Optic fiber Electronic switch the fiber serves as a transmission medium Optical networks - 1 st generation 1. Optical networks – basic notions

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 Optical networks - 2 nd generation

lightpaths ADM OADM Data in electronic form

lightpaths p1 p2 Valid coloring

Optical switch lightpath OADM (optical add-drop multiplexer) No two inputs with the same wavelength should be routed on the same edge.

Electronic device at the endpoints of lightpaths ADM (electronic add-drop multiplexer)

Where can we save? an ADM can be shared by two lightpaths 2 ADMs1 ADM

low capacity requests can be groomed into high capacity wavelengths (colors). colors can be assigned such that at most g lightpaths with the same color can share an edge g is the grooming factor Traffic grooming

lightpaths - with grooming Valid coloring g=2

Optical networks ADMs, OADMs, grooming Graph theoretical model Coloring and routing

12 W=2, ADM=8 W=3, ADM=7 2. Minimize number of ADMs

minADM Input: a graph, a set of lightpaths, t>o. Output: can the lightpath be colored such that #ADMs t ? Output: can the lightpath be colored such that #ADMs ≤ t ?

The problem is easy on a path network

k = 4 Reminder: coloring of an interval graph

Go from left to right …

2.1 minADM is NPC for a ring minADM Input: a graph, a set of lightpaths, t>o. Output: can the lightpath be colored such that #ADMs t ? Output: can the lightpath be colored such that #ADMs ≤ t ?

Coloring of a circular arc graph

Not always possible with max load

Input: circular arc graph G, k>o. Output: can the arcs be colored by k colors? Output: can the arcs be colored by ≤ k colors? Coloring of a circular arc graph

Input: circular arc graph G, k>o. Output: can the arcs be colored with k colors? Output: can the arcs be colored with ≤ k colors? minADM Input: a graph, a set of lightpaths, t>o. Output: can the lightpath be colored such that #ADMs t ? Output: can the lightpath be colored such that #ADMs ≤ t ? G

Given an instance of the circular arc graph problem, construct an instance H of minADM:

Claim: Claim: can color G with ≤ k colors iff can color H with ≤ k colors iff can color H with #ADMs ≤ N. G H

Assume a coloring with ≤ 3 colors … Claim: Claim: can color H with ≤ 3 colors iff can color H with #ADMs ≤ 13

Claim: Claim: can color with ≤ 3 colors iff ca n color the lightpaths with ≤ 13 ADMs Assume a coloring with ≤ 13 ADMs …

2.2 three basic observations

#ADMs = N + #chains N lightpaths cycles chains Cycles are good, chains are bad A. Structure of a solution

In the approximation algorithms there are two common techniques for saving ADMs: Eliminate cycles of lightpaths Find matchings of lightpaths #ADMs = N + #chains

cost(S) = N + chains=13+6=19 costs – Every path costs 1 ADM cost(S) = 2N-savings=26-7=19 saves – Every connection saves 1 ADM N lightpaths N=13

w/out grooming: ALG  2N N  OPT ALG  2 OPT N: # of lightpaths ALG: #ADMs used by algorithm OPT: #ADMs used by an optimal solution w/ grooming: ALG  2N N/g  OPT ALG  2g OPT B. The competitive ratio

Lemma: Assume that a solution ALG saves y ADMs, and OPT saves x ADMs. C. A basic lemma

Optimal solution OPT saves x ADMs a solution ALG saves y ADMs