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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
A FAST AXD SIMPLE ALGORITHM
FOR THE NLAXIMUM FLOW PROBLEM
R. K.
Ahuja
and
1.
B.
Sioan W.F. No. 19J5-S7
Orlin
June 1987
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
OCT 201987
)
A FAST AXD SIMPLE ALGORITHM
FOR THE MAXIMUM FLOW PROBLEM
R. K.
Ahuja
and
].
B.
Sioan W.F. Xo. 1905-S7
Orhn
June 1987
A
and Simple
Algorithm for the Maximum Flow Problem
Fast
Ravindra K. Ahuja* and James
B.
Orlin
Sloan School of Management
Massachusetts Institute of Technology
Cambridge, MA. 02139, USA
Abstract
We
maximum
among
that
n-^
0(nm
present a simple
log
is
n).
log
n-^
U) sequential algorithm
m
flow problem with n nodes,
arcs directed
U
+
from the source node.
polynomially bounded in n
,
Under
,
dei se networks without using any
by
complex data
of
n
of
for
in
time
0(nm
0(nm
On
A
Fast and Simple Algorithm for the
non-sparse and non-
structures.
leave from Indian
Institute
of Technology,
Maximum
Kanpur
-
+
log (n^/m),
Subject Classification
484.
U
the practical assumption
bound
a factor of log
bound
a capacity
our algorithm runs
This result improves the current best
obtained by Goldberg and Tarjan
and
arcs,
for the
Flo^v
Problem
208 016
,
INDI.A
The maximum flow problem
in
is
one of the most fundamental problems
network flow theory and has been investigated extensively. This problem
was
first
their
formulated by Ford and Fulkerson [1956]
who
also solved
well-known augmenting path algorithm. Since then
a
it
number
using
of
algorithms have been developed for this problem, as tabulated below. Here n
is
the
number
m
of nodes,
on the integral arc
the
number
of arcs, and
U
is
an upper bound
capacities.
Due
#
is
Running Time
to
0(nm U)
1
Ford and Fulkerson
2
Edmonds and Karp
3
Dinic [1970]
O(n^)
4
Karzanov [1974]
OCn^)
5
Cherkaskyll977]
©(n^m^/^)
6
Malhotra,
7
Galil[1978]
8
Galil and
9
Shiloach and Vishkin
10
Sleator and Taijan [1983]
0(nm
11
Tarjan[1984]
OCn-"*)
12
Gabow[1985]
O(nmlogU)
13
Goldberg [1985]
OCn^)
14
Goldberg and Tarjan [1986]
0(nm
from source
in
[1978]
O(n^)
0(n5/3m2.'3)
Naamad
0(nm
[1978]
showed
if
log^ n)
OCn^)
[1982]
[1972]
that the
log n)
log (n^/m))
Ford and Fulkerson [1956]
flows are augmented along shortest paths
Independently, Dinic [1970] introduced the concept of
shortest path networks, called
algorithm. This
0(nm2)
[1972]
time 0(nm-^)
to sink.
956]
Kumar and Maheshwari
Edmonds and Karp
algorithm runs
[ 1
layered networks,
bound was improved
to
and obtained an O(n-m)
O(n^) by Karzanov [1974]
who
introduced the concept of preflows in a layered network.
to a
flow except that the amount flowing into a node
flowing out of a node.
may
A preflow
is
similar
exceed the amount
Since then, researchers have improved the complexity
of Dinic's algorithm for sparse networks by devising sophisticated data
structures.
tree is the
Among
most
these contributions,
attractive
The algorithms
were
a
of
from
and Tarjan's
dynamic
[1983]
a worst case point of view.
Goldberg [1985] and of Goldberg and Tarjan [1986]
novel departure from these approaches in the sense that they did not
construct layered networks.
method but does
It
not maintain layered networks. Their algorithm maintains
To estimate which nodes
label for
contains the essence of Karzanov's preflow
and proceeds by pushing flows
a preflow
sink.
Sleator
each node that
augmenting path
is
a
to
nodes estimated
are closer to the sink,
it
to
be closer
to the
maintains a distance
lower bound on the length of a shortest
to the sink.
Distance labels are a better computational device
than layered networks since these labels are simpler to understand, easier to
manipulate, and easier to use in a parallel algorithm.
implementing the dynamic
tree data structure,
Moreover, by cleverly
Goldberg and Tarjan obtained
the best computational complexity for sparse as well as dense networks.
Bertsekas [1986]
problem
recently developed an algorithm for the
that simultaneously generalizes both the
and Bertsekas
[1979] auction algorithm for the
minimum
Goldberg-Tarjan algorithm
assignment problem.
