# A Geometric Theorem for Wireless Network Design Optimization

**ABSTRACT** Consider an infinite square grid G. How many discs of given radius r, centered at the vertices of G, are required, in the worst case, to completely cover an arbitrary disc of radius r placed on the plane? We show that this number is an integer in the set (3.4; 5.6) whose value depends on the ratio of r to the grid spacing. This result can be applied at the very early design stage of a wireless cellular network to determine, under the recent International Telecommunication Union (ITU) proposal for a traffic load model, and under the assumption that each client is able to communicate if it is within a certain range from a base station, conditions for which a grid network design is cost effective, for any expected traffic demand.

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**ABSTRACT:**Thesis (M.S.)--University of California, Los Angeles, 2002. Includes bibliographical references (leaves 42-45).01/2002; - SourceAvailable from: Martin Margala
##### Conference Paper: Control constrained resource partitioning for complex SoCs [intra-chip wireless interconnects]

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**ABSTRACT:**When moving into the billion-transistor era, the wired interconnects used in conventional SoC test control models are rather restricted in not only system performance, but also signal integrity and transmission with continued scaling of feature size. On the other hand, recent advances in silicon integrated circuit technology are making possible tiny low-cost transceivers to be integrated on chip. Based on the recent development in "radio-on-chip" technology, a new distributed multihop wireless test control network has been proposed. Under the multilevel tree structure, the system optimization is performed on control constrained resource partitioning and distribution. Several system design issues such as radio-frequency nodes placement, clustering and routing problems are studied, with the integrated resource distribution including not only the circuit blocks to perform testing, but also the on-chip radio-frequency nodes for intra-chip communication.Defect and Fault Tolerance in VLSI Systems, 2003. Proceedings. 18th IEEE International Symposium on; 12/2003

Page 1

1

A Geometric Theorem for Wireless Network

Design Optimization

Massimo Franceschetti Matthew Cook

Jehoshua Bruck

California Institute of Technology

Mail Code 136-93

Pasadena, CA 91125

Email:

?massimo,cook,bruck

? @paradise.caltech.edu

Abstract—Consider an infinite square grid

discs of given radius

quired, in the worst case, to completely cover an arbitrary

discofradius

ber is an integer in the set

on the ratio of

This resultcanbe appliedattheveryearlydesign stageof

a wireless cellular network to determine, under the recent

International Telecommunication Union (ITU) proposal for

a traffic load model, and under the assumption that each

client is able to communicate if it is within a certain range

from a base station, conditions for which a grid network

design is cost effective, for any expected traffic demand.

? . How many

? , centered at the vertices of

? , are re-

? placedontheplane? Weshowthatthisnum-

????????????????? whose value depends

? to the grid spacing.

I. INTRODUCTION

A. The disc game

Consider the following two-players game: each player

draws a disc of radius

to completely cover the disc drawn by his opponent by

new discs of the same radii

points. The objective of the game is to end up drawing

less discs than the opposite player. What is a strategy that

guarantees at least a tie with any opponent? A winning

combination example for player one is depicted in Fig. 1.

Player one’s (shaded) disc requires six discs placed at the

vertices of the grid to be fully covered, while player two’s

disc requires only four grid discs. Hence, in this case,

player one beats player two 5 to 7.

A general version of this problem can be phrased in

terms of the following fundamental question of combina-

torial geometry: how many discs of given radius

tered at the vertices of an infinite square grid, are required,

? on a square grid and then tries

? , but centered only at grid

? , cen-

This work was partially supported by the Lee Center for Advanced

Networking at Caltech.

?????????????

?????????????

?????????????

?????????????

?????????????

?????????????

?????????????

?????????????

?????????????

?????????????

?????????????

?????????????

?????????????

?????????????

???????????????

???????????????

???????????????

???????????????

???????????????

???????????????

???????????????

?????????????

?????????????

?????????????

?????????????

?????????????

?????????????

?????????????

Player one’s disc Player two’s disc

Fig. 1. The disc game. Player one’s shaded disc requires six discs

placed at the vertices of the grid to be fully covered, while player two’s

disc requires only four. In this case, player one beats player two 5 to 7.

in the worst case, to completely cover an arbitrary disc of

radius

achieved?

