Computational analysis of the random components induced by a binary equivalence relation

Thesis by G u y

de

Baibine

In P a r t i a l Fuliillmenz 05 the Requiremznts F o r tile D e g r e e af

Doctor of Phi1oso;lny

C a l i I o r n & a lii.jtiLdtc ol Technology Pasadena, C a i d o r n i a

1968

. .

.

111

A B S T R A C T

-i

-

L n e probicrn of partitionins into c l a s s e s by rncans of n binary- equivalence reiation i s i;ivt:siiigatcd. S e v e r a i a l g o r i t h m s .. for d e t e r - mining the number of components in the gralzh associated witn a

p a r t i c u l a r s e t of e i c m e n t s a r e constrlicted and compared. J I h c n tine

classification p r o c e s s o p e r a t e s on independently drawn s a m p l e s of n d i s t i n c t e l e m e n t s f r o m a popuiation, the expected number of

components i s shown to be 0bta.inabI.e recursivcLy f o r a c l a s s of p r o b l e m s called s e p a r a b l e ; i n a i l c a s e s , e s t i m a t e s a r e available to r e a c h any desired. l e v e l of a c c u r a c y . Clustering inodeis i n

Euclidean space a r e analyzed i n d e t a i l and a s y m p t o t i c forrniiias obtained to cornplomcrnt e x p e r i m e n t s . Conjeczure s c o n c e r r i r ~ g the

C h a p t e r I

I1

111

T A B L E OF COXTZYTS

Introduction

C l a s s Counting h l g o r i t i i r n s

Fiecursive D e t e r m i n a t i o n of t h e N u m b e r of Components

Estin-*ation of the N u m b e r oL Cnmponenf:i e m s D i s c r e t e a n d Continuous Adjacency P r o b l -

C omputationai Tools

Chapter X Introduction

The purpose of this s k d y i s to investigate p r o b I ~ n l s of partitioning into c i r s s e s . Given a.ny biriary relation defiiicd for a i l p a i r s of e l e m e n t s in ii? s e t to be partitioned, a cnrrespundi?ig binary eqci-ralence relat;.on can be defincu. Using a n undircctcd graph, tlie edges of which indicate that the r e l a t i o n holds for a.djaceilt v e r t i c e s , we obtain a c e r t a i n number o i components, e a c h component c o r r e - sponding to one of the equivalence c l a s s e s of the v e r t e x set. The expected n u r n j e r of componerits i s of i n t e r e s t i n p r o b l e m s in pnysical c h e m i s t r y and i n ocher fields.

Given a graph on n v e r t i c e s , thc number of components can be e x p r e s s e d i n t e r m s of the i n v e r s e of the row s u m s of tire n - 1 s t Boolean power of the graph incidence m a t r i x . This maiiiernatical lorrnliiation provides one method f o r computing the number of ccm- ponents but o t h e r s , m o r e efSicienL a l s o can be c o i ~ s t r u c t e d . A n i m l o r t a n t problem that we examine i s %he determination of the expected value of the number of comi?onerits v.<lcn the classification p r o c e s s o p e r a t e s on independently drawn s a m p l e s of n distinct e l e m e n t s f r o m a population. Of c o u r s e , f o r some r a t h e r s i m p l e e x a m ? l e s , such a s d i s c r e t e ailu contirruous occupancy problem on the l i n e , we find analytical a n s w e r s , but in the m a j o r i t y of c a s e s we

canzot hope l o r c l o s e d - f o r m solutions and m u s t r e s o r t to c o m 2 1 ~ t a - tionai experimentalion. F o r a c e r t a i n c l a s s of problems called

i f ihc s t r u c t u r e of the relation i s f a r too cornpiex o r only jtatiscbcaily hiown, our m o r e n ~ o d c s t a i m i s to d c r i v c c s l i r n a t e s I o r the probabic nii:nber of campor;cnts. Depending bpon the toil. ilia1 wc a r e ready t o

face, variorii a?;,ro:iimate merhcxls ranging f r o ~ n simple s p r i n r i e s t i m a t i o n to ienglny Monte C a r l o sampling a r e avaiiable in o r d e r to reiich any d e s i r e d l e v e l of acciiracy.

Among a i l binary equivalence r e l a t i o n s t h e r e a r e sonlc which d e s e r v e s p e c i a l attention since they a r e closely r e l a t e d to p r a c t i c a l problems. in that r e s p e c t we s l ~ a l i analyze in betail d i s c r e t e and continuous adjacency p r o b l e m s in 1, 2 o r 3 dimensions which forin silnpiified clustering m o d e l s in Euclidean space.

Throughout t h i s work g r e a t c a r e i s e x e r c i s e d i n using analyi- i c a l a n s w e r s , a s y m p t o t i c expansions and coinp.xtational e s t i m a t e s in a harmorlious conjunction. A r e s u l t of this comprehelisive study i s C .

_

inference of conjectures concerning the g e n e r a l behavior of the expected number o i components,

1. I . _" B i n a r y C i a s s i i i c a t i o n

.

B i r ~ a r

₂

E ~ ~ u i v n l e n c e Xc:l;tinn

_---

. .. 1 i

We c o n s i d e r a ;?opa.tai~.on

@

of N i n d i v i i l i a ; ~ 1 a,, a,,,

.

.

,

,

a- ;;

i i i. :t'j

:or any ? a i r oi e l e m e n t s a . ; 6 0 , let a cunnc;ct.,id !,i,n-ry r<:lation

1' &J

-

VS)

be defined. C o n n e c t e d n e s s s i m p l y i m ; ? i i c s that i?c,tweeii any two

1

d i s t i n c t & - m e m b e r s , e i t h e r

&

o r

&-

holds. We: p e r i a r r n n

i- w

e x p e r i m e n t s

Fi,.

.

',

pn

,

e a c h experirncrit b e i n g d e i i n c c a s t h e

si:iection of a m e m b e r of

@

a c c o r d i n g t o s o m e j ? r e s c r i b e d probability

%-

d

of e x p e r i m e n t s

@ .

i s s i m p l y law. T h e s p a c e

I

and t n e p r o b a b i l i t y o i a n e l e m e n t a r y event a . i s J

T h e o u t c o m e s of our combined e x p e r i m e n t

g =

El

x

P

P 2 X

...

s $ ~

a r e o r d e r e d n - t y p l e s = (ai

,

a,

,

,

.

.

,

a j f o r m i n g the s p c e

i ' 2 1 I1

The p r o b a b i l i t y a: an event

a e

@'is p ( a . , a .

,

. . .

, a . ) which, il: t h e I1 I 2 I a

e x p e r i m e n t s a r e i n d e p n i i e n t , b e c o m e s

E a c h event

&

invoives a elemesits of

&

d i s t i n c t o r not, which can be g r o u p e d into c l a s s e s by m e a n s o i the r e h t i o n

&,

T h i s

j i j f o r any p a i r a , , a

~ 8 ,

a , $ a .

,

in a c l a s s G t b e r e exists some

1 J I J

sequence

where

{

a ah2..

. .

. a i s a subset. possibly empty. oi the e~erneilcs

i

"~d

i n C

.

{ Z ) if an individ~a.1 in a c l a s s i.s in the relation

d?.

o r EE with another individual, then the second individual i s in the s a m e c l a s s a s the f i r s t one.

Tne purpose of condition ( I ) i s to accomniodate irreflexive and non-reilexivc binary relations. By irreiiexive we mean that no

rncmber oi

O)

b e a r s the relation

h?

t o itself, whereas nun-reflexive means that each @?-member does not b e a r the relation

h?

to itself, Also it e s t a b i i s h e i the chaining property among eiements of the s a m e

class. a . g i S . means c i t h e r a i@b o r bk!a o r both. The identity

1 3

binary relation E i s an equivalence relation used t o group into the

same c l a s s identicai elements. It i s convenient t o l e t

e

ja.) r e p r e -

1

sent that subset of (;i which f o r m s the c l a s s t o which a , belongs.

1

Theorem

-

F o r any binary reiation I%, under hypothesis ( 1 ) and (21, the classification i s unique and the resulting c l a s s e s a r e mutually exciusive,

?roof: a simple constructive proof proceeds a s follows; Construct

i

i n undirected graph by adding a, vertex f o r a .

ii

a.

