Managing Interconnect Resources Embedded SLIP Tutorial

1
Managing Interconnect ResourcesEmbedded SLIP
Tutorial

• Phillip Christie

2
Overview
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• Performance model
• Netlists and signatures
• Partitioning and placement
• Rent exponents
• What do you want to model today?

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Performance modelRISC/CISC
3
Instructions per second (M)IPS

Lower CPI ? more complex CPU ? internal
parallelism, branch
prediction, cache CISC CPI
? 3, RISC CPIlt1
10 reduction in CPI ? 20-40 increase in circuit
count
Larger circuits have longer cycle times
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Performance model Logic
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Pipelining
CL
CL
CL
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Performance model Electrical
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Material, geometry, placement, routing, metal
layer dependent
Reasonable assumption for logic depth gt 10
RintL1
CintL1
Cg
RintLav/(tpn-1)
Rg
Rg
RintL2
CintL2
Cg
(tpn-1) x CintLav
RintL3
CintL3
Cg
(tpn-1) X Cg
Material, geometry, placement, routing, metal
layer dependent
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Performance model Elmore delay
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Distributed elements
RintLav/(tpn -1)
Exact solution to problem in S-domain
Rg
But no known inverse Laplace Transform back to
time domain
First derivative
(tpn-1) x CintLav
?0-50
Voltage
(tpn -1) X Cg
Lumped elements
Time
?0-50 (tpn -1)RgCintLav (tpn -1) RgCg
0.5RintCintL2av RintCgLav
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Performance model Electrical optimization
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Cycle time (logic depth) x ?0-50
0.5RintCintL2max
Lav
3
10
Benchmark netlist
2
10
Lmax
Lmax
Number
1
10
0
10
0
1
2
10
10
10
Length
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Performance model Interconnect optimization
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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
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Performance model Predictive capability
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• 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?

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Netlists and signaturesformats
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Node list P1 N1 P2 N4 P3 N5 C0 N1 N2 C1 N1 N3 C2
N2 N3 N4 C3 N2 N3 N5
Net list N1 P1 C0 C1 N2 C0 C2 C3 N3 C1 C2 C3 N4
C2 P2 N5 P3C3
Net 2 has 3 terminals per net (tpn)
Cell 3 has 3 nets per cell (npc)
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Netlists and signaturesterminals per net (tpn)
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Netlists and signaturesNets per cell (npc)
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ppone, 25-Feb-2000, ltnpcgt3.6115
ibm01, 25-Feb-2000, ltnpcgt4.0237
250
4000
3500
200
3000
2500
150
Number of nets
Number of nets
2000
100
1500
1000
50
500
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
2
3
4
5
6
7
8
9
10
11
12
13
14
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Nets per cell (npc)
Nets per cell (npc)
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Netlists and signaturesTerminal counting
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Example netlist 29-Feb-2000
6
5
4
Number of terminals
3
2
1
0
1
2
3
4
5
6
7
Number of cells
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Netlists and signaturesRents rule
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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 lt ? lt1) which
indicates how self connected the netlist is
C
T
Or, if ?C, ?T are small compared with C and T
Which may be solved to yield
Statistically homogenous system
Where ltnpcgt is the average number of nets per
cell, and is generated as a constant of
integration
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Netlists and signaturesRent exponents
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ibm01, 28-Feb-2000, lt?gt 0.85
5
10
Number of terminals
Number of cells
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Netlists and signaturesSynthetic netlists
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• RMC (Random Mapped Circuit
• Darnauer and Dai
• Top-down recursive partitioning
• Allocation based on Rents rule

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

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

• Signature invariant mutants
• Brgles
• Generated my mutation of real circuits
• mutation maintains wiring signature invariance
• Rents rule observed

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

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

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Netlists and signaturesAutomatic netlist
generation-GNL
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• 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
  Rents rule.
• Constant ratio of internal to external
  connections at each level

Generates a guaranteed Rent exponent and a
realistic tpn distribution
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Netlists and signaturesparameter independence
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Recent paper shows lttpngt, ltnpcgt and lt?gt are not
independent
25
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Netlists and signaturesSummary
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• lttpngt characterizes net fan-out
• ltnpcgt characterizes cell fan-out
• lt?gt 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?

