Distance and Routing Distance and Routing Labeling Schemes in Graphs Labeling Schemes in Graphs with Applications in Cellular with Applications in Cellular

and Sensor Networksand Sensor Networks

Feodor F. DraganFeodor F. Dragan

Computer Science DepartmentKent State University

http://www.cs.kent.edu/~dragan

Talk is based onTalk is based on

• Known results for trees [Kannan&Naor&Rudich’88, Peleg’99 and ’00, Thorup&Zwick’01, Fraigniaud&Gavoille’01]

• Two recent (Fall 2003) papers (by V.D. Chepoi (U.Marseille), F. F. Dragan and Yann Vaxes (U.Marseille)

– Distance and Routing Labeling Schemes for Non-Positively Curved Plane Graphs (subm. to J. of Algorithms);

– Addressing, Distances and Routing in Triangular Systems with Applications in Cellular and Sensor Networks (to appear in Proc of 4th WNAN 04)

We start with an overview on labeling schemes …

Traditional graph representationTraditional graph representation

Store adjacency information in global data structure

– Requires much storage

– Node labels contain no information, serve only as “pointers” to global data structure

Local graph representationLocal graph representation

IDEA: Store local pieces of information, i.e.,

– Associate label Label(v) with each node v, s.t. labels allow inferring adjacency information locally, without using additional memory and information

Question: Can this be done compactly & efficiently?

““Ancient” Example: Ancient” Example: Hamming adjacency labeling

Goal: Label each node v with m-bit label Label(v) s.t. Label(v), Label (u) allow deciding if u and v are adjacent:

Question: Can this be done with short labels?

[Breuer&Folkman’67]: For any n-node graph there exists a Hamming distance adjacency labeling with m=2n, T=4 -4, where = max node degree.

Interest Revived: Interest Revived: Adjacency Labeling

[Kannan&Naor&Rudich’88] What’s in label?

Note: - Algorithm knows nothing beyond the labels (even, it does not know which graph they come from)

Adjacency-labeling (for graph family Adjacency-labeling (for graph family F ): ):

1. Function Label labeling nodes of any graph in F with short distinct labels (compactness)

2. Efficient algorithm for deciding adjacency of two nodes given their labels (efficiency) (polylogarithmic labels polylogarithmic time)

Adjacency Labeling on Trees[Kannan&Naor&Rudich’88]

Labeling: Labeling:

1. Arbitrarily root the tree and prelabel each node v with distinct integer I(v) from [1..n].

2. Label root r by Label(r)=(I(r)).

3. Label each non-root node v with parent w by

Label(v)=(I(v),I(w)).

nlog2Labels contain bitsLabels contain bits

Adjacency Labeling on Trees[Kannan&Naor&Rudich’88]

Adjacency Decoder for u,v: Check if 1st entry in Label(v) = 2nd entry in Label(u) or vice versa.

Constant time!

Labeling: Labeling: (with bits)(with bits)

1. Arbitrarily root the tree and prelabel each node v with distinct integer I(v) from [1..n].

2. Label root r by Label(r)=(I(r)).

3. Label each non-root node v with parent w by

Label(v)=(I(v),I(w)).

nlog2

Extensions

Question (compactness): Is this doable for all graphs with -bit labels?

1. Bounded arboricity graphs

– Bounded degree

– Bounded-genus graphs (e.a., planar graphs)

2. Intersection-based graphs (e.a., interval graphs)

3. c-decomposable graphs

)(log nO

Labels contain bitsLabels contain bits)(log nO

Negative results

Negative examples: Bipartite / chordal graphs

)(log nO

Note: Note: Graph classGraph class F is adjacency-labelable using -bit labels

)(log nO

each G from F is fully defined by bits )log( nnO

# of represented graphs )log(2 nnO

Therefore: Graph family F containing n-node graphs is not adjacency-labelable using -bit labels.

)log(2 nnO

Question: Can this be done for other types of information beyond adjacency?

Distance• In general graphs

• Distance • Edge / node connectivity • Routing• Flow• …

• In trees • Ancestry• Lowest common ancestor (LCA or NCA)• Depth of NCA (separation level)

Question: Can this be done for other types of information beyond adjacency?

Distance• In general graphs

• Distance • Edge / node connectivity • Routing• Flow• …

• In trees • Ancestry• Lowest common ancestor (LCA or NCA)• Depth of NCA (separation level)

Distance Labeling SchemeDistance 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)

…a method for storing and extracting distances in a “distributed” fashion…

• Labeling: v Label(v)

• Distance decoder: D(Label(v), Label(u)) dist(u,v)

Simple examplesSimple examples

Distance

Enjoy -bit distance labeling schemes with Enjoy -bit distance labeling schemes with constant-time decoderconstant-time decoder

)(log nO

General graphsGeneral graphs

Distance

Easy labeling: Label each node in n-node graph G on

with all distances

Hence, for the class of all n-node graphs

Currently:

Similar bounds apply for other large graph families (bipartite / chordal / …)

[Gavoille,&Peleg&Perennes&Raz’01]

Distance labeling on trees Distance labeling on trees [Peleg’99]

Preprocessing: Arbitrarily label each node v of T with distinct integers I(v) from [1..n]

Recursive partitioning and Recursive partitioning and decomposition tree decomposition tree

Tree separator: node v in n-node tree T, whose removal breaks T into subtrees of size at most n/2.

