4
HD28
M4 1
.
lAUG
IMPROVED TIME BOUNDS
FOR THE MAXIMUM FLOW PROBLEM
R. K.
J.
Ahuja
B. Orlin
R. E. Tarjan
Sloan W.P. No. 1966-87
December 1987
Revised: June 1988
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
23
1986
'*^
IMPROVED TIME BOUNDS
FOR THE MAXIMUM FLOW PROBLEM
R. K.
J.
B.
Ahuja
OrUn
R. E. Tarjan
Sloan W.P. No. 1966-87
December 1987
Revised: June 1988
Improved Time Bounds
for the
Ravindra
K
Maximum Flow Problem
Ahuja
James B. Orlin
'
Robert E. Tarjan
September. 1987
Revised June. 1988
ABSTRACT
Recently, Goldberg proposed a
The approach
flow problem.
time on n-vertex networks.
Sleator and Tarjan yields a
new approach
lo the
maximum
yields a ver>' simple algorithm
nmning
netv.ork
in
0{n^)
Incorporation of the dynamic tree data structure of
more complicated algorithm with
a running time
of
0{rvn log {n^ Im)) on m-arc networks. Ahuja and Orlin developed a variant of
Goldberg's algorithm that uses scaling and runs
in
works with integer arc capacities bounded by U.
ble
improvements
to the
ning lime of 0(,nm
Then we
0(,rm + n' logL') time on net-
In this
paper we explore possi-
Ahuja-Oriin algorithm. First we obtain an improved run-
n^ logL'/log logL') by using a nonconsiant scaling
-t-
obtain an even better
bound of 0{run
-t-
n^Qogt/)'^) by combining the
Ahuja-Orlin algorithm with the wave algorithm of Tarjan. Fmally.
the use of
dynamic
0{rvn log(
— Gogt/)'
factor.
we show
that
trees in the laner algorithm reduces the running time to
'
+
2)).
This result shows that the combined use of three
tn
different techniques results in speed not obtained
by using any of the techniques
alone.
MIT, Ciinbndge, MA 02139 Retearch ptrtUDy fupponed b> i Prendenti«] Yoimg InNSF. Cooiria S-JflSP ECS. ind grinu from .ViiJog DeMcts. Kpfit Compuitr Inc. «nd
Slcxn School at .Min»ganent,
vesuguor Fello»ihip from
Uie
f^ime Computer.
On
Itive from Indian Inrntuie of Technology. Kjnpur. Indu.
Dqx
Reie*n*
of
Computer Saence. Pnncaon Univemry. Pnnceuxi.
p«ru*l]y luppone^l by
NSF Gr»nt DCR-860596; md
M 085*4 ind ATAT
Office of Ntvxl
Bell Ubs. Murray Hill, N7 0^974.
Rettin* Coniiia N00014-87-K-O467.
Improved Time Bounds
for the
Maximum Flow Problem
Ravindra K. Ahuja
James
BO
'
rlin
Robert E. Tarjan
'
September, 1987
Revised June, 1988
1.
Introduction
We
We
formulate the problem as follows. Let
arc set E.
The graph
and a sink
t,
G
symmetric,
tices, the
number of
bounds,
G
iflow network
is
G = (V.E)
if
it
flow problem [5,6,13,15,20].
be a directed graph with vertex set
we assume
(v.w) e
i.e.
arcs,
£
t/
(w.v) e E.
maximum
and the
m^n and
iff
^
4.
We
denote by
n,
m, and
arc capacity, respectively.
Bounds containing
U
and
U
We
assume
number of
the
For ease
ver-
in stating time
are subject to the assumption that all
All logarithms in the paper are base
arc capacities are integral.
V
has two distinct distinguished vertices, a source s
and a non-negative real-valued capacity «(v,>v) on each arc (v,^) € E.
is
that
maximum network
consider algorithms for the classical
two unless an
explicit base is
given.
A flow f on
G
a netwoiit
is
a real-valued function
/on
the arcs satisfying the following con-
straints:
f(v,w) < u(v,w) for
f(y,w)
S
= -f(w,v)
/(v.**')
=
all
(v.w) e
for all (v.w)
for all
w
E
(capacity constraint),
e E (antisymmetry
(1)
constraint),
e V-[s.t] (conservation
(2)
constraint).
(3)
(v.w)e£
SloBi School of Msugeniail,
M J.T., Cunbndge. MA 02139.
veniguor FcDowihip from the NSF,
Conna
Re»e«rch piitudly lupponwl by a PreiidentUJ Young In-
8451517 ECS. »nd gnuiu
frcnn
Analog Devicei. Apple Computer
Inc..
ind
Prime Compuicr.
On
^
leave from Indian Intticule of Technology, Kanpui, India.
Dept
of
Compuler Saence. Pnnceloti Umveniiy, Pnnccum, NJ 08544 Bid
Researx* pamally lupponed by
NSF Grant OCR -8605962
AT&T
BeU Labi, Murray
Hill,
NI 07974.
and Office of Naval Reaearch Contract N00014-87-K-O467.
The value /I of a flow /is
I
the nel flow into the sink:
1/1=
I
/(v./).
(v.Oe£
A maximum flov.maximum
flow of
maximum
value.
The maximum
assimie that
all
arc capacities are finite.
path of infinite-capacity edges from j to
sum of the
finite capacities,
r
exists, then
some
If
each
problem
is that
of finding a
no
arc capacities are infinite but
infinite capacity
can be replaced by the
D
without affecting the problem.
The maximum flow problem has
a long and rich history, and a series of faster and faster
algorithms for the problem has been developed.
(See [10] for a brief survey.)
The previously
knowTi algorithms are that of Goldberg and Tarjan [8,10], with a running time of
fastest
0{,nm log (n^/m)), and that of Ahuja and Orlin
Both of these algorithms
we
flov,'
flow in a given network.
We
Remark.
is a
[1],
with a running time of 0{,nm
are refinements of a generic
method proposed by Goldberg
m
prefknv algorithm. For networks with
shall call the
+ n}
=
[8],
fi(n^), the Goldberg-Tarjan
logt/).
which
bound
is
O(n^), which matches the bound of several eariier algorithms [12,14,16,21]. For networks with
m = 0(n^"^)
the
for
bound of the
m
0, the
Goldberg-Tarjan bound
earlier Sleator-Tarjan algorithm [17,18].
U = 0(n*)
namely
unless
some constant e >
for
some constant
= 0(n)orm =
k,
the Ahuja-Oriin
Under
0(nm
\ogn),
the similarity
bound beats
which matches
assumption
the Goldberg-Tarjan
The former uses
a sophisticated data structure, the
and Taijan [18,19,20], whereas the
latter
their
We
dynamic
uses scaling. In this paper
we
tree structure of Sleator
explore improvements in
-»-
0(nm +
n^ logt//log logt/). In Sec-
describe an even faster variant, which uses a coristani scaling factor but combines the
scaling idea with the
0(nm
3,
a version of the Ahuja-Oriin algorithm that uses a stack-based vertex selection rule
and a nonconstanl scaling factor to obtain a time bound of
we
dynamic
begin in Section 2 by reviewing the generic preflow algorithm [8,9,10]. In Section
we develop
tion 4,
bound
speed from two different
the Ahuja-Oriin algorithm obtainable by incorporating other ideas, including the use of
trees.
[7],
Cl{n^).
The Goldberg-Tarjan and Ahuja-Orlin algorithms obtain
techniques.
is
fi^
wave algorithm of Tarjan
QogU)^'^). In Section 5
obtaining a running time of
0(nm
[21].
we add dynamic
This algorithm has a running time of
trees to the
— Oogt/)"^ +2).
The
method of Seaion
4, thereby
results in Sections 3-5 rely
on the
assumption that pointer manipulations and arithmetic operations on integers of magnitude
U take
log(
m
-3-
0(1) time. In Section 6 we consider the
namely
that arithmetic
with some
2.
final
on our time bounds of a weaker assumption,
effect
We
on integers of magnitude n takes 0(1) time.
conclude in Section 7
remarks.
