28
1414
DEWEY
JAN 10
ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
ON THE RATE OF CONVERGENCE TO OPTIMALITY
OF THE LPT RULE
by
J. B.C.
Frenk
(*)
A.H.G. Rinnooy Kan
(-)
HnrVinp Panpr
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
#
1^10-84
'
ON THE RATE OF CONVERGENCE TO OPTIMALITY
OF THE LPT RULE
by
J. B.C.
Frenk^"'^
A.H.G. Rinnooy Kan
Working Paper
#
1610-84
Abstract
The LPT rule is a heuristic method to distribute jobs among identical machines
so as to minimize the makespan of the resulting schedule.
If the processing
times of the jobs are assumed to be independent identically distributed random
variables, then (under
a
mild condition on the distribution) the absolute
error of this heuristic is known to converge to
almost surely.
In this
note we analyse the asymptotic behaviour of the absolute error and its first
and higher moments to show that under quite general assumptions the speed of
convergence is proportional to appropriate powers of
(
log log n)/n and 1/n.
Thus, we strengthen and extend earlier results obtained for the uniform
and exponential distribution.
(*)
(""•''«)
Department of Industrial Engineering and Operations Research, University
of California, Berkeley
Econometric Institute, Erasmus University, Rotterdam/Sloan School of
Management, M.I.T., Cambridge
^AW
1
1985
I
ACKNOWLEDGMENTS
The research of the first author was supported by a grant from
the Netherlands Organization for the Advancement of Pure Research
(ZWO)
and by a Fulbright scholarship.
The research of the second author was partially supported by NSF
Grant ECS 831-6224 and by a NATO Senior Scientist Fellowship.
0751253
-,4
INTRODUCTION
1.
Suppose that n jobs with processing times
among m uniform machines
Let
.
s
.
p,
,
.
.
.
,
p
have to be distributed
be the speed of machine
(i = 1,..., m).
i
If the sum of the processing times assigned to machine i is denoted by
(i)
Z
Z
n
(i =
1,..., m)
= raax.{Z
1
n
then a common objective is to minimize the makespan
,
For this NP-hard problem many heuristics have been
(i)/s.}.
1
proposed and analyzed; we refer to [Graham et al. 1979; Rinnooy Kan 1984]
for a survey.
Among them, the LPT rule in which jobs are assigned to the
first available machine in order of decreasing p
and attractive one.
The value Z
the optimal solution value Z
.
is a particularly simple
(LPT) produced by this rule is related to
(OPT)
for the case that
s
•
= 1 for all i by
[Graham 1969]
Z^-^^LPT)
—
^
-
(1)
.
^"^
^
Z^'^^OPT)
^
.
,
<
n
Computational evidence, however, suggests that this worst case analysis is
unnecessarily pessimistic in that problem instances for which
(1)
is satisfied
as an equality appear to occur only rarely.
To achieve a better understanding of this phenomenon,
processing times
random
—
^—
Z
p.
variables.
(j
=
let us assume the
1,..., n) to be independent, identically distributed
The relation between the random variables Z
-n
can then be subjected to a probabilistic analysis
(LPT)
.
In
(OPT)
and
[Frenk
&
Rinnooy Kan 1984] it was shown that (under mild conditions on the distribution
of the p
.
)
the absolute error
Z^'")(LPT)
-n
converges to
is
-
Z^™\oPT)
(2)
-n
almost surely as well as in expectation
asymptotically optimal in a strong
(
.
Thus, the heuristic
absolute rather than relative ) sense,
which provides an explanation for its excellent computational behaviour.
-1-
,
,
iFrenk
In
Rinnoov Kan 1984
c.
1
the speed at
,
the absolute error converges
was analyzed for the special oases of the uniform and exponential
to
almost sure convergence and convergence
a large class of
a
.
.:))^
.
and creneralize the results tor
Here we extend
distribution respectively.
Q
-.vhich
expectation bv showing that for
in
distributions Cessentiallv those with
F(:<)
=
x' (0
this speed is proportional to an appropriate power of
•_
'^
(log log n)/n
This implies that, although the
l/n respectively.
and
_ ^
ontimalitv of the LPT rule could only be established asymotot ically
the convergence of the absolute error to
least occurs reasonably fast.
these results are the best possible
to be explained later,
In some sense,
at
ones obtainable for this heuristic.