For problems with arc capacities polynomially bounded in n
maximum
flow algorithm
is
minimum
,
our
an improvement of Goldberg and Tarjan's
algorithm and uses concepts of scaling introduced bv
[1972] for the
cost flow
cost flow
problem and
later
Edmonds and Karp
extended by
Gabow
[1985]
for other
network optimization problems. The bottleneck operation
in the
straightforward implementation of Goldberg and Tarjan's algorithm
number
0(nm
the computational time to
the
dynamic
tree data structure.
pushes can be reduced
U
the excesses to
nodes with
become
is
U
(i.e.,
= OCn*^^^^)
<
that the
number
of
of non-saturating
scaling."
Our
each scaling iteration requires
flows from nodes with sufficiently
while never allowing
sufficiently small excesses
Consequently, the computational time of
Under
polynomial
in n)
the reasonable assumption that
the algorithm runs in time
,
<
1
.
log n for both non-sparse
a factor of
networks
E
we push
+ n^ log U).
it is
show
However, they reduce
.
+ rr log n) which improves the bound of Goldberg and Tarjan's
algorithm by
i.e.,
if
too large.
0(nm
our algorithm
We
the
by a very clever application
log (n^/m))
scaling iterations;
O(n^) non-saturating pushes
large excesses to
0(n-^)
is
0(n^ log U) by using "excess
to
algorithm performs log
0(nm
which
of "non-saturating pushes"
is
for
m
which
=
Moreover, our algorithm
more
efficient in practice), since
with
little
is
model
is
is
with
e
much
requires only elementary data structures
arc capacities are exponentially large numbers.
arguablv inappropriate.
in
number
operations take
of
bit
Odog U)
which
It is
of computation (as described
counts the
some
easy to implement (and
computationally attractive even
uniform model of computation,
steps,
is
for
computational overhead.
Our algorithm
and the
it
m= Q (n^"^^)
and
OCn^-"^^)
and non-dense networks,
all
more
if
is
not
0(n'^'^^)
In this case, the
arithmetic operations take
realistic to
In this
0(1)
adopt the logarithmic
by Cook and Reckhow
operations.
U
[1973])
which
model, most arithmetic
steps than 0(1) steps.
Using the logarithmic model of computation and modifying our
algorithm slightly to speed up arithmetic operations on large integers
,
we
.
obtain an
0(nm
log n
+ n- log n log U) algorithm for the
maximum
flow-
problem. The corresponding time bound for the Goldberg-Tarjan algorithm
0(nm
is
the non-dense
c:
our algorithm
i?
U becomes
as
exponentially large, our
T^oldberg and Tarjan's algorithm
algorithm surpa
this
Hence
log (n^/m) log U).
by
m/n
a factor of
in
n other words, the relative worst case performance of
"kingly
U increases
superior as
in size.
Our
on
results
logarithmic moaei of computation will be presented in Ahuja and Orlin
[19S7a]
Our maximum flow algorithm
since
we push
flow from one node
parallel processors
U
assumption that
is
comparable
Vishkin [19S2]
we
=
is
make "massively
difficult to
at a time.
Nevertheless, with d
==
parallel"
f
m/n1
can obtain an 0(n^ log Ud) time bound. Under the
©(n"^'^^)
,
the algorithm runs in 0(n- log n) time,
to the best available
which
time bounds obtained by Shiloach and
and Goldberg and Tarjan
[1986] using
?:
processors.
;7flrfl//f/
Thus, our algorithm has an advantage in situations for which parallel
processors are at a premium.
Our work on
presented in Ahuja and Orlin [19S7b]
.
Notation
1.
Let
G
for even,' arc
t
the parallel algorithms will be
are
= (N, A) be a directed network with a positive integer capacity
(i, j)
and
e A. Let n = INI
two distinguished nodes
m= A
I
of the network.
I
.
We
The source
that for everv arc
(i, j)
possibly with zero capacity. Let
(s,
A
e
,
an arc
U = max
j)6
A
(j, i)
(UgJ.
is
and
assume without
generality that the network does not contain multiple arcs.
assume
s
We
also contained in
sink
loss of
further
A
,
Ujj
A
flow
S
a function
is
X
-
x^i
A— R
x:
Xjj
I
(j, t)
{j:
some v >
V
A
is
{j:
forall
N-
e
i
(1)
{s, t},
(2)
,
e A)
Xjj
<
u-j
for all
,
is
(i, j)
to
e
A
(3)
,
determine a flow x
X
is
a function
A}
{j;
x:
A— R
>
which
satisfies
(2)
,
(3),
and
(1):
I
-
Xji
Xjj
>
,
for all
i
e
N-
(4)
{s, t}.
A}
(i,j)e
The algorithms described
in this
paper maintain a preflow
at
each
intermediate stage.
For
a
given preflow x
,
we
define for each node
excess
e,
for
maximized.
the following relaxation of
(j,i)€
,
The maximum flow problem
.
-prcfloiv
I
=
= V
x^t
<
which
satisfying
{j:(i,j)GA}
{j:(i,i)6A}
for
>
=
I
{j:
(j.i)e
Xjj
A}
I
{j:
(i,j)e
Xjj
A}
i
g
N-
(s, t]
,
the
A
node with positive excess
residual capacity of
any arc
=
-
positive residue
>
given by
rj:
illustrates thest
x^:
(i, j)
+ x-
cities is
tions.
e
.