C

i a i l , z a i Z , .

. .

,

v

3.

U

a,

.

If ai indeed f o r m s a new vertex, connect it by single edges

"u-

1

V _I

c l e a r l y s a t i s i i e s conditioni ( I ) a n d ( 2 ) ; each c l a s s i s obtained by a r b i t r a r i l y si:iecliiig one vertex a n d p i c k i n g out a.ii dependent

branches -anti1 thr? rioies a r e exhausted.

X ~ t i c e that, altbougii s;.;i: do nut r e q u i r e

&

t o be an equiv- alence relation, conditions ( 1 ) and ( 2 ) e n s u r e that the classification i s done in t e r m s of an equivalence reiation

4'

derived f r o m

@f

and such that

a . R a .

+==

a . R a l o r a . & a . o r both

J J . l i

0'

a . n a . and a &a.

=3

a i k f a k

1 3 j LC

0: course, if depends on the p a r t i c u l a r sample chosen

CTLAPTXR I1

C ? a s s Coiuntin:: A l g o r i t h m s

-.

In a v e r y irimited : ~ u i u b e r of c a s e s , we c a r find anal:iiiral s o i u t i o n s a n d p o s s i b l y a s y m p t o t i c e s t i m a t e s f o r the n u m b e r of c l a s s e s given a po2ula'iion

p,

a n equivalence r e l a t i o n

R'

an* a s a m p l i n g d i s t r i b u t i o n o v e r

G)

.

Most of t h e t i m e , however, t h e a l t e r n a t i v e h a s t o b e c h o s e n which r e q u i r e s computing s a m p i e s t a t i s t i c s .

S e v e r a l a l g o r i t h m s t o d e t e r m i n e t h e c u m b e r of c l a s s e s , given t h e e l e m e n t s of s o m e s a m p l e

a ,

c a n be proposed. They a l l have, a s a c o m m o n c h a r a c t e r i s t i c , t h e i r g e n e r a l i t y of u s e

t

s i n c e they a r e a p p l i c a b l e t o any eqiii-galence relati-on

6?

and f i n i t e s a m p l e

Ck

f r o m a finite o r infinite p o ~ u l a t i o n

@.

A s a m a t t e r of f a c t , their only i n t e r f a c e with t h e outaide w o r l d i s at

t h

t h e f o l i o w i i ~ g level: a r e the i and jth e l e m e n t of t h e c u r r e n t s a m p i e i n t h e r e l a t i o n

8'

with e a c h o t h e r ?

An essenciai choice to rnaice i s to select a data s t r ~ c w r e

representation f o r the connectivity inlorxn;ition among elcineni-6 of

a.

0 1

course, one may construct tlie 0 - 1 incidence rr.;itrix but this

r n j

r e q u i r e s testing a p r i o r i rill _{\ 2}

₁

p a i r s of d i s t i ~ i c l elements (ai, a;; J

$31. the rclstion

&'

to hold, Also, i f the proirability that a , d a : holds J

i s s ~ n a l i , any storage reprosentation 'ijhfcli accomn~oclates a l l o i the

jn)

e n t r i e s will appear extremely urastefui, On t k c other hand, ii 1 2

we choose t o only keep t r a c k of the l e s s liiieiy outcomes, h e r e those p a i r s f o r which a . i t ; & .

,

the r e t r i e ~ a l effort i s bound t o be

J

significantly l a r g e r than before, Yet, we can develop a n algorithm which does not n e c e s s a r i l y r e q u i r e the full incidence m a t r i x and which t e s t s a . ~ ' a . a t m o s t once, This l i s t s t r u c t u r e algorithm i s

1 ,I

presenten i a s t and is probabiy tbe most useful. K e v e r ~ h e i e s s , we f e e l that the two following algorithms have interesting c h a r a c t e r i s t i c s

in t h e i r own right which arnply justify t h e i r analysis.

2, 1. Algorithm 1: Iterative Coix13utaiion on the InciL,cnce h,ftitrIx The incidence m a t r i x A of the undirected graph Gn witn n v e r t i c e s , single edges and no loops i s

The binary relation i s an eq~uivaience relation derived f r o m { s e e section

[

1. 1

1)

and A .is t h s b r r m n ~ y - m m e t r i e rfiatr'ix with

O diagonal elements.

k . th

Let a . . oc the

ti,

3 ) element 01 the k power of A ;

1 J

k

v . t o v.

.

Since tile s h o r t e s t path between any two v e r t i c e s in G

1 J n

cannot b e u i icngrii g r e a t e r than n - 1

,

v. and v: belong t o the s a m e

1 J

c l a s s iiand only ii

f o r any s e t of coefiicients s k c h that

k

in p a r t i c u l a r , c o n s i d e r t h e m a t r i x

s t

w h e r e 1 is the identity matrix of o r d e r n

.

I t s

in-:)

power h a s t h e e x p r e s s i o n

which i s of t h e type ( 2.

2

1.

Consequently, e ( v i ) ~ e j v )

,

@ j v ) J

designating the c l a s s o r s e t of v e r t i c e s t o wnich the v e r t e x v belongs, n-:

i f a n d o n l y ii d . . 3, 0 .

* 13

L e t U be the column v e c t o r with all n cornTonenis equal t o 1.

-.e-

--

_l

F o r any v e c t o r V with n o n - z e r o components v . define a s V t h e I

i

i n v e r s e v e c t o r having components

-

_v.

.

1 -r.

7-i . ~ e o r e r n

-

T h e n u m b e r o i c l a s s e s of a grapI" - w i t h incidence

n - i

prooi: a s shown, the (i, j; cLernent oi

(A+

1)

i s e i t h e r

1

o r 0

,

depending uiron wiietl~er o r : ~ o t t h e r e exists a ~ ; ~ . t h Lram

> .I1

-

1

v; to v:

.

T a k i n g the minilntirn oi j u . .

,

i j yielids a

(0-1)

firatrix,

j 1 3

.

.

tneced

a

t r u t h m a t r i x f o r t h e r':Latinn b?' on C

.

AppL;.lng t n i s

II

a t i ,

transiorrnation t o

II

,

produces a row sun1 vector, ti-:c i corn2onev.r: of which i s the size of the c l a s s of whi.ch v . i s a rni:miser.

k~

the l a s t

1

operation t o be performed, each row i s c o ~ i n t c e v ~ i c h a vreig2:t inverseiy proporciol~al t o the s i z e of the c l a s s t o which it beioays, thereb;

yielding c by simply taking the s c a l a r product of a unit rovr vector with the inverse nf the row sum vector a s Gciir.e(i above,

n- i

R e m a r k on the computation of (/it

I)

: F u r the purpose of obtain-

2 3 n-2

i n g the number of c l a s s e s , the intermcdia.te m a t r i c e s A, /i *.A # , .

.

,PA

nee^ not be explicitly computed and a shortcut may be useci. A s a

rnztter of f a c t , the next paragraph i s not oniy relevant to m a t r i x powers but a l s o t o other transformations, proviced tiley a r e a s s o c i a t ~ u e .

Lei the binary representation of n - l be

then

operation which involves %'= mi. m

-

1

m a t r i x m-ultiplications,

S

t h e r e

I

r h c s t r a i g k i i a r w a r d approacil, Of c o u r s e , unless v ~ e a r c iri t h e m A.

s p e c i a l c a s e n - I = %

,

~ i , e p r e s e n t s c l ~ e r n e n e c e s s i t a t e s two m a t r i c e s 2k' ls:zi

t o be kept a l l along, n a m e l y , D a.nd i 3 *

.

But it i s c e r t a i n l y

i = 3

not a v e r y s e r i o u s objection, e s p e c i a l l y f o r s y m m e t r i c m a t r i c e s , in which c a s e t h e above two m a t r i c e s c a n c a n v e n i e ~ ~ t l y be h o u s e 6 i n the u p p e r a n d l o w e r h a l v e s of a n (ni-1 x n j m a t r i x .

Let u s now r e t u r n to t h e s p e c i f i c c a s e o i c l a s s c o i ~ i ~ p u t a t i o r ~ . n - l A s we h a v e a l r e a d y pointed out, we only need t o -mow w h e t h e r

d - .

+. LJ

I\ .

i s 0 o r > I o r equivalently whe2iier 1.; IS, f o r N > a - i

.