The main result for the case of almost sure converi',ence
proved in Section
in
Section
is described and
The case of convergence in expectation is dealt with
2.
where we btiund first and higher moments of the expected absolute
Some extensions and conjectures are briefly examined in Section 4.
error.
AiMOST SURE CONVERGENCE
2.
In
3,
,
[Frenk
Rinnooy Kan 1984],
&
it
is
shown that the absolute error of the
LPT rule (2) is bounded (up to a multiplicative constant) by
D
-n
where
= max^
(a)
<
o
'-l:n
p
''p
,
-
<
...
^
"^2^1
a =
times and
,,
I'-ki^n
<
a
Z'._,
p
]-l ^
'-n:n
1
m
.
.
]
:n
(3)
1
of the processing
are the order statistics
—
P
+ (m - l)s,/s
1
—
-
k:n
,
Let us assume that the distribution function
T.
of the processing times is given by
F(x)
In that
=
(0 ^ X < 1,
x'"^
case, ^n
,
k:
:n
1
~
L'!^
-K :n
<
,
a <
where
(^)
=°)
U,
-k :n
(k =
1
n)
are the order
statistics of n independent random variables uniformly distributed on
-and
b= 1/a.
-2-
(0,
1|,
.
let us define the random variable
To studv the asymptotic behaviour of D (a),
to be the index
T
-n
Hence, T
achieved.
-p:n
-iz^,u'?
a
j=l
:n
uj
>
k:n
j
(k = 1,...,
p -
1)
+ S^'^^T U^
j=k+l -j
ul"
-k:n
(a +
if'}
k=l
k - 1)
= p}
Pr (T
Thus,
(a - 1)
:n
U^
(k' =
<
-p:n
1,
.
.
.
,
p
-
1)
(a - l)(p - 1)
<
u!"
-k:n
(7)
U*"
-p:n
bounded (from above) by the probability of (7).
is
is easily verified that
it
Pr Hy^.p_^(y''))^ < x^ (k = 1,..., p - 1)}
where,
for any
statistics of
z
1
Let F (v) = ?v{\]^
-p:n
p
1
"
Pr{zf"J-
K.= i
Now Lemma
1
.
.
.
,
p - 1)
are the order
<
y}
^
103].
p.
Then (7) and (8) imply that
.
_,
,
,
<
,
?riLf_Ua +
Since (U,.
1
(8)
independent random variables, uniformly distributed on
'
J
(k =
(z)
Taylor 1981,
&
Pr {T^ = p}
U
1],
[0,
e
-
p
[Karlin
z]
(6)
(by addition of these inequalities)
so that
[0,
(5)
j
that
i.e.,
Now,
is actually
implies that
= p
a
for which the maximum in (3)
{1,..., n}
k
-izP^U*?
j=l
:n
U^
a
p
k - 1)(U,
(y^) )^/y
+ k
(a
-
£
u[^
1)
_Ay^))
.
(9)
,
ul"
,
-K :p-i
<
<
(a - l)(p - l)y}
F (dy)
(9)
is bounded by
(a - 1) (p - 1)
1
(10)
in the Appendix implies that for certain constants C=c(ct), c=c(a)
-cp
"-^^
PriT = p} < C e
-n
(11)
To derive our main result from the Bovel-Cantelli lemma, we now use (11)
ma
to bound
[d log2n 1
P'r{D^(a)
>|
^^
(12)
i
I
-3-
.
(where D is
Pr{T
-n
constant to be chosen later and
a
iog.-,n
=
by
log log n)
log n} +
^
Pr{max
+ n^r
,,b
,
L.
,
,D log.,n
,b
^-'
>
,
j=l -j :n
from (II).
*")
0(n
is
,.b
i.
a
i-k:n
The first term in (13)
„k.
1
,-U
Ilk ^ log n
(
)
^
(±2)
^
.
n
I
}
We again condition
the value
(in
corresponding to the largest order statistic being greater or
U,
smaller rhanCidog n)/n)
-log
,
to bound
the second term bv
n
n :n
'
"'
^
^
•
log n/n)''
(2
'-
•'
^
...