The Preflow Algorithm
In contrast to the classical augmenting path
flow along an entire path from ; to
f
method of Ford and Fulkerson
a time. The key concept underlying the algorithm
[12].
A
moves flow along
once, the preflow method
at
that
is
For any vertex w.
let the
flow excess of
w
be e(w)
=
"^
(v,w)€
constraint
and a relaxa-
(1), (2),
The
/(v.w).
required
E
the following:
is
e(w) ^
We
a single arc at
of a preflow, introduced by Karzanov
preflow /is a real-valued function on the arcs satisfying constraints
tion of (3).
which moves
[6],
call a vertex v active if
for all
v^ / and
w
€ V-{s} (nonnegativity constraint).
e(y)
>
0.
Observe
arc
(v,>v)
(4)
thai the nonnegativity constraint implies
thatc(5)<0.
The
uj{v,w)
= u(v,w)-f(v,w). An
an
of
capacity
residual
arc
is
saturated
if
with
ujiv,w)
respect
=
>
/
is
(The
0.)
moves flow from aaive
a preflow and
preflow
a
and unsaturated otherwise.
capacity constraint implies that any unsaturated arc (v.w) has u/v.w)
The preflow algorithm maintains
to
vertices through
unsaturated arcs toward the sink, along paths estimated to contain as few arcs as possible. Excess
flow that cannot be
moved
to the sink is returned to the source, also along estimated shortest
paths. Eventually the preflow
As an
to the
every unsaturated arc
d(v) ^ min [df{v,s)
w
(v,w)
a flow,
which
is
a
maximum
+
nonnegative integers such that dis)
(v.h').
A
n,dj{v,t)],
=
unsaturated and d{v)
n,
proof by induction shows
where dj{v,w)
is
the
= d(w) +
1.
d{t)
that,
=
0,
is
/.
We
for any valid
call
a function
and d(v) < d(w) +
minimum number of arcs on
consisting of arcs unsaturated with respect to the flow
is
flow.
estimate of path lengths, the algorithm uses a valid labeling, which
from the vertices
to
becomes
1
d
for
labeling d,
a path from v
an arc (v,>v) eligible
if
.4-
The algorilhm begins with an
preflow/and valid labeling J defined
initial
u{v,w)
if
/(V.H')=' -u(w,v) if
V
The algorithm
= m\n
[df(v,s)
consists of repeating the following
=s,
w = s,
if V
d(v)
as follows:
*
+
J
and
w
^s.
n,djiv,t)].
two
steps, in
any order,
until
no vertex
is
ac-
tive:
Push
(y,w).
Applicability: Vertex v
is
active and arc (v,w)
Action: Increase /(v.w) by
min
is
[e(v),uj{v,w)].
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 arc (y,w)
min {d(w) +1
the algorithm terminates, /is a
I
(v.w)
maximum
is
is eligible.
unsaturated).
flow.
Goldberg
arxl
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
is
No
vertex label exceeds 2n-l.
saturating pushes and at
most 0{n^m) nonsaturating
strictly increases d{v).
0(,n^).
2.2 [10]. There are at most
0{nm)
pushes.
Efficient implementations of the
and relabeling steps
to perform.
vertex, construct a (fixed)
first
on the
list
above algorithm require a mechanism for selecting pushing
Goldberg
arxl
Tarjan proposed the following method. For each
A{y) of the arcs out of
Lst, as the current
arc out of
v.
To
v.
Designate one of these arcs,
initially the
execute the algorithm, repeat the following step
-5-
no
until there
PushJRelabel
active vertices:
(v).
Applicability: Vertex v
is
active.
Action: If the current arc (v.h-) of v
is
not the last arc
on
i4(v),
is
make
wise, perform relabel (v) and
With
this
eligible,
perform push(y,w). Othenvise.
if
(v.w)
the next arc after (v.w) the current one. Other-
make
the
first
arc
on
/4(v) the current one.
implementation, the algorithm runs in 0(jvn) time plus 0{\) time per nonsaturat-
ing push. This gives an 0{,n}m) time bound for any order of selecting vertices for push/relabel
Making
steps.
the algorithm faster requires reducing the time spent
number of such pushes can be reduced by
Goldberg and Tarjan showed
that
FIFO
on rwnsaturating pushes. The
selecting vertices for push/relabel steps carefully.
selection
(first
active, first selected) reduces the
of rxjnsaturating pushes to 0{n}). Cheriyan and Maheshwari
[3]
showed
0{n}m^'^). Ahuja and Orlin proposed a
to
that highest label selec-
number of nonsaturating
tion (always pushing flow from a vertex uiih highest label) reduces the
pushes
number
which we discuss
third selection rule,
in the
next section.
3.
The Scaling Algotithm
The
behind the Ahuja-Oriin algorithm, henceforth called the scaling algo-
intuitive idea
rithm, is to
move
large
amounts of flow when possible. The same idea
capacity augmenting path method of
Gabow
[7].
One way
Edmonds and Karp
[4]
excess
Making
For
this
this
to apply this idea to the preflow algorithm
its
the algorithm proceeds, k remains fixed, but
vertex
v.
is
The
to
always push flow from a
effect of this is to reduce the
method precise requires specifying when an excess
v is said to have large excess if
power of k such
maximum
at a rapid rate.
purpose the scaling algorithm uses an excess bound
A vertex
behind the
and the capacity scaling algorithm of
vertex of large excess to a vertex of small excess, or to the sink.
maximum
is
that
AS
I/.
A
arxl
is
when
large and
an integer scaling factor
A
periodically decreases. Initially,
the invariant that e{v)
SA
A
is
for every active
Push (v.w).
active and arc (v.w)
Action: If w *r, increase /(v.w) by
increase /(v.w) by
min
min
is eligible.
(f(v).«/(v,H'), A-*(h')). Otherwise
[e{v),Uf{v,w)).
As
the smallest
This requires changing the pushing step to the following.
is
kt2.
excess exceeds A/k and small excess otherwise.
The algorithm maintains
Applicability: Vertex v
small.
(w =
r),
consists of a
The algorithm
A
stant.
number of scaling phases, during each of which A remains con-
phase consists of repeating push/relabel steps, using the following selection
active vertex has large excess, and then replacing
no
A by
rule, until
The algorithm terminates when
A/it.
there are no active vertices.
Large excess, smallest label
excess;
among such
choose one of smallest
vertices,
t/+
after [ log*
mainuins
A<
phases,
ij
integrality of vertex excesses.
number of nonsaturating pushes.
the total
tion for our
Lemma
first
3.1
The
+
<
<1>
and labeling
ing step decreases
w
<1>
most
2n^
I
initial
[
log*
Thus
When A
U+
U + 2}
in
The
.
1
total
Thus
O
call
it
provides motiva-
the
algorithm
scaling
is
the potential of the current
< d(v) < 2n
and
push
A
is
A
<I>.
nonsaturating push-
from a vertex v with excess more than A/k
w=
or
r.
The value of
A
can increase only
<I>
the total increase in
<I>
by
is at
at
most 2n^.
at
most the
This happens
over the entire algorithm
number of nonsaturating pushes
This
<1>
due to relabelings, over the entire algorithm,
increases by a factor of )!:, to
C).
for every active vertex v,
relabeling of a vertex v increases
the total increase in
<I>
We
Every pushing step decreases
changes.
changes.
times.
\j
< e(v)/A <
value of <I> and the total increase in
Choosing
\}
a flow, since the algorithm
repeat the analysis here, since
= d(v) -1, and e(w) ^ A/k
when A
increases.
logi
We
at least l/k. since the
with d(w)
most 2n^.
is at
/is
U+
acbve
the algorithm.
by
during a relabeling or
amount ^(v)
that
log*
[
Ahuja and Orlin derived a bound of 0(kn^ log* U) on
e(v)rf(v)/A.
J^
Since
d.
< 2«^ throughout
to a vertex
which implies
most
after at
l)).
V
/
1,
number of nonsaturating pushes
total
Consider the function <!)=
preflow
label.
modification of the algorithm.
[1].