Choosing

IJ

K = Z g r e a t l y o u r t a s k s i r ~ c e we c a n now d i s p e n s e

2

4 DN

wit;? one m a t r i x a n d only c a l c u l a t e D, D

,

D

,

...

,

.

F u r t h e r m o r e , t h

l e t u s a s s u m e that a t t h e ld s t a g e , i. e . , when computing D2*frorn

DZ U-

,

; 2 2- l

w e still h a v e u , . = 0

.

We m u s t th.eri f o r m t h e s c a l a r

t h

'

U - I

p r o d u c t of the i row o i 112 with i t s jth column, but a s soon a s t h i s ;?roduct becorries g r e a t e r t h a n 1 we need not go any f u r t h e r ; we

%'

just s e t d2.

=

1. A i s o , we have t a c i t l y a s s u m e d f r o m the beginning '3

tnat e v e r y m a t r i x p r o d u c t i s p e r f o r m e d in p l a c e , the updated v a l u e s d . . = i being u s e d a s soon a s t h e y a r e found. Coasequeniiy, w e n e v e r

1J

2

-4

e v e n r e a l l y compute D

,

u

,

.

. .

,

3N but at t h e completion of the &' t h

2 2"

s t a g e we obtain a rrAatrLx D

---O

So m u c h t h e bet~ei"! E'lnaily, tLe aigorirhrn t e r m i n a t e s when e i t h e r N is r e a c h e d o r a complete sxve~ep

?i- i

Q y3eas. 0% :palenl-t.na a q p a a u s:anpozd i z l e s s

an

I = ?

-

!z

N

3"

\ U

f l

z

LTUO

'

nzc

a

13nposd ayi r~ +

p

a3tie;lslp aqa :rt. 2i.e ;:a!q~

c i r c u m s t a i ~ i e m u s t c o r r e s p o n d t o a g r a p h with the s m a l i c s t n u m b e r or' c d g c s , that i s a t r e e on n v e r t i c e s , s i n c e aiiding any edge can only i e c r e s s e o r icavcl uncinangi:d the d i s t a n c e betlvcen any two

c,ges,

mat

t r e e b e a l i n e a r chain; t h e adjunction of a n e a y e ."

t o a l i n e a r chain -masirnizes ( 2. 8 ) o r (

2.

9

)

5

it i s madre zt the e n i o i the c h a i n p r o i u c i n g the s e q a e n c e of d i s t a n c e s a

=

n-

8 ,

Z

I?=

l,Z,,..,n-I;

a s

any o t h c r s e q u e n c e c a n b e c o n s t r u c t e d by rcpcatediy borroiving 3

f r o m s o m e cu and adding i t o a,, s u c h t h a t

4; =.12

b ~ t

8

1

"

2

ioa

I <

L * 2 !? li

-,

L

lo- -2

.t?2)

S O thzt the maximurn n u m b e r of s c a l a r p r o d u c t s i s e x p r e s s e d by

U s i n g

n

2

the n u d e r of s c a l a r 2 r o d u c t s wiil t h e n be at w o r s t log2" + @ i n j 2

.

E a c h s u c h operarion iiivo?ves at m o s t n t e s t s which a r c p e r i o r m e d a s l o g i c a l ' n t e r s e c t i o n (AXT,) o p e r a t i o n s r a t h e r than m u k i p l i c a t i o n s .

W e have t h a s shown 'inat in t n e w o r s t possi?iie c a s e 01 a l i n e a r re 3

chain, the n u m b e r of r e s t s t o p e r f o r m i s bounded above by 5-

_'..

loa n

.

is:* c l a s s c o u n t i n . . b y pc?:.rers of i n c i d e n c e n a t r i x 1 . 2 d o p a r t 2

-?or

h = l

1 - 3 -o t o s t e p 1 . 2 i f b i n - l ? : h = : 9 . 4 do p a r t 4 f o r i = l i I l n f o r c=C 1 . 5 t y n e c

i n

f o r a

1

2 , 1 b = b * 2 f o r e = b f o r 5=O

2 . 2

i n

p a r t

5

i f aCi,j)=0 f o r j = i + 1 ( i ) n f o r i = l ( l ) f i

3 . 2 s = - I i f a ( i , k ) " i a [ k , j ) f o r k = l ( s i n f a r

s=:

3 . 3

a(i,J9=l

f o r a ( j , i i = i f o r h = l i f s = - 1

4,: s = s + a ( i , j ) f o r j = l i i ) n f o r

s=O

4 . 2 c = c + l / s

F o r 3

I

Ziowever, in p r a c t i c e t h e a v e r a g e n u m b e r of t e s t s i s m u c h ~i:~ili'ir 1..

-

1

i

e q u a l t o

13

_E2

1

il$

9"

.

-

i 6

5

4 1

,

p

re:prescnting t h e a v e r a g e l" 4 Y n

n u m b e r of r e s t s periorxried ciaring a s c a i a r p r o d u c t , b'rlori: a m z k n

i s e n c o a n t e r e d .

h

p r a c i i t t :

p

is srsail a n d t h e aigori;hrn i s q a i t e s i m p l e s o t h a t it p r o v e s m o r e e f f i c i e n t t h a n u s i n g { W r a t h - i l l a L,

t h a t i s , p e r f o r m i n g a n a c t u a l m a t r i x i n v e r s i o n t o obtain t h e n o n - z e r o e n t r i e s in t h r f i n a l i n c i d e n c e matrix.

F i g u r e ( 2 , 1) s n o w s a n im-glemer~tation of a l g o r i t h m 1 i n CITRAI; (CIT t r a n s l a t o r ) which e x l ~ l b i t s i t s siiriplicity.

2. 2,

A l g o r i t h m 2: R . e c u r s i v e CompiAa.tion orr t h e i n c i d e n c e M a t r i x We t a k e t h e opportunity h e r e t o c o m m e n t b r i e f l y on t h e i n t e r - r e l a t i o n s h i p b e t w e e n p r o g r a m m i n g l a n g u a g e s arid the a i g o r i t h m s t h e y host. O i t e r ~ , we i i n d t h a t t h e convenient w a y of attaching a given s r o b i e r n f r o m a n u m e r i c a l , s t a n d p o i n t d o e s not c o r r c s p o n j t h e i n - t u i t i v e a p p r o a c h we urould u s e if we w e r e :iskcci t o obtain t h e s o l u t i o n ii: r; f e w s i m p l e c a s e s ,

In

s o m e i n s t a n c e s , t h i s d i s c r e p a n c y can b e

explaineci by o u r i n a b i l i t y l o think along t h e limes of a n cflicieiit b u t t o o cornpiex a l g o r i t h m . M o s t oL tile t i m e , iio-wever, we tend t o f o r m u - l a t e t h e p r o b l e m in t e r j n s of t h o s e b a s i c p r o c e s s e s w h i c n a r e m e s t n a t u r a l t o t h e p r o g r a m m i n g iaiiguagc u s e d , F o r i n s t a n c e , a language

.

.

like FORTRAN is e s p e c l a i l y s u i t e d Lor i t e ~ a t i v e c a i c i ~ i a t i n n s j u t h a s n o b u i l t - i n recuriiicin ca2abiiif-ji, w h e r e a s , t h e c n i i t r s r y i s rrGe f o r M c C a r t h y ' s LLSP 1. 0 , Consec;ucntl.y, ci3an;jir;g ~, t h e has: la11 G'.' , u c > n e

-

arid

will in general b e the ca.sc, By way of ilristration, we now exam:cie a "na.uraIv a.Igoritiirn,

,..