? placed on the plane, and how is this worst case

The answer to this question depends on the ratio of the

radius

On the left hand side of the figure six solid line discs, cen-

tered at grid vertices, are required to cover a dashed line

disc placed half way between two adjacent grid vertices.

If the ratio of

hand side of the figure), then fewer discs centered at grid

vertices are required to cover the same dashed line disc.

In Section III we present a geometric theorem that shows

that the number of discs of radius

of a square grid that are necessary and sufficient to com-

pletely cover an arbitrary disc of radius

plane is an integer

on the ratio

is proved constructively, by showing a disc placement that

requires the maximum number of

ered, thus providing the optimal strategy to play the game.

? to the grid spacing

? , as it is illustrated in Fig. 2.

? to the grid spacing

?

is increased (right

? centered at the vertices

? placed on the

??????? ?"!#?%$ ?%&(' , whose value depends

??)*? . The necessary condition of the theorem

?

grid discs to be cov-

Page 2

2

OO

Fig. 2.

the dashed line disc requires six discs centered at lattice vertices to be

fully covered. On the right hand side of the figure, it requires only

three discs.

Scaling the picture. On the left hand side of the figure,

B. Network design optimization

How can we apply this result to the design of radio

cellular networks? Let us consider a scenario in which

a telecommunication company plans the deployment of

a new network in a city. One key problem is to decide

where to position base stations, according to a distribution

of demand points, to achieve optimum quality of service,

with minimum costs. At the early design stage interfer-

ence effects are neglected, simplistic propagation models

are assumed, and the problem is to suggest initial design

strategies. We assume that, by paying alocal tax, the com-

pany can install base stations at the city traffic lights. In

this way, all base stations are placed at some vertices of

a square grid, corresponding to the street intersections, in

a way that each demand point is within a given distance

from a base station. Following this design, the company

estimates a cost of deployment of, say,

(

place the stations optimally, without constraining them to

be at the street intersections. This means identifying a

minimum cardinality set of locations of the base stations

that cover all the demand points. This strategy can po-

tentially lead to a smaller number of installations, but its

estimated cost of deployment is, for instance,

per base station, due to higher manufacturing and installa-

tion costs, and because the company may need to pay, or

to give a discount, to the owner of the building, or land,

where it intends to place a base station. Hence, the com-

pany faces the following dilemma: is it better to choose

the grid design, that may lead to a larger number of less

expensive base stations, or is it better to choose locations

optimally, potentially using less, but more expensive base

stations?

The solution to this puzzle depends on the ratio of the

communication range

length

points lie inside a circle of radius

$???? monetary units

??? ) per base station. Alternatively, the company can

???

???

???

? of a base station to the city block

? . Consider the limiting case in which all demand

? , thus requiring a single

non-grid base station of cost

serve all the points with a number

that depends on the ratio

theorem. The corresponding cost is in this case

where

ing our geometric result, we are therefore in a position to

solve the puzzle. Accordingly, we will show that the fol-

lowing holds for anydistribution ofthe demand points:

For

fective if

For

fective if

For

fective if

For

fective if

In all remaining cases a non-grid network design can be

cost effective for some distribution of demand points. By

referring to the cost values given in the example we have:

????? . Alternatively, we can

?

of grid base station

??)*? , and that is provided by our

???

?

? ,

??? is the cost of one grid base station. By apply-

?

???????

&

??? , the network grid design is cost ef-

?)*??????)?? .

?

???????

$

??? , the network grid design is cost ef-

?)*?????

?

?()?! .

?

?

???

?

!

?

? , the network grid design is cost ef-

?)*???

? .

?

???????

?

??? , the network grid design is cost ef-

?)*??? $????)?! .

???

range of the base stations to the city block length is greater

or equal to.

Note that we are comparing the best covering of the

demand points by grid discs and the best covering of the

demand points by arbitrarily located discs. In practice one

may use a polynomial time approximation algorithm to

solve any of these two problems sub-optimally (see for

example the approximation algorithms in [1] [2] [6]). In

this case the comparison can easily be done applying our

theorem in conjunction to the appropriate approximation

factors of the algorithms used.

????)?$?????????!

?

? ? hence the company should always

choose a grid design if the ratio of the communication

? ?!

"

C. Model Assumptions

We now discuss the assumptions that we make in our

model. The described scenario relies on the concept of

identifying demand nodes, on the concept of using exist-

ing traffic light poles as potential transmitting locations,

and on the assumption that a demand node can communi-

cate if it is within a given distance from a base station.

