LOCALITY AND GLOBALITY: ESTIMATIONS OF

THE ENCRYPTION COLLECTIVITIES

Cristian Lupu

Romanian Academy

Ceter of New Electronic Architectures

Tudor Niculiu

”Politehnica” University of Bucharest

Dept. Eletronics, Telecommunications and IT

Eduard Frant¸i

Microtechnology Institute of Bucharest

Bucharest, Romania

Keywords:

Self-organization, collectivity, structure, architecture, interconnection, locality, globality, symmetry.

Abstract:

In this paper we try to deﬁne a collectivity, to model and to measure it. Because N. Bourbaki names ”collec-

tivizing relation” the relation deﬁning a set, we name collectivities only the sets selected or built by the help

of the relations. The orthogonal interconnections model very well the collectivities. The behavior (structural

self-organization) around the origin is different for homogenous and non-homogenous interconnections. How

can we measure this behavior? A way is by locality and globality. The locality measures analytically by neigh-

borhoods, neighborhood reserves, Moore reserves and synthetically by diameters, degrees, average distances.

The globality is the behavior of an interconnection around a property. The globality vs. symmetry measures

by the compactity, efﬁciency and interconnecting ﬁlling. The locality and the globality are among primary

manifestations of the self-organization. In this way, collectivities modeled by self-organizing interconnections

can contribute to changing our fundamental view of computers by trying to bring them nearer to the nature.

1 INTRODUCTION STRUCTURE

AND ARCHITECTURE

A complexity system modelling means ﬁrstly the per-

ception of a self-organization of the system and then

the proper modelling. To perceive a complex, said

Wittgenstein, means to perceive the relations of its

constituent parts in a determined way. On the other

hand, one of the characteristics of the nature is the

collectivity. Through the computing terrain, Professor

Moshe Sipper said in the foreword to a recent book,

during the past few years a new wind has been sweep,

slowly changing our fundamental view of computers.

We want them, of course, to be faster, better, more

efﬁcient - and proﬁcient - at their tasks. But, more

interestingly, we are trying to imbue them with abili-

ties hitherto found only in nature, such as evolution,

learning, development, growth, and collectivity (Cas-

tro and Zuben, 2005). We can observe collectivities in

the not living world (universe galaxies, solar systems,

crystalline units) as in the living world (ant hills, bee

swarms, nations).

What properties are behind the relations who tie

the collectivities? Maybe is the gravity, the symmetry

or the survival instinct? In a word, structural self-

organization. The self-organization can be structural

and functional. Our paper refers to the structural self-

organization applied to the collectivities.

First let us deﬁne the collectivity. For this we must

answer to another question: what is a set? A set ”can

be selected by a membership or by a relation which

substantiate the membership or by bringing in the set

ﬁeld elements which fulﬁll the relation” (Dr

anescu,

1985). Because N. Bourbaki names ”collectivizing

relation” the relation deﬁning a set, we name collec-

tivities only the sets selected or built by the help of

the relations. Therefore, we exclude the sets selected

by the membership, the most general. A collectivity

not means a set made, for example, of a star, a planet,

a crystal, an ant, a bee and a man.

The relation which substantiates the membership

of a collectivity is connected with its functionality:

a collectivity is made of the least functional entities.

For example, an interconnecting is made of nodes and

486

Lupu C., Niculiu T. and Fran¸ti E. (2006).

LOCALITY AND GLOBALITY: ESTIMATIONS OF THE ENCRYPTION COLLECTIVITIES.

In Proceedings of the Third International Conference on Informatics in Control, Automation and Robotics, pages 486-493

DOI: 10.5220/0001205404860493

 SciTePress

links which is equivalent with the graph deﬁnition (a

set X of nodes and an application Γ of X in X which

gives the set of connections). The encryption collec-

tivity means a set S of signs and an application (key)

K of S in S which gives encryptions.

In this paper we try to begin to study the collec-

tivities structural and by the help of the architec-

ture concept, a connection concept toward the rela-

tion/function. We start with the deﬁnition of the con-

cept of structure (Nemoianu, 1967). The word struc-

ture comes from the Latin where there are the noun

structura, with the meaning of building, and the verb

struere (to build) with the past participle structus. In

English and French the word has the same meaning:

ediﬁce, way to build. The abstraction of the word

makes slowly: only in the XVII-XVIII

centuries ap-

pears in the sense of reciprocal relation of the parts

or the constitutive elements of a whole, determining

its nature, its organization. The initial meaning of

building maintains till now but abstracter sense will

be dominated more and more. During the XIX

cen-

tury, structure is generally opposite to function, like

static to dynamic.

