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Sirocco Peter Ruzicka

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Sirocco Results and research directions in ATM and optical networks Shmuel Zaks Technion, Israel

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Sirocco 20042

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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 …

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Sirocco 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

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Sirocco Outline Outline 4 ATM networks model 4 Optical networks model 4 Discussion – ATM networks ATM networks 4 Discussion – Optical networks Optical networks

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

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Sirocco Virtual path Virtual channel concatenation of complete paths Communication concatenation of partial paths

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

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

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Sirocco

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Sirocco Outline Outline 4 ATM networks model 4 Discussion – ATM networks ATM networks 4 Optical networks model 4 Discussion – Optical networks Optical networks

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Sirocco 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.

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Sirocco

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Sirocco 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.

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Sirocco 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.

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Sirocco 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 ?

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Sirocco P P P NP P … … ……… … … … … load hop Flammini, Eilam, Zaks

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Sirocco 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.

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

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Sirocco Use of binary trees

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

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Sirocco T(l-1,h)T(l-1,h-1)T(l,h-1) T(l-1,h)T(l-1,h-1)

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Sirocco l=3, h=2

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Sirocco Golomb

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Sirocco Use of ternary trees

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Sirocco 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

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Sirocco 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.

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Sirocco 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

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Sirocco 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

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Sirocco 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

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

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

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

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Sirocco 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

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

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Sirocco 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

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

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Sirocco 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?

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Sirocco Outline Outline 4 ATM networks model 4 Optical networks model 4 Discussion – ATM networks ATM networks 4 Discussion – Optical networks Optical networks

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

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Sirocco Optical switch 2 nd generation

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Sirocco A virtual topology

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Sirocco 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

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

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Sirocco Outline Outline 4 ATM networks model 4 Optical networks model 4 Discussion – ATM networks ATM networks 4 Discussion – Optical networks Optical networks

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Sirocco Problem 1 : minimize the number of wavelengths

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Sirocco Smallest no. of wavelengths: 2

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

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Sirocco 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.

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Sirocco Problem 2 : minimize the number of switches

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Sirocco no. of ADMs ADM

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

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Sirocco Smallest no. of ADMs: 3 wavelengths 7

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Sirocco 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.

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Sirocco clearly: result: Problem 2b : approximation algorithms

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Sirocco Calinescu, Wan Ring network Gerstel, Lin, Sasaki

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Sirocco Shalom, Zaks Ring network

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

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Sirocco 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.

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Sirocco

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Sirocco Color not valid…

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Sirocco Calinescu, Wan Use maximum matchings at each node.

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Sirocco 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

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

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Sirocco 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.

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Sirocco 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

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Sirocco d b f a g c e cost = 7

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Sirocco 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.

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Sirocco 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

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Sirocco 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.

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Sirocco 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.

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Sirocco 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

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

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

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Sirocco 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.

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

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

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

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

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Sirocco 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: =21 ADMs

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Sirocco N=

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

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Sirocco 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?

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Sirocco Questions ?

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Sirocco Thank You

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