obey capacity: for all e ! E, 0 " f(e) " c(e)
b) conservation of flow (Kirchhoff law): for all v # V except s and t: in-flow = out-flow
Introduce a feedback arc t-s of capacity w( t-s ) = \$
Df. value | f | of flow f: | f | = f( t-s )                                   s                             t

Problem: construct a flow f from s to t of maximum value | f |

Linear programming, linear optimization: maximize f = %u f( s-u) = %u f( u-t ) subject to 0 " f(e) " c(e)

Df. cut (S, T ) = a partition of V: S & T = { } , S ' T = V, s # S, t # T

Intuition: cut as a set of edges: cut ( S, T ) = { e = u-v / u # S, v # T }

2                                                                    3                2
3
s                  3                   t
s                        3                  t
1                3
1                     3
4                                       5
cuts and their capacities
6                                   6
3                   2                                                                   3                2

s                      3                                                                      s               3                      t
t
3                                                                   1                    3
1

Df. capacity w( S, T ) of cut (S, T ) = % w( u-v ), summed over all u # S , v # T

Df. flow f( S, T ) across cut( S, T) = % f( u-v ) - % f( v-u ), summed over all u # S , v # T

Max-flow min-cut theorem ( Ford & Fulkerson 1956):
In a transportation network the maximum value | f | over all flows f equals
the minimum value w( S, T ) over all cuts( S, T)

The concept of an augmenting path.

0/ 2                  the flow at left                                      0/ 2
3/ 3
leaves the
s                      3/ 3                   t       augmenting path             s                  3/ 3            t
shown at right
0/1                     3/ 3                  of capacity 1                       0/1

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 6 posted: 11/29/2010 language: Catalan pages: 1
How are you planning on using Docstoc?