Floryan These are Hortons version of the slides used in his lecture Network Flow FordFulkerson 1 In your textbook CLRS 261 and 262 Includes simple solutions to the following complications ID: 909832
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Slide1
CS4102 Algorithms Fall 2021 – Horton and Floryan
These are Horton’s version of the slides, used in his lecture.Network Flow, Ford-Fulkerson
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Slide2In your textbookCLRS 26.1 and 26.2Includes simple solutions to the following “complications”What if (u,v) and (
v,u) are in the flow graph?Called “Antiparallel” edges – easy to adjust for this, example laterWhat if we need >1 source? >1 sink?
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Slide3Flow networks3Consider a flow network, which is a specialized directed graph with:A single source node sA single terminus node tCapacities on each edge
That must be integer!What is the maximum flow you can send from s to t?
Slide4Applications4Transportation networksHow many people can be routed?Computer networksElectrical distributionWater distribution
Note that all these applications have multiple sources and multiple sinks!Whereas the flow networks we study do not, yet
Slide5Flow NetworkGraph
Source node
Sink node
Edge Capacities
Positive whole* numbers
If
then
(Note our example here violates this!)
Max flow intuition: If
is a faucet,
is a drain, and
connects to
through a network of pipes with given capacities, what is the maximum amount of water which can flow from the faucet to the drain?
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Slide6Flow Network: Antiparallel EdgesEasy adjustment to remove antiparallel edges and have equivalent flow graph: add intermediate node(Note: our later examples use graph on the left without this adjustment.)
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Flow
Assignment of values to edges
E.g.
units of water going through that pipe
Capacity constraint
Flow cannot exceed capacity
Flow constraint
,
Water going in must match water coming out
Flow of
:
Net outflow of
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Flow/
Capacity
3 in example above
Slide8Let’s Make Some Rules8Source node has NO incoming flowTerminal (sink) node has NO outgoing flow
Internal nodes has net zero flowall units of flow going in must be going out as wellNo edge is over capacityGOAL: Find the maximum flow that can be “pushed” through the networkI.e. maximize:
Slide9How to Solve This? This Greedy doesn’t work9
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Saturate Highest Capacity Path First
Slide10Greedy doesn’t work10
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Saturate Highest Capacity Path First
Slide11Greedy doesn’t work11
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Saturate Highest Capacity Path First
Overall Flow:
Slide12Greedy doesn’t work12
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Better Solution
Overall Flow:
Slide1313
Slide14Ford-Fulkerson: Algorithm overview14Iterative algorithm: push some flow along some path at each stepModel or record the
residual capacitieshow much capacity is left after taking into account how much flow is going through that edge at this timeFind a path from s to t such that the minimum residual capacity of an edge on that path is greater than zeroSince each value is an integer, it must be 1 or more
Update the residual capacities after taking into account this new flow
Repeat until no more such paths are found
Slide15Algorithm notation15f(u,v): the flow on the edge from u to vf(v,u
): the backflow on the edge from v to uc(u,v): the capacity on the edge from u to vcf(u,v): the residual
capacity on the edge from u to v
G
f
is the graph where the edges weights are the residual capacities
THIS is usually the graph we actually use when running the algorithm we are about to see.
