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Dynamic Wavelength Allocation and Wavelength Conversion

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 G xy =(V x ,V y ,E xy ) where: V x ={x 0 ,x 1 ,…x W-1 }

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Dynamic Wavelength Allocation and Wavelength Conversion

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  1. Dynamic Wavelength Allocation and Wavelength Conversion

  2. 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={x0,x1,…xW-1} • Vy={y0,y1,…yW-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 ).

  3. 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.

  4. 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 are 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.

  5. Expander Graphs • Definition: Given any , we define the neighborhood of S, namely: • Definition: A bipartite graph (V1,V2, E) is an (a, b, d)-expander if: • each node has degree at most d. • 0 < a < ½ • b > 1 • for any ,

  6. 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.

  7. Limited Wavelength Converters (Any Graph) • Theorem: There exists two constants k>1 and d>1, such that every instance can be colored with • W=kL 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.

  8. Limited Wavelength Converters (Any Graph) • Assume L <= W a(b-1) and L<=W(1-a) • Consider a path p=(e1, e2, …, el) to be colored. • For any edge e1, e2, …, 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

  9. Limited Wavelength Converters (Any Graph) • Claim:There are aW colors which are not busy (idle) in ei. (By induction on i) • i=1:L<=W(1-a), therefore there are aW colors idle in e1. • i > 1: • In edge ei+1 there are at least baW 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 baW- Wa(b-1)= a W idle colors.

  10. 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 2L. • Wline(N,L)<=L log N (prove) • Wline(2L,L)<=L log N + L • We can color the intra-segment paths with L logN + L colors with no wavelength conversion.

  11. 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. u1 u2 First segment uL v1 v2 Intermediate segment vL Last segment

  12. Incremental WLA in Rings • Claim:Any instance in Ring graphs can be colored using W <= max{L, 2L-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 doPOOL(i)={++w, ++w}

  13. 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={p1,p2,…,pi-1}

  14. 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)

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

  16. 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.

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

  18. 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:

  19. 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.

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