OC/k/j^OogilZ
Proof.
a push/relabel step to an active vertex v of large
edge capacities are integers, the algorithm terminates
If the
phases:
Apply
selection:
is at
most 2kn^
[
is at
most
most k times the sum of
log^ t/
+
3j
.
at
the
D
0(nm
-«-
n^ log U)
for the scaling algorithm, given an efficient implementation of the vertex selection rule.
One way
to
all
implement
/t
to
be
a constant
independent of n gives a
total
the rule is to maintain an array of sets indexed
time bound of
by vertex
label,
each
set containing
large excess vertices with the corresponding label, and to maintain a pointer to the
set of smallest index.
Remark. The bound
The
total
time needed to maintain
for the scaling algorithm
this structure is
can be improved slightly
0(nm + n^
if
it
is
nonempty
log U).
measured
in
terms
-7-
pushes can be reduced to
improvement
is
O
A^
phase y
-
changes
to phase ;
1
is
O
is at
(n^ logt/*).
<I>
is at
due
to
is
as follows.
most nU*
O
(nm + n} log
The algorithm maintains
Let phase j be the
.
The
).
to
bound on nonsaturating
to
phase such that
first
and including the change from
total increase in
<I>
due
to later
phase
D
time can be improved by reducing the number of nonsaturating pushes. The proof of
bounds the number of nonsaturating pushes by estimating the
that there is an
This
the invariant
Having described the scaling algorithm, we consider the question of whether
Observe
(/*).
derived for their capacity-scaling
phase changes up
most n 2) 2n/2'~* = 0(n
the
bound
the overall time
Edmonds and Karp
The argument
the total iixrcase in
Then
u(s,v)/n.
J^
U') and
excess on active vertices
Then
{/*.
(n^ log
analogous to the bound
transportation algorithm [4].
that the total
(/*=4+
Let
of a different parameter.
imbalance
changes, whereas only O(n^)
in this estimate:
due
is
0{n^
to relabelings.
log*
its
running
Lemma
3.1
total increase in the potential
U) of
Our plan
is
the increase
to
improve
is
due
to
<I>.
phase
this estimate
by
decreasing the contribution of the phase changes, at the cost of increasing the contribution of the
relabelings.
We
We make
Making
this
plan work requires changing the algorithm.
use a nonconstant scale factor and a slightly more elaborate method of vertex selection.
use of the stack-push/rtldbcl step defined below, which performs a sequence of push
and relabel steps using a stack. The stack provides an alternative way of avoiding pushes
large-€xcess vertices (other than
to
r).
Stack-Push/ReUibel(r).
Applicability: Vertex r
is
active.
Action: Initialize a stack 5 to contain
r Repeat
the following step until
Stack Step. Let v be the top vertex on S and
Apply
the appropriate
Case
let
5
is
empty:
(v,^) be the current arc out of
v.
one of the following cases:
1: (v,^) is not eligible.
Case la:
{v,-w) is not last
by the next
Case lb: {y,w)
arc
is last
on
on A(v). Replace
(v,**-)
as the current arc out of v
y4(v).
onA(v). Relabel
the current arc out of v
by the
v and
first
arc
pop
it
from
on A(v).
5.
Replace (v.w) as
-8-
Case 2:
(v.h') is eligible.
Case 2a: ^Cw) > A/2 and w*^t. Push
Case 2b: e{w) <
ning of
from
Some remarks
it
moves
at
w=
it
is
added to
and
of which can be
Our
w
begin-
= 0, pop
v
nonsaturating push big
if
*
If e{y)
r).
v while v
is
on
S, at
S
call a
During an execution of stack-
otherwise.
little
vertex v can only be popped from
Of the pushes from
reduced to zero.
if
at the
S.
units of flow
A
S.
A
maintain e(yv) <
push/relabel, every vertex pushed onto S, except possibly the
when
S.
Perform push{\,-w) (modified as
r.
this section to
onto
about stack-push! relabel are in order. Let us
A/2
least
till or
k-
first,
has an excess of
after
it is
most two
relabeled or
at least
its
A/2
excess
is
are nonsaturating, only the last
little.
variant of the scaling algorithm, called the stack scaling algorithm, consists of phases
A
just as in the scaling algorithm.
phase consists of repeatedly applying the stack-pushi relabel
when
step to a large-excess active vertex of highest label; a phase ends
there are
no large-excess
active vertices.
Lemma
3.2.
The
Oikn'^-t-n^QogtU +
To bound
Proof.
Lemma
namely
3.1, but
0=
number of nonsaturating pushes made by
total
is
l)).
number of nonsaturating pushes, we use an argument
the
with two potentials instead of one. The
e(,v)d(v)/A.
J^
the stack scaling algorithm
By
first
the analysis in the proof of
potential
Lemma
that
is
3.1,
like the
of
proof of
Lemma
3.1,
every push decreases
V tctive
<I),
the total increase in
and the
final
decreases
<I)
<1>
values of C)
by
To count
over
is
at least 1/2.
little
pushes,
all
phases
is
0(n^
logi U), and the difference between the initial
O(n^). Each big push moves
Thus
we
the
number of big pushes
at least
is
A/2
imits of flow, and hence
O(n^(\ogt
U+
divide them into two kinds, those that result in an empty stack 5,
called emptying pushes, and those that do not, called nonemptying pushes.
from a vertex v
is
such that e(v) was
We
e(v) decreasing to zero.
1/2 in
<1>
resulting from
1)).
at least
A/2 when
v
was added
to 5,
A
nonemptying push
and the push results
can charge such a push against the cimiulative decrease of
moving
the original excess
on v
to vertices
of smaller
label.
in
at least
Hence
there
can be only 0{n^ Qogt (/+!)) rwnemptying pushes.
An emptying push
namely the drop
in
<t)
from a vertex v can be associated with
a
decrease of
caused by the movement of the original excess on
v,
at least
which
is at
l/k in
least
<t>,
A/k,
But using
to smaller labeled vertices.
U+
of only Oikn^Qogt
tial, <1>2.
The
equal to the
definition of
minimum
We
1)).
count emptying pushes more carefully by using a second fwten-
involves two parameters, an integer
<1>2
during a phase. The set
that
P
is
and a
The value of
set P.
/ is
P
= 2n
/
consists of
the beginning of a phase and that
at
vertices with label greater than
all
/
nonincreasing
/ is
and
vertices with
all
from which an emptying push has been made during the current phase. Observe
/
empty
definition of
/
of 2n and the smallest label of a vertex added to S while S was empty dur-
ing the current phase. Observe that
label equal to
drop gives a bound on the number of emptying pushes
this
P
beginning of a phase and
at the
never loses a vertex during a phase. The
<I>2 is
Z
<I>2=
e(v)(rf(vH+l)/A.
= 0.
(lf/»
=
<D2
0.)
v€ P:e(v)>0
Observe
a vertex to
is
/*
< O2 ^ 2n^. Any emptying push can be associated
that
or with a decrease in
P after the
added to S but v e
I
logt (/
+
over
Ij
with decreases in
To bound
to
<1>2.
Increases in
fact that the relabeling
since
in <^2
due
The
at
when
<l>2
at
is
added
to
The number of
the niunber of
is
is
from
S and
<1>2
either v
4.
drops by
P when
at least
P
vertices added to
is
at
v
\/k
most
emptying pushes not associated
associated with decreases in
v to F.
in
Thus
/
in
relabelings contribute at
is at
most 2n^
A
per phase.
the decrease occurs every vertex in
/
[
log*
4/bi2+2/jn logifZ +
lJ
<I>2
is initially
zero.
<I)2,
we bound
the
(A vertex added
/.
A
<1>2.)
relabeling of
d(v) plus one; the "plus one" accounts for the
P
decrease in
/
most 4n^
Thus
Thus
has small excess.
U +\\
Ik over
the total
to the
by one adds
at
growth of
most n/k
to
the total increase
all [rfiases.
nimiber of emptying pushes associated with decreases in
in the
v,
are due to relabelings and to decreases in
most the increase
most 2n decreases
total increase in <I>2. since
As
v
push
push has zero excess and hence adds nothing to
may add
to decreases in
total
5.)
number of emptying pushes
a vertex v increases <&2 by
<I>2.
on
(If the
/k.