..
l-_k :
log
U^
-log n
I
'
The first term in (14)
is (Un
-k:n
n
= V-
:n
j
F,
log n
-
=
l
-j:n
n
'
(dv)
(14)'
•
Taconis 1979j).
[de Haan &
(cr
*)
a
.
.
'^
.
To bound the
second term, we observe that the term within the integral is bounded for
every
y
(0,
:
D
/
Pr'.max.^,
l::k
+
Pr' (1
so that
the
-
l)
(c"
:>y
(8))
r,,b
^
logn ,
•_
h
a
-
U,
,
k: logn ^
^"^"^" "
".1 = 1
'it
intesr-.l
1
i
H-^
-
-
1
i
>
—1 _k
^
.
,
U.
,
a
(
^ ^"g2" ^b,
n
y
j
.,b
,
1=1 -j
:
logn
-
.
^
1
l.D log^n b,
-(
j
J.
logn -
=i
1
k:logn-
-j
„k
j=l
a
1
'T,'3
1
I
log^n\b
,,b
-j:iogn-l
\-
logn
(lo;
t
The second probabiiity in (lb)
converges exponentially to
(where the maximum is attained) being greater or smaller
d
still to be chosen.
event is O((log n)
-Cdv
;.
From (11),
.•
(in log n)
We again bound the first probability bv conditioning on the index
a constant
;
itself is bounded by
-
Ilk
:^_)
(15)
D
T.
n
y
tiian
d
T(log n
log^n,
for
the probabilitv of the former
The remaining conditional probability is bounded by
-
1)
Pr{U^
-d logon :logn^
^
d = D/4,
For
(l^^^)h
>
1
(17)
log n
2
this term is O((log
n)"^'''"
)
(cf
[de Haan & Taconis
1979]).
Collecting all our upper bounds on (12), we conclude that, if
D =
max {16, 4/c}
2
Pr{D^(a) >
Define
= e
k^^
(
" ^°g2" )b} =
O(l/(logn)^)
The Borel-Cantelli lemma implies immediately that
.
k
lim sup
^
n
(18)
( ,
,
"
log2
°°
y
k^
,
D
-kn
(a)
<
-
(a.s.)
(19)
We show in the Appendix (Lemma 2) that D (a) is almost surely nonincreasing
in n.
follows that
It
lim sup
n
->
(~
— )^
log9n
°°
D (a)
-n
<
(a.s.)
c»
(20)
and we have proved the main result of this section.
Theorem
1
If the distribution function of the processing times
.
F(x) = x^
(0 < X < 1,
lim sup^^ ^
This speed
<
a < «=),
^L^)^^""
Pr^U,
-l:n
then
(Z^^'^LPT)
of convergence result is
derived from the upper bound (3)
>
It can be shown
^"§2'^
i.o.} =
,
equals
-
Z^™\oPT))
<
-
(a.s.)
the best possible one that can be
as can be seen from the fact that
(21)
1
n
[Karp 1983] that the speed of convergence to optimality for
the LPT rule is at least 1/n for the case that a = 1.
In the next section,
we shall see that this lower bound is also an upper bound when we consider
convergence in expectation.
-5-
3.
CONVERGENCE IN EXPECTATION
=x (0<x<l,
0<a<«').
-
Again, we assume that F(x)
With T as
-n
defined before,
E(D (x)^)
Pr IT = nl +
-n
<
-n
-
+ E(max-
(uj
.
< n - 1
k ^
^
<
1
,
-k:n
-
Z^
;^
a
j
})''
U^
,
=l
(22)
-j:n
As before we condition on the value of the largest order statistic to bound
the second term by
E(E(max-
E(max,
ECu''^
-n:n
ECU"!^
-n:n
Hence,
=
(n
^" +
e
+ 1)'^^E(D
-n
^^^^
h<(n+l)e
n ^
J.
(
This implies that h
"
^
n + qb
(a)*^)
is
E.
)'!