A
is
referred to as an
active node
.
The
with respect to a given preflow x
The network consisting only
referred to as the residual network.
is
of arcs with
Figure
1
Network
vsnth arc capacities.
Node
1
is
sink.
(Arcs
Node 4 is the
zero capacities are not
the source.
v»-ith
shewn)
Network with
a preflow x
The residual network with
residual arc capacities
Figure
1.
Illustrations of preflow
and residual network.
A distance
function
N—
d:
>
preflow x
for a
Z"^
function from the set of nodes to the non-negative integers
distance function
CI.
d(t
C2.
d(i;
is
valid
')
also satisfies the following
if it
+
1
,
for
Our algorithm maintains
and
is
a
labels each
This fact
for each
from
i
to
i,
is
A
with
(3, 1, 2, 0)
1(c),
d =
(0
say
,
,
,
minimum
we
)
at
.
each iteration,
The distance
to
t
d(i)
in the residual
d(i).
length of any path
the distance label exact.
call
is
the
tliat
a valid distance label,
though
represents the exact distance labels.
define the arc adjacency
list is
equals the
d(i)
residual netw^ork, then
directed out of the node
adjacency
>
r-
d(i).
i
We
.
two conditions:
vaUd distance function
a
a
easy to demonstrate using induction in the value
the distance label
in the
t
We
list
e
with the corresponding value
i
For example, in Figure
d =
(i, j)
lower-bound on the length of any path from node
network.
If
node
every arc
is
i
,
i.e.,
A(i) of a
list
A(i)
:
=
{(i,
a set of arcs rather than the
node
k) e
A
:
i
e
N
k e N}.
as the set of arcs
Note
more conventional
that
our
definition of a
as a set of nodes.
All logarithms in this paper are
otherwise.
assumed
to
be of base 2 unless stated
10
The Preflow-Push Algorithms
2.
The
preflozv-push algorithms for the
maintain a preflow
at
flow problem
every step and proceed by pushing the excess of nodes
The
closer to the sink.
maximum
first
preflow-push algorithm
is
due
to
Karzanov
[1974].
Tarjan [1984] has suggested a simplified version of this algorithm. The recent
algorithms of Goldberg [1985] and Goldberg and Tarjan [1986] are based on
ideas similar to those presented in Tarjan [1984] but use distance labels to
direct flows closer to the sink instead of constructing layered network.
thus
call their
this section,
algorithm the distance-directed preflow-push algorithm.
we
we
of brevity,
We
review the basic features of
shall
simply refer
to as the
In
algorithm, which for the sake
this
we
preflow-push algorithm. Here
describe the 1-phase version of the preflow-push algorithm presented by
Goldberg
[1987].
The
results in this section are
due
to
Goldberg and Tarjan
[1986].
All operations of the preflow-push algorithm are
performed using
only local information. At each iteration of the algorithm (except
initialization
node,
i.e.,
and
at
the termination)
a (non-source)
iterative step
is
to
If
the network contains at least one active
node with positive
excess.
choose some active node and
the sink, with closer being judged
excess flow at this
with respect
node cannot be sent
to
to
The goal of each
send
its
when
the
excess "closer" to
to the current distance labels.
nodes with smaller distance
then the method increases the distance label of the node.
terminates
at the
labels,
The algorithm
network contains no active nodes. The preflow-push
algorithm uses the following subroutines:
11
PRE-PROCESS. On
d(s) =
n
and
(Alternatively,
t.
d(t)
the
=
(s,
,
d(i)
send
=
units of flow. Let
u^:
for each
1
distance label for each node
^^
i
iit
s,
on the residual network
t
s
or
an active node
PUSH(i).
Select
an arc
(i, j)
i ^^
e A(i)
s,
with
min
We
say that a push of flow on arc
{e,
,
r^:
non-saturating
}
units of flow
r^j
is
0)
.
can be
starting at
.
and
>
from node
(i, j)
>
Cj
(i.e.,
t
6 =
RELABEL(i).
to
i
d(i)
=
d(j)
+
5 =
rjj
1.
j.
saturating
if
,
otherwise.
Replace
d(i)
by min{ dQ) +1:
(i
,
j)
g A(i)
and
r,:
>
Figure 2 contains the generic version of the preflow-push algorithm.
begin
t
)
Select
and
e A(s)
Let
.
SELECT.
Send
j)
a breadth first search
detern^int
node
each arc
}.
12
and
PUSH(i)
Figure 3 illustrates the steps
RELABEL(i)
as applied to
the network in Figure 1(a). Figure 3(a) specifies the preflow determined
The SELECT step
PRE-PROCESS.
has residual capacity
(2, 4)
=
r24
node 2
selects
1
for examination.
d(2) = d(4) +
and
1
,
by
Since arc
the algorithm
The push
performs a (saturating) push of value 5 = min{
1, 1}
reduces the excess of node 2
deleted from the residual
network and arc
an active node,
it
is
Hence
condition.
