Algorithmic Issues in Optical Network Design Lisa Zhang Bell Labs April 19, 2007
Optical Core Networks • New technologies • DWDM – Dense wavelength division multiplex • All optical transmission: • Optical amplification over long distance • Signal add/drop within optical domain • Optical core • Extremely high capacity : terabit traffic • Extremely expensive equipment : $ 100,000,000 • Optimization is key • Service providers: efficiently utilize resources • Equipment manufacturer: seek competitive advantage in bidding contract • Researcher: opportunity to explore new model, new problems
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 The Design Problem • Input • A network representing fiber connectivity • A set of demands: • src, dest, amount • protection type: 1+1, 1+0 • Output • Routing: • 1+1 demand: 2 disjoint paths between src-dest • 1+0 demand: 1 path • Wavelength assignment: • Two demands sharing a common fiber must be carried on distinct wavelengths • Can deploy multiple fibers over same link if necessary. • Objective : cost minimization SF-NYC 10 1+1 BOS-LA 5 1+0 …
3C(e) 2C(e) C(e) m 2m 3m B O A m = 2 C The Design Problem - cont • Cost function • Economy of scale • C( e ) : cost of deploying one fiber on link e • m : fiber capacity • Study routing and wavelength assignment separately • Routing: buy-at-bulk network design with protection • Disjoint paths for 1+1demands: (Pi, Qi); single paths for 1+0 : Pi • L( e ) : load on link e • Min eL( e ) / mC( e ) • Wavelength assignment • Assign wavelength to each demand path • Avoid conflict by deploying multiple fibers
Problem 1: Routing Protected Buy-at-Bulk Network Design
3C(e) 2C(e) C(e) m 2m 3m Protected Buy-at-Bulk • Protected BAB : route demands wrt protection requirement • C( e ) : cost of deploying one fiber • m : fiber capacity • L( e ) : load on link e • Min eL( e ) / mC( e ) • NP-hard • Special case 2-connected graphs is NP hard • Approximation algorithms • Approx ratio: a • For every instance of the problem, approx algo returns solution of cost at most a times the optimal.
A Brief History of Buy-at-Bulk • Only UNPROTECTED buy-at-bulk has been studied • Single-source • All demands share a common source node • Constant approximation algorithms [Salman et al 97], [Andrews-Z 98], [Guha-Meyerson-Munagala 01], [Gupta-Kumar-Roughgarden 03] • Multi-commodity • O(log n) approximation [Awerbuch-Azar 97] • Based on tree metric embedding • More flexible techniques for more general cost functions [Charikar-Karagiozava 05], [Chekuri et al 06, 07] • Hardness • Approx ratio cannot be better than O( log ¼n ) [Andrews04]
First Approx for Protected BAB • Antonakopoulos-Chekuri-Shepherd-Z 2007 • Single source • O(1) approximation • Integrality gap of a natural linear program relaxation is O(1) • Idea: clustering • Multi-commodity • O(log3n) approximation • Idea: repeatedly solve single-source BAB with min density • Edge-protection vs node-protection • All results apply to node-protection, arbitrary demands • Focus on edge-protection, unit-demands
Single-Source Protected BAB • Step 1: Connectivity • H: 2-edge-connected subgraph • OPTH : optimal cost of H, • OPTSS-BAB: optimal cost of single-source BAB • 2-approximation for OPTH can be found • Want: use O(1) copies of H for solution for BAB • Step 2: Clustering • Partition H into clusters • Each cluster has a center for aggregating traffic OPTH ≤ OPTSS-BAB center dest
Step 3: routing • From dest nodes to center: each dest can route to its own center and another center on disjoint paths • O(m) load on intra-cluster and inter-cluster edges • O(1) copies of H suffice source center dest
Step 3: routing • From dest nodes to center: each dest can route to its own center and another center on disjoint paths • O(m) load on intra-cluster and inter-cluster edges • O(1) copies of H suffice • From centers to source: route aggregated demands to source along shortest disjoint paths • Center node of each cluster aggregates ≥ m demands • O(1) copies of H suffice • Sufficient capacity for disjoint paths from destinations to source. source center dest
From Single-Source to Multicommodity • Iterative greedy approach: • Find partial solution with small “density” • Density : cost of solution / # demands connected • Idea: reduction to single source BAB via “junction structure” • Junction structure J(u,r): subgraph that contains nodes within 2-shortest-path distance from u • Low density structure exists and can be approximated by O(1) approx algo of single-source BAB. • Remove demands connected by junction structure and recurse on remaining demands • a-approx of single source BAB implies O(a log3n)- approx for multi-commodity BAB.
Problem 2: Wavelength Assignment
B O A m = 2 C Wavelength Assignment • Constraints: • Assign wavelength to each routing path • Deploy multiple fibers per link to avoid wavelength conflict • Objective: • Nl ( e ) : number of times wavelength l used on link e. • F( e ) = max lNl ( e ) : number of fibers needed on e ; • C( e ) : cost of deploying one fiber • Minimize total cost min eC ( e ) F( e ) • Variants: • Minimize maximum fiber per link, min maxeF( e ) • Combine routing with min eC ( e ) F( e ) or min maxeF( e ) • Results for unprotected routing only
Results • Network is a line (WinklerZ 03) • Optimally solvable. • f ( e )fibers necessary and sufficient on every link e • m : fiber capacity • L( e ) : load on link e • f( e) = L( e ) / m • Rings, stars, trees • NP hard • 4-approx for trees: 4 f( e) fibers sufficient on e
Results • Hard to approx for arbitrary topologies (AndrewsZ 04,05,06) Inapprox ratio SumFiber MaxFiber ChooseRoute (log n)1/4-e( log log n )1-e FixedRoute Any constant (logm)1/2-e
Results • Hard to approx for arbitrary topologies (AndrewsZ 04,05,06) Inapprox ratio SumFiber MaxFiber ChooseRoute (log n)1/4-e( log log n )1-e FixedRoute Any constant (logm)1/2-e Buy-at-bulk Congestion minimization Chromatic number 3SAT(5), Raz verifier
Results • Hard to approx for arbitrary topologies (AndrewsZ 04,05,06) Inapprox ratio SumFiber MaxFiber ChooseRoute (log n)1/4-e( log log n )1-e FixedRoute Any constant (logm)1/2-e Wavelength assignment intrinsically hard
Results • Hard to approx for arbitrary topologies (AndrewsZ 04,05,06) • Logarithmic approx (AndrewsZ) Inapprox ratio SumFiber MaxFiber ChooseRoute (log n)1/4-e( log log n)1-e FixedRoute Any constant (logm)1/2-e Approx ratio SumFiber MaxFiber ChooseRoute O(log n)O(log n) FixedRoute O (logm) O (logm) Routing from buy-at-bulk Randomized wavelength assignment
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
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.
Future Work • Many open questions regarding protected BAB • More sophisticated edge cost • Inapproximability • Combined study with wavelength assignment • How do theoretical results impact practice? step function subadditive function