Announcements

1
Announcements

• Network Flow today, depending on how far we get,
  we will do String Matching on Wednesday
• Then Review for Final next Monday
• This weeks lab will be review of Network Flow
  and other material.

2
Network Flow

• The Maximum Flow Problem
• and
• The Ford-Fulkerson Algorithm

3
Types of Networks

• Internet
• Telephone
• Cell
• Highways
• Rail
• Electrical Power
• Water
• Sewer
• Gas

4
Maximum Flow Problem

• How can we maximize the flow in a network from a
  source or set of sources to a destination of set
  of destinations?
• The problem reportedly rose to prominence in
  relation to the rail networks of the Soviet
  Union, during the 1950's. The US wanted to know
  how quickly the Soviet Union could get supplies
  through its rail network to its satellite states
  in Eastern Europe.
• In addition, the US wanted to know which rails it
  could destroy most easily to cut off the
  satellite states from the rest of the Soviet
  Union.
• It turned out that these two problems were
  closely related, and that solving the max flow
  problem also solves the min cut problem of
  figuring out the cheapest way to cut off the
  Soviet Union from its satellites.
• The first efficient algorithm for finding the
  maximum flow was conceived by two Computer
  Scientists, named Ford and Fulkerson. The
  algorithm was subsequently named the
  Ford-Fulkerson algorithm, and is one of the more
  famous algorithms in computer science.

Source lbackstrom, The Importance of
Algorithms, at www.topcoder.com
5
Network Flow

• A Network is a directed graph G
• Edges represent pipes that carry flow
• Each edge ltu,vgt has a maximum capacity cltu,vgt
• A source node s in which flow arrives
• A sink node t out which flow leaves

Goal Max Flow
6
Network Flow

• The network flow problem is as follows
• Given a connected directed graph G
• with non-negative integer weights,
• (where each edge stands for the capacity of that
  edge),
• 2 different vertices, s and t, called the source
  and the sink,
• such that the source only has out-edges and the
  sink only has in-edges,
• Find the maximum amount of some commodity that
  can flow through the network from source to sink.

12
a
b
20
16
9
4
10
7
s
t
4
c
d
14
Each edge stands for the capacity of that edge.
7
Network Flow

• One way to imagine the situation is imagining
  each edge as a pipe that allows a certain flow of
  a liquid per second.
• The source is where the liquid is pouring from,
  and the sink is where it ends up.
• Each edge weight specifies the maximal amount of
  liquid that can flow through that pipe per
  second.
• Given that information, what is the most liquid
  that can flow from source to sink per second, in
  the steady state?

12
a
b
20
16
9
4
10
7
s
t
4
c
d
14
Each edge stands for the capacity of that edge.
8
Network Flow
12
a
b
12/12
a
b
20
16
19/20
12/16
0/9
9
4
10
7
s
t
0/4
0/10
s
t
7/7
13
4
4/4
11/13
c
d
c
d
14
11/14
This graph contains the capacities of each edge
in the graph.
Here is an example of a flow in the graph.

• The flow of the network is defined as the flow
  from the source, or into the sink.
• For the situation above, the network flow is 23.

9
12
12/12
a
b
a
b
20
19/20
16
12/16
0/9
9
4
0/4
10
7
0/10
s
t
s
t
7/7
13
4
4/4
11/13
c
d
c
d
14
11/14
capacities
flow

• The Conservation Rule
• In order for the assignment of flows to be valid,
  we must have the sum of flow coming into a vertex
  equal to the flow coming out of a vertex, for
  each vertex in the graph except the source and
  the sink.
• The Capacity Rule
• Also, each flow must be less than or equal to the
  capacity of the edge.
• The flow of the network is defined as the flow
  from the source, or into the sink.
• For the situation above, the network flow is 23.

10
Network Flow

• In order to determine the maximum flow of a
  network, we will use the following terms
• Residual capacity is simply an edges unused
  capacity.
• Initially none of the capacities will have been
  used, so all of the residual capacities will be
  just the original capacity.

0/12
a
b
0/20
0/16
Using the notation used / capacity. Residual
Capacity capacity - used.
0/9
0/4
0/10
s
t
0/7
0/4
0/13
c
d
0/14
11
Network Flow

• Residual capacity of a path the minimum of the
  residual capacities of the edges on that path,
  which will end up being the max excess flow we
  can push down that path.
• Augmenting path defined as one where you have a
  path from the source to the sink where every edge
  has a non-zero residual capacity.