J
-n:n
'
-j :n ^
—
1
U
|
-7^
= ^l
- 1
^
}
^k
a
k:n -
a
)
=
])'^\
'
U
U
-n:n
)
'
=
-n:n
z'f
,
i=l
J)'^
i:n-l
U^
,„,
(23)
and (23) together imply that
(11)
'
A^V)
E(D -n
(24)
1
Then (24) implies that
+e (qb/n)
"•="
n
.
—1
-
(ul"
1
for n sufficiently large,
<-
u
-k:n
r;
.
U^
.
j=l -j:n
-n:n
,
-
e'^
a
U
k ^
< n -
^
<
1
E(D (x)^)
-n
Let h
A
1
^
k < n -
-
E(max^
)
-k:n
.
<
1
-
-
(uj
.
^
< „
n - 1
k ^
^
<
1
2
_i_
h,
n-1
(25)
bounded by a constant and we have proved the main
result of this section.
Theorem
F(x)
2
= x^
If the distribution function of the processing times
.
X <
(0 <
lim sup
n
->
°°
1,
<
a
< °°),
then
(n^/'^((Z^"'^LPT)
n
Identical results for the case that
Z^'^VoPT)
-
)'')
<
«
i
are derived in a
n
s.
=
1
for all
1
different fashion in fBoxma
1984],
Again, they are the sharpest possible
ones in the sense of the previous section.
It
is worth
noting that this is
the first time that bounds on higher moments have been derived for a heuristic
of this nature.
-6-
.
EXTENSIONS AND lONCLlDTNG RE>L\RKS
4.
Theorems
=
F(y.)
-"(x
F(x)
e
can boch be extended Co Che case chat
2
i.e.,
),
Lx"^ <
for X
and
1
<
there exisc posicive consCancs
L and U such that
(26)
[0,;)
this is done ^y showing that
one mav restrict oneself in the maximization (3)
in
,
[Jx"^
In the case of almost sure convergence,
Cas
:.
[Frenk
ti
Rinnoov Kan 1984
to ke
^l,...,[cn]}
This maximization involves only
1).
the smaller order statistics and For those we are essentially in the
situation analyzed in Section
2.
In the case of convergence
expectation, our technique requires that
„
q
(1
+ b)+
1
in
.„^.
^
for the extension of Theorem
3
to hold.
that this condition is not essential.
We strongly suspect, however.
Details of the oroofs for thes2 results
are available from the authors.
These unusually strong results, as
v;ell as
other recent ones
in
this
irea
(;BoxTTia
1984]), confirm the remarkable amenability of the LPT rule to a probabilistic
analysis.
Extensions to other prioritv rules involving order scatistizs
of processing times seem
feasible and interesting.
-7-
APPENDIX
For every
Lenrnia
1.
and
= c(g)
c
>
S
such that
Pr {I^_,
(6
+
If a >
1,
then
Proof.
-
Also,
if a
1965,
p.
there are positive constants C = C(S)
1
<
k)
\j}{^
U,
-k:m
<
Sm} i
>
U,
e"^'"
.
a.s.
-k:m
-
C
we obtain from Holder's inequality [Goffman & Pedrick
1,
that (take p = 1/a, q = l/(l-a), y^ = (6 +
2]
e"
(S + k)^ U,
-k:m
k=l
m^"^
<
-
(Z"'
+
(B
k=l
,
uy^
-k:in
k)
k)"^
)^
U^.^, x^ = 1)
a.s.
•
(A. 1)
This implies that
Pr {Zf
(6 + k) uy^" - em}
k=l
-k:m
'.
,
= Pr [ill
k=l
(B
^
+
uy^
-k.:m
k)
)"
<
-
oV)
.
<
-
Pr {e"
k=i
,
(6
+ k)^
(A. 2)
< S^'m}.
-k:in -
U,
Hence we consider the distribution of
rJl
(B
.
k= 1
Since (U,
-l:m
S
-i
=
,
y"""
+
k)"" U, ^
- k :m
U
)'^
-m,m
V and V
j=l -j
-J
103]),
w3
<
1)
(S^/S
-1 -m+i
=
with
/S
... ,S
)
-m -m+i
independent exponentially distributed random
variables with parameter
p.