P when
phases, and hence so
P because of an emptying
There are
is
1
<I>2.
the
total increase in
all
at least
push, orv e
because of pushes from v while v
n
of
<1)2
either with an addition of
<I>2 is
at
most k times the
number of such pushes
is at
most
.n
Goldberg-Tarjan and Ahuja-Oriin algorithms, the time to perform saturating
pushes and relabeling operations
is
choose vertices to add to S when S
0{nm).
is
empty.
The only
For
this
significant remaining issue
purpose,
we maintain
is
how
to
a data structure
10
consisting of a coUection of doubly linked
2n-l
j € (1,2
We
).
The
change
in label) or
N
e
('
:
e(i)> A/ k and d(i) =
phase change (by
at
most
pointer needs to be incremented or decremented
is
0(nm + kn^ + n^
thus
is
2n). Consequently, the
0(n^ + n Oog*
(log*
U
+
1)).
U+
1)).
each
listij) is
0(1) time per push operation plus time
this structure requires
to a
j] for
which
pointer can only increase due to a relabeling (by at most the
due
lime of the algorithm
=
also maintain a pointer to indicate the largest index j for
nonempty. Maintaining
tain the pointer.
/«f(/)
lists
main-
to
amount of
number of times
The
the
overall running
Choosing k ={ log
U /log log
U]
gives the following result:
Theorem
O
{run
+ n^ log
Remark.
=
2 until
A<
When
rithm.
[/'
O
to
(nm + n^ log
The algorithm must be changed
and then switch to
These phases account
4.
improved
be
we
the switch occurs,
above with
choice
appropriate
of
nms
k,
in
We
slightly.
where
f/' /log log t/*).
use a scaling factor of
E
which
the initial phases (those in
is as
an
with
time.
can
u(s,v)/n.
algorithm,
scaling
U)
bound
^
v:(i,v)€
k
stack
t//loglog
This
=4 +
t/*
The
3.3.
for only
U replaced
it
=[
A
increase
A>
U* /log
log
(if
t/') is the
log
U"\
for the remainder of the algo-
necessary) to be a power of
same
as that in the
it.
The
remark following
analysis of
Lemma
3.1.
O(n^) nonsaturating pushes. The analysis of the remaining phases
by U* .O
The Wave Scaling Algorithm
Another way
keep track of the
observation
is
to reduce the
number of nonsaturating pushes
total excess, defined to be the
that if the total
excess
is
in the scaling algorithm is to
sum of the excesses of all
sufficiently large, the algorithm
active vertices.
The key
can make significant pro-
gress by applying slack-pushJ relabel to each active vertex in turn, processing vertices in topological order with
respea
to the set
of eligible arcs. Even though some vertices of very small excess
are processed, the overall benefits of this approach yield an
the Slack scaling algorithm.
wave algorithm of Taijan
The
even better running time than
that
of
idea of processing vertices in topological order originated in the
[20,21]; therefore
we
call the
new
algorithm the
wave scaling
algo-
rithm.
The wave
factor, therefore
shall
choose
step below
is
scaling algorithm seems to derive
we
later.
fix
A
it
=
2.
no
benefit
from using a nonconstant scaling
The algorithm uses another parameter
/
S
1
whose exact value we
phase of the wave scaling algorithm consists of two pans.
repeated until the total excess
is less
than n A//.
First, the
Then stack-pusk' relabel
wave
steps are
applied to large-excess active vertices in any order until there are no large-excess active vertices.
-11
Wave;
Construct a
list
L of the
vertices in
eligible arcs. (Ordering
tex V on L,
Observe
if V is
L
V
[s,t] in
-
topological order with respect to the set of
in nonincreasing order
by vertex
active perfonn stack-pushlrelabel{v).
that the total excess is a nonincreasing function of time.
beginning of a wave takes 0{n) time using a radix sort by vertex
single
wave
wave
scaling algorithm
is
For each ver-
label suffices.)
0(n) plus time
is
0(nm)
a
the total time required by the
We
0(n) per wave plus 0(1) per nonsaturating push.
plus
the
at
The time spent during
label.
Thus
for the relabelings and pushes.
L
Constructing
complete the running time analysis with two lemmas.
Lemma
0(n^
-K
Proof.
The number of nonsaturating pushes done by
4.1.
(n^//)logt/) plus
We
separately.
analysis in the proof of
move
Lemma
at least
3.
A/2
applies to
1
with the following improvement. At the end of a phase, the
the potential at the beginning of the next phase
the old; each vertex label
A
is
is at
most
Thus
2n.)
Oi(,n^/l)logU). which implies that the
same argument gives
excess
when added
the
same bound on
at least 1/2 in
<I>.
vertex to which stack-push! relabel
is
Lemma 4 2. The number of waves
in the
Proof. Consider any
at least nA//.
the
the
wave except
The only way
causes
all
vertices to
numberof waves
is
0{,nl
O(n^).
excess
total
sum of the
increases in
big pushes
pnx)f of
The remaining
wave
is
Lemma
litUe
is
<1>
the last in a phase.
is
half of
caused by changes in
0(n^ +
(/i^//)log U).
The
vertices that have large
3.2 each such push can be
at
most one per
D
O (min(n/ + logf/,n^)).
At the end of the wave, the
on a vertex x
is
at the
total
end of a wave
excess
is
is
for x to
relabeled, e{,x) remains constant until the end of
single vertex
total
+ \ogU). Furthermore,
inactive,
is
Thus
less than nAll.
pushes consist of
scaling algorithm
for excess to remain
become
bound the number of big pushes,
number of little pushes from
must have occurred during the wave. Since the
is
is
pushes
little
applied during waves, totaling at most n per wave.
The maximum excess on any
number of waves
of flow) and
units
most 4n^/l. (The new value of A
number of
have been relabeled during the wave; once x
the wave.
is at
to the stack 5, since as in the
charged against a drop of
scaling algorithm
0(n) per wave.
coimt big pushes (those that
The
wave
the
a
is
A.
It
follows that at least nil relabelings
number of
relabelings
wave during which no
and hence terminates the
is
0{n^), the
total
relabeling occurs
entire algorithm.
Thus
the
D
Theorem 43. The mnning time of
the
wave
scaling algorithm
is
0(,nm + n^Oogt/)"^)
if / is
-12-
chosen equal
lo
Qogt/)"^.
By Lemmas
Proof.
0(n/n -t-(n^//)logt/
OCwn
-t-
-^
and 4.2,
4.1
min
n^Oogt/)"^ +
(n^/
The bound
t/*=4-»-
Theorem
in
M(j,v)/n, by doing
5^
(A > U*)
is
the
V')
is
as above with
same
=
(\ogU)^'^
algorithm
gives
is
bound of
a
min{/jlog(;.n^) ^n^Gogt/)"^.
since
D
can
reduced
be
waves only
after
as that in the remark following
A<
lo
t/*.
C>(n/n
The
-i-
where
/i^Oogt/*)"^).
analysis of the early phases
<,
5.
Use of Dynamic Trees
The approach taken
Orthogonal approach
Lemma
3.L The analysis of the
later
phases
U replaced by U* D
(A
.
in Sections 3 and 4
was
to
reduce the
number of
total
pushes.
An
to reduce the total time of the pushes without necessarily reducing their
is
This can be done by using the dynamic tree data structure of Sleator and Tarjan
We
[18,19,20].
on the
4.3
n.
l
scaling
£
(j.v)€
number.
But
min{«logt/.«^)).
wave
of the
time
Choosing
nlogt/.n^)).
