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## Dinitz's algorithm for finding a maximum flow in a network.

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Title: Dinitz's algorithm for finding a maximum flow in a network.

1
Dinitz's algorithm for finding a maximum flow
in a network.
• Presented by
• Ilan Kadar and Sivan Albagli

June 30, 2020
2
Content
• Introduction
• Ford and Fulkerson Algorithm
• Original Dinitz algorithm

2
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IntroductionThe max flow problem definition
• Capacitated directed graph G(V,E,c,s,t).
• The capacities are non-negative.

2
1
3
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IntroductionThe max flow problem definition
• A flow f, is defined as a function on the
directed edges satisfying the following
• Capacity constraint ?e?E 0?f(e)?c(e)
• Flow conservation ?v?V\s,t ?(v,u)?Ef(v,u) -
?(u,v)?Ef(u,v)0

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4
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IntroductionThe max flow problem definition
• The goal is to find the maximum flow from the
source to the sink.

Q How do we know that the flow is maximum?
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IntroductionMotivation
• Finding the maximal way to ship goods from a set
of factories to a set of stores.
• Bipartite matching

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IntroductionNaïve algorithm
A
2
2
S
t
2
4
3
B
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IntroductionNaïve algorithm
A
2
2
S
t
2
4
3
B
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8
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IntroductionNaïve algorithm
A
2/2
0/2
S
t
2/2
2/4
0/3
B
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IntroductionNaïve algorithm
A
2/2
0/2
S
t
2/2
2/4
0/3
B
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IntroductionNaïve algorithm
There is no path where one can increase the flow
Therefore, the algorithm terminates
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IntroductionNaïve algorithm
Is it the maximum flow?
A
2/2
0/2
(Flow value 4)
S
t
2/2
4/4
2/3
B
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IntroductionNaïve algorithm
Maximum flow 5
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IntroductionEquivalent flow function
• For each edge add ,
• if doesnt exists, with capacity and flow
zero.

0
0
5
0
1
8
source
sink
s
t
1
2
0
2
G
0
0
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IntroductionEquivalent flow function
• Define for each edge

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IntroductionEquivalent flow function
v
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IntroductionEquivalent flow function
• satisfy the following constraints
• Capacity constraint

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IntroductionEquivalent flow function
• satisfy the following constraints
• 2. Skew symmetry

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IntroductionEquivalent flow function
• satisfy the following constraints
• 3. Flow conservation

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IntroductionEquivalent flow function
• The net flow from v is defined as
• The flow value is defined as the net flow from s

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IntroductionEquivalent flow function
Edge is called saturated if Residual capacity (possibility to add flow)
1st representation
2nd representation
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IntroductionEquivalent flow function
Edge is called saturated if
saturated
saturated
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IntroductionEquivalent flow function
Unsaturated edges
unsaturated
unsaturated
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IntroductionEquivalent flow function
Residual capacity (possibility to add flow)
cf 5-324
cf
cf 5-14
cf
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IntroductionEquivalent flow function
We will use the second representation,
denoting it by f
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Ford and FulkersonAlgorithm
Define Gf(V,Ef) the residual network. Ef
- the unsaturated edges with capacity cf(u,v)c(
u,v)-f(u,v)gt0.
2
1
-1/0
-2/0
1
-1/0
1
0/0
-1/0
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Ford and FulkersonAlgorithm
A path is an augmenting path if it contains only
unsaturated edges. for example
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Ford and FulkersonAlgorithm
FordFulkerson(G,s,t) 1. initialize flow
f(v,u) to 0 2. while there exists an
augmenting path P do 3. augment flow
f along P 4. return f
Published in 1956
When no augmenting path exists, the current flow
is maximum (Ford Fulkerson Theorem).
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Ford and FulkersonAlgorithm
FordFulkerson(G,s,t) 1. for each edge
(u,v)? Ef do 2. f(u,v)?0 3.
f(v,u)?0 4. while there exists a path P from s
to t in the Gf do 5. cf(P) min
e?Pcf(e)gt0 6. for each edge (u,v) in P
do 7. f(u,v) ? f(u,v) cf(P)
8. f(v,u) ? f(u,v) - cf(P)
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Ford and FulkersonAlgorithm
Comments Notice that it is not certain that
E Ef, as sending flow on (u,v) might close
(u,v) (it is saturated), and may open a new edge
(v,u) in the residual network.
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Ford and FulkersonExample
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Ford and FulkersonExample
First iteration
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Ford and FulkersonExample
First iteration continue
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Ford and FulkersonExample
First iteration continue
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Ford and FulkersonExample
Second iteration
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Ford and FulkersonExample
Second iteration-continue
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Ford and FulkersonExample
There is no path at the residual network Gf The
flow is maximal
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Ford and FulkersonTime analysis
• With rational capacities, the algorithm will
always terminate.
• With irrational capacities, the algorithm may run
forever.
• Runtime is bounded by O(Ef) when the
capacities are integers.

