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Problem Statement Theoretical Results Experimental Results
Approximation Algorithms for Traffic Groomingin WDM Rings
K. Corcoran1 S. Flaxman2 M. Neyer3 C. Weidert4
P. Scherpelz5 R. Libeskind-Hadas6
1University of Oregon, USA
2Ecole Polytechnique Federale 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
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Problem Statement Theoretical Results Experimental Results
Single-Source WDM Rings
� WDM ring with given set ofwavelengths, each with fixedcapacity C
� Single source/hub from which allother destination nodes receivedata
� Source node can transmit on allwavelengths
� Each destination node has somenumber of tunable ADMs
� A path from the source to adestination has a pre-determinedroute (e.g. all clockwise)
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Problem Statement Theoretical Results Experimental Results
Single-Source WDM Rings
� WDM ring with given set ofwavelengths, each with fixedcapacity C
� Single source/hub from which allother destination nodes receivedata
� Source node can transmit on allwavelengths
� Each destination node has somenumber of tunable ADMs
� A path from the source to adestination has a pre-determinedroute (e.g. all clockwise)
![Page 4: Approximation Algorithms for Traffic Grooming in WDM Rings](https://reader033.vdocuments.us/reader033/viewer/2022052909/5598ba311a28abbc4a8b4693/html5/thumbnails/4.jpg)
Problem Statement Theoretical Results Experimental Results
Single-Source WDM Rings
� WDM ring with given set ofwavelengths, each with fixedcapacity C
� Single source/hub from which allother destination nodes receivedata
� Source node can transmit on allwavelengths
� Each destination node has somenumber of tunable ADMs
� A path from the source to adestination has a pre-determinedroute (e.g. all clockwise)
![Page 5: Approximation Algorithms for Traffic Grooming in WDM Rings](https://reader033.vdocuments.us/reader033/viewer/2022052909/5598ba311a28abbc4a8b4693/html5/thumbnails/5.jpg)
Problem Statement Theoretical Results Experimental Results
Single-Source WDM Rings
� WDM ring with given set ofwavelengths, each with fixedcapacity C
� Single source/hub from which allother destination nodes receivedata
� Source node can transmit on allwavelengths
� Each destination node has somenumber of tunable ADMs
� A path from the source to adestination has a pre-determinedroute (e.g. all clockwise)
![Page 6: Approximation Algorithms for Traffic Grooming in WDM Rings](https://reader033.vdocuments.us/reader033/viewer/2022052909/5598ba311a28abbc4a8b4693/html5/thumbnails/6.jpg)
Problem Statement Theoretical Results Experimental Results
Single-Source WDM Rings
� WDM ring with given set ofwavelengths, each with fixedcapacity C
� Single source/hub from which allother destination nodes receivedata
� Source node can transmit on allwavelengths
� Each destination node has somenumber of tunable ADMs
� A path from the source to adestination has a pre-determinedroute (e.g. all clockwise)
![Page 7: Approximation Algorithms for Traffic Grooming in WDM Rings](https://reader033.vdocuments.us/reader033/viewer/2022052909/5598ba311a28abbc4a8b4693/html5/thumbnails/7.jpg)
Problem Statement Theoretical Results Experimental Results
Single-Source WDM Rings
� WDM ring with given set ofwavelengths, each with fixedcapacity C
� Single source/hub from which allother destination nodes receivedata
� Source node can transmit on allwavelengths
� Each destination node has somenumber of tunable ADMs
� A path from the source to adestination has a pre-determinedroute (e.g. all clockwise)
![Page 8: Approximation Algorithms for Traffic Grooming in WDM Rings](https://reader033.vdocuments.us/reader033/viewer/2022052909/5598ba311a28abbc4a8b4693/html5/thumbnails/8.jpg)
Problem Statement Theoretical Results Experimental Results
The Tunable Ring Grooming Problem
� Each node may make a request r forpersonalized data to be sent from thesource
� request r consists of:� integer size: demand(r)� value: profit(r)
� A request may be partitioned ontomultiple wavelengths in integral parts
� Multiple requests (or parts of requests)can be “groomed” onto the samewavelength
� Objective: Groom requests ontowavelengths to maximize total profit of allsatisfied requests
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Problem Statement Theoretical Results Experimental Results
The Tunable Ring Grooming Problem
� Each node may make a request r forpersonalized data to be sent from thesource
� request r consists of:� integer size: demand(r)� value: profit(r)
� A request may be partitioned ontomultiple wavelengths in integral parts
� Multiple requests (or parts of requests)can be “groomed” onto the samewavelength
� Objective: Groom requests ontowavelengths to maximize total profit of allsatisfied requests
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Problem Statement Theoretical Results Experimental Results
The Tunable Ring Grooming Problem
� Each node may make a request r forpersonalized data to be sent from thesource
� request r consists of:� integer size: demand(r)� value: profit(r)
� A request may be partitioned ontomultiple wavelengths in integral parts
� Multiple requests (or parts of requests)can be “groomed” onto the samewavelength
� Objective: Groom requests ontowavelengths to maximize total profit of allsatisfied requests