0/12
a
b
0/20
0/16
Using the notation used / unused. Residual
Capacity unused - used.
0/9
0/4
0/10
s
t
0/7
0/4
0/13
c
d
0/14
12
Ford-Fulkerson Algorithm

• While there exists an augmenting path
• Add the appropriate flow to that augmenting path
• So were going to arbitrarily choose an the
  augmenting path s,c,d,t in the graph below
• And add the flow to that path.

• Residual capacity of a path the minimum of the
  residual capacities of the edges on that path.

0/12
a
b
0/20
0/16
0/9

• 4 in this case, which is the limiting factor for
  this paths flow.

0/4
0/10
s
t
0/7
4/4
0/4
4/13
0/13
c
d
4/14
0/14
13
Ford-Fulkerson Algorithm

• While there exists an augmenting path
• Add the appropriate flow to that augmenting path
• Choose another augmenting path (one where you
  have a path from the source to the sink where
  every edge has a non-zero residual capacity.)
• s,a,b,t

• Residual capacity of a path the minimum of the
  residual capacities of the edges on that path.

12/12
0/12
a
b
12/20
0/20
12/16
0/16
0/9

• 12 in this case, which is the limiting factor for
  this paths flow.

0/4
0/10
s
t
0/7
4/4
4/13
c
d
4/14
14
Ford-Fulkerson Algorithm

• While there exists an augmenting path
• Add the appropriate flow to that augmenting path
• Choose another augmenting path (one where you
  have a path from the source to the sink where
  every edge has a non-zero residual capacity.)
• s,c, d, b, t

• Residual capacity of a path the minimum of the
  residual capacities of the edges on that path.

12/12
a
b
12/20
12/16
19/20
0/9

• 7 in this case, which is the limiting factor for
  this paths flow.

0/4
0/10
s
t
0/7
7/7
4/4
4/13
11/13
c
d
4/14
11/14
15
Ford-Fulkerson Algorithm

• While there exists an augmenting path
• Add the appropriate flow to that augmenting path
• Are there any more augmenting paths?
• No! Were done
• The maximum flow 19 4 23

12/12
a
b
19/20
12/16
0/9
0/4
0/10
s
t
7/7
4/4
11/13
c
d
11/14
16
Network Flow

• Another example of the Ford Fulkerson Algorithm
  on the board

17
Cuts of Flow Networks
18
The Net Flow through a Cut (S,T)

• f(S,T) 12 4 11 19

19
The Capacity of a Cut (S,T)

• c(S,T) 12 0 14 26

20
Network Flow

• Value of the cut, example on the board

21
Augmenting Paths example

• The maximum possible flow through the cut 12
  7 4 23

Flow(2)
cut
The network has a capacity of at most 23. This
is called a minimum cut.
22
Net Flow of a Network

• The net flow across any cut is the same and equal
  to the flow of the network f.

23
Bounding the Network Flow

• The value of any flow f in a flow network G is
  bounded from above by the capacity of any cut of
  G.

24
Max-Flow Min-Cut Theorem

• If f is a flow in a flow network G(V,E), with
  source s and sink t, then the following
  conditions are equivalent
• f is a maximum flow in G.
• The residual network Gf contains no augmented
  paths.
• f f(S,T) for some cut (S,T) (a min-cut).

25
Example

• In a department there are n courses and m
  instructors.
• Every instructor has a list of courses he or she
  can teach.
• Every instructor can teach at most 3 courses
  during a year.
• The goal find an allocation of courses to the
  instructors subject to these constraints.

26
Network Flow Optional Homework Problem

• An orchestra is trying to fill all of its
  positions. For different instruments, it needs a
  different number of players.
• For example, it might need 20 violinists but only
  3 trumpet players.
• Each potential candidate for the orchestra can
  play some set of instruments.
• For example, one player might be able to play
  either a violin or a viola, while another might
  be able to play a trumpet, trombone or tuba.
• Given the following information
• A list of how many of each position the orchestra
  needs.
• A list of each candidate and which instruments
  they can play.
• Determine whether or not the orchestra can fill
  all its positions.
• Describe how to solve this problem using network
  flow.

27
References

• Slides adapted from Arup Guhas Computer Science
  II Lecture notes http//www.cs.ucf.edu/dmarino/
  ucf/cop3503/lectures/
• Additional material from the textbook
• Data Structures and Algorithm Analysis in Java
  (Second Edition) by Mark Allen Weiss
• J Elders Network Flow slides, York University