The concept of demand nodes was introduced by

Tutschku and Tran-Gia [11] [12], and the International

Telecommunication Union [7] has recently proposed its

standardization. According to their definition, a demand

node represents the center of an area with a certain traf-

fic demand and each node stands for the same portion of

traffic load. Hence, different traffic patterns correspond

to different node distributions: highly populated business

districts typically lead to dense distributions of demand

nodes, while suburban and rural areas lead to sparser dis-

tributions. More details on the identification of demand

Page 3

3

nodes can be found in the survey of Tran-Gia, Leibnitz,

and Tutschku [10].

The idea of using existing traffic light poles to place

base stations in a city has been exploited by several

telecommunication companies in the recent years, both

in U.S. and Europe, to build microcellular networks with

higher capacities than traditional cellular systems [3].

Finally, one could argue that constraining each demand

point to be within a given distance from a base station cor-

responds to assuming base station transmitters to have ro-

tational symmetric range, that is not an accurate physical

representation of what is often in practice an anisotropic

and time-varying communication range, due to shadow-

ing and fading effects. However, we argue that a circle

that bounds the maximal range can be used as a first order

approximation at the early design stage of the network, as

hexagonal cell shapes are universally adopted to approxi-

mate circular radiation patterns in the design and analysis

of cellular systems [9], and as circular radiation patterns

are assumed in the calculation of the throughput capacity

of ad-hoc wireless networks [4] [5] [8]. We also point out

that our assumption of having a city formed by regularly

spaced blocks better applies to some U.S. urban and sub-

urban areas than to older, and more irregular, European

cities.

There is no doubt that our postulated model can be im-

proved, relaxing some, or all previous assumptions. This,

however, does not lead to a simple analytical evaluation,

as it is provided in this paper, of the range of parameters

that suggest a grid or non-grid design. Our contribution is

in determining, at the early stage of the design, if there is

an indication of convenience of a grid design, that can be

later validated by more accurate numerical solvers.

II. A THEOREM IN GEOMETRY.

Theorem 1: Consider a square lattice where the dis-

tance between two neighboring lattice vertices is

a disc of fixed radius

disc. The numberof grid discs that are necessary and

sufficient to cover any disc of radius

is given by:

CASE 1. For

CASE 2. For

CASE 3. For

CASE 4. For

CASE 5. For

The rest of the paper is devoted to the proof of The-

orem 1. Conclusions and future work are discussed in

Section IV. The proofs below make use of simple geomet-

ric arguments, but are by no means trivial. They involve

? . Call

? , centered at a lattice vertex, a grid

?

? placed on the plane,

?

??)*???

?!

!

,

?

does not exist.

,

?

?!

!

?

??)*???

????

"

??& .

?

????

"

?

??)*???

? ,

,

??$ .

?

?

?

??)*??? ? ?!

"

??! .

?

??)*??

? ?!

"

,

??? .

B

C

D

FEGA

Fig. 3. CASE 2, necessary condition. The distance between the

two lattice vertices andis 1. Therefore,

??????????????

? . Since

????? ??????? ,

??????????

?

? .

finding a disc that requires the prescribed number of grid

discs to befully covered, and then finding different config-

urations of grid discs that are sufficient to cover any disc

on the plane. The number of these configurations that we

need to find increases with

?)*? .

III. NOTHING BUT PROOFS

To simplify the notation, in the following we fix

??

? .

A. Proof of CASE 1.

For

???

?!

!

the grid discs do not cover the plane com-

pactly, since they do not cover the centers of the lattice

squares. It follows that any number of grid discs is not

sufficient to cover any disc that covers the center of a lat-

tice square.

?

B. Proof of CASE 2.

The necessary condition is proven by showing that

there exists adisc that requires sixgrid discs tobecovered.

The sufficient condition is proven using a tiling argument:

we first show that there exists a triangle

any disc centered inside triangle

grid discs, and then we show that, by symmetry, we can

tile the entire plane using triangles that have this property.

?!

? , such that

?"

?

is covered by six

1) Necessary Condition: Consider

centered halfway between two neighboring lattice ver-

tices. Such disc is the dashed disc depicted in Figure 3.

