1 / 26

Scheduling Crossbar Switches

Learn about Ford-Fulkerson method for maximizing network flows in crossbar switches, including residual graph concepts and complexity analysis. Discover how to find maximum size matchings in bipartite graphs using network flows.

hlyons
Download Presentation

Scheduling Crossbar Switches

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Scheduling Crossbar Switches Who do we chose to traverse the switch in the next time slot? 1 1 N N

  2. History • [Karol et al. 1987] Throughput limited to by head-of-line blocking for Bernoulli IID uniform traffic. • [Tamir 1989] Observed that with “Virtual Output Queues” (VOQs) Head-of-Line blocking is reduced and throughput goes up.

  3. Basic Switch Model S(n) L11(n) A11(n) 1 1 D1(n) A1(n) A1N(n) AN1(n) DN(n) AN(n) N N ANN(n) LNN(n)

  4. Some definitions 3. Queue occupancies: Occupancy L11(n) LNN(n)

  5. A 1 2 B 3 C 4 D 5 E 6 F Finding a maximum size match • How do we find the maximum size (weight) match?

  6. Network flows and bipartite matching A 1 2 B Finding a maximum size bipartite matching is equivalent to solving a network flow problem with capacities and flows of size 1. Sink t Source s 3 C 4 D 5 E 6 F

  7. 10 10 10 1 10 1 10 1 Network Flows a c Source s Sink t b d • Let G = [V,E] be a directed graph with capacity cap(v,w) on edge [v,w]. • A flow is an (integer) function, f, that is chosen for each edge so that • We wish to maximize the flow allocation.

  8. 10 10 10 1 10 1 10 1 a c 10, 10 Source s Sink t 10, 10 1 10, 10 10 1 10 b d 1 Flow is of size 10 A maximum network flow exampleBy inspection a c Source s Sink t b d Step 1:

  9. Not obvious Maximum flow: a c 10, 9 Source s Sink t 10, 10 1,1 10, 10 1,1 10, 2 b d 10, 2 1, 1 Flow is of size 10+2 = 12 A maximum network flow example Step 2: a c 10, 10 Source s Sink t 10, 10 1 10, 10 1 10, 1 b d 10, 1 1, 1 Flow is of size 10+1 = 11

  10. Ford-Fulkerson method of augmenting paths • Set f(v,w) = -f(w,v) on all edges. • Define a Residual Graph, R, in which res(v,w) = cap(v,w) – f(v,w) • Find paths from s to t for which there is positive residue. • Increase the flow along the paths to augment them by the minimum residue along the path. • Keep augmenting paths until there are no more to augment.

  11. Example of Residual Graph a c 10, 10 10, 10 1 10, 10 s t 10 1 10 b d 1 Flow is of size 10 Residual Graph, R res(v,w) = cap(v,w) – f(v,w) a c 10 10 10 1 s t 10 1 10 b d 1 Augmenting path

  12. Example of Residual Graph Step 2: a c 10, 10 s t 10, 10 1 10, 10 1 10, 1 b d 10, 1 1, 1 Flow is of size 10+1 = 11 Residual Graph a c 10 s t 10 10 1 1 1 1 b d 9 1 9

  13. Complexity of network flow problems • In general, it is possible to find a solution by considering at most VE paths, by picking shortest augmenting path first. • There are many variations, such as picking most augmenting path first. • Best Algorithm based on work by Dinic.

  14. A 1 2 B 3 C 4 D 5 E 6 F Finding a maximum size match • How do we find the maximum size (weight) match?

  15. Network flows and bipartite matching A 1 2 B Finding a maximum size bipartite matching is equivalent to solving a network flow problem with capacities and flows of size 1. Sink t Source s 3 C 4 D 5 E 6 F

  16. Network flows and bipartite matchingFord-Fulkerson method Residual Graph for first three paths: A 1 2 B t s 3 C 4 D 5 E 6 F

  17. Network flows and bipartite matching Residual Graph for next two paths: A 1 2 B t s 3 C 4 D 5 E 6 F

  18. Network flows and bipartite matching Residual Graph for augmenting path: A 1 2 B t s 3 C 4 D 5 E 6 F

  19. Network flows and bipartite matching Residual Graph for last augmenting path: A 1 2 B t s 3 C 4 D 5 E 6 F Note that the path augments the match: no input and output is removed from the match during the augmenting step.

  20. Network flows and bipartite matching Maximum flow graph: A 1 2 B t s 3 C 4 D 5 E 6 F

  21. Network flows and bipartite matching Maximum Size Matching: A 1 2 B 3 C 4 D 5 E 6 F

  22. Complexity of Maximum Matchings • Maximum SizeMatchings: • Algorithm by Dinic O(N5/2) • Maximum Weight Matchings • Algorithm by Kuhn O(N3) • In general: • Hard to implement in hardware • Slooooow.

  23. Outline • Recap of LQF and proof of stability. • OCF: Weight equals the age of a cell. • Why do we need a weight to stabilize the switch for non-uniform arrivals? • Finding a maximum match. • Maximum network flow problems • Maximum size/weight matchings as examples of maximum network flows • LPF: Another algorithm.

  24. Another algorithm (LPF)

  25. LPF

  26. Properties of LPF • LPF is stable for all admissible Bernoulli traffic. • An LPF match is a maximum size match. • How can this be stable?

More Related