![Page 11: Approximation Algorithms for Traffic Grooming in WDM Rings](https://reader033.vdocuments.us/reader033/viewer/2022052909/5598ba311a28abbc4a8b4693/html5/thumbnails/11.jpg)
Problem Statement Theoretical Results Experimental Results
The Tunable Ring Grooming Problem
� Each node may make a request r forpersonalized data to be sent from thesource
� request r consists of:� integer size: demand(r)� value: profit(r)
� A request may be partitioned ontomultiple wavelengths in integral parts
� Multiple requests (or parts of requests)can be “groomed” onto the samewavelength
� Objective: Groom requests ontowavelengths to maximize total profit of allsatisfied requests
![Page 12: Approximation Algorithms for Traffic Grooming in WDM Rings](https://reader033.vdocuments.us/reader033/viewer/2022052909/5598ba311a28abbc4a8b4693/html5/thumbnails/12.jpg)
Problem Statement Theoretical Results Experimental Results
The Tunable Ring Grooming Problem
� Each node may make a request r forpersonalized data to be sent from thesource
� request r consists of:� integer size: demand(r)� value: profit(r)
� A request may be partitioned ontomultiple wavelengths in integral parts
� Multiple requests (or parts of requests)can be “groomed” onto the samewavelength
� Objective: Tune ADMs and groomrequests onto wavelengths to maximizetotal profit of all satisfied requests
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Problem Statement Theoretical Results Experimental Results
Sample Instance of the Tunable Ring Grooming Problem
Figure: Capacity C = 4 for each wavelength. Objective: Tune ADMsand groom requests onto wavelengths to maximize total profit of allsatisfied requests.
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Problem Statement Theoretical Results Experimental Results
Sample Instance of the Tunable Ring Grooming Problem
Figure: A solution. Profit = 650. Is it optimal?
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Problem Statement Theoretical Results Experimental Results
Sample Instance of the Tunable Ring Grooming Problem
Figure: Profit = 650 Figure: Profit = 950
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Problem Statement Theoretical Results Experimental Results
Overview of Results
� The Tunable Ring Grooming Problem is NP-complete in thestrong sense
� Problem remains NP-complete even for special cases
� Only one wavelength, only one ADM per node, at least twoADMs per node
� Polynomial time approximation schemes for these specialcases
� The “general case” that the number of ADMs is one or moreappears to be the most challenging
� New approximation algorithm for the general case
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Problem Statement Theoretical Results Experimental Results
Overview of Results
� The Tunable Ring Grooming Problem is NP-complete in thestrong sense
� Problem remains NP-complete even for special cases� Only one wavelength, only one ADM per node, at least two
ADMs per node
� Polynomial time approximation schemes for these specialcases
� The “general case” that the number of ADMs is one or moreappears to be the most challenging
� New approximation algorithm for the general case
![Page 18: Approximation Algorithms for Traffic Grooming in WDM Rings](https://reader033.vdocuments.us/reader033/viewer/2022052909/5598ba311a28abbc4a8b4693/html5/thumbnails/18.jpg)
Problem Statement Theoretical Results Experimental Results
Overview of Results
� The Tunable Ring Grooming Problem is NP-complete in thestrong sense
� Problem remains NP-complete even for special cases� Only one wavelength, only one ADM per node, at least two
ADMs per node
� Polynomial time approximation schemes for these specialcases
� The “general case” that the number of ADMs is one or moreappears to be the most challenging
� New approximation algorithm for the general case
![Page 19: Approximation Algorithms for Traffic Grooming in WDM Rings](https://reader033.vdocuments.us/reader033/viewer/2022052909/5598ba311a28abbc4a8b4693/html5/thumbnails/19.jpg)
Problem Statement Theoretical Results Experimental Results
Overview of Results
� The Tunable Ring Grooming Problem is NP-complete in thestrong sense
� Problem remains NP-complete even for special cases� Only one wavelength, only one ADM per node, at least two
ADMs per node
� Polynomial time approximation schemes for these specialcases
� The “general case” that the number of ADMs is one or moreappears to be the most challenging
� New approximation algorithm for the general case
![Page 20: Approximation Algorithms for Traffic Grooming in WDM Rings](https://reader033.vdocuments.us/reader033/viewer/2022052909/5598ba311a28abbc4a8b4693/html5/thumbnails/20.jpg)
Problem Statement Theoretical Results Experimental Results
Overview of Results
� The Tunable Ring Grooming Problem is NP-complete in thestrong sense
� Problem remains NP-complete even for special cases� Only one wavelength, only one ADM per node, at least two
ADMs per node
� Polynomial time approximation schemes for these specialcases
� The “general case” that the number of ADMs is one or moreappears to be the most challenging
� New approximation algorithm for the general case
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Problem Statement Theoretical Results Experimental Results
The General Case
� Let C denote the capacity of a wavelength and let q be aninteger such that every request has demand at most C
q , i.e.
