Peter Ruzicka
Sirocco 2004
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Results and research directions
in ATM and optical networks
Shmuel Zaks
Technion, Israel
[email protected]
www.cs.technion.ac.il/~zaks
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References
Works with
O. Gerstel
T. Eilam
M. Shalom
M. Feigelstein
I. Cidon
S. Moran
M. Flammini
Works of
C. Kaklamanis
E. Kranakis
D. Krizanc
A. Pelc
I. Vrt’o
V. Stacho
G. Gambossi
L. Bechetti
D. Peleg
J.C. Bermond
A. Rosenberg
L. Gargano
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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Outline




ATM networks model
Optical networks model
Discussion –
ATM networks
Discussion –
Optical networks
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ATM Asynchronous Transfer Mode
Transmission and multiplexing
technique
Industry standard for high-speed
networks
graph theoretic model
Gerstel, Cidon, Zaks
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Communication
Virtual
path
Virtual
channel
concatenation of complete paths
concatenation of partial paths
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Cost
Virtual path
Virtual channel
(space)
load = 3
hop count = 2 (time)
stretch factor = 4/3 Sirocco 2004
Other
parameters
10
Example: Find a layout, to connect a
given node with all others, with given
bounds on the load and the hop count
load  3
hop count  2
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load  3
hop count  2
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Outline




ATM networks model
Discussion –
ATM networks
Optical networks model
Discussion –
Optical networks
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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.
load  3
hop count  2
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load  3
hop count  2
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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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load
hop
1
2
3
....
1
P
P
P
…
2
P
NP
NP
…
3
NP
…
…
…
...
…
…
…
…
Flammini, Eilam, Zaks
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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.
tree, mesh
general
directed
path network
Gertsel, Wool, Zaks
Feighelstein, Zaks
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Case 1: shortest paths
(stretch factor = 1)
T(l,h)
T(l-1,h)
T(l,h-1)
l  h
| T (l , h) | 

 h 
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l 3
h2
f (3,2)  10
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l 2
h3
f (2,3)  10
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l 2
h3
l 3
h2
Use of binary trees
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l 3
h2
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l 2
h3
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l 3
h2
l 2
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h3
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Case 2: any paths
(stretch factor > 1)
TL(l,h)
TL(l-1,h)
TR(l-1,h-1)
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TL(l,h-1)
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T(l,h-1)
T(l-1,h-1) T(l-1,h)
T(l-1,h) T(l-1,h-1) T(l,h-1)
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l=3, h=2
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| T (l , h) |
min( l ,h )

i 0
 l  h 
2   
 i  i 
i
Golomb
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Use of ternary trees
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Using spheres
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
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Sp(l,r) - The l-dimensional l1Sphere of radius h : the set of lattice
points v=(x1,...,xl) with distance at most
h from the origin.
Sp(2,3): 2 - dimensional
l1-Sphere of radius 3.
point with l1-distance 3
from the origin.
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Covering Radius - Radiusl (N)
The l - dimensional Covering Radius of
N is the radius of the smallest ldimensional
sphere containing at
least N points
|Sp(2,0)| = 1
|Sp(2,1)| = 5
Radius2 ( 7)  2
Radius2 ( 23)  3
|Sp(2,2)| = 13
|Sp(2,3)| = 25
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For every ATM Chain Layouts
with N nodes and maximal load l:
Rl ( N )  Radiusl ( N )
minimal radius of
a layout with load l
and N nodes
minimal radius of an
l-dimensional sphere
with at least N internal
points
Radius2 ( 25)  3  R2 ( 25)  3
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load = 3
hop = 4
(1,0,0)
 dimension 3
 radius = 4
(1,-1,0) (1,-2,0) (1,-3,0) (0,0,0) (0,-1,0) (-1,1,0) (-1,0,0) (-1,-1,1) (-1,-1,0) (-2,0,0)
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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(2,1):
2 - dimensional
l1-Sphere of radius 1.
Sp(1,2):
1 - dimensional
l1-Sphere of radius 2.
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Using volume formulas, to
Achieve bounds on h, given N and l
For Upper Bound
( 2h  1) l
N | Sp (l , h) |
l!
1
1
1
l
 h  (l! N ) 
2
2
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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 Dl ( N ) )
between all pairs of nodes. Bounds for
Dl ( N ) in:
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:
Radiusl ( N )  Dl ( N )  2  Radiusl ( N )
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one-to-all, all-to-all, some-to-some
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)
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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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Outline