1
added
Arc
.
(2, 4)
is
to the residual
network. Since node 2
can be selected again for further pushes. The arcs
have positive residual
(2, 1)
new
(4, 2)
to
units.
min
{d(3) +
1
,
d(l) +
1)
=
min
{2, 5}
= 2
The pre-process step accomplishes several important
causes the nodes adjacent to
some node with
started.)
s
to
of the
minimum
is
distances in
d
guaranteed that
path from
t
,
there
are non-decreasing (see
in
s
.
tasks.
we
First,
it
can select
to
is
s
is
,
a
the feasibilit}' of setting
lower bound on the length
no path from
Lemma
1
s
to follow),
to
we
t
.
Since
are also
subsequent iterations the residual network will never
contain a directed path from
push flow from
to
s
again.
s
to
a
(Otherwise, the algorithm could not get
positive excess.
immediate. Third, since d(s) = n
n
and
node 2
gives
have positive excess, so that
Second, by saturating arcs incident
d(s) =
(2, 3)
satisfy the distance
RELABEL(2) ,and
the algorithm performs
distance d'(2) =
do not
capacities, but they
is still
t,
and so there can never be any need
to
13
d(3) =
l
d(l) = 4
(a)
d(4) =
Thv
.'ork
with a preflow after the pre-processing step.
d(3) =
l
d(l) = 4
d(4) =
After the execution of step
(b)
d(3) =
PUSH(2
1
d(l) = 4
d(4) =
dt2) = 2
=
e
1
-I
(c)
Fisurt
3.
After the execution of step RELABEL(2).
Illustriitioiis
of Push
and
Relabel
steps.
14
In our
improvement
the results given in Goldberg and Tarjan [1986].
proofs in order to
Lemma
make
we need
of the preflow-push algorithm,
more
this presentation
We
some
include
Moreover,
labels at each step.
at
few of
of their
self-contained.
The generic preflow-push algorithm maintains
1.
a
valid distance
each relabel step the distance label of a node
strictly increases.
First note that the pre-process step constructs valid distance
Proof.
labels.
Assume
operation,
inducti\
el}^
that the distance function
operation on the arc
may
(i, j)
an additional condition
d(j)
create an additional arc
<
d(i)
conditions remain satisfied since
A
operation.
push operation on
residual network, but this
During
relabel step, the
a
and
(i, j)
e A(i)
The
relabel step
d(i)
=
d(i)
< min{d(i) +
d(j)
+
1
>
r-
is
and
0)
+
d(i)
arc
needs
1
=
d(j)
(i, j)
new
l:
>
be
i)
A
push
with
satisfied.
r-
>
,
and
This validity
by the property of the push
might delete
this arc
from the
is
node
is
i
d'(i)
= min{d(j) +
consistent with the validity conditions.
is
no
arc
(i, j)
€ A(i)
with
Hence,
.
(i, j)
1
distance label of
performed when there
r-:
+
to
(j,
.
does not affect the validity of the distance function.
which
,
and C2
CI
satisfies the validity conditions
i.e.,
valid prior to an
is
e A(i)
and
r-
>
0}
=
d'(i)
,
thereby proving the
second part of the lemma.
Lemma
2.
At any stage of the preflow-push algorithm, each node with
positive excess
network.
is
connected
to
node
s
by
a directed path in the residual
1:
15
Proof.
By the flow decomposition theory
any preflow
x can be
decomposed with respect
into non-negative flows along
nodes
to active
pi'
active
node
in the
flow decc
and
(iii)
How
x of
in th
the flows
G
tion of x
is a
path from
Corollary
i
e
N
time that node
last
there
must be
the reversal of
in the residual
fact that
Corollary
d^^
i
was
relabeled,
it
= n and condition C2 imply that
The number
2.
one, and bv Corollarv
Corollarv
Suppose
d(i)
=
d(j)
sent back from
j
+
to
i
change cannot occur
arc
(i, j)
are both in
m
is less
of saturating pushes
(at
.
)
(i, j)
which
until
d'(j)
A
,
K
so the
(P with the
be an
from
s to
i
and hence
G',
is
most n-
1
n -
1
2n'^
=
d'(i)
at
number
most n
+
by
1
>
d(i)
+
1
(Recall that
=
and the
total
we assume
of arcs in the residual
times.
.
iteration (at
until
(i, j)
d(j)
nm
at least
+
flow
number
that
network
(i, j)
is
is
This flow
2);
Thus by Corollary
at least 2.
times,
most 2n
at
some
on
to
i
< 2n.
node by
no more than
at
from
.
label of a
sent
and
a positive excess,
at
becomes saturated
increases
d(j)
no more than nm.
is
P
had
than
Then no more flow can be
can become saturated
saturations
than
of relabel steps
that arc
1).
path P
no node can be relabeled more than
The number
3.