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Ford and FulkersonProblem 1 (Integer case)
• The algorithm runs in pseudo polynomial time

How many iterations??
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Ford and FulkersonProblem 1 (Integer case)
• The algorithm runs in pseudo polynomial time

How many iterations??
2,000,000
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Ford and FulkersonProblem 2 (General case)
• Convergence isnt guaranteed

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Ford and FulkersonOpen question?
• The question of the existence of a polynomial, or
finite algorithm, for the general case, remained
open, for almost 10 years.
• This was settled by Edmonds and Karp and by Y.
Dinitz independently, at the late 60s.
• Both algorithms are modifications of FF.

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Ford and FulkersonEdmonds-Karp(1972)
• A small fix to the Ford-Fulkerson algorithm makes
it work in polynomial time.
• Compute the augmenting path using BFS on the
residual network.
• Run in time O(VE2)

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Dinitz AlgorithmA Historical Remark
• The DA was invented in response to a
Algorithm class.

At that time, the author was not aware the basic
facts regarding FF
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Dinitz AlgorithmMotivation
• All parts of an iteration of FF except of finding
an augmenting path P cost O(P).
• Finding an augmenting path is the bottleneck of
an iteration O(E).
• Improving FF by using a smart data structure.

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Dinitz AlgorithmMotivation
• Example BFS Tree

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Dinitz AlgorithmMotivation
• Example BFS Tree

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Dinitz AlgorithmMotivation
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Dinitz AlgorithmMotivation
s and t become disconnected in the BFS tree!
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Dinitz AlgorithmExtended BFS
Dist dist from s
Dist2
Dist3
Dist4
Dist1
Dist3
Dist2
Dist3
Dist1
Dist2
Extended BFS includes the first edge found,
leading to each vertex from the previous layer.
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Dinitz AlgorithmLayered Network L(s)
V4
V0
V3
V1
V2
Vi is the set of all nodes with distance i from
s Ei is the set of all edges going from Vi-1 to
Vi
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Dinitz AlgorithmLayered Network L(s)
V4
V0
V3
V1
V2
L(s) (?Vi , ?Ei) the union of all shortest
paths from s in the graph.
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Dinitz AlgorithmLayered Network Definitions
• Vi The i layer dist(v)i ? v?Vi
• Ei Edges from Vi -1 to Vi
• L(s) (?Vi , ?Ei) the union of all
shortest paths from s in the graph.

Running time of regular BFS Running time of
extended BFS
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Dinitz AlgorithmLayered Network
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Dinitz AlgorithmLayered Network next step
Goal we want to maintain our data structure as
the union of all the shortest paths from s to t.
How can it be done? Prune into Run
extended BFS from t on L(s), in the opposite edge
direction
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Dinitz AlgorithmLayered Network next step
How can it be done? Prune into
L(s) in the opposite edge direction
Then run BFS from t on L(s)
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Dinitz AlgorithmLayered Network next step
How can it be done? Prune into
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Dinitz AlgorithmLayered Network
• Claim
• is the union of all the shortest paths
from s to t(Invariant).
• Property of
• Doesn't have vertices with no any incoming edge,
except s.
• Doesnt have vertices with no any outgoing edge,
except t.

