Addressing, Distances and Routing in Triangular Systems with Application in Cellular and Sensor...
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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
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768 9
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)(1 vv
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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
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3456
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768 9
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1T
2T
3T
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u
))(),(),(( 321 vvvv
)3,7,3(v)7,2,8(u
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))(),(),(( 321 uuuu
3
1
))(),((),(2i
iiTG uvdistuvdisti
),(212453
))(),(())(),((
))(),((
33
22
11
3
2
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uvdist
uvdistuvdist
uvdist
G
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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3456
7
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10
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3 45
768 9
11
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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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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