Proof.
which
1
a
dj < dg +
Each relabel step increases the distance
Proof.
i
s
< 2n.
d(i)
,
hence the residual network contained a path of length
The
Let
cycles.
network
G
paths from
(ii)
t,
,
in G'.
s
For each node
1.
The
Proof.
to
i
is
to
s
around directed
Thus
Then
.
orientation of each arc reversed)
there
.
[1962]
to the original net^vork
paths from source
(i)
and Fulkerson
of Ford
1
of arc
and
(j,
no more
i)
s.
.
16
Lemma
The number
3.
non-saturating pushes
of
See Goldberg and Tarjan [1986]
Proof.
Lemma
When
Proof.
maximum
a
the algorithm terminates, each
so the final preflow
excess;
labels satisfy the conditions
is
a feasible flow.
CI and C2
Zn-^m.
.
The algorithm terminates with
4.
most
is at
and
node
flow.
in
N
-
{s, t)
Further, since the distance
d(s)
= n
it
,
follows
termination the residual network contains no directed path from
This condition
is
the classical termination criteria for the
algorithm of Ford and Fulkerson [1962]
The
to
pushes
upon
to
s
maximum
t
.
flow
.
potential bottleneck operation in the preflow based algorithms
Karzanov
number
has zero
[1974]
,
Tarjan [1984], and Goldberg and Tarjan [1986]
of non-saturating pushes,
for the following reason:
which dominates the number
The saturating pushes cause
is
due
the
of saturating
structural
changes - they delete saturated arcs from the residual network. This
bound
pushes —no matter
which order they are performed. The non-saturating
in
of
0(nm) on
number
observation leads to a
the
of saturating
pushes do not change the structure of the residual network and seem more
difficult to
0(n-^)
[1986]
bound.
Goldberg
by examining the nodes
reduced the non-saturating pushes
in a specific order,
to
and Goldberg and Tarjan
reduced the average time per non-saturating push by using a
sophisticated data structure.
we
[1985]
In the next section
can dramatically reduce the
0(n2 log U)
number
we show
that
by using
of non-saturating pushes to
scaling,
17
The Scaling Algorithm
3.
Our maximum flow algorithm improves
algorithm of Section
and uses "excess
2,
non-saturating pushes from O(n-m)
push flow from
active
nodes with
sufficiently small excesses
power
iteration
is
of 2
and
satisfies
units of flow
we
K
= flog U1 +
ej
<A
is
;
that after at
we
obtain a
maximum
K
LIST(r)
lists
:
=
e
N
:
ej
too large.
e
N
Further
.
,
new
a
A decreases by
scaling
a factor of 2.
A/2
we
,
To ensure
increase.
label
most 4n-
that
consider only nodes
we
excess,
.
non-saturating pushes, the
at least 2
scaling iterations,
all
and
a
new
scaling iteration
node excesses drop
to
zero and
flow.
distance label
(i
nodes with
and among these nodes with large
In order to select an active
minimum
i
at least
excess-dominator decreases by a factor of
most
to
defined to be the least integer A that
for all
node with minimum distance
begins. After at
is
scaling iterations. For a
1
and the excess-dominator does not
with excess more than A/2
We show
basic idea
of
assure that each non-saturating push sends at least
each non-saturating push has value
select a
The
.
sufficiently large excesses to
considered to have begun whenever
In a scaling iteration
A/2
0{rr log U)
to
scaling iteration, the excess-dominator
a
number
scaling" to reduce the
while never letting the excesses become
The algorithm performs
is
the generic preflow-push
node with excess more than A/2 and with
among such
> A/2
and
nodes,
d(i)
=
r)
we
maintain the
for each r =
1,
.
a
lists
.
.
,
n-1
.
These
can bo maintained in the form of either a linked stack or a linked queue
(sec, for
example, Aho, Hopcroft and Ullman
[1974]),
which enables insertion
18
and deletion
smallest index
for
r
which LIST(r)
As per Goldberg and
Tarjan,
select efficiently the eligible arc for
with each node
arranged
The variable
of elements in 0(1) time.)
i
a
A(i)
list,
arbitrarily,
,
non-empty.
we
use the following data structure
pushing flow out of
of arcs directed out of
first
arc in
the current arc
made
is
its
arc
list.
found
to
list is
we
say that the next arc
is
This
node.
We
Arcs in each
maintain
list
has a current arc
i
again the
is
list is
i.
Initially, the
(i, j)
which
can be
null.
first
When
At
arc in
the arc
arc
the
i
examined sequentially and whenever
list is
this point, the
its
is
current arc of node
be ineligible for pushing flow, the next arc
the current arc.
arc
current arc
it.
a
to
but the order once decided remains unchanged
current candidate for pushing flow out of
the
the
is
throughout the algorithm. Each node
is
level indicates
completely examined,
node
is
relabeled
list.
