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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
A FAST AND SIMPLE ALGORITHM
FOR THE MAXIMUM FLOW PROBLEM
R. K.
Ahuja
and
James
B.
Orlin
Sloan W.P. No. 1905-87
June 1987
Revised:
March 1988
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
A FAST AND SIMPLE ALGORITHM
FOR THE MAXIMUM FLOW PROBLEM
R. K.
Ahuja
and
James
B.
Orlin
Sloan W.P. No. 1905-87
June 1987
Revised:
March 1988
M.I.T.
LIBRARIES
A Fast and Simple
Algorithm for the
Maximum Flow Problem
Ravindra K. Ahuja* and James B. Orlin
Sloan School of Management
Massachusetts Institute of Technology
Cambridge, MA. 02139 USA
,
Abstract
We
present a simple
0(nm
+ n
log U)
flow problem on a network with n
bounded by U.
n
,
Under
factor of log
nodes,
m
in
time
0(nm
0(nm
+ n
and integer arc
arcs,
the practical assumption that
our algorithm runs
previous best bound of
sequential algorithm for the
log
U
is
polynomially bounded
*
On
in
This result improves the
n).
log {vr/m)), obtained by Goldberg
and Tarjan
structures.
Subject Classification
A
capacities
,
by
a
n for networks that are both non-sparse and non-dense without using
any complex data
484.
maximum
Fast
and Simple Algorithm
for the
Maximum Flow Problem
leave from Indian Institute of Technology,
Kanpur
-
208 016
,
INDIA
The maximum flow problem
one of the most fundamental problems
is
network flow theory and has been investigated extensively. This problem was
in
first
formulated by Fulkerson and Dantzig [1955] and Dantzig and Fulkerson [1956], and
solved by Ford and Fulkerson [1956] using their well-known augmenting path
algorithm.
number
Since then, a
of algorithms
problem; some of them are tabulated below. In the
m
is
the
number
of arcs,
and
U
have been developed
table,
n
is
the
number
for this
of nodes,
an upper bound on the integral arc capacities. The
is
algorithms whose time bounds involve
U
assume
integral capacities,
whereas
others run on arbitrary rational or real capacities.
Due
#
Running Time
to
1
Ford and Fulkerson [1956]
O(nmU)
2
Edmonds and Karp
CXnm^)
3
Dinic [1970]
0(n^)
4
Karzanov [1974]
0(n3)
5
Cherkasky[1977]
CXn^^^)
6
Malhotra.
7
Galil[1980]
8
Galil and
9
Shiloach and Vishkin
10
Sleator and Taijan [1983]
0(nm
11
Tarjan[1984]
CXn^)
12
Gabow
0(nm
13
Goldberg [1985]
0{n3)
14
Goldberg and Tarjan [1986]
0(nm
15
Cheriyan and Maheshwari [1987]
16
Ahuja and Oilin [1987]
17
Ahuja, Orlin and Tarjan [1987]
Table
1.
[1972]
Kumar and Maheshwari
Naamad
0(nm log^
[1980]; Shiloach [1978]
log n)
log
U)
log (n^An))
0(nm + n^
Running times
of the
[1972]
algorithm runs in time O(nm^)
if
n)
0(n3)
[1982]
[1985]
Edmonds and Karp
source to sink.
CXn^)
[1978]
maximum flow
showed
that the
log
U)
algorithms.
Ford and Fulkerson [1956]
flows are augmented along shortest paths from
Independently, Dinic [1970] introduced the concept of shortest path
and obtained an O(n^m) algorithm. This bound
networks, called
layered networks,
was improved
O(n^) by Karzanov [1974]
in a layered
into a
to
A
network.
preflow
is
who
introduced the concept of preflows
amount flowing
similar to a flow except that the
node may exceed the amount flowing out
of a node.
Since then, researchers
have improved the complexity of Dinic's algorithm for sparse networks by devising
sophisticated data structures.
Among
dynamic
the
tree data structure
The algorithms
of
is
these contributions, Sleator
most
attractive
and
Tarjan's [1983]
from a worst case point of view.
Goldberg [1985] and of Goldberg and Tarjan [1986] were
a
novel departure from these approaches in the sense that they do not construct
Their method maintains a preflow, as per Karzanov, and
layered networks.
proceeds by pushing flows to nodes estimated to be closer to the sink.
which nodes are closer
to the sink,
lower bound on the length of
it
To estimate
maintains a distance label for each node that
a shortest
augmenting path
is
a
Distance labels
to the sink.
are a better computational device than layered networks because the labels are
simpler to understand, easier to manipulate,
algorithm.
use in a parallel
to
Moreover, by cleverly using the dynamic tree data structure, Goldberg
and Tarjan obtained the
networks.
and easier
best computational complexity for sparse as well as dense
(For applications of distance labels to
augmenting path algorithms, see
Orlin and Ahuja [1987].)
For problems with arc capacities polynomially bounded in n
flow algorithm
is
Edmonds and Karp
flow problem and later extended by
.
