Net Criticality Revisited:

1

• Net Criticality Revisited
• An Effective Method to Improve Timing
• in Physical Design
• H. Chang1, E. Shragowitz1, J. Liu1,
• H. Youssef2, B. Lu3, S. Sutanthavibul4
• 1 University of Minnesota, USA
• 2 Universite du Centre, Tunisia
• 3 Cadence Design Systems Inc., USA
• 4 Intel Corp., USA

2
Basic Ideas and Goals

• Perform timing-driven placement and routing based
  on IMP (Iterative Minimax Pert), a zero slack
  distribution algorithm for net delay bound
  calculation
• Propose new criticality metrics for placement and
  routing
• Achieve better timing results in one-pass
  physical design

3
Why IMP Algorithm?

• Review of related works on delay budgeting
  problem
• First ZSA algorithm in O(np), 1989
• IMP algorithm based on Minimax formulation with
  linear time complexity O(np), 1990
• Application of IMP to placement problem in
  O(np), 1992
• PWL-GBS algorithm solving the problem in linear
  programming formulation in O(m2logH), 1997
• MISA algorithm based on the maximal independent
  set (MIS) of a transitive slack equalization
  graph in O(kn3), 2000

4
Calculation of Net Delay Bounds by IMP Algorithm

• Path slack U? for any path ? ,
• U? Tcr - T?
• Tcr longest path delay, T? delay of the
  path ?
• U?gt0 for any noncritical path, U?0 for any
  critical path.
• Delays on non-critical paths can be increased
  without increase in a clock cycle

5
Continued..

• The IMP algorithm is based on two ideas
• a) The net delay slacks on each of the paths can
  be distributed among constituent nets, according
  to the relative weights of the nets along the
  path.

6
Continued..

• b) Each net may belong to the multiple paths.
  Therefore, the propagation delay on the net
  should not exceed the minimal value among all
  maximal delays defined on this net for each path
  separately

• This problem is NP-hard.

Reference H.Youssef, R-B. Lin and E. Shragowitz
in IEEE TCAS, 1992
7
Asymptotically Converging Approximation Algorithm

• On each step of the algorithm, a lower bound on
  the value of
• is found by a linear algorithm,

• where, both minimal path slack and
  maximal path weight
• for all the paths traversing edge e, can be
  computed in linear time by
• a pert-like algorithm.
• Repetitive application of step 1 results in
  convergence to the optimal solution of the
  initial Minimax problem,

8
Criticality Metrics

• Net delay bounds provide new opportunities for
  identification of timing-critical nets
• Net Criticality Metric

9
Probabilistic Interpretation of Criticality
Metrics

• Projected Net Delays could be considered as
  random values. Assuming Gaussian distribution
  N(mx ,?x) of net delay x,
• xmin mx- 3?x , xmax mx3?x ,
• xmax? bx , where bx is a bound on net delay
• if xmin? 0 ? xmax 2mx ? bx
• Net Criticality Metric can be rewritten as,
• Net Criticality Metric 2mx/bx
• The probability for the net delay to be below
  the bound bx is decreasing when a ratio mx/bx
  is increasing.

10
Statistical Approximation Formulas for Net
Criticality Metrics

• The ranking of nets according to criticality
  metrics is preserved when a mathematical
  expectation of a net delay is replaced by a net
  parameter.
• Criticality metrics for placement (CMP)
• Criticality metrics for routing (CMR)

11
One-Pass Flow v.s. Multiple-Pass Flow in
Physical Design
One-pass criticality-based layout flow
Traditional iterative layout flow
12
Experiments

• Environment
• Cadence Silicon Ensemble DSM Automation Layout
  System
• 4-layer, 0.18 micron technology standard cell
  library
• Integration of the proposed criticality metrics
  in layout flow of Cadence Silicon Ensemble
• Application of criticality metrics to placement
  (CMP)
• Application of criticality metrics to routing
  (CMR)
• Application of CMP and CMR to placement and
  routing in one pass of a layout process

13
Placement Results

• 26.4 improvement versus Cadence Wire Length
  Minimization mode and 13.8 improvement versus
  Cadence timing driven placement mode

WLM layout by Cadence in Wire Length
Minimization mode TDP layout by Cadence in
Timing-Driven Placement mode CPF layout by
Cadence in WLM mode with nets on Critical Paths
given higher weights during placement (2-pass
solution) CMP layout by Cadence in WLM mode
with weights derived from new Criticality Metrics
for Placement (1-pass solution).
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Routing Results

• 9.2 improvement versus Cadence Wire Length
  Minimization mode and 3.7 improvement versus
  Cadence timing driven routing mode

WLM layout by Cadence in Wire Length
Minimization mode TDR layout by Cadence in
Timing-Driven Routing mode CPF layout by
Cadence in WLM mode with nets on Critical Paths
given higher weights during routing (2-pass
solution) CMR layout by Cadence in WLM mode
with weights derived from new criticality Metrics
for Routing (1-pass solution).
15
Placement and Routing Results

• 29.5 improvement versus Cadence Wire Length
  Minimization mode and 12.4 improvement versus
  Cadence timing driven placement and routing mode

WLM layout by Cadence in Wire Length
Minimization mode TDPR layout by Cadence in
Timing-Driven Placement and Routing mode CPF
layout by Cadence in WLM mode with nets on
Critical Paths given higher weights during
placement and routing (2-pass solution) CMPCMR
layout by Cadence in WLM mode with weights
derived from new Criticality Metrics for
Placement and Criticality Metrics for Routing
(1-pass solution).
16
Conclusion

• The proposed criticality metrics
• achieve substantially better timing results in
  one pass of physical design.
• could be integrated with any layout system that
  allows weights for nets in the design.
• can be applied to timing optimization of
  placement alone or routing alone