1. 1 Slides by Yong Liu 1, Deep Medhi 2, and Michał Pióro 3 1 Polytechnic University, New York, USA 2 University of Missouri-Kansas City, USA 3 Warsaw University of Technology, Poland & Lund University, Sweden October 2007 Routing, Flow, and Capacity Design in Communication and Computer Networks Chapter 7: Networks with Shortest-Path Routing

2. 2 Outline  Shortest-path routing  MIP Formulation  Duality and Shortest-Path Routing  Heuristic Method for link weights  Examples  Extensions

3. 3 Shortest-path Routing  Take the shortest-path(s) from one point to the other  path length = summation of link weights  algorithm: Dijkstra, Bellman-Ford, extensions,  intra-domain routing: link state: OSPF, IS-IS  equal-cost multi-path split (ECMP)  Intra-domain Traffic Engineering  Good end-to-end performance for users  Efficient use of the network resources  Reliable system even in the presence of failures

4. 4 TE Optimization: The Problem  Intra-domain Traffic Engineering  Predict influence of weight changes on traffic flow  Minimize objective function (say, of link utilization)  Inputs  Networks topology: capacitated, directed graph  Routing configuration: routing weight for each link  Traffic matrix: offered load each pair of nodes  Outputs  Shortest path(s) for each node pair  Volume of traffic on each link in the graph  Value of the objective function 1 2 3 4 5 6 7

5. 5 Which link weight system to use  Link Weight can be  1, (hop count)  propagation delay (const.)  1/C (Cisco)  congestion delay (load sensitive, online update)  Objective dependent choice  hop count v.s. congestion delay  ECMP v.s. equal delay routing 1 2 3 4 100 80 1 2 3 4 100 80 100

6. 6 Shortest Path Routing: bounded link delay

7. 7 Penalty Function  use link penalty function to replace link constraints

8. 8 Shortest Path Routing: minimum average delay  load sensitive link delay   piece-wise linear approximation

9. 9 Shortest Path Routing: minimum average delay

10. 10 Minimization of Maximum Link Utilization

11. 11 MIP Formulation

12. 12 MIP Formulation

13. 13 Duality: Lagrangian Slides from Convex Optimization, Boyd & Vandenberghe

14. 14 Duality: dual function Slides from Convex Optimization, Boyd & Vandenberghe

15. 15 Dual Problem Slides from Convex Optimization, Boyd & Vandenberghe

16. 16 Duality Theorem  Weak Duality:  always hold (convex, non-convex problems)  find non-trivial lower bounds for complex problems  duality gap:  Strong Duality:  does not hold in general  hold for most convex problems, (including LP)  zero duality gap, obtain optimal solution for the original problem by solving the dual problem.  Advantages of working with Duals  less constraints  decoupling  distributed algorithms: distributed routing algorithms end system congestion control, TCP

17. 17 Duality: routing example AB h f 1 (x 1 ), x 1 f 2 (x 2 ), x 2,, h-x 1 -x 2  Lagrange dual function  Decoupling  minimal delay routing

18. 18 Duality: routing example AB h f 1 (x 1 * ), x 1 * f 2 (x 2 * ), x 2 *, *, h-x 1 * -x 2 * =0  Dual algorithm:  increase delay on virtual link if x 1 +x 2 <h, decrease delay otherwise  Dual Problem  Strong Duality

19. 19 Routing Duality: generalization  multi-demand/multi-path  routing duality  optimal flows only on shortest-paths!

20. 20 Duality and Shortest-path Routing

21. 21 Dual Formulation  Duality  optimal flows only on shortest-paths!

22. 22 Optimal Link Weights  use optimal multipliers as link weights  non-zero flows only on shortest paths  ECMP  Optimal Flow Allocation  good solution if most demand pairs only have one shortest path.

23. 23 Heuristic Methods  Weight Adjustment  iterative local search  increase weights for over-loaded links, decrease weights for under-loaded links  adjust weights for more balanced allocation  Simulated Annealing  random initial link weights  explore neighborhood: pick a random link, increase/decrease its weight by one  annealing: move to a worse weight setting with decreasing probability  Lagrangian Relaxation (LR)-Based Dual Approach  optimum Lagrange multipliers lead to optimal solution  given a set of multipliers, obtain link weights, and flow allocation  adjust multipliers according to link rates and link capacities

24. 24 Example: impact of different link weight systems  AT&T 90-node WorldNet IP Backbone  scaled up demand volumes average delay maximum link utilization

25. 25 Extensions  Uncapacitated Shortest-Path Routing  Optimizing link weights under transient failures  Selfish Routing and Optimal Routing  every user choose minimum delay path  Nash Equilibrium vs. Social Optimum AB f 1 (x 1 ), x 1 f 2 (x 2 ), x 2, f 3 (x 3 ), x 3, h

26. 26 Braess Paradox  adding a link increase user delay 1 2 3 4 x 1x 1 1 1 2 3 4 x 1x 1 1 delay=1.5 delay=2! 0