Gabow
The bottleneck operation
Goldberg and Tarjan's algorithm
O(n^)
is
[1972] for the
[1985] for other
clever application of the
the
number
dynamic
minimum
to
0(nm
We show
0(n^ log U) by using
algorithm modifies the Goldberg-Tarjan algorithm as follows.
which
sufficiently large excesses to
that the
number
excess scaling.
performs
It
nodes with
while never allowing the excesses to become too
time of our algorithm
is
0(nm
+
n'^
log U).
large.
is
log (n^/m)) by a
each scaling iteration requires O(n^) non-saturating pushes
push flows from nodes with
cost
implementation of
of non-saturating pushes
tree data structure.
non-saturating pushes can be reduced to
excesses
maximum
network optimization
in the straightforward
However, they reduce the computational time
scaling iterations;
our
an improvement of Goldberg and Tarjan's algorithm and uses
concepts of scaling introduced by
problems.
,
of
Our
log
if
U
we
sufficiently small
The computational
Under
n)
0(nm
our algorithm runs in time
,
non-dense and non-sparse,
m
i.e.,
+
= GCn'"*"^ for some
log (vr/m)) on such networks by a factor of log n.
implement and should be more
easier to
<
with
e
e
<
1,
our
0(nm), which improves Goldberg and Tarjan's bound of
algorithm runs in time
0(nm
U=0(n^^^b (i.e., it is polynomialin
n'^ log n).
On networks that are both
the reasonable assumption that
elementary data structures with
Moreover, our algorithm
efficient in practice, since
it
is
requires only
computational overheads.
little
Notation
1.
Let
every arc
G
W,
=
A) be a directed network vdth a positive integer capacity
e A. Let
(i, j)
n =
N
I
m= A
and
I
I
We
distinguished nodes of the network.
from source
of the paths
detected in
no
arcs but
0(m)
Let
e A: Uj; <
U=
"^a^
(s, j)
A
I
{j:
(j, i)
We
.
We
further
(i, j)
flow
network contains some
Xji
x
I
-
:
therefore
e A)
(j:
I
{j:
(j,
A— R
assume
have
that all arcs
finite capacity,
t)
satisfying
»
=
X|j
,
(i, j)
Xjf
for
aU
i
6
N - (s,
t),
(1)
e A)
= v
,
Ujj
,
(2)
€ A)
<
xjj
none
infinite capacity
A
a function
that
such a path can be easily
««)
is
e A, an arc
assume
{"sjl
e
are two
then the capacity of such arcs can be replaced by
infinite capacity path,
Ujj
the
if
t
no arcs directed into the
that for every arc
to sink has infinite capacity as
Observe that
time.
V
{(i, j)
assumed
and sink
s
loss of generality that the
that there are
also contained in A, possibly with zero capacity.
is
(j, i)
also
It is
The source
.
assume without
network does not contain multiple arcs and
source or directed from the sink.
I
Uj; for
<
for all
(i, j)
e
A
,
(3)
for
some
V
is
v >
The maximum flow problem
.
determine a flow x
for
which
maximized.
A
x
preflow
is
following relaxation of
I
(j:
to
is
a function
I
>
Xj|
e A)
(), i)
A— R
which
>
:
satisfies
(2)
,
(3),
and the
(1):
-
Xjj
x
(j:
,
for all
N-
e
i
(4)
(s, t).
e A}
(i, j)
The algorithms described
in
paper maintain
this
preflow
a
at
each
intermediate stage.
For a given preflow x
=
Cj
(j,i)€
{j:
A
The
given by
active.
maximum
(j,
define for each node
A)
i).
rjj
=
-
+
xj;
the
node
i,
referred to as an
to
any arc
Xjj
e
The network consisting only of
Figure
i.e.,
A(i)
:
=
{(i,
A
k) €
of
it
nodes
d
distance function
:
N^
CL
CZ
d(t)
d(i)
=
<
Z"*"
.
node
to
i
of arc
node
(i, j)
+
1
node
i
e
list
is
represents the
(i, j)
N
is
as the set of arcs directed out of
We say
(i, j)
list is
a set of arcs rather
as a set of nodes.
for a preflow
for every arc
define the
illustrates these definitions.
x
is
a function
that a distance function
two conditions:
,
We
using the arcs
j
;
d(j)
.
with respect to a given preflow x
k € N}. Note that our adjacency
to the non-negative integers
also satisfies the following
the excess
arcs with positive residual capacities
A(i) of a
list
:
1
than the more conventional definition of the
A
A
additional flow that can be sent from
define the arc adjacency
,
consequently, these nodes are never
be zero;
(i, j)
active node
The residual capacity
.
referred to as the residual network.
We
is
and sink nodes
uj;
t}
aj)e A)
{j:
residual capacity of
N - (s,
e
i
Xjj
2,
node with positive excess
excess of the source
and
-
Xjj
2-
we
,
€
A
vdth
Tj;
>
.
from the
d
is
set
valid
if
a.
Network with
Node
sink.
1
is
arc capacities.
the source
and node 4
is
the
(Arcs with zero capacities are not
shown.)
b.
c.
Figure
1.
Network with preflow x
The residual network with
residual arc capacities
Illustrations of a preflow
and the residual network
Our algorithm maintains
refer to d(i)
as the distance label of
that d(i)
a lower
is
residual network.
bound on
Let
< d(i2
any path
)
+
d(i2
1,
)
i
< d(i3
)
of length k
shortest path too.
any path from
If
to
i
-
node
for each
-
•
•
•
1(c),
-i^^
+
i,
d=
-
dCij^)
1
i\^+i
<
=
to
i
in the
t
be any path of length k
t
dUi^^;,
)
+
1
=
1.