.

u i v e n ZI. g r a i h G rvltii n v c r l i c c s , t1:i: sepai:atior 3; iii

xi

. .

v e r t e x s e t into iisjoiiii conrlectec! c:ssses can bc achic;-ed 0;. seleiti--' '-6

one of the remaining v e r t i c e s a;ld "p~liii;~ out'' a i l (ii i t s dcpa~fideot

ones, zhat i s , removing the connected subgraph of the ciast: to which i t belongs, and s o on until no vertex remains. Zi vie think in t c r m s of

zi rigid physical str-acture, t h e selection p r o c e s s i s a single o;icrstion,

a l l nodes of the subgraph b e i n g moved a.t once. E tile s t r u c t u r e i s now articalated with loose joints, tbe removai may be considered to take place in stages. r ' i r s t , a l l v e r t i c e s a t a ciistance i i r o m :hc chossri v e r t e x a r e inarlieri; then ail v e r t i c e s not ~ r e v i m . s l y marked and having an cdge in corr>rnon with one of tnc i a ~ t v e x i c e s markcii a r e themselves selected, and s o f o r t h . We have thas introduced a h i e r a r c h i c a l s t r u c t u r e with r e s p e c t to a x a r b i t r a r y node acting a s the root o.f each subgraph; each v c r t e x at a distance d f r o m the root i s responsible f o r coilecting the v e r t i c e s to whom it i s directiy connected and which have not yet been marked. This intuitive priiced-*re can be easily impierncn";ed a6 a r r c u r s i v e p r o g r a m o?era,ting on t i l e upper

-

half of tiic incidence matr'k. *. o r any a . . =

1

,

r u w i

znd

coinm j 1J

a r e scanned o m position

at

a time, always mo-ling away f r o m a , .

,

1; unaer the ioilowing r u l e s (assuming we a r e presently moving along

row i)

."

u

a . = 0

,

c e n t i m e ;

1k

i f a.

=

1 an& i s a l r e a d y r0arki.d a s a m e m b e r oi the same

;k

I.C1*

_-

_-

r e c u r s i v e

class

c o u n t i n g a l ~ o r i : b , r n

1 - i

V(kj=(k=i)

f o r U ( k ) = ( k = l ) f o r k = n ( - : ) i

1 . 2 do p a r t 2 i f

a(i,j)=l

f o r

j=i(1lri

f o r

i=1(l)n

1.3

t y p e

k-1

i n fern

2

2.1

do p a r t 3 f o r

iii)=i

f o r

J(ii=

f o r k = k + l f o r

i=i

3.05 a(ii:),J(I))=Z

3.65

go t o ste;,

5 - 2 5

15 u'=?\uq=3 f o r u=V(J(lj)

3.1 d o

4

f o r E ( - - I - \

, , - ~ < t , - l ~ s > l

for

s = - 1 3 . 2 l o

s a r t

4 f o r

E::9=i(i)+l;s)d(l)

f o r s = i

-

>.;t; r;o ;o s t e p 3 . 6 i f u - = i & " - = i j f o r

u=:;<

i ( 1 j )

3.3

do

p a r t

5

f o r

Eil:=J(ij-i(sjlill

f o r

s=-i

3 . 1 d o p a r t

5

f s r E ( i i = J ( l ) + i ( s l n f o r

s=i

3 . 6 1 x 1 - 1

4 . 0 2 done

if a(E(:),Jil

) ) = t i

4.65

c o to s t e g 4 . 5 * f o r

s=-5

i f

aiE(i),Jill)=L:

4.1

done i f

U<:4:E(i))<S

4.2

iI(E(1))=:+1

i f

tl(E(l))=O

1 . 3 do p a r t 3 f o r

I=l+1

f o r

i(l+:)=F(l)

f o r

J(l+l)=J;l)

4.5

done

5.62 done

i f

a(I(ll,E;l))=O

5.05

go

to

s t e p

5.5

F o r

s = - s i f

ailil),Eil))=k

5.1

dons i f

O<V(E(l)j<I

5 . 2

V(Eil))=l+I

i f

V(F(I))=C

5.3

do ? a r t 3

f u r

l=i+1

f o r

i(l+ij=i(l)

f o r

J(i+ij=Eil)

5.5

done

Forti? 2

if

a.; = 1 br;t colurnn ic na,s a l r e a d y been a s s i g i i e i f o r

i s

scamir.g b y s o m e tlcrrLi;nt am!:,

,

f!

-i: i

,

con.tini;e; o ~ h e r w i s e , a s s i g n coiurnn

k

f o r scanning by _'k and i n i t i a t e t h e s c z n i.n that c o i i l m ,

T h i s p r o c e s s s z a r t c w i t h a(l,

I),

proceccds t o f i n d ail of i h e v e r t i c e s i n i t s c l a s s , then e n d s up a t a i l , i) ; t h e iiiagonai el.crnerits a r e examined one a t a t i m e , u n t i l one is found wilicii has not been a s s i g n e d t o a c l a s s yet. T h e a l g o r i i h m s t a r t s a g a i n with tilis new element u n t i l ail diagonal p o s i t i o n s a r e a s s i g n e d t o s p e c i f i c

c l a s s e s ,

T o show tile c o n c i s e n e s s of t h e a l g o r i t h m when w r i t t e n in a n a a p r o a r i a t e language, f i g u r e ( 2 . 2 ) is a l i s t i n g of the corresponiiing C I T A U N (CIT t r a n s l a t o r ) p r o g r a m , p a r t s 3 , 4 and 5 being r e e n t r a n t .

The riiimber of t e s t s t o be s e r f o r m e d is a p p r o x i m a t e l y c o n s t a n t @($) but we m u s t :ake into account t h e g r e a t e r complexity of the algori-ihrn.

2.

5,

A l g o r i t h m 3: G l a s s CoufitirLg by Means oi List StriicL--Ires

Definition

-

A c o n ~ r a c t i o n I3 of a g r a p h G is a g r a p n whose

a

v e r t i c e s a r e connected s u b g r a p h s of G f o r m i n g a p a r t i t i o n i n g of the n

v e r t e x s e t of G _n

.

-

;wo v e r t i c e s of 13 a r e a d j a c e n t

if

t h e c o r r e s p x i d i n g s u b g r a p h s have a d j a c e n t v e r t i c e s in Ci

.

n

Lemma

-

T h e n u m b e r of c l a s s e s i s i r v a r i a n t n n d e r a contrbction o p e r - ation.

urooi: let ai and

a

.i

b e "Lwu d i s t i n c t v e r t i c e s of G,

.

E a c n one c o r r e s p o n d s t o unique s u b g r a p h s W; a n d 3 . i n the p r t i t i o a i a g US

J

I

I , a . e H . a h . 63 i l .

.

E i h

,

H and

3

a r e c o r ~ i ~ e c t c d sub-

1 i ' J J J 1 . j 1 j

z r a p h s with nr i c a s r cnc cdgc i;i common, thus I - tJ

1.1

is connected j

and a . f i n , ,

1 ,

.

. , . * .

i3clinition

-

A c~nt:action 1-1 oi G i s $ ~ , % n i r - a ~ xi ~ t s vertcx

n

s e t i s i n d e p n d c n t . The namber of c i a s s e i i n G i s ec,ual t o t h e

n number oi v e r t i c e s in i t s minimal eontraclioi.,.

An elementary coztraction i s tile replacement of a p a i r oi aujaceni: v e r l i c c s by a single vertex.

Any contraction of G can be reaiized a s s sequence of n

elementary c o i l t r a c t ~ o n s .

The mininxal contraction of G is odtained azter exactly n-c contractions.

7 r

v% e now rlcscribe how a v e r y sirn?iiiied ring s t r u c t u r e ca;1 ue advantageo-asiy used t o obtain the minimal contractiorr oi G , The

n >

elements a . c

a,

?I= j i , Z , .

.

.

,

nj

,

will. always be referenced by

Iu

r

means of t h e i r index z/

.

h ring i s a linked subset of

d

s ~ c h th;t

each eiement contains the index of rkc next ring element. The f i r s t element of s ring has a 'lower index than i t s p r e d e c e s s o r .

At any stage of tF~e s r o c e s s , a r i n g link and a c l a s s identifier a r e associated with cach elel-ilent. Members o i :he s a m e c l a s s a r e chzined to i o r m a riog; s t the begiming, a l l elements bcloag to single element rings (i, e ,

,

t h e i r ring link i s eqaal to t k e i r inc1ri:x). The c i a s s icientifier of an element i s tine intiex of the beginnin: of the r i n g to which it belongs.

belong t o tilc s a m e , L s s - , , - a s s r e s u l t of the t r a n s i t i v i t y p r o g c r i y 01

9

&

.

