^~
Of
r^'y
:4Mii
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
IMPROVED TIME BOUNDS
FOR THE MAXIMUM FLOW PROBLEM
R. K.
J.
B.
Ahuja
Orlin
R. E. Tarjan
Sloan W.P. No. 1966-87
December 1987
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
I
L:
IMPROVED TIME BOUNDS
FOR THE MAXIMUM FLOW PROBLEM
R. K.
}.
B.
Ahuja
Orlin
R. E. Tarjan
Sloan W.P. No. 1966-87
December 1987
^EB
8 1988
!l''i~n
Improved Time Bounds
for the
Maximum Flow Problem
Ravindra K. Ahuja
'
James B. Orlin
•a
Robert E. Tarjan
ABSTRACT
Recently, Goldberg proposed a
flow problem.
The approach
time on n-vertex networks.
new approach
maximum network
to the
yields a very simple algorithm running in
O(n^)
Incorporation of the djTiamic tree data structure of
more complicated algorithm with
Sleator and Tarjan yields a
a running time of
0(rmi log {n} Im)) on m-edge networks. Ahuja and Orlin developed a variant of
Goldberg's algorithm that uses scaling and runs in
0{nm + n?
works with integer edge capacities bounded by U. In
modification
0{ntn + n
7
of
the
lo"C/
).
Ahuja-Orlin
We
show
with
algorithm
that the use
this
a
logC/) time
paper
we
running
of dvnamic trees in
this
on
net-
obtain a
time
of
algorithm
loglogt/
reduces the time bound to 0{,nm log(
that the
^
m
combined use of scaling and dynamic
by using
+
2)).
This result demonstrates
loglogf/
trees results in
speed not obtained
either technique alone.
MA 02139. Research parliaUy supported by a Presidential Young InNSF, Contract 8451517 ECS, and grams from Analog Devices, Apple Computer Inc., and
Sloan School of Management, M.I.T., Cambridge,
vestigator Fellowship from the
I^rime
On
Computer.
leave from the Indian Institute of Technology, Kanpur, India.
DepL
tially
of C.
S.,
Princeton University, Princeton,
supported by
NSF
NJ 08544 and
AT&T Bell Labs,
Murray
Hill,
NJ 07974. Research
Grant DCR-8605962 and Office of Naval Research Contract N00014.87-K-0467.
par-
Improved Time Bounds
for the
Maximum Flow Problem
Ravindra K. Ahuja
2
1
'
James B. Orlin
Robert E. Tarjan
1.
Introduction
We
We
The graph
set E.
a source
G
is
tices, the
base
is
is
iflow network
if
it
flow problem [5,6,12,14,19].
(V,E) be a directed graph with vertex set
symmetric,
i.e.
number of
(v.h) e
E
iff
edges, and the
we assume
m
edge capacities are
integral.
(w,v) € E.
maximum
> n-\ and
U
V
and
has two distinct distinguished vertices, a sink s and
and a non-negative real-valued capacity u(v,h) on each edge (v,^) e E.
t,
time bounds,
that all
G
maximum network
G=
formulate the problem as follows. Let
edge
that
consider algorithms for the classical
We
denote by n,m, and
U the
We
assume
number of
ver-
edge capacity, respectively. For ease in stating
>4. Bounds containing
U are
subject to the assumption
All logarithms in the paper are base two unless an explicit
given.
A flcnv / on
a network
G
is
/ on
a real-valued function
the edges satisfying the following
constraints:
/(v,w) < u(v,w) for
/(v,mO = -/(w, v) for
S
v:(v,H')
/(v,w)
=
all
all
for all
(v.w) e
(v.w) g
w
E
E
(capacity constraint),
(1)
(antisymmetrj' constraint),
(2)
e V'-{j,r} (conservation constraint).
(3)
g£
Sloan School of Managemeni, M.I.T., Cambridge,
vestigator Fellowship from the
MA 02139.
Research partially supported by a Presidential Young In-
NSF, Contraa 8451517 ECS, and
grants from
Analog Devices, Apple Computer
Inc.,
and
Prime Computer.
'
On
^
DepL of C.
leave from the Indian Institute of Technology, Kanpur, India.
S.,
Princeton University. Princeton, N'J 08544 and
tjaUy supported by
NSF Grant DCR-8605962
AT&T Bell
Labs. Murray Hill,
N7 07974 Research
and Office of Naval Research Contract N0O014-87-K-O467.
par-
—
The value /I of
a flow
I
/is Ihc net flow into the sink:
Z
1/1=
/(v.f).
v:(v,()e£
A maximum flow
maximum
a flow of
maximum
value.
The maximum flow problem
is
that of finding a
flow in a given network.
We
Remark.
is
assume
that all
edge capacities are
path of infinite-capacity edges from
sum of the
finite capacities,
.s
to
f
exists,
If
finite.
some edge
D
a long and rich historj', and a series of faster and faster
The previously
algorithms for the problem have been developed. (See [10] for a brief survey.)
known
fastest
0{nm
algorithms are those of Goldberg and Tarjan [8,10], with a running time of
log (n^/m)), and that of Ahuja and Oriin
Both of these algorithms
we
no
then each infinite capacity can be replaced by the
without affecting the problem.
The maximum flow problem has
capacities are infinite but
[2],
with a running time of
method proposed by Goldberg
are refinements of a generic
shall call the preflow algorithm.