This yields
d(i)
in the
minimum
the distance label d(i) equals the
network, then
we
call
=
for
for the
length of
the distance label exact.
a valid distance label,
is
d(i)
<k
network and, hence, must also hold
(0, 0, 0, 0)
also
easy to demonstrate using induction
Then from condition C2 we have,
to the sink.
in the residual
t
It is
i.
the length of the shortest path from
i2
in the residual
For example, in Figure
0)
i-|
node
residual network from
d(i|)
=
i
We
a valid distance function at each iteration.
though d =
(3, 1, 2,
represents the exact distance labels.
d(i)
=
An
arc
d(j)
+
(i,
1.
j)
An
in the residual
arc
which
network
not admissible
is
called
is
is
admissible
if
it
satisfies
called an inadmissible arc.
The
algorithms discussed in this paper push flow only on admissible arcs.
All logarithms in this paper are
assumed
to
be of base 2
unless stated
otherwise.
Preflow-Push Algorithms
2.
The preflow-push algorithms
preflow
The
at
first
maximum
flow problem maintain a
every step and proceed by pushing the node excesses closer to the sink.
preflow-push algorithm
suggested a simplified version of
[1985]
for the
and Goldberg and Tarjan
is
due
to
Karzanov
The recent algorithms
this algorithm.
[1986] are
[1974]. Tarjan [1984]
based on ideas similar
of
has
Goldberg
to those presented in
Tarjan [1984], but use distance labels to direct flows closer to the sink instead of
constructing layered networks.
preflow-push algorithm.
We
refer to their algorithm as the (distance-directed)
In this section,
algorithm, which for the sake of brevity,
algorithm.
Here we describe the
presented by Goldberg [1987].
The
we
we
review the basic features of their
shall
simply refer to as the preflow-push
1-phase version of the preflow-push algorithm
results in this section are
due
to
Goldberg and
Tarjan [1986].
All operations of the preflow-push algorithm are performed using
information.
At each iteration of the algorithm (except
only local
at the initialization
and
at
.
8
termination)
the network contains at least one active node,
The goal
non-sink node with positive excess.
active
node and
to
send
i.e.,
of each iterative step
and
a non-source
to
is
choose some
excess "closer" to the sink, with closer being judged with
its
respect to the current distance labels.
nodes with smaller distance
If
excess flow at this node can not be sent to
then the distance label of the node
labels,
The algorithm terminates when the network contains no
increased.
is
The
active nodes.
preflow-push algorithm uses the following subroutines:
PRE-PROCESS. On each
and
d(t)
=
Let
.
arc
d(i)
labeling can be used,
=
(s, j)
1
e A(s)
for each
node
t.
Select
an active node
PUSH(i).
Select
an admissible arc
flow from node
RELABEL(i).
t
node
i
to
i
Replace
is
(i, j)
Send 6 = min
in A(i).
is
(i, j)
saturating
if
d(i)
by nunf
d(j)
+
1
The
(i
:
,
j)
e A(i)
and
(e^
,
6 =
rj:
rj:
)
,
units of
and
of the preflow-push algorithm
is
given below.
PRE-PROCESS;
an active node do begin
is
SELECT
there
{let
is
i
denote the node selected};
an admissible arc
else RELABEL(i);
>
).
is to
which further pushes can be performed.
PREFLOW-PUSH;
there
r^j
result of the relabel step
begin
end;
can be
t
i
called a relabel step.
The generic version
end;
s,
j.
create at least one admissible arc on
if
^t
otherwise.
This step
while
any valid
(Alternatively,
.
search on the residual network
first
say that a push of flow on arc
non-saturating
algorithm
i^ s or
)
SELECT.
We
send Ugj units of flow. Let d(s) = n
the distance label for each
e.g.,
determined by a backward breadth
starting at
,
in A(i)
then
PUSH(i)
d(3)=l
d(l)
=4
The
(a)
d(4)
d(2)
=
62
=2
1
residual network after the pre-processing step.
d(3)
d(l)
=
1
=4
d(4)
d(2)
=
Co =
(b)
=
=
l
1
After the execution of step PUSH(2).
d(3)
=
1
e3=4
d(l)
(c)
Figure
= 4 (T
d(4)
After the execution of step
2.
Illustrations of
=
RELABEL(2).
Push and Relabel
steps.
10
network
Figure
in
and
min{
node 2
network
2, 1}
1
,
the excess of
from the residual network and arc
node 2
3)
is still
and
an active node,
condition.
(4, 2)
min
{d(3)
+
1
,
to
s
immediate.
length of the
to the residual
1
5 =
of value
1. Arc (2,4)
to
=
r24
deleted
is
network.
Since
d(l) + 1}
= min
(2,
node 2
=2
{2, 5}
a
new
.
several important tasks.
First,
causes the
it
have positive excess, so that we can subsequently execute the
Third, since
minimum
push
RELABEL(2), and gives
Second, by saturating arcs incident
select step.
is
to
The SELECT
but they do not satisfy the distance
capacities,
The pre-process step accomplishes
nodes adjacent
node 2
added
is
Hence the algorithm performs
distance d'(2) =
residual capacity.
step.
a saturating
to the
can be selected again for further pushes. The arcs
it
have positive residual
(2, 1)
PRE-PROCESS
after the
the algorithm performs
The push reduces
units.
its
for examination. Since arc (2, 4) has residual capacity
= d(4) +
d(2)
RELABEL(i) as applied
The number beside each arc represents
1(a).
