Managing Interconnect ResourcesEmbedded SLIP Tutorial

Phillip Christie

Overview

• Performance model• Netlists and signatures• Partitioning and placement• Rent exponents• What do you want to model today?

2

Performance modelRISC/CISC

Cycle time (CT) x Cycles per instruction (CPI)

1Instructions per second (M)IPS

=

Lower CPI more complex CPU internal parallelism,

branch prediction, cacheCISC CPI 3, RISC CPI<1

10% reduction in CPI 20-40% increase in circuit count

Larger circuits have longer cycle times

3

Performance model Logic

CL

CL CL CL

Pipelining CL

CL

CL

CL

Parallelism

Logic Depth

4

Performance model Electrical

RintL1

RintL2

RintL3

Cg

Rg

Cg

Cg

CintL1

CintL2

CintL3

Rg

RintLav/(tpn-1)

(tpn-1) x CintLav

(tpn-1) X Cg

Fan-out = (terminals per net) - 1

= tpn - 1

Material, geometry, placement, routing, metal layer dependent

Material, geometry, placement, routing, metal layer dependent

Reasonable assumption for logic depth > 10

5

Performance model Elmore delay

Rg

RintLav/(tpn -1)

(tpn-1) x CintLav

(tpn -1) X Cg

0-50% = (tpn -1)RgCintLav + (tpn -1) RgCg + 0.5RintCintL2av + RintCgLav

Lumped elements

Distributed elementsExact solution to problem in S-domain

But no known inverse Laplace Transform back to time domain

Volt

age

Time

First derivative

0-50%

6

Performance model Electrical optimization

Cycle time = (logic depth) x 0-50% + 0.5RintCintL2max

Lmax

Num

ber

100

101

10210

0

101

102

103

Length

Lav

Lmax

Benchmark netlist

7

Performance model Interconnect optimization

Basic cycle time models provide insight into the complex interactions which determine cycle time.

Modelling process can also be used to optimise power dissipation in the interconnect

8

Performance model Predictive capability

• How do we know if benchmark is good?• Is geometry optimization sensitive to

netlist signature?• What if layout tools change?• What if we wish to analyse performance

of a netlist that does not yet exist?

9

Netlists and signaturesformats

Node list

P1 N1P2 N4P3 N5C0 N1 N2C1 N1 N3C2 N2 N3 N4C3 N2 N3 N5

P1C3

C1

C0

C2

P2

P3

N1

N2

N3

N4

N5

Net list

N1 P1 C0 C1N2 C0 C2 C3N3 C1 C2 C3N4 C2 P2N5 P3C3

Net 2 has 3 terminals per net (tpn)

Cell 3 has 3 nets per cell (npc)

10

Netlists and signaturesterminals per net (tpn)

0 5 10 15 20 25 300

50

100

150

200

250

300

Terminals per net (tpn)

Nu

mber

of

nets

ppone, 25-Feb-2000, <tpn>=2.8686

0 5 10 15 20 250

1000

2000

3000

4000

5000

6000

7000

8000

9000

Terminals per net (tpn)

Nu

mb

er

of

nets

ibm01, 25-Feb-2000, <tpn>=3.5834

11

Netlists and signaturesNets per cell (npc)

1 2 3 4 5 6 7 8 9 1011121314150

50

100

150

200

250

Nets per cell (npc)

Nu

mber

of

nets

ppone, 25-Feb-2000, <npc>=3.6115

1 2 3 4 5 6 7 8 9 1011121314150

500

1000

1500

2000

2500

3000

3500

4000

Nets per cell (npc)

Nu

mber

of

nets

ibm01, 25-Feb-2000, <npc>=4.0237

12

Netlists and signaturesTerminal counting

1 2 3 4 5 6 70

1

2

3

4

5

6

Number of cells

Num

ber

of

term

inals

Example netlist 29-Feb-2000

13

Netlists and signaturesRent’s rule

C

T

CC

TT

If an additional C cells are added, what is the increase in terminals T? In the absence of any other information we might guess that

But this is an overestimate since many of these T terminals may already connect into larger red structure and so do not contribute to the total.

