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Peter Ruzicka

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Results and research Results and research directions directions

in ATM and optical in ATM and optical networksnetworks

zaks@cs.technion.ac.ilwww.cs.technion.ac.il/~zaks

Shmuel ZaksTechnion,

Israel

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Works with

O. GerstelT. Eilam M. Shalom M.

Feigelstein I. Cidon S. Moran M. Flammini

References

Works ofC. Kaklamanis G. GambossiE. Kranakis L. Bechetti D. Krizanc D. PelegA. Pelc J.C. BermondI. Vrt’o A. RosenbergV. Stacho L. Gargano

and many more

and many more …

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• graph-theoretic models• algorithmic issues greedy constructions recursive constructions complexity issues approximation algorithms dynamic and fault-tolerance• combinatorial design issues• upper and lower bounds analysis• …• many open problems

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OutlineOutline

ATM networks modelATM networks model Optical networks model Optical networks model Discussion – Discussion – ATM networksATM networks Discussion – Discussion – Optical networksOptical networks

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ATM - Asynchronous Transfer Mode

Transmission and multiplexing techniqueIndustry standard for high-speed networks

graph theoretic modelGerstel, Cidon, Zaks

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Virtualpath

Virtual channel

concatenation of complete paths

Communication

concatenation of partial paths

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load = 3hop count = 2stretch factor = 4/3

(space)(time)

Virtual path Virtual channel

Other parameters

Cost

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2hop count 3load

Example: Find a layout, to connect a given node with all others, with given bounds on the load and the hop count

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3load 2hop count

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OutlineOutline

ATM networks modelATM networks model Discussion – Discussion – ATM networksATM networks Optical networks model Optical networks model Discussion – Discussion – Optical networksOptical networks

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2hop count 3load

Problem 1: Given a network, pairs of nodes and bounds h and l, find a virtual path layout to connect these nodes with the load bounded by l and the hop count bounded by h.

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3load 2hop count

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Problem 1a: Given a network and a bound on the load l and a bound h on the hop count, find a layout, to connect a given node with all others (one-to-all). a. worst-case. b. average case.Note: consider it for a given stretch factor.

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Problem 1b: Given a network and a bound on the load l and a bound h on the hop count, find a layout, to connect every two nodes (all-to-all). a. worst-case. b. average case.Note: consider it for a given stretch factor.

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Problem 2: Input: Graph G, integers h, l > 0 , and a vertex v. Question: is there a VP layout for G, by which v can reach all other nodes, with hop count bounded by h and load bounded by l ?

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P

P

P

NP

NP

P

…

NP

…

… … …

…

…

…

…

1 2 3 ....load

hop

1

2

...

3

Flammini, Eilam, Zaks

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tree, meshgeneraldirectedpath network

Gertsel, Wool, Zaks

Feighelstein, Zaks

Problem 1: Given a network, pairs of nodes and bounds h and l, find a virtual path layout to connect these nodes with the load bounded by l and the hop count bounded by h.

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T(l-1,h) T(l,h-1)

T(l,h)

Case 1: shortest paths (stretch factor = 1)

| ( , ) |l h

T l hh

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2h3l 10)2,3( f

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3h2l 10)3,2( f

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2h3l3h2l

Use of binary trees

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2h3l

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3h2l

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3h2l2h3l

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TL(l-1,h) TR(l-1,h-1) TL(l,h-1)

TL(l,h)

Case 2: any paths (stretch factor > 1)

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T(l-1,h)T(l-1,h-1)T(l,h-1) T(l,h-1)T(l-1,h) T(l-1,h-1)

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l=3, h=2

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min( , )

0

| ( , ) | 2l h

i

i

l hT l h

i i

Golomb

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Use of ternary trees

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The l1-norm |v| of an l-dimensional vector v = (x1 ,...,xl ) is defined as

|v| = |x1| + |x2| + ... + |xl|

ex: |(1,-3,0,2)| = |1|+|-3|+|0|+|2| = 6

Using spheres

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Sp(l,r) - The l-dimensional l1-

Sphere of radius h : the set of

lattice points v=(x1,...,xl) with

distance at most h from the origin.

Sp(2,3): 2 - dimensionall1-Sphere of radius 3.point with l1-distance 3from the origin.