Figiire 2(a) specifies the residual
step selects
and
PUSH(i)
illustrates the steps
Figure 2
the distance label
path from
to
s
t
,
to
,
the feasibility of setting d(s) = n
d(s) =
there
since distance labels are non-decreasing (see
s
n
is
a lower
no path from
is
Lemma
1
s
to
to follow),
bound on
t
the
Further,
.
we
are also
guaranteed that in subsequent iterations the residual network will never contain a
directed path from s
to
t,
and so there can never be any need
to
push flow from
s
again.
In our
improvement
results given in
order to
Lemma
make
1.
each step.
we need
of the preflow-push algorithm,
Goldberg and Tarjan
more
this presentation
[1986].
We
include
some
a few of the
of their proofs in
self-contained.
The generic preflow-push algorithm maintains valid distance labels at
Moreover,
at each relabel step the distance label of some node strictly
increases.
Assume
Proof. First note that the pre-process step constructs valid distance labels.
inductively that the distance function
to
be
satisfied.
(j, i)
valid prior to
an operation,
and C2 A push operation on the
with Tjj >
and an additional condition
the validity conditions Cl
additional arc
is
.
arc
,
(i, j)
d(j)
<
This validity condition remains satisfied since d(i) =
property of the push operation.
A
push operation on arc
(i, j)
i.e., it
satisfies
may
create an
d(i)
+
+
needs
1
1
by the
might delete
this arc
d(j)
1
1
from the residual network, but
During
function.
d'(i)
= min{d(j) +
1:
with
0}
=
= d(j)+l and
d(i)
d'(i)
Lemma
,
e A(i) and
relabel step
rj:
>
>
r^:
0)
which
,
l:
no arc
is
(i, j)
At any stage
2.
of the prefloiv-push algorithm, for each node
a directed path
is
from
By the flow decomposition theory
of non-negative flows along
(i)
node
to
i
of
preflow X in G.
,
each arc reversed)
in
is
Then the
i.
in the residual
i
(ii)
G
paths from s
from
>
,
with
any
,
into the
to
s
sum
to active nodes,
to the
in the flow
i
P
(P vAth the orientation of
and hence there
a path
is
from
i
to
the residual network.
Corollary
1.
Proof.
The
For each node
last
e
i
time node
i
N
,
was
d(i)
<
2n.
relabeled,
had
it
that d(s)
Lemma
= n and condition C2 imply that
d(i)
a positive excess,
n -
residual network contained a path of length at most
< d(s) + n -
The number of
2.
relabel steps is less than
2n^
Proof. Each relabel step increases the distance label of a
Corollary
1
Corollary
3.
Proof.
from
1
1
<
i
no node can be relabeled more than 2n
Suppose
The number of saturating pushes
that arc
(i, j)
becomes saturated
+1). Then no more flow can be sent on
(i, j)
and hence the
to
s.
The
fact
2n.
empty
2 also implies that a relabel step never minimizes over an
Corollary
d(j)
rj:
and flows around cycles do not
t
reversal of
network
i
be an active node relative
P
a path
since paths from s to
contribute to the excess at node
t,
Let
Then there must be
decomposition of x
s
€ A(i)
j)
in the residual network.
s
Ford and Fulkerson [1962]
paths from s to
flows around directed cycles.
(iii)
(i,
e A(i) and
preflow X can be decomposed with respect to the original network
and
is
i
again consistent with the
is
< min{d(j) +
d(i)
node
distance label of
performed when there
is
Hence,
.
new
the
step,
affect the validity of the distance
thereby proving the second part of the lemma.
positive excess, there
Proof.
(i, j)
The
validity conditions.
relabel
a
does not
this
set.
.
node by
and by
at least one,
times.
is
at
no more than
some
nm
.
iteration (at
until flow is sent
which
back from
j
d(i)
=
to
at
i,
12
which time
increases
d(j)
=
d'(j)
by
network
Lemma
that
total
+
d(i)
=
1
d(j)
+
Thus by Corollary
m
arc
1,
A
are both in
is
(i, j)
no more than nm.
so the
,
When
number
non-saturating pushes
and Tarjan
is
and
CI
conditions
[1986]
is
(Recall that
of arcs in the residual
at
2n^m.
most
.
N
C2
and
= n
d(s)
,
it
termination criterion for the
Fulkerson [1962]
follows that
to
maximum
t
,
.
upon
termination, the
This condition
of non-saturating pushes.
A
residual network.
number
in
why
of saturating pushes
—
This observation leads to a
pushes -no matter
algorithms, such as the
partial explanation of
saturating pushes cause structural changes
is
the
number
bound
0(nm)
of
on the number
their
flow
this
is
difficult to
is
tight.
dramatically reduce the
number
trees to
Cheriyan and Maheshwari [1987]
of non-saturating pushes can be decreased to
In the next section,
0{t\^)
Goldberg and Tarjan [1986]
always pushed from a node with highest distance
bound
is
network flow algorithm by using dynamic
reduce the average time per non-saturating push.
number
of
which order they are performed. The non-saturating
are examined in a first-in-first-out order.
that the
of
they delete saturated arcs from the
Goldberg [1985] showed that the number of non-saturating pushes
reduced the running time of
is
The
as follows:
pushes do not change the structure of the residual network and seem more
showed
the
is
Tarjan [1984], and Goldberg and Tarjan [1986],
non-saturating pushes dominates the
when nodes
so
flow algorithm of Ford and
many preflow based
in
algorithms due to Karzanov [1974]
saturating
has zero excess;
(s, t)
.