The algorithm can be described formally
in the
as follows:
and the
19
MAX-FLOW;
procedure
begin
PRE-FROCESS;
K: =
for
'
log
k =
U1
to
;
K do
begin
A =
2^^-'^
for each
level
:
=
while
if
1
N
e
i
do
if
> A/2
e,
then add
i
;
level <
2n do
LIST(level) = o
then
level:
= level +
1
else
begin
select a
node
i
from LIST(level);
PUSH/RELABEL(i)
end;
end;
end;
to LIST(d(i));
20
procedure PUSH/RELABEL(i);
begin
found: = false
let
(i, j)
;
be the current arc of node
while found =
if
d(i)
else
if
=
+
d(j)
and
1
^ null do
(i, j)
r-
then found: = true
>
replace the current arc of
found = true
begin
and
false
i;
node
i
by the next arc
(i, j);
on arc
(i, j)
;
then delete node
i
then
{push}
push min{ej
,
rjj
A - ej
,
units of flow
update residual capacities and the excesses;
if
(the
updated excess)
from
if
;t
j
or
s
< A/2
e^
,
LIST(d(i));
and
t
add node
(the
j
to
updated excess)
LIST(d(i))
and
e;
> A/2
set level:
,
then
= level -
end
else
begin
(relabel)
delete
node
update
d(i):
add node
the
end;
end:
i
node be
from
i
LIST(d(i));
= min{d(j) +
to
LIST(d(i))
1
;
(i, j)
and
let
the first arc of A(i);
e A(i) and r- >
0}
the current arc oi
;
1;
21
Complexity of the Algorithm
4.
In this section
,
we show
algorithm with scaling correctly computes a
0(nm
+
Lemma
n-^
log
maximum
ithm
following two conditions:
satisfies the
Each non-saturating push from
C3.
least
Ko
C4.
flow in
ne.
I"
The
5.
preflow-push
that the distance-directed
A/2
node
a
node
to a
i
j
sends
at
units of flow.
A
excess increases above
(i.e.,
.
the excess-dominator does
not increase subsequent to a push.)
For ever)' push on arc
Proof.
node
is
i
a
node with smallest distance
than A/2, and
d(j)
by sending min
{e^
in a non-saturating
Further,
the
subsequent
All
we have ej>A/2 and
(i, j)
=
,
d(i)
r-
,
-
1
A-
<
e;
by the property
d(i)
)
among nodes whose
label
> min {A/2
push the algorithm sends
push operation increases
to the push.
Then
node excesses thus remain
e':
e;
= e.+
less
r-)
,
only
from
a
A/2
min
{
Let
.
A/2
,
e':
r-
than or equal to A
the properties stated in the conditions
,
A/2
excess
,
since
more
is
push operation. Hence
units of flow,
at least
While there are other ways of ensuring
satisfies
of the
<
e:
we
ensure that
units of flow.
be the excess
A-
e;
}
at
node
< e.+ A -
e;
.
that the algorithm
is a
simple and efficient
always
C3 and C4, pushing flow
With properties C3 and C4,
the
kind of "restrained greedy approach."
way
of enforcing these conditions.
push operation may be viewed
as a
Property C3 ensures that the push
j
<A
node with excess greater than A/2 and with minimum distance
among such nodes
,
.
.
9?
from
to
i
maximum
iteration.
sufficiently large to
restraint so as to prevent
excess
is
Keeping the
be distributed
should make
if
a
number
node
much
it
maximum
easier for
of nodes
6.
If
Its
nodes
were
to
,
it
major impact
to
d(i)
is
possible
the
useful in
"encourage" flow excesses
send flow towards the
send flow
to a single
node
j
sink.
it is
,
(Alternatively,
likely that
able to send the accumulated flow directly to the sink;
node
each push
value of F
i
increase to A.)
is to
j
would have
satisfies
to
conditions
be returned from where
is
is
bounded by 2n
.
at
most
^
following two cases must apply:
came.)
Srr-
ej d(i)
.
N
bounded by 2n- A because
When
it
and C4, then the number
C3
Consider the potential function F =
initial
if
C4 may be very
i€
The
(It is
node
network. This distribution of flows
of non-saturating pushes per scaling iteration
Proof.
A.
excess,
In particular,
may
that the
during an
all its
from exceeding
excess as low as in
fairly equally in the
of the excess of
Lemma
e:
between A and A/2
would not be
j
never exceeds A
excess increases during a push.
practice as well as in theory.
to
C4 ensures
Propert}'
effective.
In particular, rather than greedily getting rid of
maximum
maximum
be
infeasibility of flow conservation
shows some
that the
is
j
ej
the algorithm examines
is
bounded by A and
node
i
,
one of the
23
Case
The algorithm
I.
pushed. In
this case
no
arc
is
(i, j)
1e
bounded by
5:
Since the
bounded by
2n,
2n2A
Case
i
le
satisfies
goes up by
distance label of
i
unable to find an arc along which flow can be
initial
5 >
d(i)
1
units.
value of
increases in F
due
=
d(i)
d(j)
+
1
and
The increase
and
in
and the
>
F
thus
is
increases are
all its
to relabelings of
Tj:
nodes are bounded by
.
