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Optimal on-line colorings for minimizing the number of ADMs in optical networks

Optimal on-line colorings for minimizing the number of ADMs in optical networks. Overview. Introduction & Definitions Lower Bound- Path Topology Lower Bound- General Topology Algorithm Online-MinADM Dual representation definitions Path Topology: Online-MinADM is 3/2 –competitive

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Optimal on-line colorings for minimizing the number of ADMs in optical networks

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  1. Optimal on-line colorings for minimizing the number of ADMs in optical networks

  2. Overview • Introduction & Definitions • Lower Bound- Path Topology • Lower Bound- General Topology • Algorithm Online-MinADM • Dual representation definitions • Path Topology: Online-MinADM is 3/2 –competitive • General Topology: Online-MinADM is 7/4 –competitive • Lower Bound- Ring Topology • Triangle Topology

  3. Introduction • Path coloring is a wavelength assignment to a lightpath– denoted by w(p). • Each lightpath requires 2 ADM’s, one at each endpoint. A total of 2*|P| ADM’s. • Two paths p=(a,…,b) and p’=(b,…,c), such that w(p)=w(p’) can share the ADM in their common endpoint b. This saves one ADM. • An ADM may be shared by at most two lightpaths.

  4. Incremental (Online) Algorithm • G = (V,E) is an known undirected graph • The input is a sequence of simple paths • G : P = {p1, p2 …pN}. • Input is supplied one element at a time. • The output corresponding to the input element is calculated w/o knowledge of the subsequent input elements. • An instance of the problem is α = (G,P).

  5. Definitions • Conflicting paths: paths that have an edge in common. Denoted by p≍p’. • Proper coloring of P: A function w: P→N, such that w(p)≠w(p’) whenever p ≍ p’. • Valid chain (resp. cycle) of α is a path formed be concatenation of distinct paths from P that do not go over the same edge twice. • Solution S of α is a set of valid chains and valid cycles of P such that each p∊P appears in exactly one of them.

  6. Lower Bound- Path Topology • Lemma: For any ε >0, there is no (3/2 – ε) - competitive deterministic on-line algorithm for path topology. • Proof: Let G be a path with 2k nodes: u1,v1, u2,v2, …, uk,vk.ALG is any deterministic algorithm.The input: • K paths of the form ai = (ui,vi). • For every 1≤i<k:If w(ai) = w(ai+1) then add bi = (u1,ui+1) and bi’ = (vi, vk).Else add ci = (vi, ui+1)

  7. Adversary Coloring – Online Example • a1..k lightpaths use 2*k ADMs • bi and b`i can’t share ADMs with no other path. (use 4 ADMs) • Let x be # of times w(ai) = w(ai+1) • Then k-1-x is # of times w(ai) ≠ w(ai+1) • Total: 2*k + 4*x + k-1-x = 3*(k +x) - 1 _ u1 v1 u2 v2 u3 v3 u4 V4

  8. Off-line solution: for consecutive ci ci+1 … ci+j color:w(bi-1)=w(ai)=w(ci)=w(ai+1)=w(ci+1)=…=w(ci+j)= w(ai+j+1)= w(b’i+j+1) _ u1 v1 u2 v2 u3 v3 u4 V4 The offline solution uses 2*k +2*x ADMs. (3*(k +x) -1)/ (2*k +2*x) ≥ 3/2 -1/2k For k>1/2ε the competitive ratio of ALG is > 3/2- ε.

  9. Lower Bound- General Topology • Lemma: There is no deterministic on-line algorithm with competitive ratio < 7/4 • Proof: Assume ALG is a deterministic on-line algorithm, with competitive ratio ρ. The following input will show that ρ ≥ 7/4 .

  10. Online Coloring – Input 1 A B C Total ADMS: 3 2 0 5 7 E D F w(EFG)=1 w(BDG)=1 G H w(EABDG)=2 w(GFEAB)=3 K M

  11. Offline Coloring – Input 1 A B C Total ADMS: 4 E D F w(EFG)=1 w(BDG)=2 G H w(EABDG)=1 w(GFEAB)=2 K M Competitive Ratio: 7/4

  12. Online Coloring – Input 2 A B C Total ADMS: 10 14 12 8 5 4 6 0 2 E w(EFG)=1 D w(BDG)=2 F w(BAE)=1 G H w(EFKMHG)=2 w(GFEAB)=3 w(EABDG)=4 w(BDGFE)=5 K M w(EABCDG)=6

  13. Offline Coloring – Input 2 A B C Total ADMS: 8 E w(EFG)=1 D w(BDG)=2 F w(BAE)=3 G H w(EFKMHG)=4 w(GFEAB)=2 w(EABDG)=1 w(BDGFE)=3 K M w(EABCDG)=4 Competitive Ratio: 14/8=7/4

  14. Online Coloring – Input 3 A B C Total ADMS: 7 0 2 4 5 9 E w(EFG)=1 D w(BDG)=2 F w(BAE)=1 G H w(EFKMHG)=3 w(EABDCHG)=4 K M