The bottleneck operation
number
-
since the distance labels satisfy
Further,
a feasible flow.
residual network contains no directed path from s
bound.
most
at
)
.
the algorithm terminates, each node in
the final preflow
the
can become saturated
The algorithm terminates with a maximum flow.
4.
classical
flow change cannot occur until
this
2;
of arc saturations
(j, i)
The number of
Proof. See Goldberg
Proof.
and
(i, j)
^
1
number
no more than
is
3.
Lemma
+
at least 2.
n times, and the
we assume
d'(i)
we show
that
label,
^)
if
and they showed
that
we
can
by using
of non-saturating pushes to
0(n^ m^'
scaling,
©(n-^ log U)
.
We
have
13
new
recently discovered a
log
UA
O fr?- iTT
non-saturating pushes to
which further reduces the number
scaling algorithm
^
^^^^ result
•
of
presented in Ahuja, Orlin and
is
Tarjan [1987].
The Scaling Algorithm
3.
Our maximum flow algorithm improves
of Section 2
by using "excess scaling"
from O(n'^m)
with
to
0(n^ log U)
sufficiently large excesses to
become
letting the excesses
reduce the number of non-saturating pushes
to
The
.
basic idea
nodes with
flog
iteration, the excess-dominator is defined to
to
and
satisfies
e^
<A
for all
i
e
N
Further
.
we
a
new
at least
we
select a
that
scaling iteration
A/2
most
Srr-
of at least 2,
and
all
a
new
to zero
distance label
{
i
€
considered
and
that each non-saturating
N
:
ej
and we obtain
a
K
After at most
among such
> A/2
elements in 0(1) time.
and
nodes,
d(i)
The variable
label.
We show
select the eligible arc for
list,
This
that
maximum
flow.
=
r)
we
maintain the
for each r
=
1,
and with
A/2
[1974]),
lists
.
.
.
,
2n-l
.
These
lists
(see, for
which enables insertion and deletion
level indicates the
a
smallest index r
for
of
which
non-empty.
As per Goldberg and
the
push has
scaling iterations,
an active node with excess more than
example, Aho, Hopcroft and Ullman
is
that the
;
can be maintained in the form of either linked stacks or linked queues
LIST(r)
we
more than A/2 and among
node with minimum distance
scaling iteration begins.
In order to select
LIST(r) =
is
non-saturating pushes, the excess-dominator decreases by a factor
node excesses drop
minimum
of
a
units of flow
choice ensures that the flow will be sent to a node with small excess.
after at
power
is
In a scaling iteration
2.
consider only nodes with excess
these nodes with large excess,
A
least integer
a factor of
To ensure
excess-dominator does not increase.
,
,
For a scaling
scaling iterations.
1
be the
guarantee that each non-saturating push sends
A/2
while never
sufficiently small excesses
U1 +
A decreases by
have begun whenever
value at least
push flow from active nodes
is to
too large.
The algorithm performs K =
2
the generic preflow-push algorithm
A(i)
,
Tarjan,
we
use the following data structure to efficiently
pushing flow out of a node.
of arcs directed out of
it.
Arcs in each
We maintain
list
with each node
can be arranged
i
arbitrarily.
1
A
but the order, once decided, remains unchanged throughout the algorithm.
arc
named
which
node
is
the
made
null arc
is
arc in
first
the current arc
list is
appended
is
end
to the
of each
list.
Each node
the current candidate for pushing flow out of
is
i
null
its
found
is
arc
to
At
When
this time, the
to the first arc in the arc
list is
the current arc of
node
relabeled
and the current
arc
list.
as follows:
MAX-FLOW;
begin
PRE-PROCESS;
K
:
=
+ riog U1
1
for k =
:
1
to
;
K do begin
A = 2K-k
i€
for each
level
:
while
=
1
do
if
ei>A/2 then add
i
to LIST(d(i));
;
level
if
N
< 2n do
LIST(level) =
then level
:
= level +
else begin
select a
node
i
from
PUSH/RELABEL(i);
end;
end;
end;
in the arc
has been completely examined,
list
The algorithm can be formally described
algorithm
Initially,
(i, j)
examined sequentially, and whenever
the arc
is
special
has a current arc
be inadmissible for pushing flow, the next arc
the current arc.
reached.
This
list.
i.
i
4
LIST(level);
1
is
the
again set
;
;
15
procedure PUSH/RELABEL(i);
begin
found =
:
let
(i, j)
false
;
be the current arc of node
while found =
if
and (i,
and
false
=
d(i)
+
d(j)
^ null do
j)
1
i;
Tj;
>
then found
replace the current arc of node
else
i
= true
:
by the next arc
found = true then begin (found an admissible
if
push min
(ej
,
rj:
A -
,
units of flow
e;}
update the residual capacity
if
if
updated excess) ej<A/2
(the
j
?t
and
r^:
or
s
and
and
t
,
set level
:
= level -
arc)
on
arc
(i, j)
the excesses ej
then delete node
(the ufxiated excess)
e;
(i, j);
> A /2
i
;
and
e;
from
LIST(d(i));
then add node]
,
to LIST(d(j))
1;
end
else begin (finished arc
delete
node
update
add node
first
from
i
d(i)
:
node
of
i)
LIST(d(i));
= min{d(j) +
to
i
list
LlST(d(i))
1
;
(i, j)
and
€ A(i)
and
>
rj;
0}
set the current arc of
node
to the
i
arc of A(i);
end;
end;
4.
