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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

M. Sc. work by Moiseev Konstantin

Supervisors: Dr. Shmuel Wimer, Dr. Avinoam Kolodny

Ri-1

Ri

Ri+1

Vcc

Si

Si+1

Vcc

L

Wi-1

Wi

Wi+1

Ci-1

Ci

Ci+1

- 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

- 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

Weak driver

Case A

Case B

Strong driver

Capacitive load

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

Circuit timing is better in case B !

- 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

Total sum of delays:

Worst wire delay:

Worst wire slack:

- 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

- 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

- 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 ???

- 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

- 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

- Proof :

- Balance the resistance of the driver and resistance of the driven line
- Mathematically:
- BMI still holds
- Simple but practical 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

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

Straight forward solution :

Our heuristic:

- 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

- Set of 6 wires
- Rdr: [0.1 ÷ 2] KΩ
(random)

- Cl: 10 fF
- Bus length: 600 μm
- Technology: 90 nm

- 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

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

- 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

- 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

- 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

- 20 sets of 5 wires
- Rdr: [0.1 ÷ 2] KΩ
(random)

- Cl: [10 ÷ 200] fF
(random)

- Bus length: 600 μm
- Technology: 90 nm

20 sets of 5 wires

Rdr: [0.1 ÷ 2] KΩ

(random)

Cl: [10 ÷ 200] fF

(random)

Bus width: 12 μm

Technology: 90 nm

- 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

- 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

- 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

- 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

- 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

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.)

- 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

Peak noise

Delay uncertainty

- 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.)

- Define new objective functions:
- Total sum of delay uncertainties:
- Worst delay uncertainty:
- Experiments show that BMI order leads to minimizing both and

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 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

- 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