We introduce a factor (0 < <1) which indicates how self connected the netlist is

CC

TT

C

dC

T

dT

Or, if C, T are small compared with C and T

Which may be solved to yieldCnpcT

Where <npc> is the average number of nets per cell, and is generated as a constant of integration

Statistically homogenous system

T

C

14

Netlists and signaturesRent exponents

100

101

102

10310

0

101

102

103

Number of cells

Num

ber

of

term

inals

ppone, 28-Feb-2000, <> =0.72

100

101

102

103

10410

0

101

102

103

104

105

Number of cellsN

um

ber

of

term

inals

ibm01, 28-Feb-2000, <> =0.85

15

Netlists and signaturesSynthetic netlists

RMC (Random Mapped CircuitDarnauer and Dai• Top-down recursive partitioning• Allocation based on Rent’s rule

GNL (Generate NetList)Stroobandt, Depreitere, and van Campenhout• Bottom-up clustering approach• allocation based on Rent’s rule• Sequential circuits possible

CIRC and GENHutton, Rose, Grossman, and Corneil• CIRC is an parameter profiler used as input for GEN• Sequential circuits generated by gluing combinational circuits• Not Rent-based

PartGenPistorius, Legai, and Monoux• Two-level hierarchical netlist generator• first level selects from 4 standard circuits• second level generates controller logic

Signature invariant mutantsBrgles• Generated my mutation of real circuits• mutation maintains wiring signature invariance• Rent’s rule observed

Random transformationsIwama, Hino, Kurokawa, and Sawada• Starts with fixed input NAND gates• Uses set of 12 transformations to generate any k-NAND functionally equivalent circuit

16

Netlists and signaturesAutomatic netlist generation-GNL

• Number of logic blocks and number of inputs/outputs specified by user

• Logic blocks are paired and (pseudo)-random connections made between blocks as determined by Rent’s rule.

• Constant ratio of internal to external connections at each level

Generates a guaranteed Rent exponent and a realistic tpn distribution

17

Netlists and signaturesparameter independence

0 5 10 15 20 250

50

100

150

200

250

300

Terminals per net

Nu

mb

er

of

nets

ppone, <npc> =3.6115, <> =0.72

25

0 5 10 15 200

1000

2000

3000

4000

5000

6000

7000

8000

9000

Terminals per net

Nu

mb

er

of

nets

ibm01, <npc> =4.0237, <> =0.85

mmmCnpcmtpn total111)(

Recent paper shows <tpn>, <npc> and <> are not independent

18

Netlists and signaturesSummary

• <tpn> characterizes net fan-out• <npc> characterizes cell fan-out• <> is the Rent exponent whose meaning is open

for discussion.• These parameters may not be independent• What happens when we embed the netlist into a

two-dimensional surface?

19

Partitioning and placementSample calculation

100

101

102

100

101

102

103

Length

Num

ber

100

101

102

103

100

101

102

103

104

Length

Num

ber

of n

ets

ibm01ppone

Partition-based placement

Lav= 4.5

Minimum length placement

Lav= 5.5

Partition-based placement

Lav= 9.8

Minimum length placement

Lav= 6.2

20

Partitioning and placementEstimation of length distribution function N(l)

B

B

BB

B B

B

A B B

BB

B

C

C

C

C

C

C

C

C

C

C

C

C

C

BA

C

BA

C

BA

C

BA

C

BA= + - -

TAC TAB TBC TB TABC= + - -

Assumption: net cannot connect A,B, and C

BAB CnpcT 1

CBBC CCnpcT

BB CnpcT

CBABC CCnpcT 1

21

Partitioning and placementConservation of terminals

CBBCBBCA CCCCCCnpcT 11

We now convert from the number of terminals to the number of nets using <tpn>

tpnTn CACA

Assumptions: all nets have <tpn> terminals per net all cells have <npc> nets per cell all terminals in net lie in region A or C

CBBCBBCA CCCCCC

tpn

npcn 11

22

Partitioning and placementEmbedding process (infinite 2D plane)

B

B

BB

B B

B

A B B

BB

B

C

C

C

C

C

C

C

C

C

C

C

C

For cells placed in infinite 2D plane

lCC 4

)1(241

1

lllCl

lB

ppppCA llllllllll

tpn

npcn 4)1(21)1(24)1(2)1(21

23

Partitioning and placementReality check

(1) All cells have <npc> nets per cell(2) All nets have <tpn> terminals per net (3) Net cannot connect A,B, and C(4) All terminals of net lie in region A or C

(2) is only consistent with (3) and (4) if <tpn> = 2, then nAC = n(l) and represents the number of 2-terminal nets of length l associated with a single cell

lllllllllltpn

npcln 4)1(21124)1(2)1(21)(

For <tpn> > 2, n(l) is internally inconsistent

24

Partitioning and placementProbability function (infinite 2D plane)

)()( lrtpn

npcln

We note

1

14)1(21124)1(2)1(21l

llllllllll

And so we can write

Where r(l) is the probability that a cell has a 2-terminal net of length l.

