Optical clocks, present and future fundamental physics tests
Fundamental Complexity of Optical Systems
description
Transcript of Fundamental Complexity of Optical Systems
Fundamental Complexity of Optical Systems
Hadas Kogan, Isaac Hadas Kogan, Isaac KeslassyKeslassy
Technion (Israel)Technion (Israel)
Router – schematic representationRouter – schematic representation
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.
Electronicto optic
Electronicto optic
…
Lookup Switching
Optic to electronic
Optic to electronic
…
Buffering
Router
Power consumption per chassisPower consumption per chassis
0
2
4
6
8
10
12
14
16
1990 1993 1996 1999 2002 2003 2004
Po
wer
(kW
)
[Nick McKeown, Stanford]
There has to be some future alternative!
How about an optical router?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?
Optical router complexityOptical 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
Main contributionsMain 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.
OutlineOutline Background Control complexity (# switch
reconfigurations) Definition Bounds
Construction complexity (# switches) Definition Optimally constructed constructions
Two possible ways to “store” lightTwo 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
An optical memory cell:
(a) writing the packet
(b) circulating the packet
(c) reading the packet
How do we store light?How do we store light?
11 1(a) (b) (c)
We’ve presented a buffer capable of storing one optical packet.
A naive optical queue with buffer A naive optical queue with buffer BB
The number of 22 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).
1 1 1 1 1
What we want: an ideal routerWhat 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.
Input 1
Input N
… …
Output 1
Output N
What we want: an ideal routerWhat 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.
How do we do it in optics?How do we do it in optics?
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]).
Input 1
Input N
…
…
Output 1
Output N
O( BlnB)
PIFO
PIFOOQ
1B
1B1
12
2
33
Output 2
Output 3
Control complexityControl complexity
Generalization to systemsGeneralization 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.
DefinitionDefinition 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?
1
2
3
4
1
2
3
4
2
1
4
3
3
4
1
2
0.5
0.25
0.25
1234
1234
21433
412
Link to codingLink to coding
Source symbols:A1 – w.p. 0.5A2 – w.p. 0.25A3 – w.p. 0.25
A 2x2 switch A binary digitState entropy Source entropy
??? Minimizing expected code length
Coding results should apply also to switching!
CodingSwitching0.
5
0.25
0.25
1234
1234
21433
412
DefinitionsDefinitions 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
C2
C1
Example:Example: ActivePassive
Active
With coding:w.p 0.5 A1 ↔0w.p 0.25 A2↔10w.p 0.25 A3↔11
0.5
0.25
0.25
1234
1234
21433
412
C1=0
C1=1, C2=0
C1=1, C2=1
Definition – control complexityDefinition – 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
* min ( )i
i
T i tt T
C P t l
itl
it
Link to codingLink to coding
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
CodingSwitching
???Control complexity
0.5
0.25
0.25
1234
1234
21433
412
Lower boundLower boundTheorem: The control complexity is lower bounded by the entropy of the states:
Proof: Similar to the proof of expected codelength lower bound
*C H
C2
C1
* 1 1 1 3*1 *2 *22 4 4 2
C H
In the previous example:
0.5
0.25
0.25
1234
1234
21433
412
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.
An upper bound on the control An upper bound on the control complexitycomplexity
* 1C H
Construction complexityConstruction complexity
DefinitionDefinition 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].812345678
N
1 2 345 6 78
12345678
1
2
3
4
5
67
8
Construction complexityConstruction 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:
* 2log KC
T
Optimally-constructed Optimally-constructed constructionsconstructions
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:
* 2log ( !)( ln )
1
NC N N C
* 2log ( !)(ln )
NC N C
N [Jordan et. al., ‘94].
[Benes, ‘65].
Conclusion – construction Conclusion – construction complexity of optical routerscomplexity of optical routers
Input 1
Input N
… …
Output 1
Output N
NxN switch: Θ(Nln(N)) PIFO buffer of sizeB: Θ(ln(B))
B
The construction complexity of an OQ-PIFO switch is Θ(Nln(N))+Θ(Nln(B)) = Θ(Nln(NB))
Thank you!Thank you!