Known fact: Every tree has a separator (can be found in linear time)

TH

nHDepth log)(

LabelingLabeling

TH

• Construct for H Depth of NCA labeling scheme ( -bit labels [Peleg’00] and constant (can be made) decision time)

• Then,

)(log2 nO

v A(v)

rc 0

1c

2c

vc 3

0...26)),(),...,,(),,(),(()( 10 hTTT cvdistcvdistcvdistvAvLabel

)(log2 nO• Hence, -bit labels

Constant Time Distance Decoder Constant Time Distance Decoder

TH

Given Label(u) and Label(v)Function distance_decoder_trees(Label(u),Label(v))• Extract A(u), A(v)• Use A(u), A(v) to find the depth k of NCA(u,v) in H 1• Output , by extracting appropriate entries from labels 2+2=4

u

v

),(),( kTkT cudistcvdist

(A(u),2,2,0))

(A(v),6,2,1,0))

u

v

Lower bounds and Planar graphs Lower bounds and Planar graphs [Gavoille,&Peleg&Perennes&Raz’01]

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

General graphsGeneral graphs

Distance

Easy labeling (full routing tables): Label each node in n-node graph G on

with full routing information

Hence, for the class of all n-node graphs one needs nlogn -bit labels

Currently: Nothing better can be done for general graphs if we insist on routing via shortest paths

)),(),...,,(),,(()( 21 niiii vvportvvportvvportvLabel

Labeling trees with -bit labels Labeling trees with -bit labels

TH

• Construct for H Depth of NCA labeling scheme ( -bit labels

• Then,

)(log2 nO

v A(v)

rc 0

1c

2c

vc 3

))),(),,(()),...,,(),,((),(),(()( 1100 vcportcvportvcportcvportvhvAvLabel hh )(log2 nO• Hence, -bit labels

)(log2 nO

Constant Time Routing Decision Constant Time Routing Decision

THGiven Label(v) and Label(u)Function routing_decision_trees(Label(u),Label(v))• Extract A(v), A(u)• Use A(v), A(u) to find the depth k of NCA(v,u) in H 1• If k<h(v) output from Label(v) port(4,12)=1 else (i.e., k=h(v)) output from Label(u)

u

v

),( kcvport

u

v

))),(),,(()),...,,(),,((),(),(()( 1100 vcportcvportvcportcvportvhvAvLabel hh

),( ucport k

1

Best routing labeling scheme for Best routing labeling scheme for trees and Planar graphs trees and Planar graphs

[Thorup&Zwick’01] For family of n-node trees there is a routing labeling scheme with labels of size O(logn)-bits per node and constant time routing decision.[Fraigniaud&Gavoille’01] For family of n-node trees there is a routing labeling scheme with labels of size -bits per node and constant time routing decision.

)loglog/(log2 nnO

[Gavoille&Hanusse’99] For family of n-node planar graphs there is a routing labeling scheme with labels of size (8n +o(n))-bits per node.

• The Duals to Benzenoid Systems

– Which is a simple circuit of the hexagonal grid and the region bounded by this circuit.

• Thus, they are subgraphs of the regular triangular grid

Triangular systemsTriangular systems

MotivationMotivationApplications 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

Our results for triangular systemsOur results for triangular systems• 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

4

6

57

8

9

12

3456

7

89

10

12

3 45

768 9

11

10

)(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

57

8

9

12

3456

7

89

10

12

3 45

768 9

11

10

1T

2T

3T

v

u

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

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

G

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

3

1

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

iiTG uvdistuvdisti

),(212453

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

))(),((

33

22

11

3

2

1

uvdist

uvdistuvdist

uvdist

G

T

T

T

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 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 Choosing direction to go from vv

)4,3(3Tport

1

23

4

6

57

8

9

12

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

11

11vQportvQport

vQvvportvQvvportvO

TT

ji

jiiTiiiTi ii

1

23

4

6

57

8

9

1T

v)4,3(

1Tport

)3,7,3(

)7,2,8(u

G

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

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

Squaregraphs: Squaregraphs: a cut and the semiplanesa cut and the semiplanes

Squaregraphs: Squaregraphs: cones and a decomposition treecones and a decomposition tree

Squaregraphs: Squaregraphs: distances, neighbor conesdistances, neighbor cones

Thank youThank you

Cellular NetworkCellular Network