On a pairing heuristic in binpacking

On a pairing heuristic in binpacking

Citation for published version (APA):

Frenk, J. B. G. (1986). On a pairing heuristic in binpacking. (Memorandum COSOR; Vol. 8613). Technische

Universiteit Eindhoven.

Document status and date:

Published: 01/01/1986

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EINDHOVEN UNIVERSITY OF TECHNOLOGY

Facult

y

of Mathematics and Computing Science

Memorandum COSOR 86-13

On

a pairing heuristic

in binpacking

by

J.B.G. Frenk

Eindhoven, the Netherlands

October 1986

ON A PAffilNG HEURISTIC IN BINPACKING

ABSTRACT

For the analysis of a pamng heuristic in binpacldng an important result is used

without proof in [I] and [2]

.

In this note we discuss this result and give

a

detailed proof of it.

Introduction

Let n

be given and suppose

(X" ..

.

•

X.) is

f

(x"

...

• x.)

Moreover assume

(

i)

OS

Ki

5 I

i

(ii)

The

stochastic vector

<X<>(I).Xo(2) •.

.

.• X<>(.»

is distributed

as

(X

"Xl

'

...• X.)

for every

per-mutation

on

(I •...•

nl

.

(iii)

f(X"Xl.

···

.x.) =f(1-x,,

'

" .x.)

Remark

Condition (ii) states that we are dealing with a finite sequence of so

-

called exchangeable random

vari-ables (cf.

[3]).

while condition (iii) is a symmelry condition.

Note that by (ii) the symmelry in (iii) holds in every component.

Before staling tbe main result introduce the following notations

t

i

if

tbe event

A

happens

1

0

otherwise

{

+

I

(i.):= _

I

if

Xi>

'2

if

lLS'2

i

I, ...

,n

i

1,' .. ,n

If we order the random variables

L

in non-decreasing order. say

r

I

S

by

tbe label of the

k

-order statistic of the sequence

(r

lr

.

l

.

Now tbe main result reads

as

follows.

2

-Theorem 1

Suppose the random variables

{K

."=1

satisfy the conditions (i), (ii) and

(iii).

Then the following results hold

a)

U:

"i

e

A}

and

i

e

A}

are independent for every subset

A

c (1,2, ..

.

,n

1

b)

/P

{(!!)

=

1t(i.),k

e

A}

=

rr

/P

((i.)

= 1t(i.)}

'eA

for every subset

A

c

{1,2, ... ,n} and

for every function 1t: {1,2," .

{-I, I}

Proof For every sequence

{Yi

1.':'1

with

Yi

e (0,

t )

and cr SQme permutation on {I, ... ,n} we obtain

/P

(Io(i)';;

Ya(i)'

(cr(i» = 1t(cr(i», i=l, ...

,k}

= /P{I-Ko(i)';; Yo(i) (i e C) "

Ka(i)';; Yo(i)

(i e (I,'"

,k}-C))

where 1';;

k ,;;

n

and C :=

[j:

1,;;

j';;

k

&

1t(crU» = I}

By (ii) and

(iii)

it follows easily

/P

(.!:.a(i)';; Yo(i),(cr(i» = 1t(cr(i» i = 1, ...

,k}

(1)

/P{Ko(i)';;

Y<>(i);i

=

1,'"

,k}

=

/P{Ki';; Y<>(i);i

=

1,'"

,k}

and this implies

/P (Ia(i)';; Yo(i);

i

=

1, ...

,k}

=

L

/P

(I<>(i)';; Yo(i), (cr(i» = 't(cr(i»;i = 1," .

,k}

=

<eD

(2)

=

L

/P (Ki';; Ya(i);

i = 1," .

,k}

=

2'/P (Ki';; Ya(i);

i = 1,'"

,k}

tED

where Disthe set of functions 't: {1,2,' .. ,n}

{-1,+1} which are different on (cr(I),' .. ,cr(k)}.

Moreover by

3

-lP

(G(i» = 7t(G(i»; i = I, ...

=

(Io(i)S

t,

(á(i» = 7t(G(i», i = I,' " ,

k}

=

(3)

=

(Xi

S

t;

i

=

I,··

·

,k}

.

Sinee the density

f

(x" ... ,x.)

symmetrie it is easy

to prove that for every I

IS n-I

and this implies using

(X

t )

=

t

that

(4)

Now by

the relations (I), (2), (3) and (4)

lP

(Io(i)S Yo(i),(G(i» = 7t(G(i»; i=I,' .

.

=

lP

(Xi

S

Yo(i);

i

=

I,···

,k}

=

lP

(G(i»

= 7t(G(i»; i=I,' .

.

,k}.1P (1:o(i)

Yo(i);

i=I," .

,k}

and so we have proved the result

in

(a)

In order

to

prove

the

result

in

b) we note !hat for every subset

A

c (1,2, .

.

.

,n)

and every funetion

7t:

(1,2,

.

.

.

,n)

-+ (-I,+I)

lP

((i.) =

7t(i.);

k

A}

=

=

L

(Io(i) S

t

S ... S

(G(k

» =

7t(G(k

»;

k

e

A}

a

=

L

lP (Io(i)

st,

Io(l)

S Io(.)}

lP (G(k»

=

7t(G(k

»;

k

A }

a

where we have used (a) to obtain the last equality.

Henee

4

-= riA I

L

i

I, ...

G

=

2-

=

TI

lP

(!!)

=

n(i.)}

Referenees

[)

[1]

Csirik,

Frenk, J.B.G., Galambos, G., Rinnooy Kan, A.H.G., A probabilistic analysis of !he dual

bin packing problem, to appear.

[2]

Karp, R.M., Lecture Notes (unpublished), Computer Science Department

,

University of California,

Berkeley, 1984

.