Net ordering for optimal circuit timing in nanometer interconnect design
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Net-Ordering for Optimal Circuit Timing in Nanometer Interconnect Design. M. Sc. work by Moiseev Konstantin Supervisors: Dr. Shmuel Wimer, Dr. Avinoam Kolodny. R i-1. R i. R i+1. V cc. S i. S i+1. V cc. L. W i-1. W i. W i+1. C i-1. C i. C i+1. Problem formulation.

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Net-Ordering for Optimal Circuit Timing in Nanometer Interconnect Design

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Net ordering for optimal circuit timing in nanometer interconnect design

Net-Ordering for Optimal Circuit Timing in Nanometer Interconnect Design

M. Sc. work by Moiseev Konstantin

Supervisors: Dr. Shmuel Wimer, Dr. Avinoam Kolodny


Problem formulation

Ri-1

Ri

Ri+1

Vcc

Si

Si+1

Vcc

L

Wi-1

Wi

Wi+1

Ci-1

Ci

Ci+1

Problem formulation

  • Minimize bus timing by ordering of wires and allocation of wire widths and inter-wire spaces

    • Total width of interconnect structure is given constant A

    • All wires have equal length L

A


Motivation

Motivation

  • Cross-capacitances between wires in interconnect structures have a major effect on circuit timing

Wires 10 years ago – area capacitance was dominant

Wires today – cross capacitance is dominant


Motivation1

Weak driver

Case A

Case B

Strong driver

Capacitive load

Motivation

  • Relative order of wire drivers in a bus influences circuit timing

Circuit timing is better in case B !


Delay model

Delay model

  • Elmore approximation for delay together with - model equivalent circuit for wire

  • Miller factor assumed 1 for all wires

  • More general case will be discussed later

Q: Is Elmore delay model good enough for state-of-the-art technology?

A: Fitted Elmore Delay model gives up to 2% error in delay estimation


Objective functions

Objective functions

Total sum of delays:

Worst wire delay:

Worst wire slack:


Agenda

Agenda

  • Solution for total sum of delays objective function

  • Solution for worst delay objective function

  • Optimization of total sum of delays with cross talk

  • Delay uncertainty issue


Solution for total sum of delays case constant wire width

Solution for total sum of delays caseconstant wire width

  • Differentiating function with respect to and area constraint and equating derivatives to zero, obtain:

  • Now assume all wires have predefined constant width and get:

This property is preserved in all kinds of optimizations discussed


Solution for total sum of delays case constant wire width1

Solution for total sum of delays caseconstant wire width

  • Substitute obtained relations for spaces to objective function, simplify and obtain:

Order-independent part

Order-dependent part

Order of wires is influenced by values of driver resistances only !

Question: Does optimal order exist ???


Bmi order

BMI order

  • Take wires sorted in descending order of drivers and put alternately to the left and right sides of the bus channel

  • Obtained permutation of wires called Balanced Monotonic Interleaved(BMI) order

BMI order

7

6

5

4

3

2

1

BMI order provides best sharing of inter-wire spaces


Optimal order theorem

Optimal order theorem

  • Define , where - non-decreasing monotonic function and - some permutation of -values

  • Theorem (optimal order):given a bus whose wires are of uniform width , the BMI order of signals in the bus yields minimum total sum of delays.

    • Proof :

      • Order-dependent part of is special case of -sum

      • Prove by induction that -sums are minimized by BMI permutation


Impedance matching

Impedance matching

  • Balance the resistance of the driver and resistance of the driven line

  • Mathematically:

  • BMI still holds

  • Simple but practical case:


Solution for general case

Solution for general case

  • In general case, wire widths are optimization variables

  • Derivatives with respect to :

  • Theorem (existence): For given set of wires , if for each pair of wires and with drivers and loads and respectively holds and , then optimal order of this set of wires is BMI, under total sum of wire delays objective function.

  • One special case: if all load capacitances are equal, then optimal order is always BMI


Minimizing total sum of delays summary

Generate all permutations of wires

For each permutation solve sizing problem

Find permutation giving minimum delay

Complexity: exponential

Number of optimization variables:

Perform impedance matching by function with parameters (if needed)

Arrange wires in BMI order

Solve sizing problem

Complexity: linear

Number of optimization variables: or

Minimizing total sum of delays - summary

Straight forward solution :

Our heuristic:


Results total sum minimization problem demonstration on random problem instances

Results: total sum minimizationproblem demonstration on random problem instances

  • 20 sets of 5 wires

  • Rdr: [0.1 ÷ 2] KΩ

    (random)

  • Cl: [10 ÷ 200] fF

    (random)

  • Bus length: 600 μm

  • Bus width: 12 μm

  • Technology: 90 nm


Results total sum minimization bus width influence

Results: total sum minimizationbus width influence

  • Set of 6 wires

  • Rdr: [0.1 ÷ 2] KΩ

    (random)

  • Cl: 10 fF

  • Bus length: 600 μm

  • Technology: 90 nm


Results total sum minimization interleaved bus

Results: total sum minimizationinterleaved bus

  • Set of 7 wires

  • Rdr: 0.1KΩ and 1.9 KΩ

  • Cl: 50 fF and 5 fF

  • Bus length: 600 μm

  • Bus width: 15 μm

  • Technology: 90 nm


Results total sum minimization comparison of heuristics on random problem instances

Results: total sum minimizationcomparison of heuristics on random problem instances

Exhaustive search best delay

Exhaustive search worst delay

16.54%

0.63%

100%

0.76%

12.60%

0.63%

16.39%

0.07%

11.28%

9.60%

0.21%

14.10%

0.20%

1st heuristic

0.42%

14.10%

Average:

2nd heuristic


Agenda1

Agenda

  • Solution for total sum of delays objective function

  • Solution for worst delay objective function

  • Optimization of total sum of delays with cross talk

  • Delay uncertainty issue


Solution for minmax case

Solution for minmax case

  • The goal: minimizing maximum wire delay (or slack)

  • Function is not differentiable

  • All wires have the same delay (S. Michaely et al.)