-^
n\ogU S n^Oogt/)"^ implies Gogt/)"^ ^
Remark.
running
the
total
conjecture that, given a version of the preflow algorithm with a bound
number of pushes,
the running time can be reduced from
0{p)
to
ofp^
0{nm log(-^
-i-
run
1))
run
by using dynamic
we have been
Although we do not know
trees.
the 0(,nm log (n^/m)) time
from using dynamic trees with the FIFO selection
case
is
Oin^). In
this section
we
shall
show
O
applied
(nm log
(
to
the
stack
——
—
m —
f
log logt/
;
-f
2));
scaling
we omit
rule; the
that the
rithm of Section 4, resulting in a time bound of
be
to
prove a general theorem to
this effect,
able to obtain such a result for each version of the preflow algorithm that
As an example,
considered.
how
Oinm
algorithm
bound of Goldberg and Tarjan
bound on
the
same idea applies
log(
of
we have
—
m
(log l/)"^
Section
number of pushes
to the
+
3,
2)).
wave
results
in this
scaling algo-
This idea can also
giving
a
bound
a description of the latter result since the former
bound
of
is
better.
The dynamic
tree data structure allows the
maintenance of a collection of vertex-disjoint
rooted trees, each arc of which has an associated real value.
directed from child to parent, and
dant of
itself.
The data
we
We
regard each tree arc as being
regard every vertex as being both an ancestor and a descen-
structure supports the following seven operations:
-13-
find-root (v):
Find and return the root of the tree containing vertex
find-size (v):
Find and return the number of vertices in the tree containing vertex
find-value (v):
Find and return the value of the tree arc leaving
returned
tie.
choose the vertex
Add
change-value (v.x):
real
root of
Combine
link (v.H'.x):
v.
If v is a tree root, the value
v.
is infinity.
Find and return the ancestor
find-min (v):
v.
w
of v with
w closest to the tree
number
j:
to the value
minimum
find-valueiyv). In case of a
root.
of every arc along the path from v
to the
its tree.
the trees containing v and
w
by making
w the
parent of v and giving
the arc (v.w) the value x. This operation does nothing if v and
w are
in the
same
tree or if v is not a tree root.
Break the
cut (v):
two
tree containing v into
trees
by deleting the
A
0{h\og
sequence of h
(z
-(
In the
the vertices.
dynamic
1))
time
Only
a time.
^
n.
We shall
bound
on the maximum
ing algorithm except that
it
A on
collection of singleton trees, takes
initial
tree size [10.18,1920].
tree arcs are a subset
The value of
a tree arc
describe a version of the
the
size of a
of the current arcs out of
is its
residual capacity.
be moved along an entire path
scaling algorithm, that uses this capability.
the algorithm, a variable
z
maximum
if z is the
eligible arcs are tree arcs.
at
an
tree operations, starting with
tree data structure allows flow to
call the tree
S
a tree root.
network flow application, the dynamic
along a single arc
z, 1
is
maximum
dynamic
Two
excess
tree.
at
wave
at
scaling algorithm, which
an active vertex, and a fixed bound
The algorithm
is
identical to the
wave
is active.
r.
we
parameters govern the behavior of
Tree-PushJRelabel{r)
Action: Initialize a stack S to contain
The
once, rather than
uses the following step in place ol stack-pushJ relabel:
Applicability: Vertex r
its
This operation breaks no arc and
parent; return the value of the deleted arc.
returns infinity if v
arc joining v to
Repeat the following step
until
S
is
empty.
scal-
-14
Stack Step. Let v be the top vertex on S and
let
(v,h) be the current arc out of
Apply
S.
the appropriate one of the follov.-ing cases:
Case
1
:
(v.h)
Case
la:
is
not eligible.
(v,h)
is
not last on
by the next
Case
arc
on
A
(v).
Replace (v,k)
y4(v).
lb: (y\V')islastonA{y). Relabel v and
as the current arc out of v by the
O'.v),
Case
2:
{v,w)
as the current arc out of v
first
pop
arc
it
from
Replace (v,h)
5.
on A (v). For every
tree arc
perform cutiy).
Ltlx= find- root(w).
is eligible.
Case 2a: e{x) >
t^ll
andx *
t.
Push x onto
S.
Case2b: e(x)<Al2orx = t.Ltlt= min{e{v),Uf{v,w),find-min(\^,)}. Lei
6 = eifx =
to
r,
6 = min{E,A-e(x))
if
x *t. Send 6
from v
uruts of flow
X by increasing /(v,h) by 6 and performing change-value (h,-5).
(This
is
scuurating
called
tree
a
The
to x.
perform cut{find-min{w)).
>
and
While
If e{y)
find-size{yv)
=
find-value
0,
+
pop
push
tree
S = min [uji\\w),find-min (w)] bcfoTZ the
if
nonsaturating otherwise.)
u/v.H')
push from v
(find-min
push
is
and
= 0,
(w))
v from S, and if in addition
find-size(v)
<
perform
z,
link
(v,w,uj(v,w)).
The
dynamic
arcs.
tion,
tree scaling algorithm stores flow in
tree arcs
and implicitly
(in the
two
dynamic
different ways: explicitly for edges that are not
tree data structure) for arcs that are
After each cut, the flow on the arc cut must be restored to
when
computed.
the algorithm terminates, the correct flow value
its
dynamic
tree
correa current value. In addi-
on each remaining
tree arc
must be
For arcs cut during the computation, the desired flow values are returned by the
corresponding cut operations. Computing correct flows on termination can be done using
We
n find-value operations.
at
most
have omitted these bookkeeping steps from our description of the
algorithm.
The algorithm maintains
tree arc
tree
is
eligible,
push big
if it
the following invariants: every active vertex
no excess exceeds A, and no
moves
given vertex v while v
is
at least
on
S, at
A/2
tree size
units of flow, and
exceeds
little
z.
Let us
otherwise.
most two are nonsaturating and
at
is
a tree root, every
call a
Of
nonsaturating
the pushes from a
most one (the
last) is small.
-15-
The
tree scaling algorithm is a variant
of the d>'namic tree algorithm of Goldberg and Tar-
jan [9,10]. Their analysis applies to give the following
Lemma
5.1 [10].
0(\og{z
+
The
time required by the tree scaling algorithm
total
is
0{nm
time per tree push plus the time needed to construct and scan the
1))
The number of links,
steps.
result:
Making
cuts,
and saturating
tree
pushes
is
+
log(z
list
plus
1))
L during
v;ave
0{nm).
the tree scaling algorithm efficient requires careful implementation of the list
used in wove steps. For the
moment we
ing issue in analyzing the algorithm
is
L
manipulating L. The remain-
shall ignore the time spent
bounding the number of nonsaturating pushes. To do
this
we need two lemmas.
Lemma 52. The number of waves
Proof. Identical to the proof of the same result for the
Lemma
0(nm
Proof.
to
The number of nonsaturating
5.3.
The
potential function
the
number of
Any
follows.
little
<I>
used
in
Section
big tree pushes by
tree push, say
event an activation
0{nm +
D
is
O
3,
(n^
from a vertex
+
v,
as applied in the
(n^//) log U).
of
We
v.
shall
derive
a
We
count
reduces ^(v) to zero.
that
made
Lemma
proof of
little
We
e(v) nonzero.
tree
shall
We
bound on the number of
3.3, serves
pushes as
count such a
call
such an
activations
of
in^/l) log t/)plus 0{nlz) per wave, thereby proving the lemma.
preflow/at the beginning of the
Initializing the
Every other activation
results in a link or cut,
larly, if the tree
push
charges account for
X = find-root{w),
is
is
we
due
to a tree
we
the push.
more than z/2
If
activa-
such a tree
charge the activation against the arc saturation.
Such
activations.
produced by a
which find-size(v) + find-sizeiw) >
z,
and T^ be the dynamic trees containing v and w, respectively.
contains
0{n) vertex
push (case 2b of tree-push' relabel).
activation, say of a vertex x, is
after
entire algorithm causes
charge the activation against the corresponding link or cut. Simi-
saturating,
0(nm)
Any remaining
to
(Lemma 4.2).
scaling algorithm
pushes done by the tree scaling algorithm
tree
push against the most recent event before the push
push
wave
(min [nl + \ogU,n^)).
-»-(n^//)log[/)plus C>(rt/2) per wave.
bound
tions.
O
in the tree scaling algorithm is
vertices and small otherwise.