(a
X
=
1
(
j
=
1
,
•.•{''"
.
.
.
,
m)
(
[Karlin & Taylor 1981,
can rewrite the right hand side of (A. 2) as
/
Now for every
m
>
>
e
there exists some m
0,
= mo(e)
such that, for every
m„(e), the above probability is bounded from above- by
,m+l ,
Pr {I^^^ (1 +
,
=
,
.
e
„
a+1 „
V^
,
+ 6)
.
P^^=I=i ^.,m+l Y,
>
. . ,
^
s
((1 - e)m)
>
a+1
S^
,
,
0}
i%rith
Clearly -1
<
2
'!
4-
<
^, m+1
(
(m
((I
((1
c„
£
,
and this implies for
and m
>
a+1
£)ni)
+ K)/(l
B)/(l - r)m:
+
=
_,, =
-£,m+l
c„
+
.\e[0,
2
(A. 4)
- 1
p+1^
+ 6)/(l - Om)''
1/2
(2(1 - e)/3)
.
-
1
(£ =
1,..., m + 1)
a+1
]
max(m (e), 2(B + 2)) that
From the Taylor expansion of log
( 1
+
x)
around x = 0, we then show
that the above term is bounded by
,-„m+l
,
^^P('^i=l "^^m+l +
,2 ^m+1
^
2
,
^1=1 "i,m+l^
•
Since
lim
m-'<»
aT]
P.=l
<^oP.
4.,)/in =
,m+I
l/((a + 1)(1 -
i)''"^^)
(A. 6)
and
-2/((a +
2)(l - O^-^') +
the desired result
values for
X
and
'^-^
1.
follows from the appropriate choice of positive
e.
-9-
1
Lemma
2
Z'°
n=
Proof.
It
?r
,
-U
>
)
(
,
-n+i
L
D
J}
(
-n
easy to verifv that
is
the new processing time
<
'°.
,,(')•'
D
unless (perhaps)
(2)
if
larger than all the previous ones.
is
Hence,
Pr {d
,(
')
>
-n+1
-n
(
')]
Pr
•-
-
•'ji
^'?'
^,
n+1
'.
'
1=1
P-.
"
0>
''.
r-
]
n*
00
=
D
((a-l)y) F(dv)
F
r
(A. 8)
"
so that
Z""
,
n=l
Pr {d
^Ai)
-n+1
-
D
-n
(Ol
<
-
7
U((x-Uy)F(dv)
(A. 9)
'
:
with 'J(x)=Z^
,
n=l
Frank 1984]).
Un
f"
function ([Feller 1971; Van Dulst
(x)the renewal
-
•.
^'
The result now follows from
-ii^
=7
an d the local boundedness of U(x)
X
F.(dx)
^-
on (0, ^)
-10-
.
V-'l
(A. 10)
REFERENCES
Boxma, O.J., 'A Probabilistic Analysis of the LPT Scheduling Rule',
Technical report. Department of Mathematics, University of Utrecht.
de Haan,L., Taconis-Haan tjes ,
Sample Quartiles , Ann. Inst.
'
E,,
'On Bahadur's Representation of
Math, (vol 31), 1979, 299-307.
Statist
.
An introduction to probability theory and its applications
Feller, W.
~.
Vol. 2., John Wiley, New York, 1971.
,
'The asymptotic optimalitv of the
Frenk, J. E.G., Rinnooy Kan, A.H.G.
L.P.T. rule'. Technical report. Econometric Institute, Erasmus
University, Rotterdam.
,
f-
Goffman, C, Pedrick, G., First course in functional analysis
Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965.
,
Karlin, S., Taylor, H.M., A second course in stochastic processes
Academic Press, New York, 1981.
Karp,
R.il.
,
Private communication,
.
1983.
'Approximation algorithms - an introduction'.
Rinnooy Kan, A.H.G.
Technical report, Erasmus University, Rotterdam, 1984.
,
Van Dulst, D.
Frenk, J.B.G., 'On Banach algebras, subexponential
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,
37Su
fu)
8
IDfi D
D
35T
BASEMENT
Date Due
Lib-26-67