2. Sirocco 20040 Peter Ruzicka

3. Sirocco 20041 Results and research directions in ATM and optical networks zaks@cs.technion.ac.il www.cs.technion.ac.il/~zaks Shmuel Zaks Technion, Israel

4. Sirocco 20042

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7. 5 Works with O. Gerstel T. Eilam M. Shalom M. Feigelstein I. Cidon S. Moran M. Flammini References Works of C. Kaklamanis G. Gambossi E. Kranakis L. Bechetti D. Krizanc D. Peleg A. Pelc J.C. Bermond I. Vrt’o A. Rosenberg V. Stacho L. Gargano and many more and many more …

8. Sirocco 20046 graph-theoretic models algorithmic issues greedy constructions recursive constructions complexity issues approximation algorithms dynamic and fault-tolerance combinatorial design issues upper and lower bounds analysis … many open problems

9. Sirocco 20047 Outline Outline 4 ATM networks model 4 Optical networks model 4 Discussion – ATM networks ATM networks 4 Discussion – Optical networks Optical networks

10. Sirocco 20048 ATM - Asynchronous Transfer Mode Transmission and multiplexing technique Industry standard for high-speed networks graph theoretic model Gerstel, Cidon, Zaks

11. Sirocco 20049 Virtual path Virtual channel concatenation of complete paths Communication concatenation of partial paths

12. Sirocco 200410 load = 3 hop count = 2 stretch factor = 4/3 (space) (time) Virtual pathVirtual channel Other parameters Cost

13. Sirocco 200411 Example : Find a layout, to connect a given node with all others, with given bounds on the load and the hop count

14. Sirocco 200412

15. Sirocco 200413 Outline Outline 4 ATM networks model 4 Discussion – ATM networks ATM networks 4 Optical networks model 4 Discussion – Optical networks Optical networks

16. Sirocco 200414 Problem 1 : Given a network, pairs of nodes and bounds h and l, find a virtual path layout to connect these nodes with the load bounded by l and the hop count bounded by h.

17. Sirocco 200415

18. Sirocco 200416 Problem 1a : Given a network and a bound on the load l and a bound h on the hop count, find a layout, to connect a given node with all others ( one-to-all ). a. worst-case. b. average case. Note: consider it for a given stretch factor.

19. Sirocco 200417 Problem 1b : Given a network and a bound on the load l and a bound h on the hop count, find a layout, to connect every two nodes ( all-to-all ). a. worst-case. b. average case. Note: consider it for a given stretch factor.

20. Sirocco 200418 Problem 2 : Input: Graph G, integers h, l > 0, and a vertex v. Question: is there a VP layout for G, by which v can reach all other nodes, with hop count bounded by h and load bounded by l ?

21. Sirocco 200419 P P P NP P … … ……… … … … … 1 2 3.... load hop 1 2... 3 Flammini, Eilam, Zaks

22. Sirocco 200420 tree, mesh general directed path network Gertsel, Wool, Zaks Feighelstein, Zaks Problem 1 : Given a network, pairs of nodes and bounds h and l, find a virtual path layout to connect these nodes with the load bounded by l and the hop count bounded by h.

23. Sirocco 200421 T(l-1,h)T(l,h-1) T(l,h) Case 1 : shortest paths (stretch factor = 1)

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26. Sirocco 200424 Use of binary trees

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29. Sirocco 200427

30. Sirocco 200428 TL(l-1,h) TR(l-1,h-1) TL(l,h-1) TL(l,h) Case 2: any paths (stretch factor > 1)

31. Sirocco 200429 T(l-1,h)T(l-1,h-1)T(l,h-1) T(l-1,h)T(l-1,h-1)

32. Sirocco 200430 l=3, h=2

33. Sirocco 200431 Golomb

34. Sirocco 200432 Use of ternary trees

35. Sirocco 200433 The l 1 -norm |v| of an l -dimensional vector v = (x 1,...,x l ) is defined as |v| = |x 1 | + |x 2 | +... + |x l | ex: |(1,-3,0,2)| = |1|+|-3|+|0|+|2| = 6 Using spheres

36. Sirocco 200434 Sp(l,r) - The l -dimensional l 1 - Sphere of radius h : the set of lattice points v=(x 1,...,x l ) with distance at most h from the origin. Sp(2,3): 2 - dimensional l 1 -Sphere of radius 3. point with l 1 -distance 3 from the origin.

