Problem Statement Theoretical Results Experimental Results Approximation Algorithms for Traffic Grooming in WDM Rings K. Corcoran 1 S. Flaxman 2 M. Neyer 3 C. Weidert 4 P. Scherpelz 5 R. Libeskind-Hadas 6 1University of Oregon, USA 2Ecole Polytechnique F´ ed´ erale de Lausanne, Switzerland 3University of North Carolina, USA 4Simon Fraser University, Canada 5University of Chicago, USA, Supported by the Hertz Foundation 6Harvey Mudd College, USA. This work was supported by the National Science Foundation under grant 0451293 to Harvey Mudd College
❼ ❼ ❼ ❼ ❼ Problem Statement Theoretical Results Experimental Results Single-Source WDM Rings
❼ ❼ ❼ ❼ Problem Statement Theoretical Results Experimental Results Single-Source WDM Rings ❼ WDM ring with given set of wavelengths, each with fixed capacity C
❼ ❼ ❼ Problem Statement Theoretical Results Experimental Results Single-Source WDM Rings ❼ WDM ring with given set of wavelengths, each with fixed capacity C ❼ Single source/hub from which all other destination nodes receive data
❼ ❼ Problem Statement Theoretical Results Experimental Results Single-Source WDM Rings ❼ WDM ring with given set of wavelengths, each with fixed capacity C ❼ Single source/hub from which all other destination nodes receive data ❼ Source node can transmit on all wavelengths
❼ Problem Statement Theoretical Results Experimental Results Single-Source WDM Rings ❼ WDM ring with given set of wavelengths, each with fixed capacity C ❼ Single source/hub from which all other destination nodes receive data ❼ Source node can transmit on all wavelengths ❼ Each destination node has some number of tunable ADMs
Problem Statement Theoretical Results Experimental Results Single-Source WDM Rings ❼ WDM ring with given set of wavelengths, each with fixed capacity C ❼ Single source/hub from which all other destination nodes receive data ❼ Source node can transmit on all wavelengths ❼ Each destination node has some number of tunable ADMs ❼ A path from the source to a destination has a pre-determined route (e.g. all clockwise)
❼ ❼ ❼ ❼ ❼ ❼ Problem Statement Theoretical Results Experimental Results The Tunable Ring Grooming Problem ❼ Each node may make a request r for personalized data to be sent from the source
❼ ❼ ❼ Problem Statement Theoretical Results Experimental Results The Tunable Ring Grooming Problem ❼ Each node may make a request r for personalized data to be sent from the source ❼ request r consists of: ❼ integer size: demand( r ) ❼ value: profit( r )
❼ ❼ Problem Statement Theoretical Results Experimental Results The Tunable Ring Grooming Problem ❼ Each node may make a request r for personalized data to be sent from the source ❼ request r consists of: ❼ integer size: demand( r ) ❼ value: profit( r ) ❼ A request may be partitioned onto multiple wavelengths in integral parts
❼ Problem Statement Theoretical Results Experimental Results The Tunable Ring Grooming Problem ❼ Each node may make a request r for personalized data to be sent from the source ❼ request r consists of: ❼ integer size: demand( r ) ❼ value: profit( r ) ❼ A request may be partitioned onto multiple wavelengths in integral parts ❼ Multiple requests (or parts of requests) can be “groomed” onto the same wavelength
Problem Statement Theoretical Results Experimental Results The Tunable Ring Grooming Problem ❼ Each node may make a request r for personalized data to be sent from the source ❼ request r consists of: ❼ integer size: demand( r ) ❼ value: profit( r ) ❼ A request may be partitioned onto multiple wavelengths in integral parts ❼ Multiple requests (or parts of requests) can be “groomed” onto the same wavelength ❼ Objective: Tune ADMs and groom requests onto wavelengths to maximize total profit of all satisfied requests
Problem Statement Theoretical Results Experimental Results Sample Instance of the Tunable Ring Grooming Problem Figure: Capacity C = 4 for each wavelength. Objective: Tune ADMs and groom requests onto wavelengths to maximize total profit of all satisfied requests.
Problem Statement Theoretical Results Experimental Results Sample Instance of the Tunable Ring Grooming Problem Figure: A solution. Profit = 650. Is it optimal?
