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Complexity of Wavelength Assignment in Optical Network Optimization

Complexity of Wavelength Assignment in Optical Network Optimization. Lisa Zhang Bell Labs April, 2006 Joint with Matthew Andrews. SEA. SAI. ALB. BUF. MIL. DET. CLE. BOS. SAL. CHI. DEN. SPR. NYC. PHI. PIT. KAN. BAL. CIN. SFO. WAS. LAS. NAS. RAL. PHO. ATL. LOS. ELP.

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Complexity of Wavelength Assignment in Optical Network Optimization

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  1. Complexity of Wavelength Assignment in Optical Network Optimization Lisa Zhang Bell Labs April, 2006 Joint with Matthew Andrews

  2. SEA SAI ALB BUF MIL DET CLE BOS SAL CHI DEN SPR NYC PHI PIT KAN BAL CIN SFO WAS LAS NAS RAL PHO ATL LOS ELP JAC HOU NOR TAM MIA Optical Network Design • Input • A network, a set of demands to be carried over network; • Output • Routing and wavelength assignment (RWA) for the demands; • Two demands sharing common fiber have distinct wavelengths. • No wavelength conversion. SF-NYC 10 BOS-LA 5 … B Fiber capacity = 2 AOB, BOC, COA A O OADM: optical add/drop multiplexer C

  3. To DCM From DCM To DCM From DCM Mesh In LDRX1 part of WR1 IPD LDTX1 4x1 WSS OMON OMON MON MON PD PD PD PD part of WR1 ... ... PD Thru In Sig Out PD ... ... IPD IPD Sig Out Thru Out Line Out Line In PD Sig In Sig In PD OSC Rx Mesh Out Add In 1 2 Splitters OSC Tx OSC IN IPD OSC OUT OSC IN Drop Out 1 2 IPD IPD IPD part of OMD1 Add Out part of OMD2 Add Out part of OMD2 Drop In part of OMD1 Drop In IPD IPD 20-l Mux +IPDs 20-l Mux +IPDs 20-l Demux 20-l Demux Chans In Chans Out What an OADM looks like

  4. Variations • Objectives: • F( e ) : number of fibers on e ; • Minimization problems: • Minimize total fiber: min eF( e ) • Minimize maximum fiber per link: min maxeF( e ) • Maximization problems: • Given F( e )as input • Maximize throughput: RWA as many demands as possible subject to F( e )fibers per link e • Constraints: • Routes fixed: study wavelength assignment in isolation • Routes not fixed • Six variations • Min-SumFiber–FixedRoutes Max-Thruput-FixedRoutes • Min-MaxFiber–FixedRoutes • Min-SumFiber–ChooseRoutesMax-Thruput-ChooseRoutes • Min-MaxFiber–ChooseRoutes

  5. Approximation • Simple topologies • Optimally solvable on a line • NP hard for rings, stars, trees • General topologies: approximation algorithms • Optimal solution: OPT; • Ana-approx (a > 1)algorithmguarantees solution • At most aOPT for minimization problem; • At least OPT/a for maximization problem. • No a-approx : there exists a family of instances for which no algo can guarantee a approx.

  6. Inapprox ratio SumFiber MaxFiber ChooseRoute (log M)1/4-e( log log M )1-e FixedRoute Any constant (logm)1/2-e M : size of network m : no. of wavelengths per fiber Hardness Results • Hardness of minimization problems

  7. Buy-at-bulk Congestion minimization Chromatic number 3SAT(5), Raz verifier Inapprox ratio SumFiber MaxFiber ChooseRoute (log M)1/4-e( log log M )1-e FixedRoute Any constant (logm)1/2-e M : size of network m : no. of wavelengths per fiber Hardness Results • Hardness of minimization problems

  8. Inapprox ratio SumFiber MaxFiber ChooseRoute (log M)1/4-e( log log M )1-e FixedRoute Any constant (logm)1/2-e M : size of network m : no. of wavelengths per fiber Hardness Results • Hardness of minimization problems Wavelength assignment intrinsically hard