0{nm +
For networks with
m = n(«^),
n'^^
[8],
the Goldberg-Tarjan
\ogU).
which
bound
is
O(n^), which matches the bound of several earlier algorithms [11,13,15,20]. For networks with
m = 0(n'^~'-)
the
for
bound of
namely
unless
U-
some
constant e
>
0, the
Goldberg-Tarjan bound
the earlier Sleator-Tarjan algorithm [16,17].
0{n'^) for
m =0(n) or m
some constant
k,
Under
a sophisticated data structure, the
and Tarjan [18,19], whereas the
alone.
to
We
0(nm +
first
n"^
results in a
log/i),
which matches
the similarity assumption [7],
the Goldberg-Tarjan
bound
= Q(n').
The former uses
strate that the use
0(nm
bound beats
the Ahuja-Orlin
The Goldberg-Tarjan and Ahuja-Orlin algorithms obtain
techniques.
is
latter
their
dynamic
Our main purpose
uses scaling.
speed from two different
tree structure of Sleator
in this
paper
is to
demon-
of both techniques results in efficiency not obtained by using either technique
modify the Ahuja-Oriin algorithm
^
loglogt/
).
Then we show
running time of
0(nm
log(
to obtain a small
that the use of
—
m —
;
r
77
dynamic
improvement
in
running time,
trees in the modified algorithm
+2)). Under the similarity assumption, this
loglogf/
bound
with
is
better than
m =0(n)
and
all
previously
U = 0(n*)
for
known bounds
some constant
for the
k,
the
maximum
bound
is
flow problem. For networks
0(nm
loglogn), which beats
both the Goldberg-Tarjan bound and the original Ahuja-Oriin bound by a factor of logn/loglogn.
Moreover, the bound rapidly approaches 0{nm) as the graph density
m/n
increases.
3-
Our paper
consists of four sections in addition to this introduction.
we review
the preflow algorithm. In Section 3
analyze our improvement of
method of Section
2.
it.
In Section 4
In Section 2
we review
the Ahuja-Oriin algorithm and descril>e and
we combine
dynamic
the use of
3 and analyze the resulting algorithm. Section 5 contains
some
trees with the
final
remarks.
The Preflow Algorithm
method of Ford and Fulkerson
In contrast to the classical augmenting path
flow along an entire path from
to
once, the preflow method
at
:
The key concept underlying
at a time.
A preflo\^' f IS
[11].
.s
For any vertex m,
tion of (3).
let
on
moves flow along
constraint
is
edge
the edges satisfying constants (1), (2),
and a relaxa-
the ^owexce^^ of
^
w be e(w) =
/(v.w).
e
E
the following:
e(w) >
We
a single
by Karsanov
is
v:(v,>v)
The required
which moves
that of a preflow, introduced
the algorithm
a real-valued function
[6],
call a vertex v active if
for all
v^
t
w
e V-{s} (nonnegativity constraint).
and e(v) >
0.
Observe
(4)
that the nonnegativity constraint implies
\ha.te{s)<0.
The
«/{v,h)
=
capacity
residual
u(v,w}-f(v,w).
An
of
edge
edge
an
is
(v.w)
saturated
if
with
Uy<v,w)
respect
=
a preflow and
a
/
preflow
is
and unsaturated otherwise. (The
capacity constraint implies that any unsaturated edge (v,w') has «/(v,w)
The preflow algorithm maintains
to
>
0.)
moves flow from
active vertices through
unsaturated edges toward the sink, along paths estimated to contain as few edges as possible.
Excess flow that cannot be moved to the sink
test paths.
is
Eventually the preflow becomes a flow, which
As an
to the
nonnegative integers such that d{s)
every unsaturated edge (v,m).
A
d(v) < min {dj{v,s)
where df{v,w)
is
a
maximum
+
n,dj{v,t)),
unsaturated and d{v)
=
n,
proof by induction shows
is
the
flow.
= d{w) +
1.
d(t)
=
that,
0,
flow
/.
We
is
a function
and d(v) < d(w)
+
1
d
for
for any valid labeling d,
minimum number
w consisting of edges unsaturated with respect to the
(v,w)
is
estimate of path lengths, the algorithm uses a valid labeling, which
from the vertices
V to
returned to the source, also along estimated shor-
call
of edges on a path from
an edge (v,w) eligible
if
4-
The algorithm begins with an
preflow/and valid labeling d defined
initial
as follows:
f
w(v,h)
if V
/(v.w) =\ -mCw.v)
w = 5,
if
if V
d(v)
The algorithm
= min
\d^ {\,s)
consists of repeating the following
= s,
7i
+
5
and
w
*
5,
/i,d/v,f)}.
two
steps, in
any order,
until
no vertex
is
ac-
tive:
push
(v,w).
Applicability: Vertex v
is
active and edge (v.w)
Action: Increase /(v.w) by
min
{e(v),uy<v,H')).
is eligible.
The push
is
saturating
if
(v.w)
is
saturated after the push and nonsaturating otherwise.
relabel (v).
Applicability: Vertex v
is
Action: Replace d{v) by
When
active and
no edge
min [d(w) +1
the algorithm temiinates,/is a
I
(v,>v) is eligible.
(v,w)
maximum
is
unsaturated}.
flow.
Goldberg and Tarjan derived the
fol-
lowing bounds on the number of steps required by the algorithm:
Lemma
The
2.1 [10].
total
Lemma
Relabeling a vertex v
number of relabelings
2.2 [10].
There arc
at
strictly increases d{\).