37. Sirocco 200435 Covering Radius - The l - dimensional Covering Radius of N is the radius of the smallest l- dimensional sphere containing at least N points |Sp(2, 0 )| = 1 |Sp(2, 1 )| = 5 |Sp(2, 2 )| = 13 |Sp(2, 3 )| = 25

38. Sirocco 200436 For every ATM Chain Layouts with N nodes and maximal load l: minimal radius of a layout with load l and N nodes minimal radius of an l-dimensional sphere with at least N internal points

39. Sirocco 200437 load = 3 (0,0,0)(1,0,0)(-1,0,0)(-2,0,0)(1,-3,0)(1,-2,0)(-1,-1,0)(0,-1,0)(-1,1,0)(1,-1,0)(-1,-1,1) hop = 4  dimension 3  radius = 4

40. Sirocco 200438 the tree T(l,h) fills the sphere Sp(l,h) !!! |T(l,h)| = |T(h,l)|, hence |Sp(l,h)| = |Sp(h,l)|

41. Sirocco 200439 Sp(1,2): 1 - dimensional l 1 -Sphere of radius 2. Sp(2,1): 2 - dimensional l 1 -Sphere of radius 1.

42. Sirocco 200440 For Upper Bound Using volume formulas, to Achieve bounds on h, given N and l

43. Sirocco 200441 Problem: Given a chain network with N nodes and a given bound on the maximum load, find an optimal layout with minimum hop count (or diameter ) between all pairs of nodes. Bounds for in : Kranakis, Krizanc, Pelc Stacho, Vrt’o Aiello, Bhatt, Chung, Rosenberg, Sitaraman

44. Sirocco 200442 For every graph G with diameter D(G) and radius R(G): R(G) D(G) 2 R(G) Then:

45. Sirocco 200443 Problem 3 : Design and analyze approximation algorithms for general network. Problem 4 : Solve these problems to other measures (like load on the vertices, or bounded stretch factor) one-to-all, all-to-all, some-to-some

46. Sirocco 200444 Problem 7 : Extend the duality results. Problem 8 : Extend the use of geometry.

47. Sirocco 200445 More Problem and parameters what are the input and the output? network: tree, mesh, general, directed cost measure average vs. worst case complexity approximation algorithms routing dynamic, distributed … cost of anarchy?

48. Sirocco 200446 Outline Outline 4 ATM networks model 4 Optical networks model 4 Discussion – ATM networks ATM networks 4 Discussion – Optical networks Optical networks

49. Sirocco 200447 the fiber serves as a transmission medium Electronic switch Optic fiber 1 st generation

50. Sirocco 200448 Optical switch 2 nd generation

51. Sirocco 200449 A virtual topology

52. Sirocco 200450 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 2 nd generation

53. Sirocco 200451 Example : Find a coloring with smallest number of wavelengths for a given set of lightpaths

54. Sirocco 200452 Outline Outline 4 ATM networks model 4 Optical networks model 4 Discussion – ATM networks ATM networks 4 Discussion – Optical networks Optical networks

55. Sirocco 200453 Problem 1 : minimize the number of wavelengths

56. Sirocco 200454 Smallest no. of wavelengths: 2

57. Sirocco 200455 Problem 1a : Complexity Problem 1b: Special networks, general networks Problem 1 : minimize the number of wavelengths

58. Sirocco 200456 Problem 1c : Given pairs to be connected, design a routing with minimal load, and then color it with minimal number of colors ……many references Problem 1d : Given pairs to be connected, design a routing and a coloring with minimal number of colors.

59. Sirocco 200457 Problem 2 : minimize the number of switches

60. Sirocco 200458 no. of ADMs ADM

61. Sirocco 200459 Recall: smallest no. of wavelengths: 2 8 ADMs

62. Sirocco 200460 Smallest no. of ADMs: 3 wavelengths 7

63. Sirocco 200461 Problem 2a : complexity Problem 2c : trees, special networks, general networks Problem 2b : approximation algorithms Problem 2 : minimize the number of switches Problem 2d : given pairs to connect, design a routing and a coloring with smallest number of ADMs.

64. Sirocco 200462 clearly: result: Problem 2b : approximation algorithms

65. Sirocco 200463 Calinescu, Wan Ring network Gerstel, Lin, Sasaki

66. Sirocco 200464 Shalom, Zaks Ring network

67. Sirocco 200465 1. Number the nodes from 0 to n-1 (how?) 2. Color all lightpaths passing through or starting at node 0. Gerstel, Lin, Sasaki

68. Sirocco 200466 3. Scan nodes from 1 to n-1. Color each lightpath starting at i: The colors of the lightpaths ending at i are used first, and only then other colors are used, from lowest numbered first. While color is not valid for a lightpath, try next color.