The end of the XIX

century brings a new mean-

ing of the structure concept. It will begin to represent

not a simple conﬁguration, a ”static” organization, but

a whole made by solidary elements, in which every-

one depends on all other ones and can not be what it

is than in and through them. Evidently, it is a step for-

ward. The connection between parts (the ﬁrst mean-

ing) is something less necessary, less outlined, more

approximately, more vaguely and more generally than

the total interdependence system of each part with all

other parts (the second meaning). If the ﬁrst meaning

is a sum, the second is a whole. This turning point

coincides with the penetration of the structure con-

cept in the humanities. The term has been changed

by a synonym, Gestalt, understood as form, pattern,

structure, the making of parts which are determined

by whole, system of its behavior can not equal with

the sum of the parts. Gestalt is not related to organi-

zation or to plan, but with an organism, a whole, an

entelechy. The entelechy is a term introduced by the

Austrian psychologist Ehrenfels appointing the fea-

tures (of geometric ﬁgures or melodies) by which they

exceed the sum characteristics. A geometric ﬁgure re-

mains itself even represented in other coordinate sys-

tem, decreased, enlarged, color modiﬁed. This invari-

ance of the transposing calls also isomorphism.

The linguistic researchers contribute resolutely to

the understanding and to the using of the structure

concept unifying both meanings: the coherent, co-

agulated globality and the relations system between

local parts or, in few words, the globality and the lo-

cality. This step in the evolution of the structure term

opens a path to the identiﬁcation between structure

and essence of an object or a phenomenon. Wittgen-

stein writes in Tractatus that the manner in which the

objects depend some on the others in the state of af-

fairs constitutes the structure of the state of affairs.

Having in view the above, the structure of a col-

lectivity can be self-organized locally and globally.

For example, an interconnecting structure estimates

locally by neighborhoods. Thus, the locality is the be-

havior (structural self-organization) of a collectivity

around an origin. The origin can be temporal or spa-

cial. The locality deﬁnition refers to the ﬁrst mean-

ing of the structure concept (the connection between

parts). The globality is the behavior (structural self-

organization) of a collectivity around a property. For

example, the interconnections can be estimated and

globality deﬁnition concerns to the second meaning

of the structure concept at which referred Wittgen-

stein (total interdependence system of each part with

all other parts).

On the other hand, the collectivity architecture,

a connection concept between the structure and the

function, gives a global meaning to the collectivity

with the aim to better understand the connection be-

tween the structure and the function of this collectiv-

ity. Thus, we can speak of the universe architecture,

a crystallographic system architecture, a house archi-

tecture, a town architecture, a computer architecture,

an interconnecting architecture, a communication ar-

chitecture. The architecture measures by the degree

of membership to global properties. The symmetry is

a global property.

Helping the interconnection as a collectivity model

we try to prove that the dichotomy locality-globality

covers mathematically one of the structural meanings

of the collectivity: the localization and the globaliza-

tion, i.e. a structural potential of a collectivity dy-

namics, a structural self-organization of a collectiv-

ity. The dynamics of an encryption collectivity can

help us to the decryption process.

2 INTERCONNECTION AS A

COLLECTIVITY MODEL

The interconnections made of N nodes and L links

model very well the collectivities. The nodes are the

members of the collectivity which are tied by links.

If there are the encryption collectivities the nodes are

signs and the links are the set of encryption keys (a

key is included in the set L). We shall limit, with-

out losing too much of generality, to the orthogonal

interconnections (Duato et al., 1997). The algebraic

representation of an orthogonal interconnection can

be made in a mixed radix number system, MRNS.

Any number N can be represented in MRNS as a

product of whole numbers, N = m

r−1

... m

LOCALITY AND GLOBALITY: ESTIMATIONS OF THE ENCRYPTION COLLECTIVITIES

487

u u u u

u u u u00 03

40 43

Figure 1: A GHT with N = m

· m

= 5 · 4.