One of
tree
no
since
We
push through an
link
call a
is
arc (v,h)
performed. Let T^
dynamic
tree large if
it
Ty and r» must be large just after
—
;
16
If the tree
push
above counting big
the push
wave.
accounts for 0(n^
+
(n^//) log
reduces e(v) to zero. There can be
it
at
U)
Otherwise,
most one such push from v per
has changed between the beginning of the most recent wave and the push,
If Ty
most one per
Similarly,
the argument
activations.
the activation of x to the link or cut that most recently changed T^. There are
at
By
charge the activauon of x against the push.
tree pushes, this
and thus
is little,
we
big,
is
if
link, at
most two per
cut, since vertex v is the root
T^ has changed between
the beginning of the
Oinm) such
when
of Ty
push occurs.
or T^
If neither Ty
large.
is
Each
nor r» has changed,
total
the claimed
Theorem
bound on vertex
The number of
5.4.
chosen equal to min
max
(n,
tree
0(nm) such
at the
all
:r
T^ when
at
our estimates gives
D
logL'}) and
/ is
chosen equal to
m/n
is
if z
=
0(nm)
1,
if z is
—
logU
m
if
<z<n.
Proof.
By Lemmas
Oinm +
in^/l)
Case
5.2
logU +
Oinm
and
— min
5.3,
[nk
the
number of
+ logt/.n^ )).
—7-logf/Sl.
1:
Then
m
0{run + {n^ /m)\ogU
is
most twice
beginning of any
m^
\
the
whichever of Ty
to
pushes done by the tree scaling algorithm
—
(1,
x
wave can be charged
0(n/z) per wave. Combining
is
activations.
the root of
fewer than 2n/z large trees
tfiere are
number of such charges
is
the activation of
large tree existing at the beginning of a
(once as a Ty, once as a r»). Since
wave, the
we charge
charges,
most recent wave and the push, we
most once per wave, and x
at
charge
the push occurs.
charge the activation of x to the link or cut that most recently changed T^. There are
charges, since a vertex x can be activated
we
+ n minim + ]ogU,
z
=
n^]).
pushes done by the algorithm
is
We consider three cases;
and
l
tree
l
= mln. The number
Since logU
^m^/n^,
the
of tree pushes
bound on
tree
is
pushes
min{m^/n, n^}) = 0(nm), because Tn}ln<,n} implies m<ji^, which means
-i-
rn^ln ^ nm, and
m}ln S n}
implies
m t n^, which means
n' ^ nm. (This analysis allows for the
possibility of multiple arcs in G.)
Case
2:
1
<
—„2r-logU <
Then
2
tree
pushes
is
C>(n/rj+(
—n
logty
0{nm + min{m^/n,
n^)), since
number of tree pushes
is
Oinm).
—ylogU
f^l
n.
z
—m
=
and
/
=
—„ \ogU.
The number of
2
2
)
min{
logl/
+ logt/,n^)) = 0(nm + min (m^/n.-
\ogU <m^/n^. Since min(m^/n,fi') ^ nm
—]) =
iogu
(see
Case
1).
the
—
17
Case
—^2
0(wn + min(
Observe
\ogU ^
r-
3:
Then
n.
logU + logU.n^
m
that if z is
time per dynamic tree operation
of
the
0(nm
log
(
— Gogt/)"^ +
and
n
0(\og(z
+
5.4,
1))
with
/
=
—„ logU.
The number of
tree
pushes
is
D
Theorem
in
algorithm,
scaling
tree
is
=
= 0(.nm).
])
chosen as
z
k
i.e.
equal to min{n,
= 0(\og
and
Qogf/)"^)).
—
m
(
chosen
z
as
—
m
f^2
max
{1,
Thus
r-
\ogU]
the
the running time
Theorem
in
),
5.4,
is
2)) (igrwring time spent manipulating L).
tn
All that remains
fonmed
list,
in
0{nm
log(
is to
show
that the manipulations of the
Gogt/)"^ +
—
m
for then the time scanning
L
2)) time.
will be
It is
in turn represented
by a balanced
minimum-cost flow algorithm
tree.
we
represent
wave
L
this
steps can be per-
L simply
as a lirJced
scanning time will dominate the
as a
list
of sublists, each of which
the
method
differs
from
theirs in that
represent the sublists by ordinary balanced search trees instead of finger search trees, and
the size of each sublist to
z.
the
maximum
is
This idea was used by Goldberg and Tarjan in a
Our version of
[11].
in
not sufficient to represent
0{n) per wave, and
time for the rest of the computation. Instead,
L
list
tree size in the tree scaling algorithm.
we
we
limit
These changes
lead to an analysis that parallels the analysis of the tree pushes.
The
details
of the data structure are as follows. The
computation so that
inates the
need
its
vertices are in topological order with
to reconstruct
L
at the
list
L
respea
beginning of each wave.
order merely requires moving each relabeled vertex to the front of
List
L is
maintained as a doubly-linked
have size
(a)
all sublists
(b)
any active vertex
The
at
is last
most
on
list
maintained throughout the
to the eligible arcs; this elim-
Maintaining
L when
it is
L
in tofwlogical
relabeled.
of sublists. These sublists have two properties:
z;
its
sublisL
sublists are represented as balanced search trees, e.g. red-black trees [20] or splay trees
[19,20]. This representation supports the follosving op>erations
(i)
is
Initialize a sublist to contain a given single vertex.
on the
sublists:
-18-
number of vertices
given sublisl.
00
Find
the
Oii)
Find
the sublist containing a given vertex.
Ov)
Find
the last vertex in a given sublist.
(v)
Split the sublist containing a given vertex just before (or just after) the vertex and return
the
two
resulting sublists.
Concatenate two
(vi)
The time required
and 0(\og(z +
in a
1)) for
sublists
and return the resulting
for these operations
each operation of types
is
0(1)
sublist.
for each operation of the types
(iii)-(vi),
assuming
that
no
and
(i)
sublist size ever
(ii),
exceeds
2.
During the
tree scaling algorithm, the
each vertex other than s and
/ is
computations done on
When
lisl
the sublist containing v
is
Each time a vertex v
is
relabeled, the sublisl containing v
containing v
is
moved from
is split
Initially,
and these sublists are
is
activated
(i.e.
This maintains property
split just after v.
a vertex v
its
are as follows.
initialized to be a single-vertex sublisl,
ordered in nonincreasing order by vertex label.
becomes nonzero),
L
e(v)
(b).
and the (singleton) sub-
just before v,
current position to the front of L.
Concatenation of sublists takes place only during waves.
vertex v to be the last vertex on the
first
sublist
A
wave
consists of initializing
of L, and repeating the following step
until v is
null:
Scan. Apply the appropriate one of the following cases:
Case
tices
1
:
V
is
Perform stack-push/relabel{v),
active.
splitting sublists at
newly
active ver-
and moving relabeled vertices (except for v) to the front of L as described above.
has been relabeled,
sublist after v, or
move
j:
by
to the front
Case 2: v
is
let
j:
be
v,
and proceed as follows: Replace v by the
null if there is
no such
sublisl; split
fl'.
If
inactive.
B
(the old v)
vertex on the
last
from
its
sublist
and
of L.
If v is in the last sublisl, replace v
be the sublists contairung v and the vertex after
tex in
x
If v
v,
by
null.
Otherwise,
respectively. Replace v
and B' together contain no more than
z vertices,
let
B
by the
concatenate
B
and B'
last ver-
and B'.
19
Note
then V
that,
during a wave,
immediately reprocessed
is
The
analysis of the
list
a stack-pushJ relabel applied to a vertex v does not relabel
if
O
Gog
(z
+
processing that takes place during waves
ing an execution of case 2
(those in which
B
B
and
'
is
0{nm
Proof.
+
log (z
5.6.
