Wavelength Assignment in Optical Network Design

Team 6: Lisa Zhang (Mentor)Brendan Farrell, Yi Huang, Mark Iwen,

Ting Wang, Jintong Zheng

Progress ReportPresenters: Mark Iwen

Wavelength Assignment

Motivated by WDM (wavelength division multiplexing) network optimization

Input 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 Assign a wavelength for each demand route Demand paths sharing same fiber have distinct

wavelengths

Example

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

Fiber capacity u = 2Demand routes:AOB, BOC, COA

A

B

C

O

converter

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

Each demand path assigned one wavelength from src to dest – no conversion

Deploy extra fibers if necessary

Min total fibers

Model 2: Min fiber

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

Each demand path assigned one wavelength from src to dest – no conversion

Deploy extra fibers if necessary

Min total fibers

Model 2: Min fiber

Extra fiber

Complexity

Perspective of worst-case analysisNP hard Cannot expect to find optimal solution

efficiently for all instancesHard 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 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-

[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 ordering2. 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:Sparse networks; average node degree

< 3PlanarSmall networks (~ 20 nodes) Large

network (~ 50 nodes)

Characteristics of traffic:Fiber Capacity ~ [20,100]Lightly loaded networks: 1 fiber per

link, fibers half fullHeavily 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)

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

load

probability

fibercap=20

Probability of No Wavelength Conflict vs. Link Load

-O(log u) approx.: choose a wavelength uniformly at random for each demand

-Birthday Paradox!

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 poorlySmall + heavy Random sampling has advantage Longest/heaviest less meaningful for

shortest paths in small networksLarge + 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) From empty network, add one fiber at a

time Compute a min conversion solution for fixed

additional fibers.

Combined minimization

QUESTIONS???