We have that the six (solid) grid discs depicted in Fig-

ure 3 are necessary to completely cover the dashed disc

if

???!

!

and a disc

??

? , i.e., when the four shaded areas in Figure 3

Page 4

4

B

A

C

Fig. 4.

triangle

CASE 2, sufficient condition. Any disc centered inside

is covered by the six grid discs.

?????

are not null. By repeatedly applying the Pythagorean the-

orem, we have:

?

?

??

?

??

?????

?

!

?

?

?

&??

!????

the area covered by

?

&??

(1)

imposing

2) Sufficient Condition:

the six grid discs in Figure 4. Any point inside triangle

has distance greater than

??

? , we obtain

??

????

"

.

Call

?

?!

and be covered by the six grid discs depicted in Figure 4.

By symmetry, we can tile the plane with triangles inside

which discs can be centered and covered by six grid discs.

?

? from the border of area

? . Therefore, a disc can be centered inside triangle

?!

?

?

C. Proof of CASE 3.

1) Necessary Condition: Consider

centered at a distance

two neighboring lattice vertices (dashed disc in figure 5).

By the same reasoning of CASE 2, we have that the five

(solid) grid discs depicted in Figure 5 are necessary to

completely cover the dashed disc, if

peatedly applying the Pythagorean theorem, we have in

this case:

??

????

"

and a disc

? to the left from halfway between

???

? . By re-

?

?

??

?

?

? ??

???

?

!

???

?

!

?

?

???

!???

!

?

?

?

!

?

?

???

!

?

(2)

imposing

??

? we obtain:

C

EFGA

B

D

Fig. 5.

the dashed disc and is shifted by

the lattice vertices A and E. Therefore, we have:

CASE 3, necessary condition. Point F is the center of

? to the left from halfway between

?????

?

? ?

?"!

????

?

?$#

? .

B

CA

O

Fig. 6.

tween lattice vertices A and C. Any disc centered inside triangle AOB

is covered by the five grid discs.

CASE 3, sufficient condition. Point O is halfway be-

?

?

!

?

?

?

?

$%

?

??

(3)

By symmetry we can restrict the

and for any

2) Sufficient Condition:

the five grid discs in Figure 6. Any point inside triangle

has distance greater than

? range:

??&?

??

!

??

? inequality (3) is verified.

Call

?

the area covered by

?('

which discs can be centered and covered by fivegrid discs.

? from the border of area

? . Therefore, a disc can be centered inside triangle

and be covered by the five grid discs depicted in Figure 6.

By symmetry, we can tile the plane with triangles inside

?('

?

Page 5

5

D. Proof of CASE 4.

1) Necessary Condition: Consider

placed at the center of a lattice square. If we place the

grid discs as depicted in the right section of Figure 7, four

grid discs are always necessary to cover the dashed disc

placed at the center of a lattice square, because:

??

? and a disc

???

??

?

?

'

?

??

?

(4)

the same holds whenever two grid discs are centered at

neighboring lattice vertices or at lattice vertices on the

same diagonal of a lattice square. If we examine the re-

maining possible placement of grid discs, depicted in the

left section of Figure 7, we have that four grid discs are

necessary only if

theorem, we have in this case:

?

?

?

?

?

? . By the Pythagorean

?

?

??

?

?

?

?

?

??

?

?

!??

?

(5)

imposing

2) Sufficient Condition:

Figure 8. The four grid discs cover any disc centered in-

side the shaded area

triangleand by the circle centered at point

circle is the locus of the centers of the discs to be covered

that touch point

ing areais not covered by the four grid discs. Con-

sider now the right section of Figure 8, the four (solid)

grid discs cover, in this case, any disc centered inside

the shaded area. This area is defined by triangle

and by the circle centered at point

is the locus of the centers of the discs to be covered that

touch point

adjoining the two shaded areas

fully cover the area of triangle

inside triangleis covered by four grid discs: the

four depicted in the left section or the four depicted in the

right section of Figure 8. By symmetry, the same holds

for triangleand we can tile the plane with triangles

inside which discs can be centered and covered by four

grid discs.

??

? we obtain

??$

?!

" .

Consider the left section of

?!

?

?

. This area is defined by the

?"

?

? . This

? . Any disc centered inside the remain-

?

?

?

?

?

???

?

?? . This circle

? . Such circle passes by point

?

, therefore,

?!

?

?

and

?