1))
1)).
more than
total
1)), if z
and
time
2>
is
1
dur-
(n/z) executions of case 2 per wave
z vertices),
/
are chosen as in
The
is
O
L
processing
spent
Theorem
(nm). So
is
each execution of case 2 causes a
to min{rt,
max{l,
—
m^
r-
log
U} and
/is
during
number of
the
splits
by
at
most
the
tree
algorithm
scaling
is
5.4.
number of tree pushes, by Theorem
the
tree scaling algorithm runs in
2
chosen equal
O
a execution of case
The time spent processing L
per relabeling.
Except for
like the analysis
most the number of tree pushes plus the number of relabelings.
is at
The number of relabelings
Theorem
+
much
the following result:
The
5.5.
+
(z
together contain
n-1, and the number of splits
Lemma
Gog
(z
The number of concatenations can exceed
sublist concatenation.
Thus we obtain
O
is
The time spent processing L during
O Gog
per tree push plus
1))
Case 2 oi scan.
in
already done to count tree pushes.
v,
—
Gog U)
m
— =
chosen equal
O
{nm log
(
to
if z
+2)) time
1
or to
n
5.4.
D
if z is
—
log
m
f/ if
1.
Immediate
Proof.
log (z
+
1)
= O Gog
Theorem
from
(— Gog Uf^
+
Lemma
and
5.4
5.5,
since
—+
2), analo-
2)).
\I/2
Remark. The bound
in
Theorem
5.6 can be improved to
O
gously to the improvements in the bounds of Sections 3 and
run the stack scaling algorithm until
6.
Bounds
in
A^
U*, and then switch
a Semilogarithmic Computation
The time bounds derived
4.
To
^
(
m
obtain this improvement,
to the tree scaling algorithm.
we
D
Model
in Sections 3-5 are all
U
(nm log
based on the assumption that addition and
U
huge, this assumption
comparison of integers of magnitude
at
may
explore the consequences to our results of using a semilog-
be unjustified. In
this section
we
most
takes 0(1) time.
arithmic computation model, in which each computer
word
any operation involving 0(1) words takes 0(1) time. In
this
is
If
is
allowed to hold
model,
all
O
(logn) bits and
the elementary graph and
.
20-
list
manipulation operations needed by our algorithms take 0(1) time each, but adding or com-
paring two capacity or flow values can take
O
Oog„ V) time
if
an exact answer
required.
is
Thus
a straightforward translation of our results into the semilogarithmic model increases each of the
time bounds by a factor of
By
Gog„
suitably modifying the algorithms, however,
of our algorithms requires
Thus
O
example
for
(nm log
(
6 (m
ties is
t/).
—
Gog f/)"^ +
m
log,
U)
running
in this
2)
+
possible to reduce the extra time each
semilogrithmic model to an additive term of
in the
the
it is
m
time
of
the
tree
scahng
O
(m log, U).
algorithm
log, U). Note that the time needed to read
all
becomes
the arc capaci-
model.
The general approach
to approximately solve a sequence of
is
O
Qog, U) closer-and-closer
approximations to the original problem. Each approximation uses as a starting point the approxi-
mate solution
O
of only
Gog n)
Making
this
bits.
approach work involves a number of messy technical detaQs. Since the result
mainly of theoretical
interest,
a
is
power of
where
=
5i
t///i*.
assume (without
G
^
We
Observe
exactly.
= log,
that for k
«t_i (v.w) -Uk(v,w)
solve problem
log, U,
we
define
problem
kfoTk =
k, all
^
5*
t/,
=
1;
to
)t
is
a
power of two and
be the
Ut(v,w)=\
thus problem log,
is
modifying the algo-
in
loss of generality) that n
with arc capacities w* (v,^) defined by
lem. Also, for a general value of
arc (v.w),
merely sketch the ideas involved
shall
For k = 1,2
n.
problem on the graph
we
We
rithms of Sections 4 and 5.
that t/
Solving each problem requires manipulation of integers
to the previous problem.
f/ is
maximum
flow
u(v.u')/5tl 5^,
the original prob-
capacities in problem k are divisible by 5*; and. for every
6i_i
1.2, ....log, (7-1
approximately and then solve problem log,
Throughout the computation we maintain a preflow / and
a valid labeling
d
U
for the
current problem. Preflow /satisfies the following constraint:
e{v)
^ min
"^
{
f(u,v), n(5t-\)
)
for all v *^ (flow holdback constraint).
(5)
/(y.v) >
(In inequality (5). the
For a vertex
from
V, is
a(v)
v,
sum over/(u,v)
is
taken to be zero
the available excess at v.
= max {0,e(v)-n(5i-l)
).
which
is
if /(u,v)
the
S
amount
for all arcs (u,v).)
actually allowed to be pushed
Throughout the computation, the available excesses
used in place of the actual excesses to determine when vertices are active and
how much
are
flow can
be pushed.
The flow holdbacks
to
one problem into
a
are
good
needed
initial
to maintain the preflow property
when
preflow for the next problem. For the
first
converting a solution
problem, the preflow
21-
/and
valid labeling
Once
a solution to problem
d
are initialized exactly as in Section 2 (using the arc capacities «i(v,w)).
k-l
is
by replacing the current preflow
computed,
/by
is
converted into an
is
obtained from
(v.w) by an amount between zero and mm[f{v,w), 5t-i
dity of the flow holdback constraint for /in
constraint
forf
in
problem
k.
For
a(v)
I
<k <
< 2n(5i-l)
f
is
for every active vertex v.
valid labeling
Thus
f
constraint implies that
a valid labeling
/ and
log, U, a preflow
decreasing the flow on each arc
=
(inclusive). Since S^.j
by construction. Since every
problem k.d
in
= min{/(v,H'),M4(v,>v)).
arc saturated
for/
d
in
problem
is
by /
in
at
problem k-l
k.
constitute a solution to
a solution to
problem log,
U
problem k
every active
k,
most 4n^5t. of which 3n^5t = 3n5t_i comes from the excess on v
initializing the preflow for
problem
bound on maximum available excess)
number of
for
problem k
is
phases, continuing until the
maximum
available excess
happens by the time
phases needed
21ogn
3.
-I-
is
at
A
is
reduced to nS* (or to 1/2
most log n +
Thus each problem
subproblems
(These are overestimates.) The
k.
is
O
(logn log,
2;
if
requires
U) =
k= log,
OQogn)
4n^5k. Problem k
if
/t
= log,
U).
initial
is
is at
Uk
<
value of
in
to
/in
A
(the
solved by performing
most 2n (5t-l), which
number of
log, U, the
U, the number of phases needed
phases, and the total
if
maximum
gives a
problem k-l (since a(v) < 2n(5t_i-l)), and n^5t = n5k-i comes from the changes made
a
f
a preflow, since
flow in the original network. After the initialization of the preflow for problem
vertex has an excess of
nS*. the vali-
problem k-l implies the validity of the flow holdback
The flow holdback
satisfies the capacity constraint
remains saturated by
)
/by
preflow for problem k
initial
the preflow/' defined by /'(v,H')
0^ M4_i(v,H')-«t((v,H') < 5i_i,/'
Since
it
is
at
most
number of phases over
all
(log U).
During the solution of problem
k,
the preflow
/is maintained so lhat/(v,H')
is
a multiple of
5i for each arc (v.w). (This happens automatically, since A. the initial flow values, and
all
capa-
multiples of 5^). Thus the flow values and capacities can be represented in units of 6*.
cities are
Furthermore, the flow values and capacities are not represented explictly, but rather as differences
from the
let
/o
(v,^).
initial
flow values.
= 0. Then
be more precise: let/4_i be the
the current preflow
The capacity fuiKtion
each arc (v.w).
To
We claim
u^
is
that the
> 9n'
represented by the value
amount by which
n
A <4n'
is
To
5* units of flow can be
prove the claim,
min
the flow
at
5*, the extra residual capacity
used and can be ignored.
preflow for problem k-l\
/is represented by the value f{v,'w)-fk-\ (v.w) for each arc
tion of problem k and subsequent problems
"i (v.H')-/t(v,H')
final
most
{«t(v,H')-/t_i(v,H'),
on an arc can change during
8/j' 5*.
due
to
is at
if
(beyond 9n' 5*) on arc (v.w) will never be
we
note that between relabelings
moved through any given
modifying /between subproblems
for
the solu-
This claim implies that
arrc,
and there are
belings over the entire algorithm. This accounts for 8n' 5* units of change.
arc
9n' S^)
most
< 2hk.\
J^ 6^
j-k-\
= 2n
at
at
most In}
most
rela-
The change on an
5*.