Complexity of the Algorithm
In this section
,
we show
that the distance-directed
excess scaling correctly computes a
Lemma
5.
C3.
The scaling algorithm
satisfies
Each non-saturating push from
A/2
C4.
maximum
hlo
0(nm
flow in
the following
a
preflow-push algorithm with
node
i
+
n'^
log U) time.
two conditions:
to a
node
j
sends at least
units of flow.
excess
increases
above
A
increase subsequent to a push).
(i.e.,
the excess-dominator does
not
6
1
push on
Proof. For every
arc
node with smallest distance
d(j)
=
(ej
rjj
,
-
d(i)
<
e;
)
A-
1
,
label
e';
remain
=
> min {A/2
+ min
e:
ej
{
,
rj:)
,
rj;
A/2
,
A-
e;
<
)
way
efficient
< A/2
e;
A-
+
e;
by sending min
,
Further,
< A
All
.
the push operation
node
at
after the push.
j
node excesses thus
that the algorithm
always
C3 and C4, pushing flow from
A/2 and with minimum
node with
a
among such nodes
distance
sufficiently large to
its
Its
i
equally in the network.
iteration.
is to
It is
likely that
to the sink, in
would have
C4 ensures
In particular,
may be
node
its
be returned.
to
i
maximum
that the
may
is
from exceeding
e:
A.
make
it
easier for
fairly
nodes
distance label
to a single
node
j
creating a large
send the accumulated flow closer
to
would
phenomenon
increase
is
and much
of
its
excess
prevented by maintaining
condition C4.
Lemma
6.
If
to
also be important because of the following
would not be able
This
j
excess
"encourage" flow excesses to be distributed
This
j
push from
a kind of
rather than greedily getting rid
This distribution of flows should
which case
to
a simple
is
very useful in practice as well as in
suppose several nodes send flow
consideration:
that the
restraint so as to prevent
lower
excess
send flow towards the sink.
excess.
Property
shows some
major impact
C3 ensures
Property
effective.
during an
maximum
Keeping the
theory.
be
node
excess,
the
satisfies
of enforcing these conditions.
"restrained greedy approach."
of all
a
more than A/2, and
is
With properties C3 and C4, the push operation may be viewed as
never exceeds A
is
i
ensure that in a non-saturating
be the excess
e';
node
since
,
.
properties stated in the conditions
and
e;
excess
units of flow.
Let
only.
j
we
units of flow,
While there are other ways of ensuring
excess greater than
and
among nodes whose
least
than or equal to A
less
> A/2
e^
by the property of the push operation. Hence
d(i)
push the algorithm sends at
increases the excess at node
Then
we have
(i, j)
C3
each push satisfies conditions
non-saturating pushes per scaling iteration
Proof. Consider the potential function F =
is
at
most
£ e^
ieN
and
C4,
8n^
d(i)/A.
then the
number
of
.
The
initial
value of F at the
1
beginning of A-scaling phase
d(i)
bounded by 2n
is
.
bounded by
is
When
because
Zn-^
the algorithm examines
ej
node
bounded by A and
is
i,
7
one of the
following
two cases must apply:
Case
The algorithm
1.
pushed. In
label of
this case
node
no
goes up by
i
the total increase in
bounded by
all
Case
2.
initial
to
A
j
and since
is
and
to relabelings of
to
and the distance
>
rj:
nodes
is
is
is
i
most
at
Irr-
which flow can be pushed
d(j)
=
d(i)
-
(i,
1, this
j)
sends
A/2 units of flow from
at least
Oirr- log U)
value of the excess-dominator
initial
-
decreases F by at least
sum
units
to at
A <
{s, t).
maximum
Theorem
2.
Oinm + n^
new
1;
.
As
the
most 4rr
,
Srr- times.
The scaling algorithm performs
1.
F
In either case,
A
is
2'
'°8
non-saturating pushes.
'-^
'
< 2U
.
the value of the excess-dominator decreases by a factor of 2
N-
Since
bounded by 2rr
node relabelings
able to identify an arc on
can not occur more than
scaling iterations,
a
F due
non-saturating push on arc
non-saturating pushes and a
e
1
This increases F by at most e units.
phase (Actually, the increase in F due
node
The
Proof.
i
+
value of F for a scaling phase plus the increases in F
Theorem
,
units.
1
d(j)
performs either a saturating or a non-saturating push.
it
this case
6
>
d(i)
along which flow can be
an arc
scaling phases).
decreases.
i
=
throughout the running of the algorithm for each
The algorithm
and so
node
d(i)
(i, j)
e
to find
2n, the total increase in
in the scaling
over
arc
unable
satisfies
is
scaling iteration begins.
and by the
The algorithm thus obtains a
After
1
integrality of the flows
feasible flow,
which by
By Lemma
8n^
within
+ flog
e^
U
=
Lemma
1
such
for all
4
must be
flow.