ATM networks model
Optical networks model
Discussion –
ATM networks
Discussion –
Optical networks
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1st generation
the fiber serves as a transmission medium
Electronic
switch
Optic
fiber
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2nd generation
Optical
switch
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A virtual
topology
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2nd generation
Routing in the optical domain
Two 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
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Example: Find a coloring with
smallest number of wavelengths
for a given set of lightpaths
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Outline




ATM networks model
Optical networks model
Discussion –
ATM networks
Discussion –
Optical networks
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Problem 1 :
minimize the number of wavelengths
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Smallest no. of wavelengths:
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Problem 1 :
minimize the number of wavelengths
Problem 1a : Complexity
Problem 1b: Special networks, general
networks
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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
Problem 1d : Given pairs to be connected,
design a routing and a coloring with
minimal number of colors.
……many references
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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:
7
3 wavelengths
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Problem 2 :
minimize the number of switches
Problem 2a : complexity
Problem 2b : approximation algorithms
Problem 2c : trees, special networks,
general networks
Problem 2d : given pairs to connect,
design a routing and a coloring with
smallest number of ADMs.
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Problem 2b : approximation algorithms
clearly:
alg
1
2
opt
result:
alg
1
?
opt
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Ring network
alg
1
2
opt
Gerstel, Lin, Sasaki
Calinescu, Wan
alg 3

opt 2
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Ring network
Shalom, Zaks
alg 11 7 3
  
opt 8 5 2
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Gerstel, Lin, Sasaki
1. Number the nodes from 0 to n-1
(how?)
2. Color all lightpaths passing
through or starting at node 0.
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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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1
0
2
14
3
13
4
12
5
11
6
10
7
9
8
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Color not valid…
1
0
2
14
3
13
4
12
5
11
6
10
7
9
8
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Calinescu, Wan
Use maximum matchings at
each node.
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Shalom, Zaks
Combine ideas from
Gerstel, Lin, Sasaki
Calinescu, Wan
together with preprocessing
of removing cycles, which uses an
approximation algorithm
Hurkens, Schrijver
to find all cycles up to a given size.
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alg 11 7 3
  
opt 8 5 2
Analysis:
Use of linear programming
to show alg  7 (11 )
opt
5
8
we show a set of constraints that,
together with alg  7 (11)
opt
5
8
cannot be satisfied.
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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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e
d
g
c
a
b
f
cost = 7
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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.
Problem 4a: Characterize the networks
topologies G, in which any simple path can
be extended to a simple cycle.
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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
Korach, Ostfeld
Chartrand, Kronk
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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.
Problem 4b :
Input: A Graph G, a set of lightpaths
in G, a number k.
Question : is there a ring partition of
cost  k ?
Liu, Li, Wan, Frieder
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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.
Problem 4c: Design and analyze an
approximation algorithm.
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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
or:
( |D|=n )
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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Problem 4c: Design and analyze an
approximation algorithm.
We showed the measure of total
number of switches, thus :
1
alg
2
opt
Problem 4d : What about the saving in
alg vs the saving in opt in the number of
switches?
Note: 0  alg, opt  n
o  savings_of_alg  saving_of_opt  n
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Problem 5 : find a routing with
linear filters.
One-band routers:
DEMUX
DEMUX
DEMUX
Received
Forwarded
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:
u1
v1
w1
z1
u2
v2
w2
z2
u3
v3
w3
z3
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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.
N=13
j
i
13 12  156
Units of flow
Cost:
13+5+3=21 ADMs
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2
1
1
3
5
1
4
2
1
2
N=13
1
2
1
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Problem 6a :
What can be said about
simple polygons? about non-simple
polygons?
Shalom, Zaks
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More Problem and parameters









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
cost of anarchy?
…
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Questions ?
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Thank You
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