Thus
the claim
22
is true.
Storing
/and
ut in such a difference form, in units of 6*, allows the algorithm to manipulate
numbers consisting of only 0(\ogn)
and
bits,
necessary additions and comparisons of flows
all
and capacities take 0(1) time. Constructing the
final
preflow
is
when
together the successive flow differences /i-/o,/2-/i.-- computed
By
lems.
and the
adding these differences from right to
time
total
for each problem
0(m
log^ U). This
number of
scaling algorithm
relabelings
nonsaturaiing pushes
0(min
[nl
+
log U, n^
is
O
is
becomes n^ +log;, U;
also the total time required for initializing the preflow
is
is
used, the bounds in Section 4 remain valid almost without
+
(n^
+ log,
this is
bound of 0{nm + n^
Oinm
of
log(
the
U)
(n^/l) log
Gog
—
m
t/)"^
U problems.
is
the only
0(nm),
is
the
one
Specifically,
the
number of
number of waves
that changes, in that
is
n^
because a wave without relabelings terminates only the solution of
m
+
+ 2) + m
log,
Chooseing
L and
list
Section
of
To obuin
log, U).
the
for tree pushes associated with vertices that
/
= Gog U)^'^ gives an
overall
time.
if)
algorithm
scaling
time required for initializing the
0(m)
log„
0(n) per wave, and
plus
Tht bound on waves
U]).
(log l/)"^
tree
all
O(n^), the number of saturating pushes
the current problem, and rwt the entire algorithm.
Use
made 0(m),
the time per addition can be
left,
modification, and apply to the calculations involved in solving
the
solving successive prob-
(Oim) time per problem).
wave
If the
is
merely a matter of adding
this
dynamic
become
5
results
a
in
bound
time
of
bound, we must verify that the extra
trees for each problem,
and the extra time
active at the beginning of a
new problem,
is
per problem.
We
need two extra
data structure, the flow values for
ermore, a
new
set
dynamic
facts about the
all
of flow values for
the
all
dynamic
tree data structure.
tree arcs
By
can be computed
traversing the entire
in
0{n)
time. Furth-
these arcs can be installed in the data structure in
0(n)
time. These facts can easily be verified by checking the description of the data structure [19.20].
When
the flow /is modified at the beginning of a
new problem, new
excess are created, possibly as
many
part of the initialization of the
new problem. To begin
dynamic
tree arcs is
computed
explicitly.
and capacities are modified to their
available excess
and
is
is set
as n-2.
initial
greater than the total excess
Next, the vertices of
C
are
on
all
A
Such
a step either relabels v or reduces
vertex v
is
scanned by applying
new problem,
0(n) time
values for the
is
a
is
the hardest
the current flow
as noted above.)
Then
on
all
the flow
new problem. The bound on maximum
n times the value proposed eariier in this section,
active nodes.)
scanned
arcs.
Handling these newly active vertices
(This takes
equal to 4n' 6*. (This
vertices with available
to
its
in topological order with respect to the set
it
of eligible
a pushlrelabel step of the kind defined in Section 2.
available excess to zero.
After
all
vertices have been
23
scanned,
all
the available excess
is
on vertices
that
have been relabeled during the scanning. The
current flow values for dynamic tree arcs are reiosuied in the d>-namic tree data structure, and
every dyiiamic tree arc that has been saturated during the scanning or that has had one of
vertices relabeled
cut Finally, the
is
list
L
is reinitialized
by sorting the vertices
order with respect to the set of eligible arcs and constructing one sublisi in
tex
with V as
V.
tex.
Constructing
can be
0{n)
ning in the initialization
plus (9Gog(r
-K
A
Then
the tree scaling algorithm
problem
for each
is
0{m). The time
The time
to initialize
timing analysis for the tree scaling algorithm
a total time
L
Goldberg and Tarjan
7.
increased, the
The time
-fm
[8,10],
log^,
resulting
0{m). Once
is
number of phases
to carry out all the scan-
0{jn)
(
same
—m Gog t/)"^) +
a
bound
time
+
m
log^,
maximum
the
in
initialization,
and the
as the analysis in Section 5.
2)
also be applied to the
in
been com-
the initialization has
been relabeled during the
virtually the
is
bound of O (nm log
The techniques discussed above can
{run log(/i^/m)
overall.
begun.
to initialize the d>Tiamic tree data structure is
pleted, the only active vertices are those that have
O
is
is
per cut; the time for a cut can be charged against the corresponding arc satura-
1))
tion or vertex relabeling.
Thus we obtain
that foUov^' the last active ver-
pairs of
by \ogn per subproblem, or logl^
increases, but only
for each active ver-
0{x) time by doing repeated pair-wise concatenanon (concatenating
starting value of
the
in topological
x items
singletons, then pairs of pairs, and so on).
Because
end
time, since a binar>- search tree containing
the sublists takes
all
initialized in
L
vertex, and one sublist for any vertices in
its last
L
its
U).
flow algorithm of
semilogarithmic model of
U). The details are straightforward.
Remarks
The algorithms of Section
may
be very efficient
these algorithms
Section 5
chance
is
is
in practice.
and 4 not only have a good theoretical time bounds, but they
We
hope
to
perhaps mainly of theoretical
that the use
for the
of dvnamic trees
maximum
cost flow problem.
conduct experiments to determine whether either of
competitive with previously
The two obvious open
bound
3
It is
may
known methods. The
interest,
although for huge nenAorks there
is
is
of
some
be practical.
theoretical questions are
flow problem
tree scaling algorithm
whether further improvement
in the time
possible and whether our ideas extend to the
not unreasonable to hope for a bound of
0{nm)
for the
minimum
maximum
flow
problem; note that the bounds of Theorems 4.3 and 5.6 are 0{.nm) for graphs that are not too
sparse and
whose
better than
0{nm) would seem
arc capacities are not too large,
to require
i.e.
major new
Oogf.')'
ideas.
*
= 0{m
'n).
Obtaining a bound
24-
As
0(nm
a pania]
loglog
assuming
all
U
answer
to the
log (nC)) for the
minimum
we have been
/
^ C
US
able to obtain a time
cost flow problem, where
arc costs are integral [2]. This result uses
additional ones.
b
second question,
some
C is the maximum
bound of
arc cost,
ideas in the present paper and
some
-25-
8.
Acknowledgements
We
9.
Andrew Goldberg
thank
for inspiring ideas and stimulating conversations.
References
[I]
R.K. Ahuja and J.B. Orlin.
"A
and simple algorithm for the
fast
maximum
flow problem."
Technical Report 1905-87, Sloan School of Management, M.I.T., 1987.
[2]
R.K. Ahuja, A.V. Goldberg, J.B. Oriin, and R.E. Tarjan, "Finding minimum-cost flows by
double scaling.*'
[3]
J.
To
appear.
Cheriyan and S.N. Maheshwari. "Analysis of preflow push algorithms for
work flow." Department of Computer Science and Engineering, Indian
nology,
[4]
J.
New
maximum
Institute
net-
of Tech-
Delhi, India, 1987.
Edmonds and R.M. Karp.
"Theoretical improvements in algorithmic efficiency for net-
work flow problems." Journal of the ACM, 19:248-264, 1972.
Even. Graph Algorithms. Computer Science Press, Potomac,
[5]
S.
[6]
R.L. Ford,
Jr.
and D.R. Fulkerson. Flows
in
MD,
1979.
Networks. Princeton University Press, Prince-
ton. NJ, 1962.
[7]
H.N. Gabow.
"Scaling algorithms for networic problems."
J.
of Comp. and Sys.
Sci.,
31:148-168, 1985.
[8]
A.V. Goldberg. "Efficient Graph Algorithms for Sequential and Parallel Computers."
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