The
log
W
complexity
.
of
the
maximum
flow
scaling
algorithm
is
18
The complexity
Proof:
the while
loop
PUSH/RELABEL(i)
main program.
the
in
depends upon the number
of the algorithm
step
In
of executions of
such execution either
each
a
performed or the value of the variable level increases.
is
Each execution of the procedure PUSH/RELABEL(i) results
in
one of the following
outcomes:
A
push
Case
1.
the
number
0(nm
Case
performed. Since the number of saturating pushes
is
+ n^ log U)
label of
0(n) times for each node
i
calls the
The
to find
number
I
needed
A(i)
node
and
Thus the algorithm
effort
an arc
perform the push operation
to
node
V
is
operations. This
i€
I
A(i)
i
2n
I
A(i)
I
is
log U)
times.
plus the
0(1)
label of
node
After
i
goes
= 0(nm) plus the number of
N
is
clearly
0(nm
+ n^ log U).
requires examining arcs in
= 0(nm) time for
I
case occurs
replaced by the next arc in A(i).
Computing
consider the time needed for relabel operations.
distance label of node
2n
is
i
Case 2 occurs and distance
i.
ie
PUSH/RELABEL(i)
0(nm + n^
procedure PUSH/RELABEL(i)
up. Thus, the total effort needed
V
this
OCn'^) in total.
of times the current arc of
Now
,
goes up. By Corollary!, this outcome can occur
i
such replacements for node
I
0(n^ log U)
is
0(nm) and
times.
The distance
2.
pushes
non-saturating
of
is
all
The
new
This yields a total of
A(i).
relabel operations.
the
lists
LIST(r) are stored
N
as linked stacks
time.
and queues, hence addition and deletion
Consequently, updating these
Finally,
we need
each scaling iteration,
Hence
its
number
decreases plus 2n
such a case
it
bounded by
0(nm
to
level is
the
number
of increases of the variable
bounded above by 2n -
Further, level can decrease only
.
number
the
.
1.
of
Hence
its
any element takes 0(1)
not a bottleneck operation.
of increases per scaling iteration
decreases by
+ n^ log U)
bound
lists is
of
is
level.
and bounded below by
1
bounded by the number
when
a
increases over
push
all
is
performed and
In
1.
of
in
scaling iterations are
pushes plus 2n(l + riogUl),
which
is
again
9
1
Refinements
5.
As
its
a practical matter, several modifications of the algorithm might improve
actual execution time without affecting
worst case complexity.
its
We
suggest
three modifications:
1.
Modify the
2.
Allow some non-saturating pushes of small amount.
3.
Try
The
scale factor.
nodes disconnected from the
to locate
suggestion
first
is
no
less
2,
however, some other fixed integer scaling factor
The excess-dominator
will then be the least
power
of
than the excess at any node, and property C3 becomes
Each non-saturating push from a node
C3'.
in the
reduces the excess-dominator by a factor 2
i.e., it
In practice,
P > 2 might yield better results.
P that
The algorithm
to consider the scale factor.
is
present form uses a scale factor of
in each scaling iteration.
sink.
node
to a
i
j
sends
at least
A/p
units of flow.
The
scaling algorithm presented earlier can easily be altered to incorporate the
p scale factor by letting LIST(r) =
shown
run
to
in
fixed value of p
0(nm +
is
pn'^
(i
e
N
ej
:
logg U)
>A/p and
time.
optimum; the best choice
From
d(i)
=
r
).
The algorithm can be
the worst case point of view any
p in practice should be
for the value of
determined empirically.
The second suggestion focuses on
as stated selects a
push.
We
node uith
e^
variation might produce
pushes after
A/2
its
many
A/2
or reduce
saturating pushes from the
we
its
excess to zero. This
node and even allow
excess has decreased below A/2. Also, the algorithm as stated sends at
is
obtained
if
for
some
non-saturating pushes sends at least
number
at least
uruts of flow during every non-saturating push.
the algorithm
The
Our algorithm
a saturating or a non-saturating
could, however, keep pushing the flow out of this node until either
perform a non-saturating push of value
least
the non-saturating pushes.
and performs
> A/2
fixed
A/2
>
1
,
one out of every
r
of
>
1
units of flow.
third suggestion recognizes that
of relabels.
r
The same complexity
one potential bottleneck
in practice is the
In particular, the algorithm "recognizes" that the residual
network contains no path from node
i
to
node
t
only
when
d(i)
exceeds
n - 2
.
20
Goldberg [1987] suggested
search so as to
first
of breadth
An
distance
make
that
with distance greater than k
path from
until all
is
to
j
is to
is
after
is
up
discovered that a judicious use
the algorithm.
any
relabel for
of nodes
nj^
some
whose
then each node
k,
disconnected from the sink in the residual network.
disconnected from the sink,
t
He
keep track of the number
decreases to
nj^
If
k.
j
desirable to occasionally perform a breadth
the distance labels exact.
alternative approach
(Once node
may be
search could dramatically speed
first
is
it
it
stays disconnected since the shortest
We
nondecreasing in length.)
would avoid
selecting such
nodes vdth positive excess become disconnected from the
sink.
At
nodes
this time,
the excesses of nodes are sent back to the source. This approach essentially yields the
two phase approach
Tarjan [1986]
maximum
.
The
to solve the
first
maximum
phase constructs a
flow problem as outined in Goldberg and
maximum
preflow which
is
converted to a
flow in the second phase.
Fuhire Directions
6.
Our improvement
of the distance directed preflow-push algorithm has
maximum flow problem. Our
for the maximum flow problem
several advantages over other algorithms for the
algorithm
is
superior to
all
previous algorithms
under the reasonable assumption
that
U
is
polynomially bounded in
the algorithm utilizes very simple data structures
which makes
it
n.
attractive
Further,
from an
implementation viewpoint.