?irkenevcr a a a t c h i n , g ~c;erneni; i s foiind, thc :iew c l a s s obt&ined i s

-

. .

a s s i g n e d a c i a s s i d e n t i f i e r cc4.ua.i t o the i r ~ i n i r n i r ~ r of thc p z i r a: :uenti- i i e r s . Tt,c n e s t stej? of the eiementa,ry c o n t r a c t i s n i s to m e r g e both

r i n g s while a s s i g n i n g tiic new c i a s s i d e n t i f i e r t o a l i e l e m e n t s e n - c o u n t e r e d ,

A i r e r r e p e a t i n g t h i s p r o c e d u r e n t i m e s , we obtain c cciasses, c being i:c,iiai t o n m i n u s the n u m b e r ef m e r g e r s p e r f o r m e d . The a c t u a l c l a s s e s c a n be t r i v i a l l y obtained by waiking a r o u n d the r i n g s ,

s t a r t i n g with t h e i r e1ernen.t of l o w e s t index.

X o t i c c t h a t the l i n k s w e r e c a r e f u l l y chosen iii o r d e r t o m a k e t h e m e r g i n g p r o c e s s simp1.e and efficient; t h e new c i a s s i d e n t i f i e r is t h e index a t which m c r g i n g i s t a s t a r t ; it thexi p r o c e e d s by updating ring l i n k s and c l a s s i d e n t i f i e r s oi both r i n g s until t h e i r f i r s r e l e m e n t i s reached.

A l g o r i t h m 3 d Z f e r s f r o m t h e p r e v i o u s o n e s in t h e i a c t t h a t it

I

m a k e s u s e of tile t r a n s i t i v i t y 2 r o p e r t y of

b?

t o p e r t o r r n a s feiv t e s t s

t

a s p o s s i b i e . T h i s may b e of s p e c i a l i m p o r t a n c e i C t h e reLation

k

i s c o s t l y t o evaluate. A t t h e s a m e t i m e , s i n c e the whole incidence m a t r i x i s i i e v e r ili?cciea, t h e a m o u n t c f s t o r a g e r e q u i r e d is kept t o a s m a l l value, namely, 3 n w o r d s .

T~ A n e num.ber of opera ti or:^ invoi,tped c a n h e a n a l y z e d a s l o i i o v ~ s :

in the w o r s t p o s s i b l e c a s e , e v e r y c o n i r a c t i e n a s s o c i a t e s a sZagLe

m>

v e r t e x of

G

_n ~ 4 t h a s u b g r a p h

H

₇₄ of C n

.

i n e corres;3onding rizg m e r g i n g r c q u i r e s t h e r e f o r e Z ( V f 1) o p e r a t i o n s , c c l a s s e s of s i z e s n i , n 2 , .

.

.

, x i t h e ring operz-iion count i s Sonnded by

1,1* Ciass countina

by

3ist

a r o c t s s l n z

1 . 3 C ( j ) = j f s r ; . { j j = , j f o r j = l . ( l ) n

1 . 4 do p a r t 2

o r

i = i . i i i i i n f o r k = l i % ) n - 1 f o r

c = n

1 . 5 t y p e c i n f o r m

4

2 . 1 3 3 0 e s r e : a t i o n

_.

hold?

-

2 . 1 2 d o n e

: r

""- t, t.j

=C

1 )

- .

2 . 1 7 * p a r t

2 C

s e t s ; = i i f

R

holds,

L o t h e r w i s e ;

2.14

do

? a r t

23

2

...

T ;j ,,: "oiie i i f = O

2 . 2 p = : C ( k j = - l i f o r .:1=::1in;";k),C(lN

f o r

c = c - 1

2 - 2 2

R!p)=C!ij f o r Ril-pj=C(kj

2 . 2 3 R ; 0 j = L ( c t O j )

for

S = R ( O )

for

p = O

2 . 2 4 p = i - ; > i f f i ( s ) i R : l - p )

2 - 2 6 z ( F : ( 1 ) j = l ? i j f ~ = l

C

2 . 2 8 t = L ! d ) f o r d = ? ( p )

l o r

L(d)=R(?)

2 . 3 go

co

s t c ~

2 . 2 4 f o r R ( p l = : i f r j d 2 . 3 2 L ( d ) = i l ( p )

f o r

p = i - 2

2 . 3 4 d o n e i f p=O

2 . 3 6 t = L [ ; : ( p ) l f o r C i R : p > i = m

2 . 1 8 go

t o

s t e p 2 . 3 6

f o r

R ( p ) = t i f t > R ( p ) 2 . k ? ( R ( p ) ) = r n

Fori;] ii

-

c l a s s e s f o u n d

r i e l a t i o n

R:

2

iri;egcrs

a r e

r;iul t i p l e s ( 1 0 n u , a h e r s i

5 c l a s s e s f o ~ . r , d

C l a s s

1:

$ 4 2 7 3 7 2 8

4

C i a s s 2 :

*

6

Class

5:

1 7

C;ass

I :

2

0

10

. .

Z c s i d e s , r; numiscr of ? . i : s t s a.

(2

a; a,re pcriormeci; i3 wc. distingti.;s;;

i a

1 berw-een positive aiid negative t e s t s , the t e s t being positive if a . g a,.,

1 j

negaiivc ii

aitx' a . 3 a l l oi the negative t e s t s have ro be c a r r i e d mi-

whereas the n u r n l ~ e r a: positive t e s t s i s dependezit u2oa tire ordering

o i the eiemerits in the sarzple

a

{ w e a s s u n e h e r o that a;(,.d7' a , cannot

J

be deduced f r o m the auicome of previous t e s t s involving a. o r a .

i J

s e p r a t e l y ) . The usi-t-her of negative t e s t s performed i s always

whereas the minimum nar,iber of positive t e s t s corresponds to t'nc total n-uinber of branch.(:s n e s e s s b r y t o s2an each c l z s s w i t h a t r e e , that i s jn-c) rests. The mii-iimiim number of testa i s therefore

o r a s a fraetlon of the : o a l ncrrioer of t e s t s

-

b ~ t in g e n e r a l it will be xr~ucli s m z l i e r . .Endeed ti;e l*;wer So-anti of

t h e ring operation count is

wbicn r e s t r i c t s

r

to be in the range

A point of i n t e r e s t i s

the

variation of o: wit11 the ordering u i

sample elements. F o r a i ~ x e c i number of c i a s s e s c

,

i* i s minimum

C c

T-

when L_ n,' i s maximum subject t o ni

=

n

.

Ln

c-dimensional

i=

l

i 1= I

Euciiciean space, the solutions are points w i t n integer coordinates

c

n r e a i e r than o r eyual t o 1 in the hyperplane ni

-

n = 0

.

They

0 _"-"

1 ' 1

a r e o: tile f o r m

(1,

1,

. . .

,

1, a - c + 1; 0;- ar,y permnut;cion of t h e vector elements by symmetry; the corresponding c.

The zlgrsrithrn vzill 2 a y 3 i E i f GI i s c l o s e t o a

.

anri v is i l s c i f

irnm rnin

s - - : ~

,

which r e q u i r e s having z s few c l a s s e s a s p o s s i b i e , t h e c l a s s e s being a s i a r g e a s 23ss?oie. C o n s e q ~ e n t l y ,

ii

we have t::c p o s s i b i l i t y o i o r d e r i n g t h e e i e m c n t s within t h e s a m p l e d r a w n , we should r a n k

I

t h e m accorciing t o t h e ?robzbiltty t h a t t h e y a r e i n t h e r e l a t i o n

&?

\with a l l of t h e o t h e r s a m p l e e l e m e n t s . Notice by the way, that tke o r d e r k g only m a t t e r s wiihin e a c h c l a s s ; a s we a l r e a d y pointed out, we cannot avoid p e r f o r m i n g all negative z e s t s anyway. T h a t p r o b l e m is s o a e - viilat a k i n t o t h e c o n n e c t o r probierri of comrCunica:ion theory, Between any two n o d e s of a given network, a d i r e c t coixnecting l i n e c a n b e

e s t a b l i s h e d at a given c o s t and t h e p r o b l e m i s t o d r a w the c h e a p e s t n e t w o r k , the c o s t of a t r e e being the siim of t h e c o s t s of i t s e d g e s ,

A

n i n i i n a l c o s t t r e e c a n b e s i m p l y c o n s t r u c t e d b y choosing a t e a c h s t e 2 , the cnea;lest c o n n e c t o r until a spanning t r e e i s obtained, This spanning t r e e i s c a l l e d an e c o n a m y t r e e and can b e shown t o be of m i n i m a l cost,

-.