No
vertex label exceeds 2n-\.
at
most 0{n^m) nonsaturating
0{n^).
is
most 0{nm) saturating pushes and
pushes.
Efficient implementations of the
and relabeling steps
to perform.
vertex, construct a (fixed)
the
first
on the
list,
list
above algorithm require a mechanism for selecting pushing
Goldberg and Tarjan proposed the following method. For each
A{\') of the
as the current
edges out of
edge out of
v.
To
v.
Designate one of these edges,
initially
execute the algorithm, repeat the following
5-
step until there no active vertices:
push/relabel
(v).
Applicability: Vertex v
is active.
Action: If the current edge (v.w) of v
is
not the last edge on A{v),
make
With
0{n^m) time bound
ing push. This gives an
Making
steps.
the
implementation, the algorithm runs in
this
first
edge onA(v) the current one.
0(nm)
time plus 0(1) time per nonsaturat-
for any order of selecting vertices for push/relabel
the algorithm faster requires reducing the time spent
number of such pushes can be reduced by
Goldberg and Tarjan showed
that
FIFO
on nonsaturating pushes. The
selecting vertices for push/relabel steps carefully.
selection
[3]
showed
tion (always pushing flow from a vertex with highest label) reduces the
Ahuja and Orlin proposed
to 0{n''m^'").
number
active, first selected) reduces the
(first
of nonsaturating pushes to 0{rr'). Cheriyan and Maheshwari
pushes
if (v,iv)
the next edge after (v.w) the current one. Other-
make
wise, perform relabel (v) and
perform pay/2(v,w). Otherwise,
is eligible,
that highest label selec-
number of nonsaturating
which we discuss
a third selection rule,
in the
next section.
3.
The Scaling Preflow Algorithm
The
algorithm,
imum
behind the Ahuja-Orlin algorithm, henceforth called the scaling preflow
intuitive idea
is to
move
large
amounts of flow when possible. The same idea
capacity augmenting path method of
rithm of
Gabow
[7].
One way
from a vertex of large excess
reduce the
maximum
Making
For
tor
method
[4]
to apply this idea to the preflow algorithm is to
to a vertex of
smaU
k>2. A
As
smallest
vertex v
is
excess, or to the sink.
The
always push flow
effect of this is to
at a rapid rate.
precise requires specifying
said to
have large excess
if its
the algorithm proceeds, k remains fixed, but
power of k such
behind the max-
and the capacity scaling algo-
when an excess
purpose the scaling preflow algorithm uses an excess bound
this
wise.
this
excess
Edmonds and Karp
is
that
A>U.
is
A and
large and
when
small.
an integer scaling fac-
excess exceeds A/k and small excess other-
A
periodically decreases. Initially,
The algorithm maintains
the invariant that e(v)
<A
A
is
for every
active vertex v. This requires changing the pushing step to the following.
push (v,w).
Applicability: Vertex v
Action:
Uw^^t,
is
active and edge (v,w)
increase /(v,w) by
min
is
eligible.
{e(v),Uf (v,w), A-e(w)). Otherwise (w
the
=
t).
6-
incrcasc/(v,w) by min
The algorithm
A
stant.
{
e(v),uy (v,w)}.
number of scaling phases, during each of which A remains con-
consists of a
phase consists of repeating push/relabel steps, using the following selection
A by
no active vertex has large excess, and then replacing
A/k.
Large excess, smallest label selection: Apply a push/relabel step
among such
choose one of smallest
vertices,
edge capacities are
If the
phases: after [log^
+
^7
integers,
A <
phases,
ij
1,
to a vertex v
number of nonsaturating pushes.
of large excess;
label.
algorithm terminates after
tlie
which implies
We repeat the
most [log^
at
U+
that /is a flow, since the algorithm
Ahuja and Orlin derived a bound of 0(kn^
tains integrality of excesses.
rule, until
analysis here, since
log;^
U) on
ij
main-
the total
provides motivation for our
it
modification of the algorithm.
Lemma
The
3.1 [2].
0{kn'^ Gog*
U+
total
number of nonsaturating pushes
preflow algorithm
is
1)).
0= ^
Consider the function
Proof.
in the scaling
We
e{v) d{\-)IA..
call
<t>
the potential of the current
V active
preflow
/
and labeling
Q<^<2n~
0<e(v)/A<
O by at least
w with
d(yv)
ing a relabeling or
=
\lk, since the
2/z^.
[log;:
U
+
Llog;t t/
-(-
,
when A changes. A
value of
When A
times.
\}
2j
O
Q<d{\)<2n
and
.
The
and the
Choosing
)t
changes,
Thus
total
O
is
for every active vertex v,
A
orw =
t.
The value of <t> can
relabeling of a vertex v increases
A
due
the total increase in
is at
by
at
most
2n'^.
is at
at
is
total
is
at
most
at
most 2«'
most k times the sum of the
most 2kn^ [^og*
be constant independent of n gives a
most the amount
This happens
over the entire algorithm
number of nonsaturating pushes
O. This
k, to at
<I>
increase only dur-
over the entire algorithm,
to relabelings,
^
nonsaturating push-
from a vertex v with excess more than Alk
increases by a factor of
total increase in
to
push
d(v) -1 and e(w)< A/k
d{v) increases. Thus the total increase in
most
1
throughout the algorithm. Every pushing step decreases O.
ing step decreases
to a vertex
Since
d.
initial
^ + 3j D
.
time bound of
Oinm +
n'^
\ogU)
for the scaling preflow algorithm, given an efficient implementation of the vertex selection rule.