  9. Inapprox ratio SumFiber MaxFiber ChooseRoute (log M)1/4-e( log log M )1-e FixedRoute Any constant (logm)1/2-e M : size of network m : no. of wavelengths per fiber Hardness Results • Hardness of minimization problems • Hardness of maximization problems Inapprox ratio Throughput ChooseRoute (log M)1/2 FixedRoute M (1-e)/2

  10. Inapprox ratio SumFiber MaxFiber ChooseRoute (log M)1/4-e( log log M )1-e FixedRoute Any constant (logm)1/2-e M : size of network m : no. of wavelengths per fiber Hardness Results • Hardness of minimization problems • Hardness of maximization problems Inapprox ratio Throughput ChooseRoute (log M)1/2 FixedRoute M (1-e)/2 Edge-disjoint paths Chromatic number

  11. Inapprox ratio SumFiber MaxFiber ChooseRoute (log M)1/4-e( log log M )1-e FixedRoute Any constant (logm)1/2-e M : size of network m : no. of wavelengths per fiber Hardness Results • Hardness of minimization problems • Hardness of maximization problems Inapprox ratio Throughput ChooseRoute (log M)1/2 FixedRoute M (1-e)/2 New results in this paper

  12. 4 1 3 2 Hardness of Min-SumFiber-FixedRoute • Graph coloring • Input: graph G and c • Output: node color G s.t. no neighboring nodes share a common color • NP hard to decide if c colors sufficient. • NP-hardness reduction from graph coloring • Create an instance of Min-SumFiber-FixedRoute from graph coloring • Every node of G corresponds to a demand • Two nodes of G are neighbors iff two demand paths intersect • Fiber capacity m = c • G is c colorable iff 1 fiber per link suffices for all demands Graph coloring : G Min-SumFiber-FixedRoute 1 3 2 4

  13. Hardness of Min-SumFiber-FixedRoute • Strategy for showing inapproximability • Reduction from a hard problem X to create an instance Y • If X is “yes”, then Y has a low value < l • If X is “no”, then Y has a high value > h. • Y cannot be approximated within approx ratio h / l • Run any (h / l)-approx algo on Y • If solution > h, then X is “no” • If solution < h, then X is “yes”. • No constant approx for Min-SumFiber-FxiedRoute • Reduction from 3SAT • If f satisfiable, M total fibers sufficient; • If f not satisfiable, need more than cM fibers for any constant c • Use probabilistic checkable proofs.

  14. Transform P to a fiber instance I • Demands in P 1-1 corresponds to demands in I • Random mapping: demand routes sharing common link in P do not share common link in I Graph P that describes PCP for instance f

  15. Inapprox ratio SumFiber MaxFiber ChooseRoute (log M)1/4-e( log logM )1-e FixedRoute Any constant (logm)1/2-e approx ratio SumFiber MaxFiber ChooseRoute log Mlog M FixedRoute logmlogm M : size of network m : no. of wavelengths per fiber Approximation Results • Hardness of minimization problems • Approximation of minimization problems

  16. Inapprox ratio SumFiber MaxFiber ChooseRoute (log M)1/4-e( log logM )1-e FixedRoute Any constant (logm)1/2-e approx ratio SumFiber MaxFiber ChooseRoute log Mlog M FixedRoute logmlogm M : size of network m : no. of wavelengths per fiber Approximation Results • Hardness of minimization problems • Approximation of minimization problems Routing from buy-at-bulk Randomized wavelength assignment

  17. Heuristics for Wavelength Assignment • Greedy • Demands ordered in arbitrarily • For each demand choose a wavelength that increases fiber count least • Longest first • Like greedy; Demands with more hops dealt with first • Most congested first • Like greedy; Demands with congested routes dealt with first • Randomized assignment • Choose a wavelength [1, m ] uniformly at random for each demand; • O(log m ) approx • Optimal solution via integer programming

  18. Performance on 3 large carriers’ networks • Why not randomization? • Birthday paradox: • If load >m, some wavelength chosen twice with prob > ½ • If load = m, some wavelength chosen logm time whp.

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