Dynamic Wavelength Allocation and Wavelength Conversion Wavelength Converters

Dynamic Wavelength Allocation and Wavelength Conversion

Wavelength Converters • A wavelength converter is modeled by a bipartite graph. • For any two adjacent edges x, y, we define the corresponding conversion graph Gxy=(Vx, Vy, Exy) where: • Vx={x 0, x 1, …x. W-1} • Vy={y 0, y 1, …y. W-1} • if and only if wavelength i on endpoint x can be converted to wavelength j on endpoint y (i is compatible with j on x-y ).

Wavelength Converters • A wavelength converter is symmetric if: • A full wavelength converter corresponds to a complete bipartite graph. • The degree of a wavelength converter is the maximum degree of a node in the bipartite graph.

Full Wavelength Converters • Any instance can be colored using W=L wavelengths. (In fact we need only d>=L) • Proof: – Given a path , we can color its edges independently of each other, because the full conversion capability. – Consider any edge : There at most L 1 paths other than p traversing this edge. They use at most L-1=W-1 colors. We can use any one from the remaining colors for p.

Expander Graphs • Definition: Given any , we define the neighborhood of S, namely: • Definition: A bipartite graph (V 1, V 2, E) is an (a, b, d)-expander if: – each node has degree at most d. – 0<a<½ –b>1 – for any ,

Expander Graphs • Lemma: There is a triple (a, b, d), such that: for every sufficiently large n, there is an (a, b, d)-expander with n nodes.

Limited Wavelength Converters (Any Graph) • Theorem: There exists two constants k>1 and d>1, such that every instance can be colored with – W=k. L colors – Using wavelength converters with degree d. • Proof: Between each two adjacent edges we use the converter which correspond to the (a, b, d)expander whose existence is guaranteed by the previous lemma. • Let k=1/min{a(b-1), 1 -a} • We will prove that as long as L <= W min{a(b 1), 1 -a}, any path can be colored.

Limited Wavelength Converters (Any Graph) • Assume L <= W a(b-1) and L<=W(1 -a) • Consider a path p=(e 1, e 2, …, el) to be colored. • For any edge e 1, e 2, …, el a color is said to be busy if it is used by another path. • For any edge ei, (i>1) a color c is said to be busy also if all the colors compatible with c are busy in ei-1

Limited Wavelength Converters (Any Graph) • Claim: There a. W colors which are not busy (idle) in ei. (By induction on i) – i=1: L<=W(1 -a), therefore there a. W colors idle in e 1. – i > 1: • In edge ei+1 there at least ba. W colors compatible with the idle colors of ei • At most L<W a(b-1) of them are used by other lightpaths. • We are left with at least ba. W- Wa(b-1)= a W idle colors.

Limited Wavelength Converters (Rings) • Theorem: Any instance of ring graph can be colored with – W=L log L + 4 L colors (independent of N !!) – using converters of degree 2. • Proof: – Divide the ring into segments of length at least L, but less than 2 L. • Wline(N, L)<=L log N (prove) • Wline(2 L, L)<=L log N + L • We can color the intra-segment paths with L log. N + L colors with no wavelength conversion.

Limited Wavelength Converters (Rings) – Use the following graph to color inter-segment paths: • An edge of the graph is a color. • A vertex joins compatible colors. u 1 u 2 u. L v 1 v 2 v. L First segment Intermediate segment Last segment

Incremental WLA in Rings • Claim: Any instance in Ring graphs can be colored using W <= max{L, 2 L-d}colors. • Algorithm: – Initialization: • • • M = max {0, L-d} for i=0 to M do POOL(0)={1, …, min{L, d}} w=d for i=1 to M do POOL(i)={++w, ++w}

Incremental WLA in Rings Notation: • l(e/S) -Load induced on edge e by paths in S, namely: • Note that l(e)=l(e/P). • • Fi the set of paths received before path i, namely: Fi={p 1, p 2, …, pi-1}

Incremental WLA in Rings • Algorithm (path p) – i=0; – While L(p/Si) >= d+i do i++ – – Color the edges of p using wavelengths from SHELF(i)

Incremental WLA in Rings Lemma: Let then Proof: Assume and , therefore contradicting to the fact that .

Incremental WLA in Rings Lemma: Let then Proof: w. l. o. g. x<y. • Assume , then by previous lemma The algorithm would place py in SHELF(j) for some j<i.

Incremental WLA in Rings • Assume , therefore The algorithm would place py in SHELF(j) for some j>i.

Incremental WLA in Rings • The maximum load induced by the paths of SHELF(0) is d. By code inspection. • The maximum load induced by the paths of SHELF(i) (i>0) is 2. – Assume otherwise. There is an edge with three paths traversing it. By previous lemma, none of them contains the other. W. l. o. g assume they are sorted by their starting points:

Incremental WLA in Rings By the above picture, for any set S of paths: By the first lemma: Load is non decreasing: Combining, we get: The algorithm would place py in SHELF(j) for some j<i.