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Covering Radius-

The l - dimensional Covering Radius of N is the radius of the smallest l-dimensional sphere containing at least N points

lRadius(N)

|Sp(2,0)| = 1

|Sp(2,1)| = 5|Sp(2,2)| = 13

|Sp(2,3)| = 25 Radius2 23 3( ) Radius2 7 2( )

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For every ATM Chain Layouts with N nodes and maximal load l:

R N Radius Nl l( ) ( )

Radius R2 225 3 25 3( ) ( )

minimal radius of a layout with load l and N nodes

minimal radius of an l-dimensional spherewith at least N internal points

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load = 3

(0,0,0)

(1,0,0)

-(1,0,0) -(2,0,0)

(1-,3,0)(1-,2,0)

-(1-,1,0)(0-,1,0) -(1,1,0)(1-,1,0)

-(1-,1,1)

hop = 4

dimension 3

radius = 4

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the tree T(l,h) fills the sphere Sp(l,h) !!!

|T(l,h)| = |T(h,l)| , hence

|Sp(l,h)| = |Sp(h,l)|

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Sp(1,2): 1 - dimensionall1-Sphere of radius 2.

Sp(2,1): 2 - dimensionall1-Sphere of radius 1.

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For Upper Bound

21

)!(21

!)12(

|),(|

1

l

l

Nlh

lh

hlSpN

Using volume formulas, to Achieve bounds on h, given N and l

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Problem: Given a chain network with N nodes and a given bound on the maximum load, find an optimal layout with minimum hop count (or diameter ) between all pairs of nodes. Bounds for in:

D Nl ( )

D Nl ( )

Kranakis, Krizanc, Pelc

Stacho, Vrt’o

Aiello, Bhatt, Chung, Rosenberg, Sitaraman

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For every graph G with diameter D(G) and radius R(G): R(G) D(G) 2 R(G) Then:

Radius N D N Radius Nl l l( ) ( ) ( ) 2

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Problem 3: Design and analyze approximation algorithms for general network.

Problem 4: Solve these problems to other measures (like load on the vertices, or bounded stretch factor)

one-to-all, all-to-all, some-to-some

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Problem 7: Extend the duality results.

Problem 8: Extend the use of geometry.

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More Problem and parameters

what are the input and the output? network: tree, mesh, general,

directed cost measure average vs. worst case complexity approximation algorithms routing dynamic, distributed …

cost of anarchy?

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OutlineOutline

ATM networks modelATM networks model Optical networks modelOptical networks model Discussion – Discussion – ATM networksATM networks Discussion – Discussion – Optical networksOptical networks

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the fiber serves as a transmission medium

Electronic switch

Optic fiber

1st generation

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Optical switch

2nd generation

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A virtual topology

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Routing in the optical domainTwo complementing technologies:- Wavelength Division Multiplexing (WDM):

Transmission of data simultaneously at multiple wavelengths over same fiber- Optical switches: the output port is determined according to the input port and the wavelength

2nd generation

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Example: Find a coloring with smallest number of wavelengths for a given set of lightpaths

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OutlineOutline

ATM networks ATM networks modelmodel

Optical networks Optical networks modelmodel

Discussion – Discussion – ATM networksATM networks Discussion – Discussion – Optical networksOptical networks

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Problem 1 : minimize the number of wavelengths

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Smallest no. of wavelengths: 2

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Problem 1a : Complexity

Problem 1b: Special networks, general networks

Problem 1 : minimize the number of wavelengths

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Problem 1c : Given pairs to be connected, design a routing with minimal load, and then color it with minimal number of colors

……many references

Problem 1d : Given pairs to be connected, design a routing and a coloring with minimal number of colors.

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Problem 2 : minimize the number of switches

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no. of ADMs

ADM

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Recall: smallest no. of wavelengths: 2

8 ADMs

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Smallest no. of ADMs:

3 wavelengths

7

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Problem 2a : complexity

Problem 2c : trees, special networks, general networks

Problem 2b : approximation algorithms

Problem 2 : minimize the number of switches

Problem 2d : given pairs to connect, design a routing and a coloring with smallest number of ADMs.

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clearly:

result:

1 2 algopt

1 ? algopt

Problem 2b : approximation algorithms

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Calinescu, Wan

Ring network

Gerstel, Lin, Sasaki

1 2 algopt

3

2

algopt

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11 7 3

8 5 2

algopt

Shalom, Zaks

Ring network

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1. Number the nodes from 0 to n-1 (how?)