I

i n e s z m e meti-od col;id be zppiied

S

we w e r e to _{evaiuate a , & a.}

i j

r )

. .

t i

f o r all a < a , a s s l g n l x g ;ic o s t 1 if a . f i a., 0 i f z . 87 a:

.

3ikt, i f we

"

=

J

t r y zo m i n i m i z e the n u m b e r of t i m e s &I h a s t o b e c o r n y ~ t e d , only a

p r o b a b i l i s t i c c o s t c s n be a s s i g n e d t o e a c h e l e m e n t a . t:

2,

_{T h i s c o s t}

1

In the g r a p h Gx

,

ahis i s cquiva?cnt t o a s s i g n i n g :G e a c h v c r i e x a

.

.

c c r z ~ i n wezgst, depending upon which v e r t i c e s i t b a s e d g e s in

c o m m o n with. Lf t h e s a m 2 i e

a

is obttiined

by

u n i i o r m l y san;piir~g

i o c a i d e g r e e (I; of a , by 1

and the e l e n x n t s of

a

shoulri the;-' b e o r d c r e d by decreasing weinht L

-t o m a k e a optimal.

Ln

p r a c t i c e , e v e n though we m a y n o t explicitly .know the l o c a l d e g r e e s 'in the g r a p h G

-

,

we m a y s t i l l b e a b l e t o estirnz.te u;:

.

F o r

h

- 7 .

i n s t a n c e , take t h e s i r L p l e c a s e w h e r e t h e r e l a t i o n Pk 1s:two i n t e g e r s i n the r a n g e

[

1 ,

]

a,-e mukipies of e a c h o t h e r . w . i s then a

de-

i

c r e a s i n g monotone sequence i f tne i n t e g e r s in the siirnpl,e

a

f o r m a n a s c e n d i n g seciuence,

liJlltiple Size Sampling 2.4.

Ln

t n e p r e v i o u s p a r a g r a p h s , we d e s c r i b e d how the ring a i p r i t h m c a n be u s e d =ost efiicienrly by o r d e r i n g t h e sample e l e m e n t s . K i t c r - natively,

i

f

we do not ; i c r f o r i ~ ~ any rearrangement, t h e a l g o r i t n r n c a n b e used t o compute a t t b c s a m e time t h c m n b c r of c l a s s e s in s x n p l e s

fi

GI,

%ii'

' .

.

4

* t h e 7/"\sample being a s u b s e t oi vc w i t h elcrnenrs

n

-

a a .

,

.

*his c o n s t i t u t e s "ie g r e a t e s t zrivaricage of algo-

1

11 '2

. .

LC:

G _ be a c n r n ; i i e e grapli on i? vcrliccs w i t h zlc slan.r;i ( a

A A c.

sling i s ari edge c o n n e c t i ~ g a v e r t e x t o itsel;) a n 6 single e d g c s (no e d g e s in p a r s l i c l ) , G f o r m s a n n - c l i q ~ i ? i n B e r g e ' s notation. L e t

n

V(Gnj b e t h e v e r t e x s e t of CI1

,

-

:&(G j i t s s e t of edges; e ( G j ciesig-

n n

n a t e s the nunlber oi e l e m e n t s i n E ( G n j , i, e . , t h e n u m b e r oi edges. Q a r e x p e r i m e n t c o n s i s t s of selecl'mg a spanning subgraph H of Gn s u c h that

n

tne probability of choosing any ? a r t i c u l a r s u b g r a p h G being on1.y n

dependent upon i t s n u m b e r of e6,ges

-- ii:e outcome oi the e x p e r i m e n t is thc nurnbor of components in - -

.

t h e s u b g r a p h ki

,

I. e ,

,

the m i n i n r n i*u;ni>cr oi connected s u b g r a p h s

-.

- 3 0 -

A p a r z i c u l a r c a s e of i n t e r e s t c o r r e s ; l o n d s t o t h e choice

wSLcii a r i s e s when s e l e c t i n g k d i s t i n c t e d g e s out n i S ( G )

,

r.

each edge having rhe s a m e p r o b a b i l i t y o i selection, 'J'e define Definition

-

A r a n d o m n - g r a p h with e eriges is a subgrzph of a c o m p l e t e graph 6- with exactly e distinct edges obtained by

A ,

sarcplic;; u G i l o r x i y and without r e p l a c e m e n t E(Cli)

.

We fiow p r e s e n t a computational method t o I i n d t h e

p r o b a b i l i t y d i s t r i b x t i o n of the n u m b e r of c o m p n e n t s i n a r a n d o m g r a p h .

m~

i c e o r e r n

-

The

p r o b a b i l i t y p ( n , c ) t h a t a r a n d o m n - g r a p h h a s e components i s given by t h e r e c u r r e n c e f o r m u l a

&/("

-

g)

c r o s s e s the p a r s t i o n , wilich occilrs wit11 probability (1-pj

For

the

farrriZy of p;rtit:i;>us satisfying the sarre constrairit we get

after rnuleipiying by the number of s u b s e t s of U v e r t i c e s . These p a r t i d r e s u l t s s d l have to be s u ~ n r n e d o v e r all possiide U's

,

however in the p r o c e s s of s i n g i i ~ t g out every connected s - ~ i ? g r a p h

we count each configuration a s many t i m e s as i t Itas componei?ts, that i s exactly c t i m e s , The t h e o r e m ioliov-.s,

It

i s interesting to notice that k e s e r e c u r r e n c e i c ~ r ~ i i l a s railst be used in a p r e c i s e seq-uence, The computation of the prob- ability that a random graph ;le strongly connected i s based lipon

the knowledge of the probability distribution f o r m o r e than one cornponerit. T h e r e f o r e , we p r o c e e d a s follow-, a.

i ) a s s c m e

d l

t e r m s pjk, r/j known f o r

I

6

k

<

V S nn- l i i ) cornTute p ( n , t i ) 2

<

v ' ~ o - i

i i i j subsequently obtain pjn, 3 )

tv)

r e p e a t s t e p s i) rhrougln iii) i u r ri : iii'l

proof: :his f o r m u l a i s m o s t e a s i l y deduced Eroim f o r m ~ u l a (3.43, since the probability p(n, c , e ) is in fact equal to t i e coefficient of

~ ~ ( 1 - p ) i n

p(n,

c ) divided by t h e sum of these coefiicients f o r

c = l , 2 ,

...,

n - 1 , i . e . , : Alternatively,

To r e l i e v e the b-urden of notatioc, l e t u s define

Another sirn$iiica:ion yet cam be aclliaved i f we use the *lev$ integer iuncrion

which s a t i s f i e s the reciirrence formula

It should be c i e a r that those integer coefficients g(ri, c , e ) actually r e p r e s e n t the number of graphs with exactly e edges and

N

c components among all possibie 2 subgraphs of G

.

F r o m a

ri

numericai staniipoii~t, they a r e m o s t conveaicaziy evaluate6 for

reasonable 1; a s long a s we stay within the rafige of integer

arithmetic. Table ( 3 , 1) shows the values of ?jn, c ) f o r n = 1 , Z , ,

.

,

,

6.

Lemma

-

F o r any raindom n-gra?h, the foliowing relationship holds

n-c S e

<

(3. 12)

proof: i ~ r s t , examine "Le inequality on zhe ?eft; l e t n be the nurriber 1

.

tX

of v e r t l c e s i n the i component, t h ~ s

-, . i h

any tree on n. vertices, the niin~ber of edges being then n . - 1 ;

1 1

si;a:xni.zg over all co,%ponents -we get

- .

l a y lor the inequality On tie right, the rna;;ti*r,um nn!mber of

ti5

.

.

edges in the i cor~~ponerit corresponds t o a11 n.-clique anc nas a

/'%I

\

2,eciges so that

2

--,

c

Xow, t h e ;r&xirnmn of ,L," n, subject to the constraint n.=n

r = i i 1:1 1

is attzined f o r a vector of the form

(1,

1,.

. .