One way
to
containing
implement the
all
by vertex
label,
each
set
large excess vertices with the corresponding label, and to maintain a pointer to the
nonempty
set
Oimn
\ogU).
-(-«'
rule is to maintain an array of sets indexed
of smallest
index.
The
total
time
needed
to
maintain
this
structure
is
-7
Remark. The bound for the scaling preflow algorithm can be improved
terms of a different parameter. Let U*
in
=4 +
'^
O
+
(nni
can
pushes
turating
be
reduced
logU*). This improvement
n'^
O
to
is
and
{n^ iogU*)
the
to
/
be the
first
phase such that
A<U*
and including the change from phase
.
/
Then
-
1
time
overall
The argument
[4].
algorithm maintains the invariant that the total excess on active vertices
phase
bound on nonsa-
the
improvement Aho and Li
similar to the
measured
it is
E
minimum-cost flow algorithm of Edmonds and Karp
the
Then
u {s,v)/n.
v:(i,v)€
slightly if
obtained for
The
as follows.
most nU'
.
^ InlT'^ = O(n^).
most n
is at
i
is at
to
Let
O due to phase changes up
the total increase in
to phase
is
[1]
bound
The
7=0
due
total increase in
to later
Having described
phase changes
is
O
(«^ logt/*).
the Ahuja-Orlin algorithm,
we
D
consider the question of whether
ning time can be improved by reducing the number of nonsaturating pushes.
Lemma
bounds the number of nonsamrating pushes by estimating the
3.1
potential O.
due
to
Observe
that there is
an imbalance in
phase changes, whereas only 0(n^)
is
due
to relabelings.
mate by decreasing the contribution of the phase changes,
tion of the relabelings.
We
Making
0{n^
this estimate:
log^
Our plan
at the cost
its
run-
The proof of
total increase in the
U) of
is
to
the increase
improve
is
this esti-
of increasing the contribu-
plan work requires changing the algorithm.
this
use a larger scale factor and a slightly more elaborate method of vertex selection.
push flow from a vertex of large excess, choosing among such vertices one with highest
We
label.
This vertex selection rule does not guarantee that a nonsaturating push moves enough flow, and
we need
a stack-based implementation to ensure this.
tices,
with the property that
of
At the beginning of
V.
if
w
is
just
on top of
v
The algorithm maintains
on the
a stack 5 of ver-
stack, then (v,wO is the current
a scaling phase, the stack is initialized to
be empty. The phase consists
of repeating the following step until no vertex has large excess, and then replacing
stack push/relabel: If S
S
of
Apply the appropriate one of the following three
V.
Case
definitely
1
:
(v,w)
this section).
while S
Case
is
nonempty. Let
is eligible.
If the
is
v be the top vertex
A/k.
on 5 and (v,w) the current edge out
cases:
Perform push (v,w) (modified as described
push
nonempty and
2: (v.w)
A by
empty, push onto 5 any large-excess active vertex of largest label.
is
Now
is
edge out
is
its
nonsaturating and e(v)
top vertex has small excess,
not eligible and
out of V by the next edge on
>
A
is
(v).
not last on
A
(v).
0,
push
pop
w
at the
onto
S.
beginning of
Otherwise,
S.
Replace (v,w) as the current edge
-8-
Case
and
3: (v,w) is not eligible
is last
(v.w) as the current edge out of v by the
its
A
on
(v).
first
Relabel v and pop
from
it
edge on A(v). While S
is
Replace
S.
nonempty and
top vertex has small excess, pop S.
Observe
that the stack push/relabel step
is
augmented with some
just a push/relabel step
manipulation of the stacks.
Before carrying out a detailed analysis of
Every push made by the algorithm
results in
tion of
move
we make
A
from a vertex of large excess.
is
several observations.
push along an edge (v.w)
one of three events: saturation of (v.w), conversion of v into an inactive vertex, or addi-
w
to
S with an excess of A.
In the third case (and only in the third case), the
push
may
zero flow.
Lemma
3.2.
algorithm
is
The
Lemma
0{nm + kn^ +
3.1, but
Y,
number of nonsaturating pushes made by
total
To bound
Proof.
<S?=
this algorithm,
n^ Oogt
U+
the stack-based scaling prcfiow
1)).
number of nonsaturating pushes, we use an argument
the
with two potentials instead of one. The
By
^(^) '^(v)/A.
the analysis in the proof of
first
Lemma
potential
that of
is
3.1, ever>'
proof of
like the
Lemma
push decreases
<i>,
3.1,
and
V active
O over all
the total increase in
or decreases the size of
ing flow from V to
unstacked
until
it
w
is
S.
phases
(9(n^ log^U). Every nonsaturating push either increases
is
Consider a nonsaturating push
and stacking w.
relabeled or until
When w
its
w to
the relabeling of
w
after
w is
k>
2, there are a total
decreases in
stacked,
excess decreases to
charge the push from v to
caused by pushes from
is
that increases the size
h',
it
at
has an excess of A.
it
is
It
cannot be
most A/k. In the former case we
in the latter, to the at least
stacked and before
of S, say by push-
unstacked. Since
1-1 Ik decrease in
1
-1 /i >
1
/2 for
<i>
all
of O(n^) pushes charged to relabelings and Oin^ \ogkU) pushes charged to
(t>.