Our algorithm
even
if
U
is
computationally attractive from a worst-case perspective
not 0(n^'^'); that
In this case, the
take 0(1) steps,
model
is
is, if
uniform model of computation,
is
arguably inappropriate.
of computation (as described
number
of bit operations.
modifying our algorithm
It is
in
more
which
all
realistic to
more
U
)
which counts the
most arithmetic operations take OQog U)
slightly to
speed up arithmetic operations on large integers,
0(nm log n + n^
log n log U) time.
corresponding time bound for the Goldberg-Tarjan algorithm
Hence, as
adopt the logarithmic
Using the logarithmic model of computation and
claim that our algorithm would run in
log U).
arithmetic operations
by Cook and Reckhow [1973]
In this model,
steps rather than 0(1) steps.
we
the arc capacities are exponentially large numbers.
becomes exponentially
large,
is
0(nm
log
The
{vr/m)
our algorithm becomes more and
attractive relative to the Goldberg-Tarjan algorithm.
Our
logarithmic model of computation will be presented in a future paper.
results in the
—
21
CXir algorithm
is
a novel
approach
previous scaling algorithms developed by
Gabow
Edmonds and Karp
and right-hand-sides,
coefficients or the capacities
w^ith the costs
approximated by C/2^
reoptimize so as to solve the problem w^ith
method does not
C
some
for
[1980]
would
,
and
integer T.
We
framework.
and so
,
v^ould then
,
The recent
of relaxation.
Tarjan [1987] for the
minimum
problem
is
Our
Rather, our algorithm
cost scaling algorithm of
cost flow
and then
forth.)
works with true data, relaxes the flow conservations constraints and
maximum amount
solve the
first
approximated by C/2-^~^
into this standard
fit
(e.g., vs'e
C approximated by C/2^~^
reoptimize for the problem with
scaling
Rock
[1972],
In the
scaling involved a sequential approximation of either the cost
[1985],
problem
to combinatorial scaling algorithms.
scales the
Goldberg and
similar in nature
~
this
algorithm scales the relaxation of the complementary slackness conditions.
The
scaling algorithm for the
by using more clever rules
improvements
factor
in
to
>
flow problem can be improved further
push flow or by using dynamic
Ahuja, Orlin and Tarjan [1987].
We show
algorithm runs in
Use
4.).
of the
/
algorithm to
We
dynamic
f
V
Q nm
T\
log
U
describe such
by using
a larger scale
label
—U
n^ log
+
tree data structure further
Oj^nm logj^—
We
trees.
that
and pushing flow from a node with highest distance
sufficiently large excess, the
U
maximum
|
among nodes having
A
tj
-,
time (Assume that
improves the complexity of
this
\\
,^^-f^^+2jJ.
have also undertaken an extensive empirical study
to
assess
the
computational merits of the preflow-push algorithms versus the previous best
algorithms, those of Dinic and Karzanov.
Our experiments
so far suggest that
preflow-push algorithms are substantially faster than Dinic's and Karzanov's
algorithms.
Our algorithms and
those due to Goldberg and Tarjan suggest the superiority
of distance label based approaches over the layered
improvements we have obtained do not seem
utilizing layered networks.
The distance
layered networks and hence are
in
to
network based approaches. The
be possible for the algorithms
labels implicitly store
more powerful.
We show
dynamically changing
the use of distance labels
augmenting path algorithms, capacity scaling algorithms and
networks
in Orlin
and Ahuja
[1987].
for unit capacity
22
The maximum flow problem on
maximum
bipartite
Ac
bipartite
networks
an important class of the
is
flow problem (see Gusfield, Martel and Fernandez-Baca [1985]).
network
N2
N-j X
maximum
Let
.
a
is
n-j
G
network
=
I
N-j
I
and n2 =
I
N2
flow algorithm can be modified
resulting in significant
network flows
speedup over the
appear
will
in a future
Our maximum flow algorithm
we push
= (N, A)
flow from one node
processors
we
to
•
I
run
jointly
is difficult
in
at a time.
with
to
= N|
For cases where
0(n|m
original algorithm.
paper
N
such that
Our
n-j
u N2
The
and
< < n2, our
2
+ n, log U), thus
results
on
bipartite
C Stein and R. Tarjan.
make
"massively parallel" since
d = fm/nl parallel
Nevertheless, with
can obtain an 0(n^ log Ud) time bound. Under the assumption that
= O(n^^^^0, the algorithms runs
in
0(n^ log
n) time,
which
available time
bounds obtained by Shiloach and Vishkin
Tarjan [1986]
using
situations for
which
n parallel processors.
is
comparable
[1982]
and
U
to the best
Goldberg and
Thus, our algorithm has an advantage in
parallel processors are at a
premium.
Our work on
the parallel
algorithms will also appear in a future paper.
Acknowledgements
We
wish
thank John Bartholdi,
to
Tom
Magnanti, and Hershel Safer for their
suggestions which led to improvements in the presentation.
supported
in part
This research was
by the Presidential Young Investigator Grant 8451517-ECS of
the
National Science Foundation, by Grant AFOSR-88-0088 from the Air Force Office of
Scientific Research,
Prime Computer.
and by Grants from Analog Devices, Apple Computer,
Inc.,
and
23
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\\f
,
I
Date Due
SEP.
1
7
1992
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