An Efficient Surface-Based Low-Power Buffer Insertion Algorithm

1
An Efficient Surface-Based Low-Power Buffer
Insertion Algorithm

• Rajeev R. Rao, David Blaauw, Dennis Sylvester,
  Charles Alpert, Sani Nassif
• Department of EECS, University of Michigan, Ann
  Arbor, MI
• IBM Austin Research Laboratory, Austin, TX
• rrrao, blaauw, dennis_at_eecs.umich.edu, alpert,
  nassif_at_us.ibm.com

2
Interconnect Trends

• Interconnect power a major issue
• Huge power consumption in both global and local
  signal nets
• Repeater counts increasing drastically
• IBM 50 of leakage in inverters/buffers
• Assuming continuation of current design styles,
  dramatic projections for the 32nm technology node
• 70 of cell count repeaters
• 65-80 of dynamic power due to interconnects
• Leakage increasing exponentially
• Require Optimal repeater usage with the
  objective of total power minimization

3
Outline

• Introduction
• Delay and Buffer models
• Previous Work
• Proposed Algorithm
• Library characterization
• Generation of different types of candidates
• Merging, Propagation, Snapping
• Results
• Conclusion

4
Introduction

• Wire RC delay is quadratic function of wire
  length
• Segmenting wires decreases delay
• Same idea applicable for interconnect tree
  structures
• Buffers inserted for delay management
• Additional benefit Buffers/Inverters decouple
  large output loads

1
1
Repeater
Receiver
Driver
Wire Length 2, Wire Delay ? (1)2(1)2 2
5
Elmore Delay model

• Represent interconnect tree with a lumped RC
  model
• Assume binary tree topology is fixed with an
  initial Steiner tree estimation
• n vertices (branch points) and (n-1) edges (ie.,
  wires)
• For a wire e connecting vertices (u, v) the
  Elmore delay is
• where T(v) is the maximal subtree rooted at v
  that does not contain buffers
• The total delay from a vertex v to a sink node si
  is

6
Buffer model

• Linear gate delay model used for the buffers
• Assumption Delay is a linear function of output
  capacitance
• Isolation Property Buffer devices decouple
  downstream output loads from the parent trees
• Assumption Miller effect (bootstrapping) due
  to Cgd is negligible

Dbuffer Dintrinsic-delay Rintrinsic-resistance
Coutput-load
v
Cload
7
Buffer Insertion Problem
BufLib b1 b2 b3

• Timing Metrics
• Required Arrival Time (RAT)
• Each sink specified a given RAT(si) value and
  source is fixed as RAT(so)0
• Delay minimization ? Maximize slack at source
  q(so)
• Subtree Delay (SD)
• SD(si) RATmax(si) RAT(si)
• Delay minimization ? Minimize SD(so)
• Advantage Unlike RAT, equations using SD are
  additive
• Our approach
• Tradeoff surfaces in 3D space of delay,
  capacitance and power
• Continuously-sized buffer libraries

8
Outline

• Introduction
• Delay and Buffer models
• Previous Work
• Proposed Algorithm
• Library characterization
• Generation of different types of candidates
• Merging, Propagation, Snapping
• Results
• Conclusion

9
Previous Work

• L. P. P. P. van Ginneken (VG) ISCAS90
• Two phase dynamic programming algorithm
• Backward traversal up the interconnect tree to
  compute of load and delay values
• Forward solution pass to reconstruct best
  candidate

Function BOTTOM_UP (v) 1. If v e sink return
(Cv, SDv) Else 2. / compute options for
subtrees / 3. BOTTOM_UP( left(v) ) 4. BOTTOM_UP(
right(v) ) 5. Join pairs of subtrees by a merge
operation 6. Find best cnd among merged cnds to
add a buffer 7. Add parent wire to both types of
cnds 8. Prune inferior cnds from set of cnds 9.
Store cnd list for node v and return
Post-order DFS traversal
Merge operation Cparent Cleft
CrightSDparent max(SDleft, SDright)
Buffer candidate creation
Pruning provably inferior candidates
10
VG Algorithm

• Candidate Format 2-tuple (Load, Subtree Delay)
  (c,s)
• Recursive forumulas for two possible cases
• Pruning Criteria (c1,s1) better than (c2,s2)
  if both load and subtree delay values are lower
  i.e., c1ltc2 and s1lts2
• Merge operation linear
• Complexity O(n2) where n number of buffer
  locations
• Additional objective Minimize buffer count ??
  Complexity is non-polynomial

Only a wire is added at root of subtree A buffer and a wire added at root of subtree
c1 c0 cwire s1 s0 dwire c1 cbuf cwire (Isolation Property) s1 s0 dint rbufc0 dwire

(c1.s1)
(c1.s1)
(c0.s0)
(c0.s0)
11
Previous Work

• Extensions to VG by Lillis et. al. ICCAD95,
  JSSC96
• A buffer library B can be used during buffer
  insertion ? Complexity O(n2B2)
• Simultaneous wire sizing and buffer insertion
• Incorporate signal slew into buffer delay model
• Dynamic power minimization subject to timing
  constraints
• Candidate Format 3-tuple (Load, Subtree Delay,
  Power) (c,s,p)
• Equate power with effective total capacitance
• Assumption All capacitive values can be linearly
  mapped onto a polynomially-bounded integer domain
  (cmax max cap value)
• Sophisticated pruning mechanism using orthogonal
  range query
• Complexity O(n3Bc2maxlog(ncmax)) based on the
  assumption

