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Peter Ruzicka

Peter Ruzicka. Results and research directions in ATM and optical networks. Shmuel Zaks. Technion, Israel. zaks@cs.technion.ac.il www.cs.technion.ac.il/~zaks. References. Works of C. Kaklamanis G. Gambossi E. Kranakis L. Bechetti D. Krizanc D. Peleg

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Peter Ruzicka

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  1. Peter Ruzicka Sirocco 2004

  2. Results and research directions in ATM and optical networks Shmuel Zaks Technion, Israel zaks@cs.technion.ac.il www.cs.technion.ac.il/~zaks Sirocco 2004

  3. Sirocco 2004

  4. Sirocco 2004

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  6. 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 … Works with O. Gerstel T. Eilam M. Shalom M. Feigelstein I. Cidon S. Moran M. Flammini Sirocco 2004

  7. 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 Sirocco 2004

  8. Outline • ATM networks model • Optical networks model • Discussion – ATM networks • Discussion – Optical networks Sirocco 2004

  9. ATM - Asynchronous Transfer Mode Transmission and multiplexing technique Industry standard for high-speed networks graph theoretic model Gerstel, Cidon, Zaks Sirocco 2004

  10. Communication • Virtual • path • Virtual channel concatenation of complete paths concatenation of partial paths Sirocco 2004

  11. Cost • Virtual path • Virtual channel • load = 3 (space) Other parameters hop count = 2 (time) stretch factor = 4/3 Sirocco 2004

  12. Example: Find a layout, to connect a given node with all others, with given bounds on the load and the hop count Sirocco 2004

  13. Sirocco 2004

  14. Outline • ATM networks model • Discussion – ATM networks • Optical networks model • Discussion – Optical networks Sirocco 2004

  15. Problem 1: Given a network, pairs of nodes and bounds h andl, find a virtual path layout to connect these nodes with the load bounded by l and the hop count bounded by h. Sirocco 2004

  16. Sirocco 2004

  17. 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. Sirocco 2004

  18. 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. Sirocco 2004

  19. 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 ? Sirocco 2004

  20. load hop 12 3 .... P P P … 1 P NP NP … 2 NP … … … 3 … … … … ... Flammini, Eilam, Zaks Sirocco 2004

  21. Problem 1: Given a network, pairs of nodes and bounds h andl, find a virtual path layout to connect these nodes with the load bounded by l and the hop count bounded by h. tree, mesh general directed path network Gertsel, Wool, Zaks Feighelstein, Zaks Sirocco 2004

  22. Case 1: shortest paths (stretch factor = 1) T(l,h) T(l-1,h) T(l,h-1) Sirocco 2004

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  25. Use of binary trees Sirocco 2004

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  29. Case 2:any paths (stretch factor > 1) TL(l,h) TL(l-1,h) TL(l,h-1) TR(l-1,h-1) Sirocco 2004

  30. T(l,h-1) T(l,h-1) T(l-1,h-1) T(l-1,h) T(l-1,h) T(l-1,h-1) Sirocco 2004

  31. l=3, h=2 Sirocco 2004

  32. Golomb Sirocco 2004

  33. Use of ternary trees Sirocco 2004

  34. Using spheres The l1-norm|v|of an l-dimensional vectorv = (x1 ,...,xl ) is defined as |v| = |x1| + |x2| + ... + |xl| ex: |(1,-3,0,2)| = |1|+|-3|+|0|+|2| = 6 Sirocco 2004

  35. Sp(l,r) - The l-dimensional l1-Sphere of radius h: the set of lattice points v=(x1,...,xl) with distance at most h from the origin. Sp(2,3):2-dimensional l1-Sphere of radius3. point with l1-distance 3 from the origin. Sirocco 2004

  36. Covering Radius - The l - dimensional Covering Radius of N is the radius of the smallest l-dimensional sphere containing at least Npoints |Sp(2,0)| = 1 |Sp(2,1)| = 5 |Sp(2,2)| = 13 |Sp(2,3)| = 25 Sirocco 2004

  37. 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 Sirocco 2004

  38. load = 3  dimension 3  radius = 4 hop = 4 (1,0,0) (1,-1,0) (1,-2,0) (1,-3,0) (0,0,0) (0,-1,0) (-1,1,0) (-1,0,0) (-1,-1,1) (-1,-1,0) (-2,0,0) Sirocco 2004

  39. the tree T(l,h) fills the sphere Sp(l,h)!!! |T(l,h)| = |T(h,l)| , hence |Sp(l,h)| = |Sp(h,l)| Sirocco 2004

  40. Sp(2,1): 2-dimensional l1-Sphere of radius1. Sp(1,2): 1-dimensional l1-Sphere of radius2. Sirocco 2004

  41. For Upper Bound Using volume formulas, to Achieve bounds on h, given N and l Sirocco 2004

  42. 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 Sirocco 2004

  43. For every graphG with diameter D(G) and radius R(G): R(G) D(G) 2 R(G) Then: Sirocco 2004

  44. one-to-all, all-to-all, some-to-some Problem 3: Design and analyze approximation algorithms for general network. Problem 4: Solvethese problems to other measures (like load on the vertices, or bounded stretch factor) Sirocco 2004

  45. Problem 7: Extend the duality results. Problem 8: Extend the use of geometry. Sirocco 2004

  46. what are the input and the output? • network: tree, mesh, general, directed • cost measure • average vs. worst case • complexity • approximation algorithms • routing • dynamic, distributed • … More Problem and parameters cost of anarchy? Sirocco 2004

  47. Outline • ATM networks model • Optical networks model • Discussion – ATM networks • Discussion – Optical networks Sirocco 2004

  48. 1st generation the fiber serves as a transmission medium Electronic switch Optic fiber Sirocco 2004

  49. 2nd generation Optical switch Sirocco 2004

  50. A virtual topology Sirocco 2004

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