We now
count the nonsaturating pushes that reduce the size of
pushes that begin with more than one vertex 5
ceded by a nonsaturating push
is
0{n^
that adds a vertex to 5,
logytt/), since
The number of such
S.
each such push must be pre-
and the number of these
is
0{n'^ logit/) by
the above argument.
The remaining pushes
end with none on
5.
We
decrease in 0, namely
that
call
\/k\
are those that begin with
such pushes emptying.
thus using
0(^71^ \ogkU) on their number.
potential,
must be counted
We
O
to
An emptying
one vertex on S and
push can cause a rather small
bound such pushes only gives a bound of
count emptying pushes more carefully by using a second
^2- The definition of O2 involves two parameters, an integer
/
and a
set P.
The value
of
/
is
equal to the
empty during
minimum
with label equal to
The
that
Observe
the current phase.
The
creasing during a phase.
Observe
of 2« and the smallest label of a vertex added to S while S was
P
P
=
/
the beginning of a phase and
2/i at
consists of all vertices with label greater than
and
/
is
/
all
nonin-
vertices
from which an emptying push has been made during the current phase.
empty
is
definition of
/
set
that
P never loses
the beginning of a phase and
at
a vertex during a phase.
<I>2 is
Z
<E>2=
+
e(v) (rf(v) -/
1)/A.
(If/'
= 0,
<I>2=0.)
ve P :e{v)>0
Observe
by
at least
hence so
1
is
The number of vertices added
Ik.
the
number of emptying pushes
To bound
the
total increase in
to
P
< O2 < 2«^. Any emptying push
that
P
to
^2
in
are
due
is at
that decrease
to relabelings
O2 by
fact that the relabeling
There are
due
at
may add
v to P.
most 2n decreases
to decreases in
The
total
is at
/
in
Thus
/
4kn^ + 2n^llogi,U +
As
most 2«^ [logt
O2
in the
is initially
by
at least 1/^,
and to decreases
{
5 when S
of doubly
1,2,. ...2/1-1
}.
We
in
P has
that decrease
zero.
Thus
is
is
0(nm). The only
empty. For
linked
hsts
the
(A vertex added
A
<I>2)
relabeling of
one" accounts
at
for the
growth of
to the
by one adds
<[>2
all
most n/k
to
phases.
by
the total
at least 1//: is at
most k times the
number of such pushes
purpose,
this
list(r)
=
to
is at
most
perform saturating
significant remaining issue
{i
this structure requires (9(1)
The pointer can only
change in label) or due
/.
we bound
small excess. Thus the total increase in
/k over
Ij
/
phases, and
/k.
we
e N:
is
to a
how
e(i)>A/k and
d(i)
=
at
most
pointer needs to be incremented or decremented
is
choose
for
r}
which
each
list (r) is
time per push operation plus time to main-
increase due to a relabeling (by at most the
phase change (by
to
maintain a data structure consisting of
also maintain a pointer to indicate the largest index r for
non-empty. Maintaining
tain the pointer.
+
f/
decrease in
1
most 4n^
Goldberg -Taij an and Ahuja-Oriin algorithms, the time
vertices to add to
€
at least
all
O2
.
\\
pushes and relabeling operations
collection
A
or decreases
by
relabelings contribute at
per phase.
number of emptying pushes
total increase in (i>2, since
r
<I>2
P
over
\j
in d(v) plus one; the "plus
most the increase
at
O2, since when the decrease occurs every vertex in
a
<I>2
U+
because of an emptying push has zero excess and hence adds nothing to
a vertex v increases
<I>2
most n [logi
do not decrease
that
number of emptying pushes
O2. Increases
either adds a vertex to
2n).
Oin'^
amount of
ConsequenUy, the number of times the
+ nQogk U +
1)).
The
overall running
10
time of the algorithm
thus
is
0(nm + kn^ + n^ilog/^ U +
l)).
Choosing
k=l
log t//log log
Ul
gives the following result:
Theorem
The stack-based
3.3.
0(nm + n^
^°f^,, )time.
,
log
Remark.
\ogU
=
2 until
the switch occurs,
4.
we
O
to
U replaced
O
6'*)
is
\ogU' /log
A
increase
which A >
These phases account for only
as above with
to ^ =1"
and then switch
the initial phases (those in
is
improved
be
(nw + n' logt/*/loglogf/*),
We
where
use a scaling factor of
E
A<U'
When
rithm.
can
" {s,v)/n. The algorithm niust be changed slightly.
Y,
v:(j,v) e
k
bound
This
U' =4 +
scaling preflow algorithm, with an appropriate choice of ^, runs in
(if
the
necessary) to be a power of
same
as that in the
(n^) nonsaturating pushes.
by U'
.
for the remainder of the algo-
logL'*"j
The
The
k.
remark following
analysis of
Lemma
3.1.
analysis of the remaining phases
D
Use of Dynamic Trees
The approach taken
in Section 3
tices for push/relabel steps carefully.
number of pushes by choosing
was
to reduce the total
An
orthogonal approach
is to
ver-
reduce the total lime of the
pushes without necessarily reducing their number. This can be done by using the dynamic tree
data structure of Slcator and Tarjan [18,19].
algorithm with a bound of p>
from 0(p)
to
0(nm log(-^
mn
prove a general theorem to
of the preflow algorithm
mn on
-i-
1))
the total
conjecture that, given a version of the preflow
number of pushes,
by using dynamic
this effect,
we have
We
we have been
considered.
the running time can be reduced
Although we do not know
trees.
applies
the
to
o(^iog(
scaling
log iogU
m :'°f,,
The dynamic
in this case is
As an example,
preflow
0{n^). In
algorithm
the
0(nm
log {n^ Im)) time bound
this section
of Section
3,
we
shall
resulting
show
The data
find root (v):
time
bound
same idea
bound
of
^2)).
tree data structure allows the
directed from child to parent.
itself.
rule; the
that the
a
in
maintenance of a collection of vertex-disjoint
rooted trees, each edge of which has an associated real value. Each tree edge
of
to
able to obtain such a result for each version
of Goldberg and Tarjan results from using d>Tiamic trees with the FIFO selection
on the number of pushes
how
We
is
regarded as being
regard every vertex as being both an ancestor and a descendant
structure supports the following seven operations:
Find and return the root of the trce containing vertex
v.
