Maximum Flow CSC 172 SPRING 2002 LECTURE 27
Maximum Flow CSC 172 SPRING 2002 LECTURE 27. Flow Networks Digraph Weights, called capacities, on edges Two distinct veticies Source, “s” (no incoming.
Dec 21, 2015
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Maximum Flow
CSC 172
SPRING 2002
LECTURE 27
Flow Networks
Digraph
Weights, called capacities, on edges
Two distinct veticiesSource, “s” (no incoming edges)
Sink, “t” (no outgoing edges)
Example
Source
s
t
Sink
2
2
2
2
2
3
1
21
1
4
2
Capacity and Flow
Edge Capacities
Non-negative weights on network edges
Flow
function on network edges
0 <= flow <= capacity
flow into vertex == flow out of vertex
Value
combined flow into the sink
Example
Source
s
t
Sink
2
2
2
2
2
3
1
21
1
4
1 21
0 21
0
11
1
1
22
FormallyFlow(u,v) edge(u,v)
Capacity rule edge(u,v)
0 <= flow <= capacity
Conservation rule
uin(v) flow(u,v) = uout(v) flow(v,w)
Value rule
|f| = wout(s) flow(s,w) = uin(t) flow(u,t)
Maximum Flow Problem
Given a network N, find a flow of maximum value
ApplicationsTraffic movement
Hydraulic systems
Electrical circuits
Layout
Example
Source
s
t
Sink
2
2
2
2
2
3
1
21
1
4
2 21
1 11
1
21
0
1
22
Augmenting Path
t
s
2
2
1
12
2
2
12
1
Network with flow 3
Augmenting Path
t
s
2
2
1
12
2
2
12
1
Network with flow 3
t
s
2
2
2
10
2
Augmenting Path
Augmenting Path
t
s
2
2
1
12
2
2
12
1
Network with flow 3
t
s
2
2
2
10
2
Augmenting Path
t
s
2
2
2
12
2
2
02
2Result
Augmenting Path
Forward Edges
flow(u,v) < capacity
we can increase flow
Backward edges
flor(u,v) > 0
flow can be decreased
Max Flow Theorem
A flow has maximum value if and only if it has no augmenting path
Ford & Fulkerson Flow Algorithm
Initialize network with null flow
Method FindFlow
if augmenting path exists then
find augmenting path
increase flow
recursively call FindFlow
Finding Flow
t
s
2
2
0
12
2
0
00
0
Initialize Network
Finding Flow
t
s
2
2
0
12
2
1
11
0
Send one unit of flow through
Finding Flow
t
s
2
2
0
12
2
2
11
1
Send another unit of flow through
Finding Flow
t
s
2
2
1
12
2
2
12
1
Send another unit of flow through
Finding Flow
t
s
2
2
2
12
2
2
02
2
Send another unit of flow through
Note that there is still an augmentingPath that can proceed backwards
Finding Flow
t
s
2
2
2
12
2
2
02
2
Send another unit of flow through
Note that there are no moreAugmenting paths
Residual Network Nf
v
u c(u,v)
f(u,v)
Residual Network Nf
v
u c(u,v)
f(u,v)
v
ucf(u,v)=f(u,v)
cf(u,v)=c(u,v)-f(u,v)
Residual Network
In Nf all edges (w,z) with capacity cf(w,z)=0 are removed
Augmenting Path in N is a direct path in Nf
Augmenting paths can be found by performing a depth-first search on the residual network Nf
Finding Augmenting Path
t
s
2
2
1
12
2
2
12
1
Finding Augmenting Path
t
s
2
2
1
12
2
2
12
1
t
s
1
1
12
2
1
1
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