?

???

, we

?

? . Any disc centered

?

?

?" ??

?

E. Proof of CASE 5

1) Necessary Condition: By symmetry, any two discs

with the same radius, centered far apart by an arbitrary

?

?

? , cover less than half of each other’s perimeter,

therefore, any grid disc must cover less than half of the

perimeter of any other disc not centered at a lattice point.

It follows that any two grid discs must cover less than the

O

D

B

A

O

C

E

F

D

Fig. 7. CASE 4, necessary condition. Point O isat the center of a

lattice square. In the left section of the figure, the dashed disc centered

at point O requires one more grid discs to be covered, if

In the right section of the figure, the dashed disc centered at point O

always requires one more grid disc to be covered.

??????? .

AD

B

E

F

B

A

P

EC

C

Q

Fig. 8.

any disc centered inside the shaded areas. Intersecting the two shaded

areas, we obtain triangle

CASE 4, sufficient condition. Four grid discs cover

????? .

entire perimeter of any other disc not centered at a lattice

point. Hence, any disc not centered at a lattice point re-

quires at least three grid discs to be covered.

2) Sufficient Condition: It is enough to prove this con-

dition when

culations fixing

. We consider three different

placements of three grid discs on the lattice and show that

they are enough to cover any disc arbitrarily placed on the

plane. Consider the upper left section of Figure 9. Taking

pointas the origin of the coordinate system, the three

grid discs are placed at points:

The coordinates of pointare calculated by applying the

Pythagorean theorem to triangle

? is minimum, therefore, we carry out cal-

??

? ?!

"

'

?

?

?

? ??????

?

?

???????

?

? ??? .

?

?

'??

obtaining:

??

?

??

?

??

"

"?? . Therefore, the locus of the centers of the

discs to be covered that touch point

? , given by the circle

Page 6

6

OA

B

Q

H

R

A

P

C

H

O

B

AO

H

E

D

P’’

C’’

Q’

B

P’

C’

Fig. 9. CASE 5, sufficient condition. Three grid discs cover any

disc centered inside the shaded areas. Intersecting the three shaded

areas, we obtain triangle

.

?????

centered at point

? , is defined by the equation:

?

!

?

???

?

?

?*!

!

?

!

??

!

?

(6)

this circle passes by point

at point

see that any disc centered inside the shaded area

is covered by the three grid discs centered at points:

?

and intersects segment

It is easy to

??

?

?

?

?

?

!

???

"????

??

"

"

? .

?"

?

?

?

?

? ??????

?

?

???????

?

? ??? . We now consider the three grid

? . Its coordinates are calculated by apply-

ing the Pythagorean theorem to triangle

discs depicted in the upper right section of Figure 9, cen-

tered at points:

cus on point

?

?

?

? ????????

?

??????

?

?

? . First, let us fo-

'???? , obtaining:

??

?

??

"

"

? ?

? . Since

?

??

?

'

?

'????

!

?

??

"

"

?

? ,

the locus of the centers of the discs to be covered that

touch point

does not intersect triangle

point

? , given by the circle centered at point

? ,

?" ?? . Now, let us focus on

??? . The coordinates of point

??? are calculated by

intersecting the two grid circles:

?

?

?

?

?

?

!

?

?

!

??

!

?

!

?

?

?

?

?

?

!

??

!

?

(7)

obtaining:

of the centers of the discs to be covered that touch point

?

?

??

?

!

?

?!?

"

?

!??

?!?

"

? . Therefore, the locus

??? , given by the circle centered at point

??? , is defined by

the equation:

?

?

?

?

?

??

?

!

?

!??

???

?

?

?

??

?

!

?

!

??

!

?

(8)

this circle passes by point

at point

see that any disc centered in the shaded area

is covered by the three grid discs centered at points:

?

and intersects segment

It is easy to

??

?

?

?

?

???

!

?

!??

?!?

?

?!??

"

? .

?

?

???

?

?

?

? ????????

?

??????

?

?

? .

Intersecting the two circles centered at points

defined by equations (6) and (8) respectively, we obtain

two points:

?

and

?

? ,

?

?

?

?

?

?

???

?

?

? ???

?

?

?

?

?

?

??

?

?

?

??

?

!

?

?

?

?

?

???

??*!

!

?

depicted in the left section of Figure 10. By the coor-

dinates of these points,

?