,

l , n-c-ilj or any perr=utarion oi' its elements. See a m o r e general proof in section ( 4

1)

Finaiiy

?I.,

lnis :zrarna has a direct ~racricai application to r-ic

tor. ,I. put- ,A t: .on

or

recurrefice formulas (3.

12

f o r g(n, c, el. indeed,

by stsrpenir,g 5 ; e i i i m i t s ai s i i ~ n r n a t i o n , it becones no longer cecessary to store and refer to iidi values of gjn, c, e j ; i i i e s e

Aicer ixitroducing tile bounds just cornpileu, w e obtain

g(n, c, e )

=

0

,

otherwise

The great savings in com~puting t i m e and storage space, definitely justify

the

slight increase in compiexity,

3 , 2 . Expected K u m b e r oL Coz~~3oner1ts

-, a $

ine po1ynomia:s ( 3 , 9 ) involve monomials oZ the i o r m p i, which can b e transformed to n singie va-;able p or c,

, simply by

wnich, a f i e r seversin: i:l+: o r d s r ol sumr?la;ion, yields

In :he v a r i a . d e q , t h e compu:a:ion p r o c e e r i s ir, an i d c n t i c d fashion t o give t h e polynomial

whicil could. zictvally b e d e r t v e d d i r e c t l y f r o m (3,16) since exchanging t i l e r o l e s of p a n d

q

i s e q - ~ i v a l e n r t o p e r r r u c n g the ~ m o n o m i a i s

a T-CY x-a a

2 (; a n d

p

q

,

which in ti;rn m e a n s exchanging t h e c u e f f i c i e n t s g j i ~ , c , Y ) a n 2 g ( n , c , X-jii

.

The c o r r e s p o n d i n g graph inierpretatiori i s

-" .

simpl:-, to r e p l a c e i d oy i t s con3plernent gra2'n i n G

n n '

If we ~ x s e the s i n g l e v a r i a b l e poiyiiorrxiais, we yet f o r the e x - p e c t e d n c m b e r of cornponecis in a ranciom n - g r a p h the foliowing e x -

F o r t>a2 actual f i - x . - - a-ic;l

~ A a ~ ~ t , h e evnl;:aZion of these res-elis, -+-$e

vvoulh o'bvioasly a;;sl:r tile last ;eix~nz, as w e f o r m c r i y

tiis,

and \isc tiii: fokio;sJing

T- i r e

--

polynomials pjn, c ) can be founci i n table

i

3 . 1 ) and ti:.?

expected value of t h e nlir;*oer of com2onents is ziven in ~ a 5 ? c s

!

3. 2 ) an6 ( 3 . 3 ). F i g u r e ( 3. i ) shows t h e variazion of tne ex2ecied v s h e i o r v a r i o u s probability l e v e l s f r o r n 0 to

I

by i n c r e m e n t s ol

,05.

It

i s c o n v e n i e n t t o uefine a c l u i x e r i n g coefficient p by

I.

f'

=

--yT"-r

..

.

When

eiiker

p

or

c;

lieconni:~ sma;;, the v- a = ~ a ~ l o n * ' - * - - oi the

. .

rt::nl?~er of c l a s s e s c;;n be ezsiiy es.tirnsred using the foi:ossikg

l e m m a ,

LeirArna

-

The e x ~ e c t c d ririnber of coil-,po;lerits ir. a raadorn n-gra;i;i is

proof: as a starzing ~ o i n t we u s e e x p r e s s i o n s (3.2 0 j ar,d (3.2 1 j f o r the e x ~ e c t e d -values. 3 r o m our lemma, together w i t h s o m e sirnpie georr'etrical consideratio~~s w e know that

X

X

m ( X i

/'x-

I\

L _{2=n\ N-2, )gin, n, 0 ) - j n - l j i k 7 ].g(n, x-i,}

₁

_j<~(c- t c r ( ~ ~ p n-2, 2 )

I

-'

,

but this z e r m i s m e r e ; y the n~,;ni^uer o: triangles which cafi

be

f o r m e d

-

/il \

with t h r e e e d g e s an.6 n v e r t i c e s , i. e.

,

C

? , 3

- j 3 j

.

i;l";hogk, vuie

do cot explicitly co-i^?ute G _{2,"; *} i t is c l e a r frox-XI t h e s t r ~ c t i i r e of

ro', 4 ,

[ 3 . 2 0 j that C

=

;in

r

whick yieliis i u r r n d a (5.2Z),

P,

4

-.,- previui;s r c s - i i r s

.

do not exciusively q;>;y to the random

thaz the probability o: s:ly given coafiyuratfon be .tinic;~cly d e t e i -

--

' , ' i ~ e i i by its n i i ~ x b e r of edges, al: 2

iV

c o ~ . i i g c r a r i o n s being feasib;e, Icdeed, knowing g j n , c , c ) vjhich i s the nu,mber of ~ o n f i ; i ; ~ r ~ t i o ~ j f~i.rniir*g C c l a s s e s f o i i'ixcd e

,

i t i s a s i n ; j i e rriatrer "i obtain

1

.

.

?jn, c ) for some edge 2iatribr;tinr. f(n, .J) such rhat _id i j n , v ) = l .

7 l ~ O 7

The probability of c components 'becomes

and the averzge nurnbcr oi components i s

3. 4< :>scrc.';e S(;$o l--.-. :;: '3

-...

:,I ..:,qs

.!&A'-L,.e L &,.,,,.<c--.

-

-

-.

_{ne appror-:h t;;ier i z :.he c s s c of t h e rancioz~ gra??. car,} be exteraecd to erbcir-~n:;yj w i d e r c l a s s

nros;errs

.;Ila? X Z J ~ jlr,a;i

A '- *

I I ~ ~

me

~ ~a p : - i l i t a b i : c d - y ~ ~ ..a.b ~ ~-' ~ n e ~method " . i s base6 q o n our

abiiity t o GxTress any coniigcration w'ih c c l a s s e s a s the s u m of

. .

p a i r s oi configi;ration w:t~, i e s l e c t i v e i y , ir and c - t / c i a s s e s ,

-.

1

e -

.

I n e set s f n points i s thh-;s broken into subsets of k and

x-k

p o i ~ ~ t s forming

7~

and c - zi c l a s s e s .

All

other

.,

.

p a r a m e t e r s d e s c n a i n g the configuration m u s t be s - m i n e d and the method w-ill succeed i f all of these sums can 'se c s p r e s c e d i n z e r z r ~ s

"~

of sirLlpler co---- AzA,b~A,,Atioas .r-.w.; exciusiveiy, i, e,

,

cofifigurations already evaI-~azc<. Xotice that the prolbabiiity of having a single c l a s s i s

iDW1& I;>,~L, by ;-.icing 2 the 6 i i ~ e r e n c c

.

- a 5eb.veen the t o t a l nurnScr

of configsirations and the ni;rI(ije: oi configu+a%iuns i~avir,g 2, 3,

.

,

.

o r n c l a s s e s for a fixed s e t

d

p a r a m e t e r s .

i i i e now in-~estigaze in some d e t a i l ar, example of se?ara.ble problem,

3. 5. Xocks P r o t j i r n o n a C&ss'bi,srd

Given an n x m r e c i a x ~ g ~ d a r board

I3

w i t h j?osi";icns n, I n

b, ., i S1 < n ,

I

S j S nnl

,

w e c a n speak of r.:;ni.be+ o i r o o k s -'J

c l a s s e s i o r r ~ e i ; by k m a r k s an 13 ' z r y "i.;hpo ;narr>:s b. n* r* 11, j. '

-

sequer;ce oP marks

Tor which eicker ip i ,

.

and jy

:k

j - 5

p i IJ.+.L

i

.

, . _{a "}

i

1 s i / < e or ":Fiv+l nQ J v -

-

J ~ ~ + ~

i

Let g(n, ;^?, c, k ) be t>;e nwxber of distinct cofifigurario~s

of

k

marks or, E forming c rool-s classes. Let &in2 m, C, k j n A?

be the correspor~dir,g courit whec the co~iiigurati~n~ are c o ~ ~ s t r a i s i e d r:o occ-2,ny exactly n rows an2 rm ccr1111:xn:s; si~ck conf",u.:atians

are called dense, It is cLvlous f i a t

by

symmetry

Or,

3 the Aiurnsar of cor,fig-iizatioas can be ex2ressed in terms nm '

of derlse coniigurziiions by

L

~

"-mi." A >s

~

e y T - -

. ~ .