On the basis of this representation, to each node of

an interconnection we can associate an address X,

0 ≤ X ≤ N − 1, made of r digits. Afterwards,

we present some orthogonal interconnections as col-

lectivities, i.e. sets selected or built by relations.

A generalized hypercube, GHC, is a collectivity

with N = m

r−1

... m

nodes interconnected in

r dimensions. In every dimension i, i = 1, 2, ..., r,

the m

nodes are interconnected all by all, i.e. ev-

ery node X = (x

r−1

... x

i+1

i−1

... x

)

is connected with the nodes addressed by X

′

r−1

... x

i+1

′

i−1

... x

), where 1 ≤ i ≤ r,

0 ≤ x

′

≤ m

i−1

and x

′

6= x

. From GHC de-

rives the hypercube, HC, with N = m

, the bi-

nary hypercube, BHC, with N = 2

nodes, and the

completely connected structure, CCS, with N = m

nodes. A generalized hypertorus, GHT, have N =

r−1

... m

nodes in r dimensions, in every di-

mension i, i = 1, 2, ..., r, the m

nodes being inter-

connected in a torus, i.e. every node X is connected

with the nearest neighbor nodes addressed by X

′

r−1

... x

i+1

′

i−1

... x

), where 1 ≤ i ≤ r,

′

= |x

± 1|

modulo m

. From GHT derives the hy-

pertorus, HT, with N = m

, the binary hypercubes

also, and the torus, T, with N = m. A generalized

hypergrid have N = m

r−1

... m

nodes in r di-

mensions, in every dimension i, i = 1, 2, ..., r, the m

nodes being interconnected in a chain, i.e. every node

X is connected in a grid with the nodes addressed

by X

′

= (x

r−1

... x

i+1

′

i−1

... x

), where

1 ≤ i ≤ r; x

′

= x

± 1|x

6= 0 and x

6= m

− 1;

′

= x

+ 1|x

= 0; x

′

= x

− 1|x

= m

− 1. From

GHG derives the hypergrid, HG, with N = m

, the

chain, C, with N = m nodes and BHC again.

These are homogenous (at links) interconnections.

As example of non homogenous interconnections

we gave a variation of non-homogenous orthogo-

nal interconnections, the generalized hyper struc-

tures, GHS (Lupu, 2002). A GHS is an inter-

connection in which every node X is connected in

the dimension i, 1 ≤ i ≤ r, to the nodes ad-

dressed by an interconnecting vector



∪

j=1



u u u u

00 03

40 43

Figure 2: A GHS with N = m

· m

= 5 · 4. The intercon-

necting vector is

, X

and GHS is coded (CCS, T ).

r−1

... x

i+1

′

i−1

... x



∪

j=1



spec-

iﬁes that a node of GHS is connected by a vector of

unions of elementary interconnection structures, in-

stead of a single elementary interconnection struc-

ture in the homogeneous interconnections. This in-

terconnecting vector has r elements, 1 ≤ i ≤ r. So,

this interconnecting vector is deﬁned, on one hand,

by the number of dimensions, r, and, on the other

hand, by k

elementary interconnection structures,

i = 1, 2, ..., r, for which the unions



∪

j=1



are

speciﬁed, j = 1, 2, ..., k

. X

are homogeneous in-

terconnections, like tori, T, grids, G, and completely

connected structures, CCS, and must not be disjoint

for a dimension.

In the ﬁgures 1 and 2 we give two examples of

simple homogenous and non-homogenous intercon-

nections. At homogenous regular interconnections,

as the GHC or HT, the origin position does not mat-

ter. The interconnections are spherical, the diameter

is the same. At irregular networks, as the general-

ized hypergrids and other non-homogenous intercon-

nections, it matters where the position of the origin is.

The ”structural” behavior around the origin is differ-

ent for homogenous and non-homogenous intercon-

nections. How can we measure this behavior? One

way is by locality and globality.

3 LOCALITY: A FIRST SENSE OF

COLLECTIVITY STRUCTURE

The collectivities having as a model the interconnec-

tions made of nodes and links can be estimated by

locality and globality. The locality is the spatial be-

havior of interconnection around an origin. As in

physics, where the gravity characterizes attraction of

the objects, the locality deﬁnes the interconnection:

nearer objects communicate better or nearer nodes in-

terconnect easier. As we told above, the locality deﬁ-

nition refers to the ﬁrst meaning of the structure con-

ICINCO 2006 - ROBOTICS AND AUTOMATION

488

cept, the connection between parts (links of nodes).