Slide16Backflow16Each edge has forward flow and backflowThe two must always be “inverses” of each other!I.e. they sum to the total capacity for that edgeThis allows for modeling of flow “returning” along a given edge
One way to think about this:How much of the forward flow we could “un-do”
Slide17Residual Graph
Keep track of net available flow along each edge
“
Forward edges
”: weight is equal to
available flow
along that edge in the flow graph
“
Back edges
”: weight is equal to
backflow
along that edge in the flow graph
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Flow Graph
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Residual Graph
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Flow I
could
add
Flow I
could
remove
Slide18Residual Graphs Example
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Flow Graph
Residual Graph
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Slide19Ford-Fulkerson Algorithm19
Define an augmenting path to be a path from
in the residual graph
(using edges of non-zero weight)
Overview: Repeatedly add the flow of any augmenting path
Ford-Fulkerson max-flow algorithm:
Initialize
for all
Construct the residual network
While there is an augmenting path
in
:
Let
Add
units of flow to
based on the augmenting path
Update the residual network
for the updated flow
Slide20Define an augmenting path to be a path from
in the residual graph
(using edges of non-zero weight)
Overview: Repeatedly add the flow of any augmenting path
Ford-Fulkerson max-flow algorithm:
Initialize
for all
Construct the residual network
While there is an augmenting path
in
:
Let
Add
units of flow to
based on the augmenting path
Update the residual network
for the updated flow
Ford-Fulkerson Algorithm
20
Ford-Fulkerson approach:
take
any
augmenting path
(will revisit this later)
Slide21Define an augmenting path to be a path from
in the residual graph
(using edges of non-zero weight)
Overview: Repeatedly add the flow of any augmenting path
Ford-Fulkerson max-flow algorithm:
Initialize
for all
Construct the residual network
While there is an augmenting path
in
:
Let
Add
units of flow to
based on the augmenting path
Update the residual network
for the updated flow
Ford-Fulkerson Algorithm
21
(
is the weight of edge
in the residual network
)
Ford-Fulkerson approach:
take
any
augmenting path
(will revisit this later)
Slide22Ford-Fulkerson Algorithm:Updating Gf22
f(u,v) = 0 for all edges (u,v)While there is an “augmenting” path p from s to t in Gf such that
c
f
(
u,v
) > 0 for all edges (
u,v
)
p
Find
c
f
(p) = min{
cf(u,v) | (u,v) p}For each edge (u,v) pf(u,v) = f(u,v) + cf(p) send flow along the pathf(v,u) = f(v,u) - cf(p) send backflow the other way
Slide23Ford-Fulkerson Example23
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Slide24Ford-Fulkerson Example24
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Slide25Ford-Fulkerson Example25
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Slide26Ford-Fulkerson Example26
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Slide27Ford-Fulkerson Example27
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Slide28Ford-Fulkerson Example28
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Slide29Ford-Fulkerson Example29
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Slide30Ford-Fulkerson Example30
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Slide31Ford-Fulkerson Example31
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Slide32Ford-Fulkerson Example32
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Slide33Ford-Fulkerson Example33
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Slide34Ford-Fulkerson Example34
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Slide35Ford-Fulkerson Example35
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Slide36Ford-Fulkerson Example36
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Maximum flow:
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Slide37Our example37
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Slide38Define an augmenting path to be a path from
in the residual graph
(using edges of non-zero weight)
Overview: Repeatedly add the flow of any augmenting path
Ford-Fulkerson max-flow algorithm:
Initialize
for all
Construct the residual network
While there is an augmenting path
in
:
Let
Add
units of flow to
based on the augmenting path
Update the residual network
for the updated flow
Ford-Fulkerson Algorithm - Runtime
38
Time to find an augmenting path:
Number of iterations of While loop:
Slide39Define an augmenting path to be a path from
in the residual graph
(using edges of non-zero weight)
Overview: Repeatedly add the flow of any augmenting path
Ford-Fulkerson max-flow algorithm:
Initialize
for all
Construct the residual network
While there is an augmenting path
in
:
Let
Add
units of flow to
based on the augmenting path
Update the residual network
for the updated flow
Ford-Fulkerson Algorithm - Runtime
39
Time to find an augmenting path:
Number of iterations of While loop:
Slide40What type of search?40“While there is an augmenting path
in ”
Using a depth-first search is the Ford-Fulkerson algorithm
Each augmenting path can be found in O(E) time
And there can be
|𝑓|
paths
So the running time is O
(𝐸⋅|𝑓|)
Will not terminate with irrational edge values
Using a breadth-first search is the Edmonds-Karp algorithm
Runs in O(V
⋅
E
2
)Total number of augmentations is O(V⋅E)And finding each augmentation takes O(E)Guaranteed termination with irrational edge valuesRun-time is independent of the maximum flow of the graph