  15. Offline Coloring – Input 3 A B C Total ADMS: 0 5 E w(EFG)=1 D w(BDG)=2 F w(BAE)=1 G H w(EFKMHG)=3 w(EABDCHG)=4 K M Competitive Ratio: 9/5 >7/4

  16. Online Coloring – Input 4 A B C Total ADMS: 14 10 12 8 5 6 2 0 4 E w(EFG)=1 D w(BDG)=2 F w(BAE)=2 G H w(BDCHG)=1 w(EABDG)=3 w(GFEAB)=4 w(GKFEAB)=5 K M w(EFGDB)=6

  17. Offline Coloring – Input 4 A B C Total ADMS: 8 E w(EFG)=1 D w(BDG)= 2 F w(BAE)=3 G H w(BDCHG)=4 w(EABDG)=1 w(GFEAB)=4 w(GKFEAB)=2 K M w(EFGDB)=3 Competitive Ratio: 14/8=7/4

  18. Online Coloring – Input 5 A B C Total ADMS: 7 0 2 4 5 9 E w(EFG)=1 D w(BDG)=2 F w(BAE)=2 G H w(BDCHG)=3 w(GHMKFEAB)=4 K M

  19. Offline Coloring – Input 5 A B C Total ADMS: 5 E w(EFG)=1 D w(BDG)=2 F w(BAE)=1 G H w(BDCHG)=1 w(GHMKFEAB)=2 K M Competitive Ratio: 9/5>7/4

  20. Algorithm Online-MinADM • Definition feasible: A color λ is feasible for a new path p if there is no existing path p’ with w(p’)=λ such that p≍p’. • Algorithm: when lightpath p with endpoints u and v arrives: • If there exists a chain of lightpaths of some color λ with endpoints u and v and λ is feasible for p, then w(p):= λ. • Otherwise, if there exists a chain of lightpaths of some color λ with one of the endpoints u or v and λ is feasible for p, then w(p):= λ. • Otherwise, w(p):= λ’, where λ’ is an unused color.

  21. Dual representation definitions • Shareability graph of α = (G,P), is the edge labeled multi-graph Gα= (P,Eα). There is an edge e=(p,q) labeled u in Eαiff p≭q and u is a common endpoint of p and q in G. Network with lightpaths 1 2 Shareability graph 3 4 A A 1 2 C D B D 3 E C 3 1 4 C

  22. Dual representation definitions (cont) • Valid chain (resp. cycle) of Gαis a simple path, such that any two consecutive edges have distinct labels and it’s node set is properly colorable with one color (in G). • The sharing graph of a solution S of α, is the subgraph of Gα : GS = (P,ES). Two lightpaths p,q∊P are connected with an edge labeled u in ES iff they are consecutive in a chain or cycle in S, and their common endpoint is u. • Number of ADMs used in solution S: cost(S) = 2*|P| − |ES| = 2*N − |ES| .

  23. Online-MinADM is 3/2 -competitive in Path Topology • In path topology any optimal solution connects maximum number of paths in each node. • Any solution S can be made into some optimal solution S* by adding edges to Es. • For solution S found by Online-MinADM there exists one to one function f:( Es*\Es)-> Es. s t p q r

  24. Online-MinADM is 3/2 -competitive in Path Topology (cont) • For solution S found by Online-MinADM |Es|>= 1/2*|Es*|. • cost(S)-cost(S*) =|Es*| - |Es|≤ |Es*|- ½*|Es*| ≤N/2 ≤ cost(S*) /2 • cost(S) ≤ 3/2*cost(S*)

  25. Online-MinADM in General Topology Lemma: The competitive ratio of Online-MinADM is at least 7/4. Optimal solution Online-MinADM solution A A 4 4 3 3 2 2 1 1 B 3 C B C 3 4 4 Optimal uses 4 ADMs, Online-MinADM uses 7ADMs

  26. Online-MinADM in General Topology (cont) • Denote by d(p) the degree of a node in GS (sharing graph of a solution S ). • For solution S the set of lightpaths P is partitioned into disjoint subsets depending on the degree of the corresponding node in GS (the number of ADMs it shares).∀i ∈ {0, 1, 2} ,Di(S) = {p ∈ P | d(p) = i}di(S)= | Di(S) | d0(S) + d1(S) + d2(S) = |P| = N.