The locality measures analytically by neighborhoods,

neighborhood reserves, Moore reserves and syntheti-

cally by diameters, degrees, average distances (Lupu,

2004a). We consider the locality to be classiﬁed

ﬁrstly as structural (topological), and, secondly, as

functional. Therefore, the locality of an interconnec-

tion will be deﬁned by two localities: a structural lo-

cality and a functional locality.

The structural localities can be appreciated by

neighborhoods. The neighborhoods can be classiﬁed

as surface (radial) neighborhoods and volume (spher-

ical) neighborhoods. The surface neighborhood of

an interconnection is the number of nodes at a dis-

tance d, SN

(O) = N

(O), where O is the ori-

gin chosen arbitrarily. The volume neighborhood is

V N

(O) =

i=1

(O). By neighborhoods, the

structural locality can be evaluated analytically. An-

other measure, more synthetically, of the structural lo-

cality is the diameter: at the same number of nodes,

the smaller diameter is the bigger locality is.

A problem, as we told above, is that the neighbor-

hoods and the diameters depend on the origin posi-

tions. At homogenous regular interconnections, as

the generalized hypercubes or hypertori, the origin

position does not matter. At irregular interconnec-

tions, as the generalized hypergrids and other non-

homogenous structures, it matters where the position

of the origin is. The topographic model presented in

(Lupu, 2004b) helped us to study the description and

the behavior of the direct interconnections, homoge-

nous and, especially, non-homogenous. The proper-

ties of interconnecting locality can be better ”read” by

the diameter contour patterns in the structural relief

of the interconnection.

We introduced a measure that helps us to reveal

the interconnection relief, the state of agglomeration.

The structural localities are more or less agglomer-

ated, as in reality. The depth of the valley (minimum

diameter) informs us about maximum agglomerated

locality, and the height of the peak (maximum diame-

ter) about the minimum agglomerated locality. Thus,

structural state of agglomeration of an interconnec-

tion node is given by the interconnection diameter

computed with the origin in the corresponding node.

The contour patterns of structural states of agglom-

eration (of the diameters computed with the origin in

every node) constitute a map with the structural relief

of the interconnection.

The structural locality is an invariable information

depending on the topology. A functional point of view

on the interconnection locality can take into consider-

ation the message routing distributions, Φ

(d), where

O is the origin and d is the distance.

As the structural locality, the functional locality

measures also by neighborhoods: a functional sur-

face neighborhood, F SN

(O) = Φ

(d) × N

(O),

and a functional volume neighborhood, F V N

(O) =

i=1

(i) × N

(O). For the functional locality,

there is also a synthetic measure like diameter, the

functional average distance. The functional average

distance helps the next deﬁnition: the functional state

of agglomeration of an interconnection node is given

by the functional average distance of the intercon-

nection computed with the origin in the correspond-

ing node. Shorter the functional average distance is,

greater the state of functional agglomeration is! Us-

ing the contour patterns of the functional states of ag-

glomeration we can draw a map depicting the func-

tional relief of the interconnection (see next section).

The surface and volume neighborhoods, on the

one hand, and the diameter or degree, on the other

hand, are analytical and synthetic evaluation means

of the intercommunication capability of interconnec-

tions, measuring the structural locality. By functional

neighborhoods and, indirectly, by functional average

distance, it expresses which part of the structural lo-

cality is used by communication process implemented

on the network. In other words, the functional neigh-

borhoods and the functional average distances express

the functional locality of the interconnections.

Obviously, for a given interconnection, SN

≥

F SN

and V N

≥ F V N

. The difference between

the two types of neighborhoods represents what we

named the neighborhood reserve. The neighborhood

reserve is of surface, SNR

= SN

− F SN

, or of

volume, V NR

= V N

− F V N

. Using the neigh-

borhood reserve, we introduced a design/evaluation

criterion of a topology by enunciating the following

conjecture: the intercommunication structural poten-

tial of an interconnection is optimally used in a com-

munication process characterized by a routing distri-

bution Φ if the neighborhood reserve is minimal.

