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 …
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
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.
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.
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 ?
Sirocco 200419 P P P NP P … … ……… … … … … 1 2 3.... load hop 1 2... 3 Flammini, Eilam, Zaks
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.
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
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.
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
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
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)|
Sirocco 200439 Sp(1,2): 1 - dimensional l 1 -Sphere of radius 2. Sp(2,1): 2 - dimensional l 1 -Sphere of radius 1.
Sirocco 200440 For Upper Bound Using volume formulas, to Achieve bounds on h, given N and l
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
Sirocco 200442 For every graph G with diameter D(G) and radius R(G): R(G) D(G) 2 R(G) Then:
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
Sirocco 200444 Problem 7 : Extend the duality results. Problem 8 : Extend the use of geometry.
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?
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
Sirocco 200451 Example : Find a coloring with smallest number of wavelengths for a given set of lightpaths
Sirocco 200455 Problem 1a : Complexity Problem 1b: Special networks, general networks Problem 1 : minimize the number of wavelengths
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.
Sirocco 200457 Problem 2 : minimize the number of switches
Sirocco 200460 Smallest no. of ADMs: 3 wavelengths 7
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.
Sirocco 200462 clearly: result: Problem 2b : approximation algorithms
Sirocco 200463 Calinescu, Wan Ring network Gerstel, Lin, Sasaki
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
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.
Sirocco 200468 2 3 4 0 14 13 12 11 10 98 5 6 7 1 Color not valid…
Sirocco 200469 Calinescu, Wan Use maximum matchings at each node.
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
Sirocco 200471 Analysis: Use of linear programming to show we show a set of constraints that, together with cannot be satisfied.
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.
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
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.
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
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.
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.
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
Sirocco 200480 question: is there a heuristics for which cost = opt + n ( < 1 ) ? answer: no.
Sirocco 200481 question: is there a heuristics for which cost opt + k n (k < 1 ) ? answer: yes. cost opt + 3/5 n
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.
Sirocco 200483 One-band routers: DEMUX Received Forwarded Problem 5 : find a routing with linear filters. Flammini, Navara
Sirocco 200484 Problem 5 : find a routing with linear filters. Problem 5a : Is it always possible to find a routing?
Sirocco 200489 Shalom, Zaks Problem 6a : What can be said about simple polygons? about non-simple polygons?
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?