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Partitioning and placementSample calculation
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ppone
ibm01
Partition-based placement Lav 4.5
Partition-based placement Lav 9.8
Minimum length placement Lav 5.5
Minimum length placement Lav 6.2
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Partitioning and placementEstimation of length
distribution function N(l)
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-
-
C
B
C
C
TA?C

-
-
TAB
TBC
TB
TABC

B
B
B
C
C
B
B
A
B
B
C
C
Assumption net cannot connect A,B, and C
B
B
B
C
C
B
C
C
C
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Partitioning and placementConservation of
terminals
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We now convert from the number of terminals to
the number of nets using lttpngt
Assumptions all nets have lttpngt terminals per
net all cells have ltnpcgt
nets per cell all terminals
in net lie in region A or C
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Partitioning and placementEmbedding process
(infinite 2D plane)
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For cells placed in infinite 2D plane
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Partitioning and placementReality check
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(1) All cells have ltnpcgt nets per cell (2) All
nets have lttpngt 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 lttpngt
2, then nA?C n(l) and represents the number
of 2-terminal nets of length l associated with a
single cell
For lttpngt gt 2, n(l) is internally inconsistent
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Partitioning and placementProbability function
(infinite 2D plane)
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We note
And so we can write
Where r(l) is the probability that a cell has a
2-terminal net of length l.
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Partitioning and placementApproximate form for
r(l) (infinite 2D plane)
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By expanding individual terms in r(l) as binomial
series we observe the underlying form
r?(l)
Where K is determined by the requirement that
r(l)
And so we may write
Riemann zeta function
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Partitioning and placementSite densities and
occupancies (infinite 2D plane)
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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
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Partitioning and placementPlanar model A
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Finite system, CtotL2, 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
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Partitioning and placementPlanar model B
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L
Finite system, CtotL2, includes edge effects,
use q(l)
Assume q(l) retains functional form from infinite
plane but now use site density function Db(l) and
appropriate normalization
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Partitioning and placementPlanar model comparison
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Ctot 1024 lttpngt 2 ltnpcgt 4 lt?gt 0.66
Model A
Model A Lav 4.53 Model B Lav 2.27
Model B
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Partitioning and placementHierarchical model C
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L
h1
h2
L
h3
At level h there are 4(H-h) equivalent partitions
of side Lh2h
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Partitioning and placementRelationship between
Model B and C
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Db(l)
Dc(l,h)
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Partitioning and placementIntra-layer model C
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As before, within each level
where
Net distribution for system is given by sum over
hierarchies
Only remaining problem is to estimate number of
nets in each level, Nhtot
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Partitioning and placementInter-layer model C
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Number of nets in each layer may be determined by
another application of Rents rule. Consider
single partition at level h
One group of 4h cells generate
nets
nets
Four groups of 4h-1 cells generate
Total number of nets in level h partition is
Since there are 4H-h equivalent partitions
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Partitioning and placementHierarchical model D
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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
Net distribution for system is given by sum over
hierarchies
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Partitioning and placementModel D average wire
length
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Simpler mathematical form for Model D enables
rare analytical expressions
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Partitioning and placementHierarchical model
comparison
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Ctot 1024 lttpngt 2 ltnpcgt 4 lt?gt 0.66
Model D
Model C Lav 2.05 Model D Lav 5.14
Model C
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Partitioning and placementPlanar and
hierarchical model comparison
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Model A
Model D
Model B
Model C
Models B (planar) and C (hierarchical ) are
sometimes equivalent
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Rent exponentsTopology versus Geometry
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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
But how do we estimate the geometrical Rent
exponent?
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Rent exponentsWiring cell analysis
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Let us consider a simple two-level circuit,
optimized for placement
With reference to Nhtot from inter-layer model C
we note that
or
also
For the above example N1tot11, N2tot5 and lttpngt
2.0. Therefore pG 0.431 ltnpcgt122.0
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Rent exponentsDilation of wiring cell
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Two level system
H level system
? is constant if pG is constant
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Rent exponentsMonte Carlo sampling
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Therefore
and so
System defined if we know ? and ltnpcgt1
For the example wiring cell ?0.455 ltnpcgt11.375.
For a circuit of size Ctot106 (H9.966)
Known a priori
pG0.431 ltnpcgt 2.52
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Rent exponentsSampling applet
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Rent exponentsDilational filter
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In calculating the Rent exponent we are only
interested in details which are dilationally
invariant.
q1
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
Probability of connection between groups of four
cells at level 2 is
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Rent exponentsNon-linear functionality
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? and ltnpcgt expressed parametrically in terms of
q1
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Rent exponentsTheory versus experiment
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Majority rule Renormalization group
MBC algorithm 1024 cell netsists
MBC algorithm Monte Carlo Sampling
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What do you want to model today?Cycle time
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• Layer assignment
• Optimal repeater insertion
• Optimal power dissipation
• Effects of placement
• Effects of wiring signature

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What do you want to model today?Wiring Signatures
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What are sufficient parameters to characterize
netlists
lttpngt, ltnpcgt, and pT are not independent
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What do you want to model today?Universal
placement model
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What do you want to model today?Rent exponents
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What do you want to model today?Heterogeneous
systems
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Object oriented approach to system-on-a-chip
integration
Extremely difficult to predict interconnect
resources required to implement global wiring
between inhomogeneous system blocks
Global nets require different modeling techniques