2. Color all lightpaths passing through or starting at node 0.

Gerstel, Lin, Sasaki

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3. Scan nodes from 1 to n-1. Color each lightpath starting at i: The colors of the lightpaths ending at i are used first, and only then other colors are used, from lowest numbered first. While color is not valid for a lightpath, try next color .

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23

4

014

13

12

11

109 8

5

6

7

1

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23

4

014

13

12

11

109 8

5

6

7

1

Color not valid…

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Calinescu, Wan

Use maximum matchings at each node.

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Combine ideas from

together with preprocessingof removing cycles, which uses an approximation algorithm

to find all cycles up to a given size.

Shalom, Zaks

Calinescu, WanGerstel, Lin, Sasaki

Hurkens, Schrijver

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Analysis:

Use of linear programming

to show

we show a set of constraints that, together with

cannot be satisfied .

7 11( )

5 8

algopt

7 11( )

5 8

algopt

11 7 3

8 5 2

algopt

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Problem 1 : minimize the number of wavelengths.Problem 2 : minimize the number of switches.

Problem 3 : find trade-offs between the two measures of number of switches and number of colors.

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Problem 4 : Given a set of lightpaths, add a minimal number of lightpaths and color all lightpaths, such that all lightpaths will be partitioned into cycles.

Eilam, Moran, Zaks

fast and simple protection mehanism

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d

b

f

a

g c

e

cost = 7

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Problem 4a: Characterize the networks topologies G, in which any simple path can be extended to a simple cycle.

Problem 4 : Given a set of lightpaths, add a minimal number of lightpaths and color all lightpaths, such that all lightpaths will be partitioned into cycles.

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Answer: iff

- G is randomly Hamltonian ( = each DFS tree is a path) , or

- G is a ring, a complete graph, or a complete balanced bipartite graph

Chartrand, Kronk

Korach, Ostfeld

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Liu, Li, Wan, Frieder

Problem 4b : Input: A Graph G, a set of lightpaths in G, a number k. Question : is there a ring partition of cost k ?

Problem 4 : Given a set of lightpaths, add a minimal number of lightpaths and color all lightpaths, such that all lightpaths will be partitioned into cycles.

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Problem 4c: Design and analyze an approximation algorithm.

Problem 4 : Given a set of lightpaths, add a minimal number of lightpaths and color all lightpaths, such that all lightpaths will be partitioned into cycles.

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A trivial heuristics:

Given a set of lightpaths D, extend each lightpath to a cycle by adding one lightpath.

cost = 2 n ( |D|=n )

or: cost opt + n

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question: is there a heuristics

for which

cost = opt + n ( < 1 ) ?

answer: no.

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question: is there a heuristics

for which

cost opt + k n (k < 1 ) ?

answer: yes.

cost opt + 3/5 n

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We showed the measure of total

number of switches, thus :

Note:

1 2 algopt

0 alg, opt n

Problem 4d : What about the saving in alg vs the saving in opt in the number of switches?

o savings_of _alg saving_of _opt n

Problem 4c: Design and analyze an approximation algorithm.

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One-band routers :

DE

MU

X

DE

MU

X

DE

MU

X

Received

Forwarded

Problem 5 : find a routing with linear filters.

Flammini, Navara

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Problem 5 : find a routing with linear filters.

Problem 5a : Is it always possible to find a routing?

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No: One-band routers are not universal:

ru2

u3

u1

v2

v1

v3

z1

z2

z3

w1

w2

w3

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Problem 5 : find a routing with linear filters.

Problem 5b : Define other routers and explor etheir capabilities.

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Problem 6 : Find a uniform all-to-all routing in a ring, using a minimum number of ADMs.

i

j

N=13

13 12 156 Units of flow

Cost:

13+5+3=21 ADMs

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5

1

3

N=13

2

1

1

4

221

21 1

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Shalom, Zaks

Problem 6a : What can be said about simple polygons? about non-simple polygons?

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what are the input and the output? cost measure, worst case vs. average

case. coloring and routing Wavelength convertion networks: specific, general complexity approximation algorithms Dynamic …

More Problem and parameters

cost of anarchy?

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Questions ?

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Thank You