To evaluate the structural locality of an intercon-

nection, besides the neighborhoods and neighborhood

reserves, we proposed a simple measure: the Moore

reserve based on the Moore bound. As it is known,

the Moore bound is given as the maximum number

of nodes which can be present in a graph of given

degree l and diameter D: N

Moore

= 1 + l(((l −

−1)/(l−2)). This bound is deduced from a com-

plete l-tree with diameter D and is an absolute limit

for a diametrical volume neighborhood, V N

(O) =

i=1

(O), in any graph (interconnection) of l de-

gree and D diameter. Except for the complete l-ary

trees, this bound is rarely reached. Petersen graph,

completely connected structures and rings with odd

number of nodes are interconnections that reach the

Moore bound. Therefore, it makes sense to com-

pute for an interconnection how far is this bound: the

farther away the Moore bound, the structural local-

ity properties are worse. This is implemented by the

Moore reserves.

LOCALITY AND GLOBALITY: ESTIMATIONS OF THE ENCRYPTION COLLECTIVITIES

489

The surface Moore reserve is deﬁned by the differ-

ence between the number of nodes in a correspond-

ing Moore tree at the distance d, with the degree in

considered interconnection, and the surface neigh-

borhood in considered interconnection: SM R

l(l − 1)

d−1

− N

. The Moore reserve is deﬁned

by the difference between the Moore bound at the

distance d and the volume neighborhood: MR

Moore

(d) − V N

4 HOMOGENEITY AND

SYMMETRY

Based on the topographic model we estimate three

bidimensional interconnections more and more non-

homogenous and asymmetrical. Let us draw, in the

ﬁrst example, the functional relief for uniform dis-

tribution of bidimensional interconnection having 20

nodes on a dimension.

The unidimensional elementary interconnection

structure, non-homogenous, EIS1, is the same in both

dimensions being composed of a completely con-

nected structure (nodes 0 ÷ 8), a grid (nodes 8 ÷

11) and, again, of a completely connected structure

(nodes 11÷19). EIS1 has, in this way, 20 nodes ”sym-

metrically arranged”.

In the ﬁgure 3 we give the contour patterns for the

uniform distribution. First, we notice the perfect sym-

metry in both dimensions thanks to the symmetry of

the EIS, the same in both dimensions. According to

this symmetry, we observe that the biggest part of the

functional relief is formed of four tablelands having

the same height, 5.5 nodes, orientated to the four car-

dinal points. In the middle of the interconnection, like

a cross 4 nodes wide, four canyons deepen, with the

average distance of 4.5 nodes. Right in the intercon-

nection center there is a valley, the most agglomer-

ated part of the structure, with a depth of 3.5 nodes.

The biggest slope of the average distance

(O), to

the interconnection middle, is 2 nodes, and the slopes

crossing the canyons are 1 node.

The functional reliefs for the other distributions

(structural and exponential) look likewise. The

heights or the slopes are the difference.

Let us draw, in the second example, the functional

relief of a bidimensional non-homogenous structure

which has in the ﬁrst dimension an elementary in-

terconnection structure EIS2 being composed of a

completely connected structure (nodes 0 ÷ 8), of a

grid (nodes 8 ÷ 11) and, again, of a completely con-

nected structure (nodes 11 ÷ 19) and in the second

dimension, the elementary structure EIS3 being com-

posed of a torus (nodes 0 ÷ 8), of a completely con-

nected structure (nodes 8 ÷ 11) and, again, of a torus

(nodes 11 ÷ 19). In the ﬁgure 4 we give the con-

q q q q q q q q q q q q q q q q q q q q

5.5

5.0

4.5

5.0

5.5

5.0

4.5

5.0

5.5

5.0

4.5

5.0

5.5

5.0

4.5

5.0

5.5

5.0 4.5 4.5 5.0

4.0 3.5







Figure 3: The functional relief for the bidimensional inter-

connection with the non-homogenous EIS1 for the uniform

distribution. The contour patterns of the functional average

distance

(O) are drawn.

tour patterns of this interconnection for uniform dis-

tribution. The bidimensional interconnection is sym-

metrical too, though it has in the making of the el-

ementary interconnection structures, EIS2 and EIS3,

different homogenous sub-interconnections. The re-

lief of this interconnection is more varied: four peaks,

rather small tablelands, 7.5 nodes height, and a larger

valley, of four nodes, separating the network in two

along x

dimension and in the middle of x

dimen-

sion, 5.5 nodes depth. Still there are two saddles 6.5

nodes height between the peaks and, in the middle of

the network, as in the previous example, the deepest

valley (the most agglomerated part), 4.5 nodes depth.