  27. Online-MinADM in General Topology (cont) • The connected components of GS are paths or cycles • Cs - the set of connected components which are paths • |Es| = N-|Cs| • cost(S) = 2*N - |Es| = N+ |Cs| • The sum of the degrees of the nodes in GS : 2*|Es| = d1(S) + 2*d2(S) = N - d0(S) + d2(S) • cost(S)-cost(S*) = |Es*| - |Es| = N-|Cs*|-(N - d0(S) + d2(S) )/2 = N/2 + (d0(S) - d2(S)-2*|Cs|)/2

  28. Online-MinADM in General Topology (cont) • Lemma: The competitive ratio of Online-MinADM is at most 7/4. • Proof: Direct each path of Gs* in arbitrary direction. LAST* = {p ∈ P|dout(p) = 0} Is the set of nodes that don’t have a successor in directed Gs*. | LAST*| = |Cs|

  29. Classification of d0(S) • CLASSIFY(p ∈ D0(S)) { if p ∈ LAST * then { p ∈ A; fA(p) = p } else { q = Next*(p); if q ∈ D2(S) then { p ∈ B; fB(p) = q } if q ∈ D1(S) then {p ∈ C; fC(p) = {p, q}} if q ∈ D0(S) { p ∈ D } } } • D = ∅ • fA(p) = p is one to one =>|A|≤|LAST*| = |Cs| • fB: B→D2(S) • fC: C→2P , |C| ≤ N/2. • d0(S) ≤ |Cs| + d2(S) + N/2

  30. Online-MinADM in General Topology (cont) • cost(S)-cost(S*) = N/2 + (d0(S) - d2(S)-2*|Cs|)/2 ≤ N/2 + (N/2-|Cs|)/2 ≤ ¾*N • cost(S) ≤ cost(S*) + ¾*N ≤ 7/4 * cost(S*)

  31. Lower Bound- Ring Topology • Lemma: No deterministic on-line algorithm has a competitive ratio better than 7/4, even for the ring topology.

  32. Online Ring Coloring When algorithm assigns A & C same color - total ADMS: 0 2 4 6 8 A B w(A)=1 w(C)=1 w(DAB)=2 w(BCD)=3 D C

  33. Offline Ring Coloring Total ADMS: 0 4 A B w(A)=1 w(C)=2 w(DAB)=2 w(BCD)=1 D C Competitive Ratio: 8/4>7/4

  34. Online Ring Coloring When algorithm assigns A, C and B different colors - Total ADMS: 7 6 0 2 4 A B w(A)=1 w(C)=2 w(B)=3 w(D)=2 D C Competitive Ratio: 7/4

  35. Online Ring Coloring – problematic case When algorithm assigns A & C same color - 5 4 2 0 7 9 total ADMS: A B w(A)=1 w(C)=2 w(B)=1 w(BCD)=3 w(CDA)=4 D C

  36. Offline Ring Coloring – problematic case When algorithm assigns A & C same color - 6 total ADMS: A B w(A)=1 w(C)=2 w(B)=1 w(BCD)=3 w(CDA)=4 D C

  37. Ring Coloring – The Adversary • Ring with n nodes, named 0, 1, · · · , n − 1 • The adversary runs in stages, requesting lightpaths which are segments of the ring. • For 0 ≤ i, j < n/4 : Ai,j = <i, i + 1, · · · , n/2−1− i> Bi,j = <n/2−1− i, n/2−i, · · · , n/2 + j> Ci,j = <n/2+j, n/2+j+1, · · · , n−1−j> Di,j = <n−1−j, n−j, · · · , i> • For any lightpath X, define X’ (so they don’t overlap and together cover the ring).

  38. Ring Coloring – The Adversary • Request A i,j and C i,j if not released so far • if w(A i,j) = w(C i,j ) • Request lightpaths A’ i,j and C’i,j • Mark A i,j, C i,j , A’ i,j and C’i,j • Increment both i and j by 1 • if w(A i,j) != w(C i,j ) • Request B i,j • if w(A i,j) w(B i,j ) w(C i,j ) are all different • Request Di,j • Mark A i,j, Bi,j, , C i,j and D i,j • Increment both i and j by 1

  39. Ring Coloring – The Adversary • if w(A i,j) = w(B i,j) • Request A’ i,j and B’ i,j • Mark A i,j, B i,j, A‘ i,j and B‘ i,j • Increment i by 1 and keep j unchanged • if w(B i,j) = w(C i,j) • Request B’ i,j and C’ i,j • Mark B i,j, C i,j, B‘ i,j and C‘ i,j • Increment j by 1 and keep i unchanged • Repeat if i < n/4 and j < n/4.

  40. Ring Coloring – increasing i and j A B i,j i+1,j+1 i+1 i D C

  41. Ring Coloring – increasing only i A B i,j i+1,j i+1 i D C

  42. Ring Coloring – increasing only j A B i,j i,j+1 i D C

  43. Ring Coloring – The Adversary • The adversary has between n/4 and n/2 stages. • At the end of the adversary, at most one unmarked lightpath remains. • Paths from different stages can’t share ADMs. • At any stage cost(S) ≤ (7/4)cost*(S), S are the marked passes.

  44. Triangle Topology • For triangle topology the 7/4 -competitive ratio is not true because it is a bounded size ring . • The article shows a tight bound of 5/3 for this topology and gives an optimal algorithm for it.

  45. Reference • Optimal on-line colorings for minimizing the number of ADMs in optical networks. Mordechai Shalom, Prudence W.H. Wong, Shmuel Zaks

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