11
find size (v):
Find and retuna the number of vertices in the tree containing vertex
find value (v):
Find and return the value of the
the value returned
find min (v):
edge leaving
tree
v.
If v is a tree root,
is infinity.
w of v with minimum find
choose the vertex w closest to the tree root.
Find and return the ancestor
case of a
change value
Add
(v,x):
tie,
real
v.
number x
value (w). In
of every edge along the path from v to
to the value
find root (v).
Combine
link (v,w,xy.
the trees containing v and
giving the edge (v,w) the value
w are
in the
Break the
cut (v):
to
same
sequence of
(9(/log (2
+
1))
time
/
if
tree containing v into
two
trees
by deleting
the edge joining v
if v is
This operation
a tree root.
tree operations, starting with an initial collection of singleton trees, takes
if z is
maximum
the
tree size [18,19].
In our application, the dynamic tree edges form a subset of the current edges out of
vertices.
dynamic
Every dynamic
tree
A on
size of a
an eligible edge;
is
the
maximum
dynamic
tree.
Two
ability.
excess
at
no active
residual capacity.
tree push/ relabel (v):
active.
the
The
the preflow algorithm, called the tree
parameters govern the behavior of the algorithm, a
z, 1
<
z
<
n,
on the maximum
consists of beginning with an initial preflow
vertices:
is
is its
all
entire path at once, rather than
an active vertex, and a bound
The algorithm
Applicability: Vertex v
value
moved along an
labeling d, and with every vertex as a one-vertex
until there are
its
The following version of
at a time.
preflow algorithm, uses this
bound
edge
tree data structure allows flow to be
along a single edge
v and
tree or if v is not a tree root.
parent; return the value of the deleted edge.
its
parent of v and
This operation does nothing
x.
breaks no edge and returns infinity
A
w by making w the
dynamic
tree,
/and
a valid
and repeating the following step
12
Action: Let (v.w) be the current edge out of
Apply
v.
one of the
the appropriate
following cases:
Case
J: (v,w) is eligible.
and 5 = eifx =
5 = min
r,
x=find
Let
A
(e,
root (w), e
-e(x)}
x ^
if
= min
{e(v), Uf (v.w), find
Send 5
t.
from V to
X.
The
tree
push
is
saturating if 5
push) and nonsaturating otherwise.
push), and find size (v)
Case
2; (v,w)
=
3: (v,w)
is
push
is
called a tree
push
min
Uf (v,w),find
is
x by
(v)} (before the
nonsaturating, e(v)
perform link (v,w,Uf
(v,h')).
=
(after the
Otherwise, while
and not
not eligible and
current edge out of v by the
last
on A(v). Replace (v.w)
last
on A(v).
as the current
edge
A (v).
out of V by the next edge on
Case
z,
(
to
perform cut (find min (w)).
not eligible
is
If the tree
+ find size (w) <
find value (find min (w))
= min
from v
units of flow
increasing /(v.w) by 5 and performing change value (h',-5). This
min (w)),
first
is
Relabel
edge onA(v). For every
tree
Replace (v,w) as the
v.
edge
(}',v)
perform
cut(y).
This algorithm stores flow in two different ways: explicitly for edges that are not d>'namic
tree
edges and implicitly
in the
dynamic
tree data structure for
After each cut, the flow on the edge cut must be restored to
when
the algorithm terminates, the correct flow value
its
edges that are dynamic tree edges.
correct current value.
on each remaining
In addition,
edge must be com-
tree
puted. For edges cut during the computation, the desired flow values are easily computed, since
the current residual capacity of an
The algorithm
just presented
Goldberg and Taijan
[9,10].
is
0(nm
tree
log
(z
pushes
The
in turn
[10].
+
is
1))
The
plus
tree size
total
OOog
returned by the cut operation.
Computing
most nfind value operations.
a generic, modified version of the
Goldberg and Tarjan gives the following
4.1
at
is
dynamic
tree algorithm of
maintains the following invariants: every active vertex
It
no excess exceeds A; no
Lemma
that is cut
on termination can be done using
correct flows
root;
edge
(z
exceeds
z;
every tree edge
eligible.
The
a tree
analysis of
result:
by
time
required
+
per tree push.
1))
is
is
the
dynamic
tree
The number of
prcflow
links, cuts,
algorithm
is
and saturating
0(nm).
efficiency of the algorithm depends on the
depends on
how
number of nonsaturating
tree push/relabel steps are selected.
