Fundamental Complexity of Optical Systems

1. Fundamental Complexity of Optical Systems Hadas Kogan, Isaac Keslassy Technion (Israel)

2. Lookup Switching Buffering Router – schematic representation Router Problem - electronic routers do not scale to optical speeds: • Access to electronic memory is slow and power consuming. • Data conversions are power consuming as well. Opticto electronic Electronic to optic … … Optic to electronic Electronic to optic

3. Power consumption per chassis There has to be some future alternative! [Nick McKeown, Stanford]

4. How about an optical router? • No electronic memory bottleneck • No O/E/O conversions BUT: An optical router is thought to be too complex. Is it?

5. Optical router complexity Objective: quantify the fundamental complexity of an optical router Two types of fundamental complexity: • Construction complexity: number of basic optical components needed (e.g., 2x2 optical switches) • Control complexity: frequency of optical switch reconfigurations

6. Main contributions • Define fundamental complexity in general optical constructions: • Control complexity • Construction complexity • Find lower and upper bounds on these costs. • Construct optical router with minimum complexity.

7. Outline • Background • Control complexity (# switch reconfigurations) • Definition • Bounds • Construction complexity (# switches) • Definition • Optimally constructed constructions

8. Two possible ways to “store” light • To slow/stop light. BUT: requires gas environments with tight temperature and pressure constraints, and currently seems impractical. • Use optical switches and fiber delay lines. . Buffer Buffer

9. How do we store light? An optical memory cell: (a) writing the packet (b) circulating the packet (c) reading the packet (a) (b) (c) 1 1 1 We’ve presented a buffer capable of storing one optical packet.

10. A naive optical queue with buffer B 1 1 1 1 1 • The number of 22 switches needed for the naive construction is B. • Could be less than B when several packets can share the same line (with different line lengths).

11. Input 1 Output 1 … … Input N Output N What we want: an ideal router • An output-queued push-in-first-out (OQ-PIFO) switch. • OQ - Arriving packets are placed immediately in the queue of size B at their destination output. • PIFO – packets departure ordering is according to their priority.

12. What we want: an ideal router • Why it is ideal: • OQ: Work conserving implies best throughput and minimal delay. • PIFO: Enables FIFO, strict priorities, WFQ… • But – up to N packets destined to the same output: • Speed-up for switch • Speed-up for queue • PIFO is hard to implement.

13. How do we do it in optics? PIFO 1 B OQ If packets are destined to different outputs: • Switching: optical switch NxN with O(NlnN) 2x2 optical switches ([Shannon ’49], [Benes ’67]). • Buffering: optical PIFO queue B 2x2 optical switches ([Sarwate & Anantharam ’04]). 1 Input 1 Output 1 2 Output 2 1 Output 3 … 3 … 3 Input N Output N 2 1 B PIFO

14. Control complexity

15. Generalization to systems • An optical system - a network element that has input links, output links and inner states, and is built with optical 2x2 switches and FDLs. • Inner states - the different settings of the system elements.External states – distinguishable possible system outputs.

16. 3 4 1 2 1 2 3 4 1 2 3 4 2 1 4 3 0.5 1 1 2 3 2 2 4 1 0.25 3 3 4 1 0.25 4 4 2 3 Definition • Control complexity – a measure of the minimal expected number of switch reconfigurations. Example: • 4 inputs, 4 outputs, 3 external states: What is the control complexity of an optical system with these states?

17. 1 2 3 4 1 2 3 4 2 1 4 3 3 4 1 2 0.5 0.25 0.25 Link to coding Coding Switching Source symbols: A1 – w.p. 0.5 A2 – w.p. 0.25 A3 – w.p. 0.25 A 2x2 switch A binary digit State entropy Source entropy ??? Minimizing expected code length Coding results should apply also to switching!

18. Definitions • A super switch: • Passive and active controls – for each state, a control is called passive if its value is irrelevant for setting that state. Otherwise, it is called active. C

19. Active Active Passive C1 3 4 1 2 2 1 4 3 1 2 3 4 1 2 3 4 0.5 0.25 0.25 C2 Example: C1=0 C1=1, C2=0 C1=1, C2=1 With coding: w.p 0.5 A1 ↔0 w.p 0.25A2↔10 w.p 0.25A3↔11

20. Definition – control complexity • Definition: the control complexity of an optical system is its minimal expected number of active controls, T – states space, - number of active controls per state

21. 1 2 3 4 1 2 3 4 2 1 4 3 3 4 1 2 0.5 0.25 0.25 Link to coding Coding Switching Source symbols: A1 – w.p. 0.5 A2 – w.p. 0.25 A3 – w.p. 0.25 A 2x2 switch A binary digit. States entropy Source entropy Minimized expected code length ??? Control complexity

22. C1 1 2 3 4 1 2 3 4 2 1 4 3 3 4 1 2 0.5 0.25 0.25 C2 Lower bound Theorem: The control complexity is lower bounded by the entropy of the states: Proof: Similar to the proof of expected code length lower bound In the previous example:

23. An upper bound on the control complexity Theorem: The control complexity is upper bounded as follows: Stages of proof: • Generate Huffman coding (expected code length ≤ H+1) . • There exists a construction (using multiplexers and distributers) of a memoryless system such that the active controls for each state are the Huffman coding of that state • A system with memory can be composed from a memoryless system using a time-space transformation.

24. Construction complexity

25. 8 7 6 5 4 3 2 1 5 1 4 2 8 3 6 7 N 7 1 6 2 3 3 8 4 2 5 4 6 1 7 5 8 Definition • Construction complexity: the minimal possible number of 2x2 switches in the construction. • Examples: • An NxN switch: N! states, O(NlnN) switches [Shannon, ‘49], [Benes, ‘65]. • A Time Slot Interchange (TSI) with time frame N: N! states - O(lnN) switches [Jordan et. al., ‘94].

26. Construction complexity • Intuition: With C 2x2 switches during T time slots, the possible number of resulting states K is upper bounded by 2CT. • Therefore: to get K states in state duration T, a lower bound on the construction complexity is given by:

27. Optimally-constructed constructions • A construction algorithm is optimally constructed if its number of 2x2 switches is equal in growth to the construction complexity. • Examples: • An NxN switch: • A TSI: [Benes, ‘65]. [Jordan et. al., ‘94].

28. Input 1 Output 1 … … Input N Output N Conclusion – construction complexity of optical routers B The construction complexity of an OQ-PIFO switch is Θ(Nln(N))+Θ(Nln(B)) = Θ(Nln(NB)) NxN switch: Θ(Nln(N)) PIFO buffer of sizeB: Θ(ln(B))

29. Thank you!