peter ruzicka
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Sirocco 2004 1
Peter Ruzicka
Sirocco 2004 2
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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Sirocco 2004 4
Sirocco 2004 5
Sirocco 2004 6
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
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