The algorithm
2.
pushed and so
it
is
able to identify an arc on
performs either a saturating or
A/2 units of flow to a
decrease in F of
node with smaller distance
at least
A/2 units.
F cannot exceed
increases in
4n-^
A
Since the
,
this
(A more careful analysis leads
Theorem
a non-saturating push.
F decreases. Moreover, each non-saturating push sends
either case,
times.
which flow can be
with a corresponding
more than
case cannot occur
to a
bound
at least
value of F plus the
initial
The procedure Max-Flow performs
1.
label
In
of 4n'^
©(n-^ log
8n-^
)
.
U) non-saturating
pushes.
The
Proof.
Lemma
5
,
initial
value of the excess-dominator A
the value of the excess-dominator decreases
8n- non-saturating pushes and a
U1
for
such scaling iterations,
all
Lemma
i
e
4
is 2'"^°S
N-
{s, t}.
must be
a
A <
1
new
and by the
flow.
By
"1
.
a factor of 2
scaling iteration begins.
After
1
within
+ flog
integrality of the flows e^ =
The algorithm thus obtains
maximum
by
^ < 2U
a feasible flow,
which by
24
Theorem
0(nm
The complexity of
2.
+ n^ log U)
flow algorithm
of the algorithm
depends upon the number
executions of the while loop in the main program.
PUSH/RELABEL(i)
either a
PUSH/RELABEL(i)
A
1.
push
is
results in
Case
performed or the value of the variable
one of the following outcomes:
the non-saturating pushes are Oirr log
0(nm
The distance
2.
is
+ n^ log U)
label of
node
label requires
i
goes up. By Corollary
examining arcs
examined 0(n)
increases
is
Thus the algorithm
The
the
number
After
i
I
A(i)
effort
A(i).
to find
an arc
to
number
of
the
Since each arc in
all
this
outcome
new
A
distance
is
of the 0(n-^) distance
total effort
needed
is
0(nm
+ n^ log U)
i
is
is
Zj
0(1) plus
replaced by the next arc in A(i).
Case 2 occurs and distance
i.
iG
plus the
this case
perform the push operation
node
such replacements for node
goes up. Thus, the
Computing
2,
procedure PUSH/RELABEL(i)
of times the current arc of
I
,
.
calls the
needed
in
i.
times, the total effort for
0(nm)
U)
times.
can occur 0(n) times for each node
times.
In each such execution
performed. Since the number of saturating pushes are
0(nm) and
occurs
step
of
Each execution of the procedure
level increases.
Case
is
.
The complexity
Proof:
maximum
the
2n lA(i)l
= 0(nm) times
N
PUSIi/RELABEL(i) operations. This
label of
is
clearly
node
25
0(nm
+ n- log U). The
lists
LIST(i) are stored as linked stacks
hence addition and deletion of any element takes Oil) time.
updating these
the
number
of increases of the variable
bounded above by 2n -
1
level
decreases by
number
5,
can decrease only
1.
Hence
its
bounded by
when
a
increases over
the
push
all
which
of pushes plus (2n log U),
is
Hence
1.
number
its
number
performed and
0(nm
again
level is
of
of decreases plus 2n
scaling iterations are
is
bound
+
n-^
in
improve
a practical matter, several modifications of the
its
actual execution time without affecting
such a case
bounded by
log U)
its
algorithm might
worst case complexity.
1.
Modify the
2.
Allow some non-saturating pushes of small amount.
3.
Try
scale factor.
to locate
nodes disconnected from the
in the present
sink.
form uses a scaling factor of
2,
i.e., it
reduces the excess-dominator by a factor 2 in the consecutive scaling
iterations.
However,
in practice
it
might be better
dominator by some other fixed integer factor
be the
least
power
of
and property C3 becomes
P that
is
no
less
P >
to scale the excess2.
The excess-dominator
than the excess flow
at
any node,
it
the
.
characterize the modifications as one of the following three types:
The algorithm
will
.
Refinements
As
We
is
to
In each scaling iteration,
and bounded below by
increases per scaling iteration
Further,
level.
Consequently,
we need
not a bottleneck operation. Finally,
lists is
and queues,
26
Each non-saturating push from a node
C3'.
i
node
to a
j
sends
at least
A/p units of flow.
The procedure maxflow can
by
factor
letting LIST(j) =
0(nm
run in
+
(i:
logo U)
pn-^
any fixed value of P
is
> A/p
d(i)
time.
be altered
easily
to incorporate the P scale
The algorithm can be shown
}.
Clearly,
to
from the worst case point of view
optimum. The best choice of the value of
P in
practice should be determined empirically.