69. Sirocco 200467 2 3 4 0 14 13 12 11 10 98 5 6 7 1

70. Sirocco 200468 2 3 4 0 14 13 12 11 10 98 5 6 7 1 Color not valid…

71. Sirocco 200469 Calinescu, Wan Use maximum matchings at each node.

72. Sirocco 200470 Combine ideas from together with preprocessing of removing cycles, which uses an approximation algorithm to find all cycles up to a given size. Shalom, Zaks Calinescu, WanGerstel, Lin, Sasaki Hurkens, Schrijver

73. Sirocco 200471 Analysis: Use of linear programming to show we show a set of constraints that, together with cannot be satisfied.

74. Sirocco 200472 Problem 1 : minimize the number of wavelengths. Problem 2 : minimize the number of switches. Problem 3 : find trade-offs between the two measures of number of switches and number of colors.

75. Sirocco 200473 Problem 4 : Given a set of lightpaths, add a minimal number of lightpaths and color all lightpaths, such that all lightpaths will be partitioned into cycles. Eilam, Moran, Zaks fast and simple protection mehanism

76. Sirocco 200474 d b f a g c e cost = 7

77. Sirocco 200475 Problem 4a: Characterize the networks topologies G, in which any simple path can be extended to a simple cycle. Problem 4 : Given a set of lightpaths, add a minimal number of lightpaths and color all lightpaths, such that all lightpaths will be partitioned into cycles.

78. Sirocco 200476 Answer: iff - G is randomly Hamltonian ( = each DFS tree is a path), or - G is a ring, a complete graph, or a complete balanced bipartite graph Chartrand, Kronk Korach, Ostfeld

79. Sirocco 200477 Liu, Li, Wan, Frieder Problem 4b : Input: A Graph G, a set of lightpaths in G, a number k. Question : is there a ring partition of cost  k ? Problem 4 : Given a set of lightpaths, add a minimal number of lightpaths and color all lightpaths, such that all lightpaths will be partitioned into cycles.

80. Sirocco 200478 Problem 4c: Design and analyze an approximation algorithm. Problem 4 : Given a set of lightpaths, add a minimal number of lightpaths and color all lightpaths, such that all lightpaths will be partitioned into cycles.

81. Sirocco 200479 A trivial heuristics: Given a set of lightpaths D, extend each lightpath to a cycle by adding one lightpath. cost = 2 n ( |D|=n ) or: cost  opt + n

82. Sirocco 200480 question: is there a heuristics for which cost = opt + n  (  < 1 ) ? answer: no.

83. Sirocco 200481 question: is there a heuristics for which cost  opt + k n (k < 1 ) ? answer: yes. cost  opt + 3/5 n

84. Sirocco 200482 We showed the measure of total number of switches, thus : Note: Problem 4d : What about the saving in alg vs the saving in opt in the number of switches? Problem 4c: Design and analyze an approximation algorithm.

85. Sirocco 200483 One-band routers: DEMUX Received Forwarded Problem 5 : find a routing with linear filters. Flammini, Navara

86. Sirocco 200484 Problem 5 : find a routing with linear filters. Problem 5a : Is it always possible to find a routing?

87. Sirocco 200485 No: One-band routers are not universal: ru2u2 u3u3 u1u1 v2v2 v1v1 v3v3 z1z1 z2z2 z3z3 w1w1 w2w2 w3w3

88. Sirocco 200486 Problem 5 : find a routing with linear filters. Problem 5b : Define other routers and explor etheir capabilities.

89. Sirocco 200487 Problem 6 : Find a uniform all-to-all routing in a ring, using a minimum number of ADMs. i j N=13 Units of flow Cost: 13+5+3=21 ADMs

90. Sirocco 200488 5 1 3 N=13 2 1 1 4 2 2 1 2 11

91. Sirocco 200489 Shalom, Zaks Problem 6a : What can be said about simple polygons? about non-simple polygons?

92. Sirocco 200490 what are the input and the output? cost measure, worst case vs. average case. coloring and routing Wavelength convertion networks: specific, general complexity approximation algorithms Dynamic … More Problem and parameters cost of anarchy?

93. Sirocco 200491 Questions ?

94. Sirocco 200492 Thank You