?

?

?

?

?

? is placed inside triangle

?" ?? . Hence, any disc centered inside the shaded area

left section of Figure 10 is covered by neither the three

grid discs depicted in the upper left section, nor by the

three grid discs depicted in the upper right section of Fig-

ure 9.

In order to cover such a disc, we consider the three

grid discs depicted in the lower section of Figure 9,

placed at points:

the coordinates of point

the Pythagorean theorem to triangle

given by the intersection of the two discs depicted in the

?

?

?

? ??????

?

?

????????? ??? .In this case,

??? are calculated by applying

?

?

'?? , obtaining:

?

?

?

?

?

?

!

?

???

!?

?

? . Therefore, the locus of the centers

of the discs to be covered that touch point

circle centered at

??? , given by the

?

? , is defined by the equation:

?

?

?

?

?

?

!

?

???

?

?

???

%

?

!

??

!

?

(9)

this circle passes by points

mentat point

nates of point

grid circles:

?

and

'

and intersects seg-

??

??

?

?

?

!

?

? ?!

?

?

"??

"? . The coordi-

?

?? are calculated by intersecting the two

?

?

?

?

?

?

!

?

?

?

?

???

!

??

!

?

!

?

?

!

??

!

?

(10)

Page 7

7

obtaining:

of the centers of the discs to be covered that touch point

?

??

?

?

?

??

?

!

???

?

?

"

"

? . Therefore, the locus

???? , given by the circle centered in

equation:

???? , is defined by the

?

?

?

?

&

?

?

?

?

!??

?

?

?

?

&

?

!

!

?

!

??

!

?

(11)

this circle passes by point

at point

that any disc centered inside the shaded area

is covered by the three grid discs centered at points:

'

and intersects segment

It is easy to see

??

?

??

?

?

?

?

!

?

"

?

?

???

?!

?

"? .

?

?

?

??

?

?

?

?

? ??????

?

?

????????? ??? . In order to check that this place-

ment of grid discs covers any disc centered inside the

shaded area given by the intersection of the two discs de-

picted in the left section of Figure 10, we intersect the two

circles centered at points

equations (8) and (9) respectively, obtaining two points:

?

? and

?

? in Figure 9, defined by

?

?

?

?

?

?

???

?

?

? ???

?

?

!

?

?

!

??

?

?

?

?

?

!

?

?

?

?

?

???

?!?&

!

?

depicted in the right section of Figure 10. Given the co-

ordinates of these points, since

the area of triangle

discs placed at points:

disc centered inside the shaded area given by the intersec-

tion of the two discs depicted in the left section of Fig-

ure 10. It follows that intersecting the three shaded ar-

eas of Figure 9:

completely. Hence, any disc centered inside trian-

gleis covered by three grid discs, placed in one of

the three configurations depicted in Figure 9. By symme-

try, the same holds for triangle

plane with triangles inside which discs can be centered

and covered by three grid discs.

?

?

!

?

?

!

? is placed outside

?! ?? , we conclude that the three grid

?

?

?

? ??????

?

?

????????? ??? , cover any

?!

? ,

?

?

?

? ,

?

?

?

??

? , we cover triangle

?! ??

?! ??

'

??

and we can tile the

?

IV. CONCLUSIONS

We presented a basic theorem in combinatorial geome-

try that can be applied in the design of a wireless cellular

network. Using the concept of demand nodes to model

the expected network traffic, this theorem can be useful to

give an indication of a cost effective choice at a very early

design stage of the network planning, for any expected

traffic load.

In the future, we plan to generalize the theorem to other

lattice structures. We also plan to study a related problem

also inspired by wireless communication networks. That

is the problem of placing a minimum number of discs of

(X1,Y1)

B

O

A

(X0,Y0)

(X2,Y2)

B

A

O

(X0,Y0)

H

H

Fig. 10.

triangle

CASE 5, sufficient condition. Point

, point is outside triangle

???

?

!??

?

?

is inside

???

?

???

?

!??

?

?

???

?

.

given radius

of given points, and that the resulting graph, which has the

centers of the discs as vertices and vertices joined by an

edge if the corresponding discs intersect, is connected.

? on the plane, in a way that they cover a set

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[10] P. Tran-Gia, K. Leibnitz, and K. Tutscku. “Teletraffic issues in

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