Wavelength Assignment in
Optical Network Design
Team 6: Lisa Zhang (Mentor)
Brendan Farrell, Yi Huang, Mark Iwen,
Ting Wang, Jintong Zheng
Progress Report
Presenters: Mark Iwen
Wavelength Assignment
Motivated by WDM (wavelength division multiplexing)
network optimization
Input
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

A network G=(V,E)
A set of demands with specified src, dest and routes
demand di = (si, ti, Ri)
WDM fibers
U: fiber capacity, number of wavelengths per fiber
Output
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
Assign a wavelength for each demand route
Demand paths sharing same fiber have distinct wavelengths
Example
Model 1:
Min conversion

Routes given




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L(e): load on link e
u: fiber capacity
f(e) =  L(e) / u 
Deploy f(e) fibers on link e :
no extra fibers
Use converters if necessary
Min number of converters
converter
Fiber capacity u = 2
Demand routes:
AOB, BOC, COA
B
A
O
C
Model 1:
Min conversion

Routes given






L(e): load on link e
u: fiber capacity
f(e) =  L(e) / u 
Deploy f(e) fibers on link e :
no extra fibers
Use converters if necessary
Min number of converters
converter
Model 2:
Min fiber



Each demand path
assigned one
wavelength from src to
dest – no conversion
Deploy extra fibers if
necessary
Min total fibers
Model 1:
Min conversion

Routes given






L(e): load on link e
u: fiber capacity
f(e) =  L(e) / u 
Deploy f(e) fibers on link e :
no extra fibers
Use converters if necessary
Min number of converters
converter
Model 2:
Min fiber



Each demand path
assigned one
wavelength from src to
dest – no conversion
Deploy extra fibers if
necessary
Min total fibers
Extra fiber
Complexity
Perspective of worst-case analysis
NP hard

Cannot expect to find optimal solution
efficiently for all instances
Hard to approximate

Cannot approximate within any constant
[AndrewsZhang]

For any algorithm, there exist instances for
which the algo returns a solution more than
any constant factor larger than the optimal.
Heuristics
Focus:


Simple/flexible/scalable heuristics
“Typical” input instances: not worst-case analysis
A greedy heuristic

For every demand d in an ordered demand set:
Choose a locally optimal solution for d
Why greedy?
Viable approach for many hard problems

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Set Cover Problem (NP-hard)
SAT solving (NP-hard)
Planning Problems (PSPACE-hard)
Vertex Coloring (NP-hard)
…
Vertex coloring:
A closely related problem
A classic problem from combinatorial optimization
and graph theory
Problem statement

Graph D

Color each vertex of D such that neighboring
vertices have distinct colors

Minimize the total number of colors needed
Connection to vertex coloring
Create a demand graph D from wavelength
assignment instance G:

One vertex for each demand

Two demands vertices adjacent iff demand
routes share common link
Demand graph D is u colorable iff wavelength
assignment feasible with 0 extra fibers and
0 conversion.
What we know about vertex coloring
Complexity – worst case

NP-hard

Hard to approximate: cannot be approximated to
within a factor of n1-e [FeigeKilian][KnotPonnuswami]
Heuristic solutions – common cases

Greedy approaches extremely effective
For vertex v in an ordering of vertices:
Color v with smallest color not used by v’s neighbors

Example: Brelaz’s algorithm [Turner] gives priority
to“most constrained” vertex
Try greedy wavelength assignment
For every demand d in an ordered demand set:
Choose a locally optimal solution for d
- Is there good ordering?
- Is it easy to find a good ordering?
- Local optimality is easy!
Local optimality for model 1 :
min conversion
1. Starting at first link, assign wavelength available
for greatest number of consecutive links.
2. Convert and continue on a different wavelength
until the entire demand path is assigned
wavelength(s).
Strategy locally optimal
Local optimality for model 1 :
min conversion
1. Starting at first link, assign wavelength available
for greatest number of consecutive links.
2. Convert and continue on a different wavelength
until the entire demand path is assigned
wavelength(s).
Strategy locally optimal
Local optimality for model 2:
Min fiber
1. Choose wavelength w such that assigning w to
demand d requires minimum number of extra
fibers
Local optimality for model 2:
Min fiber
1. Choose wavelength w such that assigning w to
demand d requires minimum number of extra
fibers
Extra fiber on first link
Ordering in Greedy approach
Global ordering:
1. Longest first : Order demands according to
number of links each demand travels.
2. Heaviest : Weigh each link according to the
number of demands that traverse it. Sum the
weights on each link of a demand.
3. Ordering suggested by vertex coloring on
demand graph
4. Random sampling: choose a random
permutation.
Ordering in Greedy approach
Local perturbation: d1, d2, d3, d4, …
1. Coin toss :
- Reshuffle initial demand ordering by:
- Flipping a coin for each entry in order
- With a success, remove the demand and
move it to new ordering
2. Top-n :
- Reshuffle initial demand ordering by:
- Randomly choosing a first n demands
- Removing the demand to new ordering
Iterative refinement
Global ordering
Greedy
Local perturbation
Greedy
Generating instances
Characteristics of network topology:
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Sparse networks; average node degree < 3
Planar
Small networks (~ 20 nodes) Large network
(~ 50 nodes)
Characteristics of traffic:
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Fiber Capacity ~ [20,100]
Lightly loaded networks: 1 fiber per link,
fibers half full
Heavily loaded networks: ~ 2 fibers per links
Topologies of real networks
Topologies of real networks
Topologies of real networks
Experimental data
Group 1: real networks (light load)
Experimental data
Group 1: real networks (light load)
Probability of No Wavelength Conflict vs. Link Load
1
-O(log u) approx.:
choose a wavelength
uniformly at random
for each demand
fibercap=20
0.9
0.8
probability
0.7
0.6
-Birthday Paradox!
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
10
load
12
14
16
18
20
Experimental data
Group 2: simulated networks (heavy + small)
Experimental data
Group 2: simulated networks (heavy + small)
Experimental data: Large Networks
Group 3: simulated networks (heavy + large)
Experimental data
Group 3: simulated networks (heavy + large)
Summary – Preliminary observations
Small + light (real networks)

All greedy solutions close to optimal

Log approx behaves poorly
Small + heavy

Random sampling has advantage

Longest/heaviest less meaningful for shortest
paths in small networks
Large + heavy

Longest/heaviest more meaningful
Combined minimization
New territory:

Ultimate cost optimization

Combined minimization of fiber and conversion
Proposed approach

Compute a min fiber solution (x extra fibers)
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From empty network, add one fiber at a time

Compute a min conversion solution for fixed
additional fibers.
Combined minimization
QUESTIONS???