� If a request can demand as much as capacity C , then q = 1� If every request demands at most 1
2 of C , then q = 2
� Main Result: A polynomial time approximation algorithmthat guarantees solutions within q
q+1 of optimal, i.e.
� If q = 1, profit is guaranteed to be within 1/2 of optimal� If q = 2, profit is guaranteed to be within 2/3 of optimal� If q = 10, profit is guaranteed to be within 10/11 of optimal
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Problem Statement Theoretical Results Experimental Results
The General Case
� Let C denote the capacity of a wavelength and let q be aninteger such that every request has demand at most C
q , i.e.
� If a request can demand as much as capacity C , then q = 1
� If every request demands at most 12 of C , then q = 2
� Main Result: A polynomial time approximation algorithmthat guarantees solutions within q
q+1 of optimal, i.e.
� If q = 1, profit is guaranteed to be within 1/2 of optimal� If q = 2, profit is guaranteed to be within 2/3 of optimal� If q = 10, profit is guaranteed to be within 10/11 of optimal
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Problem Statement Theoretical Results Experimental Results
The General Case
� Let C denote the capacity of a wavelength and let q be aninteger such that every request has demand at most C
q , i.e.
� If a request can demand as much as capacity C , then q = 1� If every request demands at most 1
2 of C , then q = 2
� Main Result: A polynomial time approximation algorithmthat guarantees solutions within q
q+1 of optimal, i.e.
� If q = 1, profit is guaranteed to be within 1/2 of optimal� If q = 2, profit is guaranteed to be within 2/3 of optimal� If q = 10, profit is guaranteed to be within 10/11 of optimal
![Page 24: Approximation Algorithms for Traffic Grooming in WDM Rings](https://reader033.vdocuments.us/reader033/viewer/2022052909/5598ba311a28abbc4a8b4693/html5/thumbnails/24.jpg)
Problem Statement Theoretical Results Experimental Results
The General Case
� Let C denote the capacity of a wavelength and let q be aninteger such that every request has demand at most C
q , i.e.
� If a request can demand as much as capacity C , then q = 1� If every request demands at most 1
2 of C , then q = 2
� Main Result: A polynomial time approximation algorithmthat guarantees solutions within q
q+1 of optimal, i.e.
� If q = 1, profit is guaranteed to be within 1/2 of optimal� If q = 2, profit is guaranteed to be within 2/3 of optimal� If q = 10, profit is guaranteed to be within 10/11 of optimal
![Page 25: Approximation Algorithms for Traffic Grooming in WDM Rings](https://reader033.vdocuments.us/reader033/viewer/2022052909/5598ba311a28abbc4a8b4693/html5/thumbnails/25.jpg)
Problem Statement Theoretical Results Experimental Results
The General Case
� Let C denote the capacity of a wavelength and let q be aninteger such that every request has demand at most C
q , i.e.
� If a request can demand as much as capacity C , then q = 1� If every request demands at most 1
2 of C , then q = 2
� Main Result: A polynomial time approximation algorithmthat guarantees solutions within q
q+1 of optimal, i.e.