We
shall
show
-I
analogous to that used in Section 3 results in a bound of
0(nm +
kn^ +
—
tree pushes,
that a selection order
fl"
z
which
(logi t/
+
1
))
on the
-13-
number of nonsaturating
of
Oinm
log
We
—
—
m —
f
-7
log log (7
+
;
(
A
In addition to parameters
maximum
(the
that
A>
and
A
maximum
z (the
this a total
time bound
2.
active vertex v
dynamic
tree
is
tree size), the algorithm
Whereas
=
decreases. Initially S
and
(J)
and k remain fixed
z
A
is
the smallest integer
said to have large excess if e(v)
is
said to be large if
The tree/scalmg algorithm
otherwise.
srrujll
An
U.
A
small excess otherwise.
excess) and
the scaling factor, and a stack S.
k,
From
the scaling factor.
by an appropriate choice of k and
2)) follows
throughout the running of the algorithm,
power of k such
is
version of the pre flow algorithm the tree/scaling prefiow algorithm.
call the resulting
uses a third parameter
where k
tree pushes,
consists
> A/k and
contains more than z/2 venices
it
of repeatedly applying the tree
push/relabel step to the top vertex on S, while manipulating S according to the following rules:
Rule
1
.
If
S
is
tices are roots
empty, add
of large
S any
to
add
trees,
active vertex that
to
vertices are roots of large trees and
S
a large-excess active vertex
have small excess, replace
all
minates
when A <
Rule
After a tree push from a vertex
2.
push X onto
Rule
3.
If all active ver-
of largest label.
A by
If all active
Alk. (The algorithm ter-
1 .)
Otherwise, while S
S.
the root of a small tree.
is
v
to a vertex x, if the
nonempty and
is
After relabeling a vertex, pop
it
from
S,
its
push
is
>
0,
nonempty or
its
nonsaturating and e{\)
top vertex has small excess, pop
and continue
pop S
to
until
it is
S.
top vertex has large excess.
Every
tree
push made by the tree/scaling algorithm
a small tree or has large excess.
Rules 1-3
tree
0{n} +
is
nlogk,
{U
-^
As discussed
\))
plus
(9(
1
)
from
is
in Section 3, the
a vertex that
is
either the root of
time necessary for implementing
per tree push, which
is
dominated by the bound on
pushes derived below.
—
2
Lemma
4.2.
The
tree/scaling
algonihm makes Oinm
-1-
kn^ +
(log^
U
-t-
1))
nonsaturating
tree pushes.
The argument
Proof.
defined by
0= ^
V
and changes
after a
analogous
We
e{v) d{v)! A.
10 the
proof of
have 0<<t><^^.
Lemma
3.2.
Increases in
Let the potential
O
are
due
O
be
to relabelings
acuve
in the value
change
is
of A. The
in the value of A,
<t>
total increase in
< 2n-
:,
<i>
due
to relabelings is Oin'^).
Immediately
because every active vertex must be the root of a
large tree, and only Iniz large trees can coexist.
Thus
the total increase in
O
caused by changes
14-
—
2
inAisOC
Gogi
+
f/
l)).
z
Consider nonsaturating
tex
added
least 1-1
We
to S.
/it
is
O caused by pushes from x before x
decrease in
n'
begin
with
0(n^
—
-^
0(n'
thus
pushes that add a vertex to
(logic ^'
h
+
Similarly, the
!))•
let
U+
is
popped. The
x be the ver-
at
one
least
pushes
tree
from
vertex
at
number of such
total
number of nonsaturating
more than one vertex on S and pop
(logt
For such a push,
5.
charge the push either to the relabeling of x that pops x from S or to the
->
pushes
tree
S
that
also
is
\)).
z
The
on 5 and
the only vertex
An emptying
them
push from
We call
after the push.
push from a vertex
a
v that is
such a push an emptying push.
To count emptying
a vertex v reduces e(v) to zero.
trees
is
and those from roots of large
we
pushes,
divide
trees.
charge an emptying push from the root v of a small tree to the event that either created
made
the tree or last
active,
popped from 5
is
from roots of small
into those
We
we must count
kind of nonsaturating tree push that
last
we charge
the
push
most one push, and there
If the tree
whichever occurred
v active,
Each
to the link or cut that created the tree.
0{nm)
are
was created before
v
links
became
and
cuts,
was created
If the tree
first.
link or cut
became
after v
is
charged
to at
which gives an 0{nm) bound on such pushes.
active, then v
became
active as a result of a sauirating
push, of which there are 0(nm), or as a result of a nonsaturating push from the root of a large
tree.
to
We
have already counted the number of such pushes that are nonemptying. Thus
count emptying pushes from the roots of large
We
integer
/
set P, as in the
proof of
Lemma
3.2.
^2
To
(As
in Section 3,
which A stays constant.) The
with label equal to
definition of
Oj
/
The value of
/ is
the
minimum
X
P
Tlie total increase in
due
consists of
all
an
of 2n and the
was empty dunng
the
maximal period of time during
vertices with label greater than
/
and
all
vertices
is
v €
is
P
define a phase to consist of a
to define
from which an emptying push has been made during the current phase. The
^2=
tion
set
we
we need
define Ot,
smallest label of a large-excess root of a large tree added to S while S
current phase.
suffices
trees.
count such pushes using a second potential
and a
it
to relabelings.
e(v)(d{v)-l + l)/A.
(If
P
=0,^2=0)
eiv) >
O; over
all
The 0{
phases
—
kz
Gog*
is
0(n'
U+
1))
-t-
—
Oog^
contribution
U+
is
1)).
due
The 0{n')
contribu-
to decreases in
/.