The symmetry is not the same on the two intercon-

nection axes, like in the ﬁrst example. The symmetry,

in present example, differs from an axis to the other

and, therefore, is weaker.

In the last example is given a non-homogenous in-

terconnection with a marked characteristic of asym-

metry. Let us draw the functional relief of a non-

homogenous bidimensional interconnection with 20

nodes per dimension. On the ﬁrst dimension there

is an elementary interconnecting structure EIS4 be-

ing composed of a completely connected structure

(nodes 0 ÷ 5), a grid (nodes 5 ÷ 12) and a torus

(nodes 12÷19). On the second dimension the elemen-

tary interconnecting structure EIS5 is composed of a

torus (nodes 0÷10), a completely connected structure

(nodes 10÷15) and, again, a torus (nodes 15÷19). In

the ﬁgure 5 we give the contour patterns of this asym-

metrical on both axes interconnection. The structure

presents only partial symmetries on certain areas.

We presented three bidimensional interconnections

with the same number of nodes per dimension and

ICINCO 2006 - ROBOTICS AND AUTOMATION

490

q q q q q q q q q q q q q q q q q q q q

6 6.5 7 7.5 7.5 7 6.5 6 5.5 5.5 5.5 5.5 6 6.5 7 7.5 7.5 7 6.5 6

5.5 6 6.5 7 7 6.5 6 5.5 5 5 5 5 5.5 6 6.5 7 7 6.5 6 5.5

5 5.5 6 6.5 6.5 6 5.5 5 4.5 4.5 4.5 4.5 5 5.5 6 6.5 6.5 6 5.5 5

5.5 6 6.5 7 7 6.5 6 5.5 5 5 5 5 5.5 6 6.5 7 7 6.5 6 5.5

6 6.5 7 7.5 7.5 7 6.5 6 5.5 5.5 5.5 5.5 6 6.5 7 7.5 7.5 7 6.5 6





 @@

@@ 



 @@

@ 



 @

@ 





Figure 4: The functional relief for the bidimensional non-

homogenous interconnection with the elementary structures

EIS2 and EIS3. The contour patterns of the functional aver-

age distance

(O) for the uniform distribution are drawn.

with elementary interconnection structures more and

more non-homogenous. The functional reliefs proved

these three interconnections have a more and more

marked asymmetry, the structures having a more and

more emphasized ”structural dynamism”, structural

self-organization. This structural dynamism leads to a

more and more powerful structural self-organization

property. Therefore, the non-homogeneity leads, on

the one hand, to the asymmetry, and, on the other

hand, to the more intense structural self-organization.

5 GLOBALITY: A WAY FROM

THE STRUCTURE TO THE

ARCHITECTURE

One of the most important properties of any physical

space structure is the symmetry. The transformation

that keeps the structure of the space is named auto-

morphism. Giving a space conﬁguration, a structure,

a form, an interconnection, we can emphasize a set

of space automorphisms, which leave unchangeable

this interconnection. Thus, the emphasizing automor-

phisms form a group which describes precisely the

symmetry of the giving conﬁguration.

The amorphous space has a total symmetry corre-

sponding to the group of all automorphisms. The

symmetry of an interconnection will be described, as

we have told, by a subgroup of all automorphisms.

The total symmetry of the space deﬁned by n points

(nodes, permutations) will be described by S

, while

a partial symmetry is expressed by a subgroup (of per-

q q q q q q q q q q q q q q q q q q q q

13.012.512.011.511.010.510.010.010.511.011.512.012.513.012.512.012.512.512.512.5

12.512.011.511.010.510.09.5 9.5 10.010.511.011.512.012.512.011.512.012.012.012.0

12.011.511.010.510.09.5 9.0 9.0 9.5 10.010.511.011.512.011.511.011.511.511.511.5

11.511.010.510.09.5 9.0 8.5 8.5 9.0 9.5 10.010.511.011.511.010.511.011.011.011.0

11.010.510.09.5 9.0 8.5 8.0 8.0 8.5 9.0 9.5 10.010.511.010.510.010.510.510.510.5

10.510.09.5 9.0 8.5 8.0 7.5 7.5 8.0 8.5 9.0 9.5 10.010.510.09.5 10.010.010.010.0

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