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Dinitz AlgorithmAugmenting path finding
• Just walk from s over . After l steps
t is reached.

1
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2
2
1
1
1
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Dinitz AlgorithmAugmenting path finding
• Remove the saturated edges.

1
1
2
2
1
1
1
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Dinitz Algorithm Augmenting path finding
• Just walk from s over . After l steps
t is reached

1
1
1
1
1
1
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• We might have vertices without outgoing edges
• We wish to remove them, so the next augmenting
path finding from s wont get stuck and will work
in O(l) time.

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• Initialize two queues Ql and Qr by the list of
saturated edges sat that were removed.
• Define two procedures
• Left pass
• Right Pass

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Dinitz AlgorithmCleaning dead ends Left Pass
1
1
2
2
1
1
1
(d,t)
Ql
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Dinitz Algorithm Cleaning dead ends Left Pass
1
1
2
2
1
(c,d)
1
(a,d)
Ql
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Dinitz Algorithm Cleaning dead ends Left Pass
1
1
2
1
(s,c)
Ql
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Dinitz Algorithm Cleaning dead ends Left Pass
1
1
2
(s,c)
Ql
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• After applying LeftPass all the vertices(except
t) have outgoing edge.
• After applying RightPass all the vertices(except
s) have incoming edge.
• Why we need both procedures???

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• No path in the layered network from s to t could
be removed.
• Next augmenting path of length l can be found in
O(l) time incase the cleaned layer is not
vanished.

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Dinitz AlgorithmThe original algorithm
Phase
Iteration
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Dinitz AlgorithmThe original algorithm
Constructing of the layered network using two
BFS-es, at the beginning of a phase costs
O(E).
Phase
Iteration
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Dinitz AlgorithmThe original algorithm
Phase
Augmenting path finding using a walk from s over
the layered network costs O(l)O(V) every time
thanks to the cleaning maintenance .
Iteration
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Dinitz AlgorithmThe original algorithm
Phase
Flow updating costs O(l)O(V) as well.
Iteration
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Dinitz AlgorithmThe original algorithm
Phase
Iteration
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Dinitz AlgorithmThe original algorithm
Phase
In practice a single iteration might cost
O(EV) because we should remove every edge
and vertex from the graph.
Iteration
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Dinitz AlgorithmTime analysis
• Cost of the algorithm

O(iterations V Phases E)
E
V
How many Iterations? How many Phases?
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Dinitz AlgorithmTime analysis
• Intuition After any iteration, there is no
augmenting path of length less then l.

s
t
Augmenting path
New path
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Dinitz AlgorithmTime analysis
• If there are paths of length l they are contained
in

s
t
Augmenting path
New path
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Dinitz AlgorithmTime analysis
• In the end of the phase(when the layer vanish),
doesnt contain paths of length l.
• We can prove that the residual network doesnt
contain paths of length l.
• Conclusion The length of the layered network
grows from phase to phase.

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Dinitz AlgorithmTime analysis
- In every iteration at least one edge is being
removed (min edge of path augmenting). - At
most E edges are being removed during a phase
? There are at most E iterations during a
phase.
Phase
Iteration
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Dinitz Algorithm Time analysis
Constructing of the layered network
O(E) Path finding and capacity updating
O(l)O(V). It is done at most E times during
a phase. Maintenance of the layered network
O(E). So, a phase costs O(EEVE)O(E
V).
Phase
Phase
Iteration
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Dinitz AlgorithmTime analysis
- Distance(s,t)?V-1 ? There are at most V-1
phases in DA
Phase
Iteration
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Dinitz AlgorithmTime analysis
DA running time is O(EVV) O(V2E)
Phase
Iteration
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Further Progress
Time Method Discover Year
O(V E2) Shortest Path Edmonds and Karp 1972
O(V3) Preflow-push Karzanovs Algorithm 1974
Dynamic trees Sleator-Tarjan 1983
FIFO preflow-push Goldberg-Tarjan 1986
Length function Goldberg-Rao 1997