Addressing, Distances and Routing in Addressing, Distances and Routing in Triangular Systems with Application in Triangular Systems with Application in

Cellular and Sensor NetworksCellular and Sensor Networks

Victor Chepoi, Feodor Dragan, Yan Vaxes

University of Marseille, FranceKent State University, Ohio, USA

Cellular Network Cellular Network

• Benzenoid Systems: is a simple circuit of the hexagonal grid and the region bounded by this circuit.

• The Duals to Benzenoid Systems are Triangular Systems

Benzenoid and Triangular SystemsBenzenoid and Triangular Systems

Addressing, Distances and Routing Addressing, Distances and Routing in Triangular Systems:in Triangular Systems: MotivationMotivation

Applications in cellular networks

• Identification code (CIC) for tracking mobile users

• Dynamic location update (or registration) scheme• time based

• movement based

• distance based (cell-distance based is best, according to [Bar-Noy&Kessler&Sidi’94])

Distances

•Routing protocol

• Current cellular networks do not provide information that can be used to derive cell distances

– It is hard to compute the distances between cells (claim from [Bar-Noy&Kessler&Sidi’94])

– It requires a lot of storage to maintain the distance information among cells (claim from [Akyildiz&Ho&Lin’96] and [Li&Kameda&Li’00])

Current situationCurrent situation

[Nocetti&Stojmenovic&Zhang’02] recently considered isometric subgraphs of the regular triangular grid and give among others – A new cell addressing scheme using only three small

integers, one of them being zero

– A very simple method to compute the distance between two sells

– A short and elegant routing protocol

isometric not isometric

Recent resultsRecent results

[Nocetti&Stojmenovic&Zhang’02] recently considered isometric subgraphs of the regular triangular grid and give among others – A new cell addressing scheme using only three small

integers, one of them being zero

– A very simple method to compute the distance between two sells

– A short and elegant routing protocol

isometric not isometric

Recent resultsRecent results

Our results for triangular Our results for triangular systemssystems

• Scale 2 isometric embedding into Cartesian product of 3 trees

cell addressing scheme using only three small integers

distance labeling scheme with labels of size -bits per node and constant time distance decoder

routing labeling scheme with labels of size O(logn)-bits per node and constant time routing decision.

)(log2 nO

Three edge directionsThree edge directionsthree treesthree trees

Three edge directionsThree edge directionsthree treesthree trees

Three edge directionsThree edge directionsthree treesthree trees

Three edge directionsThree edge directionsthree treesthree trees

Addressing Addressing

1

23

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768 9

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)(1 vv

1T

2T

3T

)(3 vv

)(2 vv

v

u

))(),(),(( 321 vvvv

)3,7,3(v

)7,2,8(u

G

Scale 2 embedding into 3 trees Scale 2 embedding into 3 trees

1

23

4

6

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8

9

12

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7

89

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3 45

768 9

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1T

2T

3T

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u

))(),(),(( 321 vvvv

)3,7,3(v)7,2,8(u

G

))(),(),(( 321 uuuu

3

1

))(),((),(2i

iiTG uvdistuvdisti

),(212453

))(),(())(),((

))(),((

33

22

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3

2

1

uvdist

uvdistuvdist

uvdist

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T

T

T

Distance Labeling Scheme Distance Labeling Scheme

Distance

Goal:Goal: Short labels that encode distances and distance decoder, an algorithm for inferring the distance between two nodes only from their labels (in time polynomial in the label length)

• Labeling: v Label(v)

( for trees, bits per node [Peleg’99])

• Distance decoder: D(Label(v), Label(u)) dist(u,v) (for trees, constant decision time)

)(log2 nO

Distance labeling scheme for Distance labeling scheme for triangular systemstriangular systems

• Given G, find three corresponding trees and addressing (O(n) time)

• Construct distance labeling scheme for each tree (O(nlogn) time)

• Then, set

• -bit labels and constructible in total time

))(),(),(( 321 vvvv 321 ,, TTT

))(()( vLabelv ii

)))(()),(()),((()( 321 vLabelvLabelvLabelvLabel

)(log2 nO)log( nnO

Distance decoder for triangular Distance decoder for triangular systemssystems

Given Label(u) and Label(v)

Function distance_decoder_triang_syst(Label(u),Label(v))

• Output ½(distance_decoder_trees( )))(()),(( 11 uLabelvLabel

Thm: The family of n-node triangular systems enjoys a distance labeling scheme with -bit labels and a constant time distance decoder.

)(log2 nO

+(distance_decoder_trees( )))(()),(( 22 uLabelvLabel

+(distance_decoder_trees( ))))(()),(( 33 uLabelvLabel

Routing Labeling SchemeRouting Labeling Scheme

Distance

Goal:Goal: Short labels that encode the routing information and routing protocol, an algorithm for inferring port number of the first edge on a shortest path from source to destination, giving only labels and nothing else

• Labeling: v Label(v)

• Distance decoder: R(Label(v), Label(u)) port(v,u)

vnode

1

2

3

d(for trees, bits per node and constant time decision [Thorup&Zwick’01])

)(lognO

Routing labeling scheme for Routing labeling scheme for triangular systemstriangular systems

• Given G, find three corresponding trees and addressing

• Construct routing labeling scheme for each tree using Thorup&Zwick method (log n bit labels)

• Then, set

))(),(),(( 321 vvvv 321 ,, TTT

))(()( vLabelv ii

)),....)(()),(()),((()( 321 vLabelvLabelvLabelvLabel

Something more

Choosing direction to go from v Choosing direction to go from v

)4,3(3Tport

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23

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57

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3456

7

89

10

12

3 45

768 9

11

10

1T

2T

3T

v)4,3(

1Tport

)3,7,3(

)7,2,8(u

G

)6,7(2Tport

Direction seen twice is good

Mapping tree ports to graph ports Mapping tree ports to graph ports )4()4,2())(),...,(()( 1 jvvv j

iii

),()( 21GG

ji portportvQ

))(),4,3(()),...,(),2,3(((

))()),(),((()),...,()),(),(((()(21

11

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11vQportvQport

vQvvportvQvvportvO

TT

ji

jiiTiiiTi ii

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v)4,3(

1Tport

)3,7,3(

)7,2,8(u

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Then, )),....)(()),(()),((()( 321 vLabelvLabelvLabelvLabel

(i.e., 3xlog n+3x4x3xlogn bit labels)

Routing Decision for triangular Routing Decision for triangular systemssystems

Given Label(u) and Label(v)

Thm: The family of n-node triangular systems enjoys a routing labeling scheme with -bit labels and a constant time routing decision.

)(lognO

We showed for triangular systemsWe showed for triangular systems• Scale 2 isometric embedding into

Cartesian product of 3 trees cell addressing scheme using only three

small integers

cell-distance labeling scheme with labels of size -bits per node and constant time distance decoder

routing labeling scheme with labels of size O(logn)-bits per node and constant time routing decision.

)(log2 nO

Extensions, Open Problems Extensions, Open Problems - BS at arbitrary positions- BS at arbitrary positions

BS determine

– Delaunay triangulation = dual Voronoi diagram

- Not-Simply Connected Cellular Networks

Other ResultsOther ResultsThm: The families of n-node (6,3)-,(4,4)-,(3,6)-planar graphs enjoy distance and routing labeling schemes with -bit labels and constant time distance decoder and routing decision.

)(log2 nO

(p,q)-planar graphs:

• inner faces of length at least p • inner vertices of degree at least q

Thank youThank you