A
15-
decrease in
/
by one causes an 0(
—
)
increase in O2, since, just after such a decrease in
occurs,
/
kz
P
every vertex in
has level greater than
/,
P
and every active vertex in
has small excess and
the
is
root of a large tree.
Every emptying push from the root of
<^2'
which can happen Oikn^ +
U+
can happen OinQog^
Combining
Remark.
A
—
U+
Gog*
Since
])) times.
a large tree causes either a decrease
<
2
\))
at least
1/^ in
times, or an addition of a vertex to P,
which
a choice of
—+
=!"
/t
preflow algorithm runs in 0{nm\oz
and
2]
^
z
Lemmas
—
4.1
+
(
"
By
on
1))
the
argument using only
= min
{[
n
Proof.
the former.
number of nonsa-
<I>,
analogous
to the
D
3.1.
With
4.3.
+
(log2 t/
turating tree pushes can be obtained with a simpler
Theorem
dominated by
*"
kn
1
proof of Lemma
is
D
weaker bound of 0(nm + kn' +
slightly
bound
n, the latter
our estimates gives the lemma.
all
of
—
m
log^ t/
+
,n) the tree/scaling
l]
2)) time.
mlog \ogU
and
running
the
4.2,
of
time
algorithm
the
is
->
OCfrini +/tn-
+
Gogi
+ l))log(z +
f.'
This
D).
O(Aimlogn)
is
if
z
=[
z
On
the other hand,
the algorithm
if z
again
is
Case
=
0(nm
\ogU
1:
the
ai
I
compulation
logn).
We
complete
is
after
—
m
+
logt t/
<
l]
n.
one phase, and the running time of
consider two cases.
\og\ogU > ml\n. Then
^
log(
+
2)
=
Q.(\ogn),
and the
mloglogu
theorem
true.
is
logU
log \ogi' <
Case
2:
log2[/
= 0{m). Let
/
m
/
\n
.
Then log2t//log:log;t/ < m^"*, which implies
—
^2
us estimate the quantity
= 0{nm).
n (\ogt,U
+
\)
—
U
+ \)<
On
Gog^
hand,
other
the
U
+
\).
If r
if
z
=
n, this
=\
—
m
quantity
log^ t/
+
is
l]
,
^2
(log;,
2 run.
follows
Ii
that
the
running
time
of the
algorithm
is
z
—
nl02tL'
0{nm\ogiz +
^
^^—
\))
= 0(nm\0'ci
"log*
log
(
m
+
2)
- 0(log
m
+2)).
^locL''
(
r
m loglogc/
,,
+
2)).
It
We
remains
have log*
U
prove
to
= logU
I
log
that
m
(—
n
+
2).
——
16
-
Again
the proof divides into
Casel.l: \ogU
>{—
two cases:
+ 2)\ Then
loglogt/>4,
log
—r~77 +
mlog
then log
Case
log
which
(—7
>
(—
log
+
(-
implies
log log
logt-^
—
<
— log \ogU.
> log logt/ - log log \ogU - log (m/n) >
\ogU <
+
^
2)
(
2)
< log Qog^ +
—n +
=
2)^.
= 0(log(
2)
In this case log
^
log
+
(
mlog(— +
Also,
2)
-^^^
m
—+
log
(
2)), as desired.
—n +
—^—
-
(
But
'^
="
\ogU
log
\ogU < 4
< log
Hence
log logL^.
4
+
m
> log(m/n).
2)
n
logt/
2.2:
(
2)
log logf/
-J-
4
n
m\og\ogU
which implies
2),
+
= OC
2)
f-— +
—
m —
;
log logL
2)
as desired.
Thus
in all cases the algorithm runs in the claimed time bound.
D
1
Remark. The bound
in the
theorem can be improved
gously to the improvements in the bounds of Section
of the algorithms of Section 3 with k
A<
T
!*
+
(
m
obtain this improvement,
U' and then switch
,
2),
analo-
log logL'
we
run one
to the d>Tiamic tree algo-
Remarks
The
is
2 until
To
log
n
rithm,
5.
=
3.
0{nm
to
scaling preflow algorithm of Section 3 not only has a
We
likely to be very efficient in practice.
this is true.
The algorithm of Section 4
huge networks there
some chance
is
The two obvious open
bound
for the
maximum
cost flow problem.
It
is
is
perhaps mainly of theoretical interest, although for
of dynamic trees
theoretical questions are
is
not unreasonable to hope for a bound of
capacities arc not too large,
^=
i.e.
0{nm) would seem
to require
major new
ideas.
0(nm)
0(nm)
for the
in the time
minimum
maximum
flow
for graphs that are not too
—— = Oimin).
loglogf/
better than
be practical.
possible and whether our ideas extend to the
I02L'
whose edge
may
whether funher improvement
problem; note that the bounds of Theorems 3.3 and 4.3 are
sparse and
theoretical time bound, but
intend to conduct experiments to detenmine whether
that the use
flow problem
good
Obtaining a bound
2),
17
As
a partial answer to the secxind question,
Oirvn loglog
cost,
U
assuming
additional ones.
log (nC)) for the
all
It
edge costs are
will
minimum
integral.
we have been
able to obtain a time
cost flow problem, where
C
is
the
bound of
maximum edge
This result uses ideas in the present paper and some
appear in a future paper.
-18-
6.
Acknowledgements
We thank
7.
Andrew Goldberg
for inspiring ideas
and stimulating conversations.
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Operations
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107
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