25

Partitioning and placementApproximate form for r(l) (infinite 2D plane)

By expanding individual terms in r(l) as binomial series we observe the underlying form

23)( lKlr

Where K is determined by the requirement that

1

231l

lK

And so we may write

231

K

Riemann zeta function

100

101

10-3

10-2

10-1

100

Length (gate pitches)

Pro

babi

lity

r(l)

r(l)

26

Partitioning and placementSite densities and occupancies (infinite 2D

plane)

llqlr 4)()(

In this context r(l) is interpreted as the probability that a cell has a net of length l. We factor it into two parts

where 4l is the number of available wire sites per cell of length l and q(l) is the expectation number of nets occupying that site. Since q(l) can never be greater than 1, it may also be interpreted as an occupation probability

lllllllllll

lq 4)1(21124)1(2)1(2141

)(

27

Partitioning and placementPlanar model A

L

L

Finite system, Ctot=L2, no edges, approximate form for q(l)

Assume q(l) retains functional form from infinite plane but now use site density function for finite cyclic system and appropriate normalization

)()( lrNlN tot

tottottot CC

tpn

npcN

22)( lLlDa

)()()( lqlDKlr a

L

l

lr2

1

)(1

28

Partitioning and placementPlanar model B

L

L

else0

2for 312212

1for 361

)(

2

LlLlLlLlL

LllLLll

lDb

Finite system, Ctot=L2, includes edge effects, use q(l)

)()( lrNlN tot

tottottot CC

tpn

npcN

Assume q(l) retains functional form from infinite plane but now use site density function Db(l) and appropriate normalization

)()()( lqlDKlr a

L

l

lr2

1

)(1

29

Partitioning and placementPlanar model comparison

100

101

102

100

101

102

103

104

Length

Num

ber

of n

ets

Model A

Model B

Ctot = 1024<tpn> = 2<npc> = 4<> = 0.66

Model A: Lav = 4.53Model B: Lav = 2.27

30

Partitioning and placementHierarchical model C

L

L

else0

2for 312212

2for 3181514

21for 12

)(22

LlLlLlLlL

LlLlLLlllLL

LllLll

lDc

At level h there are 4(H-h) equivalent partitions of side Lh=2h

h=1

h=2

h=3

31

Partitioning and placementRelationship between Model B and C

0 10 20 30 40 50 60 700

0.5

1

1.5

2x 10

4

Length

Nu

mb

er

H

hc

hHb hlDlD

1

),(4)(Dc(l,h)

Db(l)

32

Partitioning and placementIntra-layer model C

),(),( hlrNhlNtoth

)(),(),( lqhlDKhlr c

hL

l

hlr2

1

),(1

As before, within each level

where

Only remaining problem is to estimate number of nets in each level, Nhtot

Net distribution for system is given by sum over hierarchies

H

h

hlNlN1

),()(

33

Partitioning and placementInter-layer model C

Number of nets in each layer may be determined by another application of Rent’s rule. Consider single partition at level h

4h-1 cells 144 h

tpn

npc

h

tpn

npc4

Four groups of 4h-1 cells generate

One group of 4h cells generate

hh

tpn

npc

tpn

npc444 1

Total number of nets in level h partition is

Since there are 4H-h equivalent partitions

hhhH

h tpn

npc

tpn

npcN

tot4444 1

nets

nets

34

Partitioning and placementHierarchical model D

Essentially same as Model C but with no intra-layer optimization. Then site occupancy probability is independent of length and equal to a constant, set q(l)=K, which is determined by normalization.