,A.L.J;oc

~

"-

~

i~ tiirlzs

,

of

dense configurziions

will

be

Jmy configuration formirig c c:asses, c 2

2

,

can ;ic';-ual?y be decomposed into a

pair

of p a T t e r n s or, sma2,icr 'mzrds 13

n. rnT

I

*

;,rd i3 called sub-'t>oaxis, which together comprise aLi n - n q rn-:~~.

i

. .

is con-~2;e~eiy - - - ! . - - . r , ~ ~ . ~ , ~ e d :r, aa1y s n e of

the

two sub-boards.

LE

marks forming c slavses on 73 is

giver,

by t h e rcc~rrcficc

fir.

- 1

r n - 1 1.-i n-n, :x~-m?

T - 7

r L

r a

i

g(n, m, c, k ) =

-

a _{- .}

L _ r L i ,

i

fiq=1 m l = l

x7

= I

n

-1

-'

i A

,.

2 -

2-"

proof: let us 5rsz astabfish how to put 3 i n a canonical order. nrn

Pick t h c itrst mark 5.

.

encoantered w h e ~ s c m n i a g sitccessive'~~

I' ; -"I

i

-.

_{Ire mar;< s. .} is a. z;,emjer of s o m e class w t i c i l h a s r o w set 1 .I.

\ I i _I

!.

.

:

1

---~.r"-,

2 :

:

.

8

(1.1. . & . . . i n - i L;irC,. ~ l l ) i i ~ iiii O C C s b?jropiatc per-

* *

\ J - i ? ' - ' d m i -r-

1"

- . .

n;i:a:ioris are :hi:;; ; i p p ~ r e o l o :'ie rows rr,d crilurrns in order to

The su3-ooard 9 noiv conlairLs a unique c ~ a s s forming a

*

jm3

".

.

.

dense coi.,LigcraZ;ori,

If

we r e p a t t k i s proccss for

5

n-nlm-rn, until all classes

have

been trz>sfarx;?ed into their d z f i s e equivalent, w e say that

B

,

,

?

.

;,c n o w in canonical form. Designate -by

. a

3 t h e dense equiva.ient o f 3 Eacli dense C O G -

n ~ L I u-n3 m - m 7 '

2 2 i i

i i g i r a i i o n can n o w

be

eqsnlied i a l o a -~-uiiiSer of eq~iivaieni con- fig~ra"ions by apprcpria2cl.y permuting rows azd colcmns

in-

.

-

-.

dependently. Hw+.rever, to f a r b i d genc:rati;,g ';ke s a m e coniroara- C)

tion in several ways, tl:eperrsl-;zatiuns ,mAclsi. p r e s e r v e the or6er of the rows and c o i ~ m n s of

B

amo:~g themselves; t h i s also

nq iTi? i r

h n l a s f o r those of I3 and fur the crn2ty rows a7,d c o l i i z m s n, rn

&.

2

.

,

B

,

The

coe5licieni in t i l e srimArnatiion is then n-n, -n2' rri-n;,

i A - r n ~

simply the prod-cc; of the n-amber of ordered -;ia-+'"; r i-~aoli~ l i ~ ,

n

*

2'

" .

n-al-n2) of the :ow and colux~n s e t o f

aZll+

wzzch is

Still, s a m e irlentical configurations a r e m u l t i p l y counted since B i s going to r e p r e s e n t successively each one of the

n m

1 1

c c l a s s e s on B

nm' Thus, f o r m u l a s (3, 2 7 ) and ( 3 . 2 8 ) follow immediately. Formula (3.29) derives the numher of dense con- figurations on B by removing f r o m the total number oi con-

nm

figurations of k m a r k s forming c c l a s s e s , those configurations which have a dense representation on p r o p e r sub-boards of

Brim.

L e m m a

-

The only possibly non-vanishing t e r m s g(n, m, c , k ) and d(n, m, c, k ) occur under the following conditions

g(n, m, c , k ) if c S k S n m

-

( c - 1 ) ( n + m - c )

1

<

c S min (n, m )

d(n9 m, c ,

k)

if n+m-c 6 k

<

nm

-

( c - l ) ( n t m - c ) 1

<

c < m i n (n, m )

proof: obvious f r o m geometrical considerations.

Using that lemma, f o r m u l a s ( 3 . 27) through (3. 29) can be made m o r e efficient computationally by sharpening the l i m i t s of summation, thus reducing greatly the numerical toil. We obtain

n-1 m - 1 n m c-1Ck-k - n )

1" 1 2

g(n, m, c, k)

= -

C

n -1 m , = i k -n irn -1 n - c - l m -c-1

1- A 1- 1 I 2-

2-

These f o r m u l a s m u s t of c o u r s e be used in a definite sequence. Assume all values g(n, m, c ,

k)

and d(n, m, c , k ) known f o r

2

nm 4 h-

.

Using formula (3. 3 i ) , we can compute g(n, rn, c , k j for n

=

E+l, m

=

1, 2, ...,Hi 1 and a l l p e r m i s s i b l e values of c and

2

k

.

This p r o c e s s r e q u i r e s only sub-boards nm 4 N and i s

t h e r e f o r e successful.

Then ( 3.32) gives g(n, m, 1, k ) f o r all the newly computed nxm boards. Finally (3.331 produces d(n, n ~ , c , k ) which will be

r e q u i r e d in the computation of l a r g e r boards. All values of d 2

and g f o r nm 4 (NS1) a r e now known, which completes the induction proof,

It is c l e a r that by symmetry we only need to compute ex- plicitly g(n, rn, c , k ) and d(n, rn, c , k ) for n a m

.

Of course, the previous r e s u l t s can be i n t e r p r e t e d a s the probability that k m a r k s distributed at random on B

and i o r m u l a s (3. 31) through (3. 34) c a n tnen he rewritten i n t e r m s of p(n, m, c , k )

.

\\*r t . . 1 1 1 . t l s u procccd as i n Lhc random graph c a s e and derive c l a s s polynomials by assigning to each configuration of

k

k m a r k s a probability

r?)

_{( I}

_-p) nm-k

.

Let q = l -p, we get

n m - ( c - l ) ( n + m - c )

-

-

1

k

n m - k

P q g(n, m, c , k ) (3. 36)

k = c

Transformation t o a single variable polynomial i s s t r a i g h t - f o r w a r d by means of

-

-

f

Pi a ( - i ) i - j n-1 i = O j = O n - j (i-

i)

T a b l e 3 . 5 a

RECTANGULAR BOARD, ROOKS POLYiiOMiALS

~ ( 5 , 1, 01 = q 5

4 2 3 3 2 4 ~ ( 5 , 1 9 1 )

=

5pq + l o p q + l o p

q

+5p q t p 5 pi53 290) = q 10

9

2 8 3 7 4

6

5 5

6

4 7 3

p ( 5 , 2 , 1 ) = lOpq +25p q +bop q +140p q iZZ2p q +ZlOp q t l 2 0 p q

8 2

9

10

145p q + l o p q t p

Table 3.

6

Table 3 . 7

Tables ( 3 . 5 ) and (3, 5 a ) l i s t the rooks polynomials f o r

each c l a s s while tables ( 3 . 6 ) and ( 3 . 7 ) contains the c o r r e s - ponding c l a s s polynomials.

Often however, the foregoing method may not be applicabie i f the problem i s not of the s e p a r a b l e type. Notwithstanding, even i f an elegant a n - ~ l y t i c a l approach s e e m s remote, numerical t r e a t - ment can yield f a s t and accurately, a n s w e r s f o r s m a l l c a s e s ;

some insight into the behavior of the general solutions may hope- fully be gained f r o m these r e s u l t s , We i l l u s t r a t e these r e m a r k s with some rooks problem.

Suppose that the hoard Bnrn has a s e t of a r b i t r a r y r e s t r i c t e d positions and the rooks relationship holds a c r o s s them. We can think of building

Brim

by successively appending

1 row t o B. 1 i - 1 ; the c h a r a c t e r i s t i c f e a t u r e of t h i s xm *

problem i s that, for the purpose of c l a s s counting, configurations of k m a r k s forming c c l a s s e s on B can be described in