Our algorithm
as stated selects a
saturating or a non-saturating push.
flow out of
A/2
this
or reduce
node
its
until either
node with
We
ej
every non-saturating push.
some
at least
fixed
A/2
One
r
>
1
,
we perform
a non-saturating
node and even decrease
its
push
r
>
1
is
the
of value
many
excess below A/2.
A/2
units of flow during
The same complexity of the algorithm
one out of every
a
could, however, keep pushing the
Also, the algorithm as stated sends at least
for
and performs
excess to zero. This variation might produce
saturating pushes from the
if
> A/2
is
obtained
non-saturating pushes sends
units of flo\v.
potential bottleneck in practice
number
particular, the algorithm "recognizes" that the residual
path from node
i
to
node
may
suggested that
it
search so as to
make
use of breadth
first
t
only
when
d(i)
of relabels.
network contains no
exceeds n - 2
be desirable occasionally
the distance labels exact.
to
.
Goldberg [1987]
perform a breadth
He
In
first
discovered that a judicious
search could dramatically speed up the algorithm.
27
An
whose
alternative
distance
is
approach
k.
with distance greater than k
network. (Once node
since the shortest path
j
is
At
from
j
to
flo\s'
in
(We can
.
We
would
all
nodes become disconnected from the
nodes are sent back
to the source.
of the distance-based preflow-push algorithm has
is
the
most
relax this
efficient
in
[19S7a].
realistic
maximum
flow problem.
algorithm for the
U
that
assumption on
algorithm slightly and adopt the more
computation as discussed
stays disconnected
Summary
our algorithm
n
it
nondecreasing in length.)
problem under the reasonable assumption
bounded
each node
relabel, then
is
several advantages over other algorithms for the
First,
any
of nodes
t
this time, the excesses of active
Our improvement
after
disconnected from the sinks,
is
Conclusions and
6.
keep track of the number n^
disconnected from the sink in the residual
avoid selecting such nodes until
sink.
to
decreases to
n^,
If
is
U
is
if
maximum
polynomially
we modify
the
logarithmic model of
)
Second, our model has several intuitively appealing features that are
not guaranteed in other preflow-push methods.
from
a
node with
a large excess.
We
always
try to
push
flo^v
(Gabow's scaling version of Dinic's
algorithm always selects an augmentation along a path with a large capacity.)
This feature accounts for
never allow too
much
much
of the efficiency of his
excess to accumulate at a node.
algorithm relies on verv simple data structures with
method.
Also,
we
In addition, our
little
computational
28
This feature contrasts dramatically with the algorithms that rely
overhead.
on dynamic
trees.
Third, our algorithm
[1972]
a novel approach to combinatorial scaling
In the previous scaling algorithms developed
algorithms.
Karp
is
Rock
,
[1980]
,
and Gabow
[1985]
,
by Edmonds and
scaling involved a sequential
approximation of either the cost coefficients or the capacities and
right-hand-sides,
approximated by c/2^
to solve the
we would
(e.g.,
for
problem with
some
c
first
solve the problem with the costs
integer T.
We would
approximated by
c/2'^~^
then reoptimize so as
.
We would
reoptimize for the problem with c approximated by c/2^~^, and so
Our
scaling
method does not
Orlin[1987a],
we
which includes
fit
into this standard
then
forth.)
framework. In Ahuja and
describe an alternate framework for scaling algorithms
of the previous scaling algorithms as well as the algorithm
all
described here and >he algorithms of Goldberg and Tarjan [1987] for the
minimum
cost flow problem,
and the algorithms
of
Gabow and
Tarjan [1987]
assignment and related problems.
for the
Fourth, our algorithm appears to be quite efficient in practice.
Ahuja and Orlin
for the
[1987c],
maximum
we
empirically test a
flow problem.
preliminary results
to
number
In
of different algorithms
The algorithm presented here appears from
be comparable
to the best other
algorithm for a range of distributions.
only simple data structures so that
it is
maximum
f\o\v
In addition, our algorithm involves
relatively easy to encode.
29
Fifth,
our algorithm can be implemented efficiently on a parallel
computer with
that
d =
a small
m/n and
,
0(n-^ log n) time
on
number
that
a
log
parallel
of parallel processors.
U
= O(log
random
d parallel processors. This time bound
n).
Then our algorithm runs
access
is
In particular, suppose
machine (PRAM) using only
comparable
achieved by Goldberg and Tarjan [1986] using
in
to the
time bound
n parallel processors. Thus
introduce a n/d improvement in utilization of computational resources.
addition, our
model
of
computation
is
discussed in Ahuia and Orlin [1987b].
less restrictive.
These results are
we
In
30
Acknowledgements
We
wish
to
thank John Bartholdi and
Tom Magnanti
suggestions which led to improvements in the presentation.
was supported
in part
for their
This research
by the Presidential Young Investigator Grant
8451517-ECS of the National Science Foundation.
31
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,
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Work
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Orlin.
J.B.
A
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