� If q = 1, profit is guaranteed to be within 1/2 of optimal� If q = 2, profit is guaranteed to be within 2/3 of optimal� If q = 10, profit is guaranteed to be within 10/11 of optimal
![Page 26: Approximation Algorithms for Traffic Grooming in WDM Rings](https://reader033.vdocuments.us/reader033/viewer/2022052909/5598ba311a28abbc4a8b4693/html5/thumbnails/26.jpg)
Problem Statement Theoretical Results Experimental Results
The General Case: The Algorithm
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Problem Statement Theoretical Results Experimental Results
The General Case: The Algorithm
1 Sort requests by non-increasing density into a list S
2 Let A = S if total demand ≤ CW qq+1 , otherwise let A be the
minimal prefix of S with total demand > CW qq+1
3 Pack A onto wavelengths with First Fit Decreasing (FFD)
4 if some request in A was not packed then
5 Let r denote first request not packed by FFD6 Let B be the set containing r and all requests which were
packed with demand ≥ demand(r)7 Discard the request with the least profit from B8 if r was not discarded then
9 Pack r in place of the discarded request
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Problem Statement Theoretical Results Experimental Results
The General Case: The Algorithm
1 Sort requests by non-increasing density into a list S
2 Let A = S if total demand ≤ CW qq+1 , otherwise let A be the
minimal prefix of S with total demand > CW qq+1
3 Pack A onto wavelengths with First Fit Decreasing (FFD)
4 if some request in A was not packed then
5 Let r denote first request not packed by FFD6 Let B be the set containing r and all requests which were
packed with demand ≥ demand(r)7 Discard the request with the least profit from B8 if r was not discarded then
9 Pack r in place of the discarded request
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Problem Statement Theoretical Results Experimental Results
The General Case: The Algorithm
1 Sort requests by non-increasing density into a list S
2 Let A = S if total demand ≤ CW qq+1 , otherwise let A be the
minimal prefix of S with total demand > CW qq+1
3 Pack A onto wavelengths with First Fit Decreasing (FFD)
4 if some request in A was not packed then
5 Let r denote first request not packed by FFD6 Let B be the set containing r and all requests which were
packed with demand ≥ demand(r)7 Discard the request with the least profit from B8 if r was not discarded then
9 Pack r in place of the discarded request
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Problem Statement Theoretical Results Experimental Results
The General Case: The Algorithm
1 Sort requests by non-increasing density into a list S
2 Let A = S if total demand ≤ CW qq+1 , otherwise let A be the
minimal prefix of S with total demand > CW qq+1
3 Pack A onto wavelengths with First Fit Decreasing (FFD)
4 if some request in A was not packed then5 Let r denote first request not packed by FFD6 Let B be the set containing r and all requests which were
packed with demand ≥ demand(r)7 Discard the request with the least profit from B8 if r was not discarded then
9 Pack r in place of the discarded request
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Problem Statement Theoretical Results Experimental Results
The General Case: Analysis
� The approximation algorithm is proved correct and analyzed inthe paper
� The running time is O(R log R + RW ) where R is the numberof requests and W is the number of wavelengths
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Problem Statement Theoretical Results Experimental Results
The General Case: Analysis
� The approximation algorithm is proved correct and analyzed inthe paper
� The running time is O(R log R + RW ) where R is the numberof requests and W is the number of wavelengths
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Problem Statement Theoretical Results Experimental Results
Heuristics and Experiments
� Heuristic “on top” of approximation algorithm� Performs q/(q + 1)-approximation algorithm for general case� Attempts to improve solution using heuristic rules, including
splitting
� Experiments using heuristic
� Heuristic profit divided by optimal profit� Optimal found with linear programming
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Problem Statement Theoretical Results Experimental Results
Heuristics and Experiments
� Heuristic “on top” of approximation algorithm� Performs q/(q + 1)-approximation algorithm for general case� Attempts to improve solution using heuristic rules, including
splitting
� Experiments using heuristic� Heuristic profit divided by optimal profit� Optimal found with linear programming
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Problem Statement Theoretical Results Experimental Results
Experimental Results: Parameters
Parameter Possible valuesWavelength capacity C 4, 8, 16Number of wavelengths 5Number of requests 16, 32
Probability α that a request hastwo ADMs (one ADM other-wise)
α = 0, 14, 1
2, 3
4, 1
Demand limited to fraction 1/qof capacity
q = 1, 2
Density of request Constant or variable(∈ U[1/2, 2))
Table: Parameters used in generating random instances
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Problem Statement Theoretical Results Experimental Results
Sample Results
� When q = 1, approximation algorithm guarantees ratio of 12
0
0.1
0.2
0.3
0.4
0.5
0.6
.99-1.00.96-.97.94-.95.92-.93.90-.91.88-.89<= .87
Fra
ctio
n of
inst
ance
s
Approximation ratio
Tunable Results: Worst Case
Figure: Worst ratios found in experiments. Parameters: 5 wavelengths,wavelength capacity C = 16, q = 1, 1
2 of nodes have 1 ADM andremaining have 2 ADMs
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Problem Statement Theoretical Results Experimental Results
Future Work
� Generalizing to allow requests to demand more than awavelength’s capacity
� Tighter approximation bounds
� What if the direction of travel for a request is notpre-determined? Can we still find good approximationalgorithms?
� Using splitting in algorithm, not just heuristic