  • Assumptions:

    • wire width is convex monotonic decreasing in driver resistance (impedance matching)

    • Drivers and loads satisfy existence theorem


Solution for minmax case1

Solution for minmax case

  • Supposition: In minimization of maximum wire delay, optimal order of wires is BMI

    • Under assumptions of previous slide delay expression can be written as:

  • Edge effects (S. Michaely et. al) can break down optimality of BMI


Results minmax optimization bus width influence

Results: minmax optimizationbus width influence

  • 20 sets of 5 wires

  • Rdr: [0.1 ÷ 2] KΩ

    (random)

  • Cl: [10 ÷ 200] fF

    (random)

  • Bus length: 600 μm

  • Technology: 90 nm


Results minmax optimization bus length influence

20 sets of 5 wires

Rdr: [0.1 ÷ 2] KΩ

(random)

Cl: [10 ÷ 200] fF

(random)

Bus width: 12 μm

Technology: 90 nm

Results: minmax optimizationbus length influence


Results minmax optimization interleaved bus

Results: minmax optimizationinterleaved bus

  • Set of 7 wires

  • Rdr: 0.1KΩ and 1.9 KΩ

  • Cl: 50 fF and 5 fF

  • Bus length: 600 μm

  • Bus width: 15 μm

  • Technology: 90 nm


Agenda2

Agenda

  • Solution for total sum of delays objective function

  • Solution for worst delay objective function

  • Optimization of total sum of delays with cross talk

  • Delay uncertainty issue


Crosstalk issue

Crosstalk issue

  • So far: we assumed Miller factor 1

  • In practice: can be 0, 1 or 2

  • Introducing Miller factor changes wire delay equation:

  • The solution will be different according to three cases:

    • Miller factor is equal for all pairs of wires

    • Miller factor different only near walls

    • Each pair of wires has its own different Miller factor


1 st case uniform miller factor

1st case: uniform Miller factor

  • The order-dependent part of objective function is given as:

  • When all Miller coefficients are equal, above expression changes to:

  • Conclusion:

    • Uniform Miller factor doesn’t affect functional form of delay function and therefore optimal order will be BMI

    • Impact of wire ordering emphasized even more


2 nd case almost uniform miller factor

2nd case: almost uniform Miller factor

  • All Miller coefficients in internal inter-wire spaces are equal to

  • Miller coefficients near the walls are

  • Order-dependent part of objective function can be written as:

  • BMI order remains optimal if

  • In other cases order is monotonic but not always BMI

  • Minmax optimization gives the same results


3 rd case non uniform miller factor

3rd case: non-uniform Miller factor

Miller coefficients can be presented by the matrix

Minimization problem then is equivalent to :

Where and

  • Proved to be NP-complete (A. Vittal et al.)


Agenda3

Agenda

  • Solution for total sum of delays objective function

  • Solution for worst delay objective function

  • Optimization of total sum of delays with cross talk

  • Delay uncertainty issue


Delay uncertainty issue

Peak noise

Delay uncertainty

Delay uncertainty issue

  • Due to difference in arrival times of signals transmitted by neighbor wires, crosstalk noise is created

  • Crosstalk noise is characterized by two main parameters: peak noise and delay uncertainty

  • Maximum delay uncertainty for a signal in a bus can be expressed as follows:

(A. Vittal et al.,

T. Sato et al.)


Minimization of delay uncertainty

Minimization of delay uncertainty

  • Define new objective functions:

    • Total sum of delay uncertainties:

    • Worst delay uncertainty:

    • Experiments show that BMI order leads to minimizing both and


Results minimization of delay uncertainty

Results: minimization of delay uncertainty

Total sum

  • 20 sets of 5 wires

  • Rdr: [0.1 ÷ 2] KΩ

    (random)

  • Cl: [10 ÷ 200] fF

    (random)

  • Bus length: 600 μm

  • Bus width: 15 μm

  • Technology: 90 nm

Minmax

  • Average improvement:

    • Total sum of delay uncertainties: about 27 %

    • Worst delay uncertainty:

      about 43%


Monotony in ordering optimizations

Monotony in ordering optimizations

  • Monotony is most important property of solutions of ordering optimization problems

    • Total sum of delays: optimal order is monotonic, BMI

    • Maximum delay: optimal order is monotonic, BMI

    • Optimization with crosstalk: optimal order is monotonic

    • Delay uncertainty optimization: optimal order is monotonic, BMI

  • Generally, all above problems can be solved on cyclic bus and obtained optimal order will be monotonic

  • BMI and other monotonic orders are special cases and defined by edge conditions only


Conclusions

Conclusions

  • Problem of optimal simultaneous wire sizing and ordering was presented and solved

  • Effects of crosstalk on nominal delays and delay uncertainty are examined

  • Monotonic ordering according to driver strength is shown to be advantageous for the various objective functions

  • Examples for 90-nanometer technology are analyzed and discussed


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