Problem Statement Theoretical Results Experimental Results Sample Instance of the Tunable Ring Grooming Problem Figure: Profit = 650 Figure: Profit = 950
❼ ❼ ❼ ❼ ❼ Problem Statement Theoretical Results Experimental Results Overview of Results ❼ The Tunable Ring Grooming Problem is NP-complete in the strong sense
❼ ❼ ❼ Problem Statement Theoretical Results Experimental Results Overview of Results ❼ The Tunable Ring Grooming Problem is NP-complete in the strong sense ❼ Problem remains NP-complete even for special cases ❼ Only one wavelength, only one ADM per node, at least two ADMs per node
❼ ❼ Problem Statement Theoretical Results Experimental Results Overview of Results ❼ The Tunable Ring Grooming Problem is NP-complete in the strong sense ❼ Problem remains NP-complete even for special cases ❼ Only one wavelength, only one ADM per node, at least two ADMs per node ❼ Polynomial time approximation schemes for these special cases
❼ Problem Statement Theoretical Results Experimental Results Overview of Results ❼ The Tunable Ring Grooming Problem is NP-complete in the strong sense ❼ Problem remains NP-complete even for special cases ❼ Only one wavelength, only one ADM per node, at least two ADMs per node ❼ Polynomial time approximation schemes for these special cases ❼ The “general case” that the number of ADMs is one or more appears to be the most challenging
Problem Statement Theoretical Results Experimental Results Overview of Results ❼ The Tunable Ring Grooming Problem is NP-complete in the strong sense ❼ Problem remains NP-complete even for special cases ❼ Only one wavelength, only one ADM per node, at least two ADMs per node ❼ Polynomial time approximation schemes for these special cases ❼ The “general case” that the number of ADMs is one or more appears to be the most challenging ❼ New approximation algorithm for the general case
❼ ❼ ❼ ❼ ❼ ❼ Problem Statement Theoretical Results Experimental Results The General Case ❼ Let C denote the capacity of a wavelength and let q be an integer such that every request has demand at most C q , i.e.
❼ ❼ ❼ ❼ ❼ Problem Statement Theoretical Results Experimental Results The General Case ❼ Let C denote the capacity of a wavelength and let q be an integer such that every request has demand at most C q , i.e. ❼ If a request can demand as much as capacity C , then q = 1
❼ ❼ ❼ ❼ Problem Statement Theoretical Results Experimental Results The General Case ❼ Let C denote the capacity of a wavelength and let q be an integer such that every request has demand at most C q , i.e. ❼ If a request can demand as much as capacity C , then q = 1 ❼ If every request demands at most 1 2 of C , then q = 2
❼ ❼ ❼ Problem Statement Theoretical Results Experimental Results The General Case ❼ Let C denote the capacity of a wavelength and let q be an integer such that every request has demand at most C q , i.e. ❼ If a request can demand as much as capacity C , then q = 1 ❼ If every request demands at most 1 2 of C , then q = 2 ❼ Main Result: A polynomial time approximation algorithm q that guarantees solutions within q +1 of optimal, i.e.
Problem Statement Theoretical Results Experimental Results The General Case ❼ Let C denote the capacity of a wavelength and let q be an integer such that every request has demand at most C q , i.e. ❼ If a request can demand as much as capacity C , then q = 1 ❼ If every request demands at most 1 2 of C , then q = 2 ❼ Main Result: A polynomial time approximation algorithm q that guarantees solutions within q +1 of optimal, i.e. ❼ If q = 1, profit is guaranteed to be within 1 / 2 of optimal ❼ If q = 2, profit is guaranteed to be within 2 / 3 of optimal ❼ If q = 10, profit is guaranteed to be within 10 / 11 of optimal
Problem Statement Theoretical Results Experimental Results The General Case: The Algorithm
Problem Statement Theoretical Results Experimental Results The General Case: The Algorithm 1 Sort requests by non-increasing density into a list S
Problem Statement Theoretical Results Experimental Results The General Case: The Algorithm 1 Sort requests by non-increasing density into a list S q 2 Let A = S if total demand ≤ CW q +1 , otherwise let A be the q minimal prefix of S with total demand > CW q +1
Problem Statement Theoretical Results Experimental Results The General Case: The Algorithm 1 Sort requests by non-increasing density into a list S q 2 Let A = S if total demand ≤ CW q +1 , otherwise let A be the q minimal prefix of S with total demand > CW q +1 3 Pack A onto wavelengths with First Fit Decreasing (FFD)
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