As before, within each level

where

),(),( hlrNhlNtoth

hL

l

hlr2

1

),(1),(),( hlDKhlr c

Net distribution for system is given by sum over hierarchies

H

h

hlNlN1

),()(

35

Partitioning and placementModel D average wire length

Simpler mathematical form for Model D enables rare analytical expressions

hhL

lavh

h

hlrll

294

297

),(2

1

14

1

14

17

1

4192

1

5.1

5.1

5.0

5.0

1

1

1

tottot

tot

H

hhav

totav

CC

C

NlN

ltot

36

Partitioning and placementHierarchical model comparison

100

101

102

100

101

102

103

104

Length

Num

ber

Model D

Model C

Ctot = 1024<tpn> = 2<npc> = 4<> = 0.66

Model C: Lav = 2.05Model D: Lav = 5.14

37

Partitioning and placementPlanar and hierarchical model comparison

100

101

102

100

101

102

103

104

LengthN

umbe

r

Model D

Model C

100

101

102

100

101

102

103

104

Length

Num

ber

of n

ets

Model A

Model B

Models B (planar) and C (hierarchical ) are sometimes equivalent

38

Rent exponentsTopology versus Geometry

mmmCnpcmtpn TT pptotal

111)(

The first use of the Rent exponent was to estimate the distribution of m-pin nets

Topological Rent exponent, now written as pT

Topological Rent exponent inappropriate. Define geometrical Rent exponent pG . Measure of placement optimization, not an intrinsic netlist property

GGGG pppp lllllllllll

lq 4)1(21124)1(2)1(2141

)(

But how do we estimate the geometrical Rent exponent?

39

Rent exponentsWiring cell analysis

Let us consider a simple two-level circuit, optimized for placement

With reference to Nhtot from inter-layer model C we note that

For the above example N1tot=11, N2tot=5 and <tpn> =2.0. Therefore

pG =0.431 <npc>1+2=2.0

pG 1 4log

npc tpnN N tpn

Ntot tot

tothh

1 2

1 2

21

2

16 4

or

also

40

G

tot

tot p

h

h

N

N 11 4

Rent exponentsDilation of wiring cell

1 21

npctpn

Nhh

tot1 2 21

2

4

pG 1 4log pG 1 4log

H level systemTwo level system

is constant if pG is constant

41

H

hh

hhHH tottot

NNtpn

npcnpc3

2

11 4

Rent exponentsMonte Carlo sampling

ThereforeN N N N

tot tot tot tot2 1 32

1 , , etc.

npctpn N

npctot

Hh

h

H H

1 1

114

1

1

For the example wiring cell =0.455 <npc>1=1.375. For a circuit of size Ctot=106 (H=9.966)

pG=0.431 <npc> =2.52

and so

System defined if we know and <npc>1

Known a priori

42

Rent exponentsSampling applet

43

Rent exponentsDilational filter

In calculating the Rent exponent we are only interested in details which are dilationally invariant.

Let probability that a single cell is connected to another cell at lowest level be q1

Probability of there being a majority of nets within group of four cells is

q1

Probability of connection between groups of four cells at level 2 is

44

21

411

51

61 11516 qqqqq

221

411

51

6112 11516 qqqqqqq

Rent exponentsNon-linear functionality

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

q1

q 2

N

N

D

D

q

q

q

q

q

q

tot

tot

tot

tot

2

1

2

1

2

1

2

1

1

1

4 4

npc tpn q1 1

6

4

and <npc> expressed parametrically in terms of q1

45

Rent exponentsTheory versus experiment

MBC algorithm1024 cell netsists

MBC algorithmMonte Carlo Sampling

Majority ruleRenormalization group

46

What do you want to model today?Cycle time

47

Num

ber

100

101

10210

0

101

102

103

Length

Lav

Lmax

Universal placementmodel •Layer assignment

•Optimal repeater insertion•Optimal power dissipation•Effects of placement•Effects of wiring signature

What do you want to model today?

Wiring Signatures

0 5 10 15 20 250

50

100

150

200

250

300

Terminals per net

Nu

mb

er

of

nets

ppone, <npc> =3.6115, <> =0.72

mmmCnpcmtpn total111)(

What are sufficient parameters to characterize netlists

<tpn>, <npc>, and pT are not independent

48

What do you want to model today?

Universal placement model

GG

tot

phphhHh tpn

npc

tpn

npcN 4444 1

H

hch lqhlDNKlN

tot1

)(),()(

GGGG pppp lllllllllll

lq 4)1(21124)1(2)1(2141

)(

B Model 'GG pp

C Model 'GG pp

D Model )(1 KqpG

49

What do you want to model today?

Rent exponents

50

Object oriented approach to system-on-a-chip integration

Extremely difficult to predict interconnect resources required to implement global wiring between inhomogeneous system blocks

What do you want to model today?

Heterogeneous systems

51

Global nets require different modeling techniques