12
Previous Work

• Several approaches presented in literature to
  target power minimization in conjunction with
  buffer insertion. Examples
• Quadratic programming Chu et. al. TCAD99
• Lagrangian relaxation C.-P.Chen et. al. TCAD99
• ClockTune J.-L.Tsai et. al. TCAD04
• Associate total power with effective capacitive
  area of wires devices
• Area minimization ? Power minimization
• Ignores the contribution of static leakage power
• Inclusion of this component results in
  non-polynomial complexity
• Addition of extra components in candidates
  generally leads to exponential complexity for
  dynamic programming

13
Contributions of this paper

• Novel continuous buffer insertion algorithm
  with total power minimization
• Inclusive of both dynamic and leakage power
• Generate tradeoff surfaces in the 3D DCP (Delay,
  Capacitance, Power) space
• User is able to pick any desired point on this 3D
  surface
• Easy to explore trade-offs between the 3
  variables
• Ability to handle arbitrarily large buffer
  libraries
• Continuously sized cell libraries with numerous
  buffer sizes
• Capable of snapping to discrete buffer sizes if
  necessary
• Worst-case polynomial complexity O(n2)
• Similar to basic VG algorithm

14
Outline

• Introduction
• Delay and Buffer models
• Previous Work
• Proposed Algorithm
• Library characterization
• Generation of different types of candidates
• Merging, Propagation, Snapping
• Results
• Conclusion

15
Library Characterization

• Buffer library with a set of continuously sized
  buffers
• Let S sizing factor of the library. Express
  delay (db), capacitance (cb) and leakage (lb) in
  terms of S.
• Determine c0, c1, l0, l1, d0, d1 through
  empirical fitting constants
• Equations combine discrete buffer sizes
  approximate the ideal of continuous buffer sizing

cb ? Buffer Area ? cb c0 c1S lb ? Device
width ? lb l0 l1S db? Linear gate delay
model ? db d0 d1(Cout/S)
16
Generation of candidates
b2
b1
b3
b4
lw1
lw2
lw3
o
u
v
t

• Point Candidate
• Candidate Format 3-tuple (Do, Co, Po)
• Node has point candidate ? there are no buffers
  in subtree rooted at that node
• All sinks have point candidates
• Write equations to determine candidate at u

17
Generation of candidates
(D0, C0, P0)
b2
b1
b3
b4
lw1
lw2
lw3
o
u
v
t
?
Variable S

• Curve Candidate
• Candidate Format Dumin,Dumax, (gi, ki)
  i0,2
• Node has curve candidate ? Exactly one buffer in
  subtree rooted at node

18
Generation of candidates
(Du, Cu, Pu)
b2
b1
b3
b4
lw1
lw2
lw3
o
u
v
t
For a given S, Cv fixed, Dv, Pv vary based on Du
C-plane with discrete Cv

• Surface Candidate
• C-plane Format Cv, Dmin,Dmax, (ki) i0,2
• Candidate Format vectorltCPlanegt

19
Generation of candidates
(Du, Cu, Pu)
(Dv, Cv, Pv)
b2
b1
b3
b4
lw1
lw2
lw3
o
u
v
t

• Similar equations can be written to determine
  candidate at t
• Ct ? S but Dt, Pt ? Cv, Dv, S
• New set of C-planes.
• ? C-plane, Lower envelope ? Power optimal
  solution
• Surface candidate ? Surface candidate

20
Design Choices

• Wire network is a binary tree
• Zero-length wires, dummy nodes
• Ignore signal polarity on buffers
• Pair of solution sets (similar to Lillis)
• Number of surface candidates per node 2
  (Buffered/Non-buffered)
• Trade-off between more fine grained solutions and
  efficiency
• No impact on optimality or complexity

21
Merging and Implicit Pruning

• First, merge left and right candidate
• Compare equal delay points by checking 4
  combinations of left and right candidates
• Create P/C curves and extract the lower envelope
  ? Pruning
• Translate P/C curves with fixed D value into P/D
  curves with fixed C values ? Creation of C-planes
  for 4 different surface candidates
• Next, recombine these 4 surfaces into single
  candidate
• Map P/D curves from one C-plane to another using
  linear interpolation
• ? (D,C) value pick lowest power value ? Pruning
• Use composite surface to create the
  buffered/non-buffered candidate

22
Reconstruction and Snapping

• Pair of candidate solutions created for source
• Any trade-off point in the DCP surface can be
  picked
• Forward solution pass to reconstruct the tree
  structure with buffer locations
• Snapping If required size is unavailable then
  buffer with nearest size value is chosen
• Problem Discrepancies in D, C, P values ?
  Solution Local refinements in the C-planes
• Single pass through the RC tree
• Complexity O(n2) where n number of possible
  buffer locations

23
Outline

• Introduction
• Delay and Buffer models
• Previous Work
• Proposed Algorithm
• Library characterization
• Generation of different types of candidates
• Merging, Propagation, Snapping
• Results
• Conclusion

24
Results

• Benchmarks C-tree nets
• TSMC 0.13um buffer library
• Number of discrete buffer choices 9
• Multilinear fitting models using GNU Scientific
  Library
• Example 3D surface

25
Results Snapping
26
Results Comparison

• Implementation of Lillis algorithm with leakage
  included
• Pruning less effective

27
Conclusion

• Buffer insertion algorithm with total power (Pdyn
  Pstat) minimization as objective
• Generate 3D surfaces in Delay, Capacitance and
  Power space
• Ability to explore different types of trade-offs
• Able to handle large buffer libraries with
  continuous sizes
• Worst case polynomial complexity