1096
IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 28, NO. 7, JULY 2009
Short Papers
Feasible Aggressor-Set Identification Under
Constraints for Maximum Coupling Noise
Debjit Sinha, Member, IEEE, Alex Rubin,
Chandu Visweswariah, Fellow, IEEE, Frank Borkam,
Gregory Schaeffer, and Soroush Abbaspour, Member, IEEE
Abstract—In this paper, we consider the problem of identifying a feasible set of aggressor nets that would induce maximum crosstalk noise or
delay pushout on a coupled victim net, under given logical constraints. We
present a novel mathematical formulation of this problem and propose a
Lagrangian relaxation-based approach for solving it efficiently and optimally. Experimental results show that the proposed approach is run-time
efficient by a factor of up to 800 times in comparison to an exhaustive
search approach and reduces timing pessimism by up to 36%. We also
formulate and solve this problem while considering the noise susceptibility
of the victim’s receiving gate.
Index Terms—Aggressor-set identification, coupling noise, crosstalk,
Lagrangian relaxation (LR), optimization, signal integrity.
I. I NTRODUCTION
Traditional coupling-aware timing or noise analysis on a given
(victim) net v in the absence of any functional information assumes
a pessimistic situation. All (aggressor) nets coupled to v whose
switching windows overlap with that of v are considered switching1 in
certain directions to compute the noise that may be induced on v in the
worst case. However, given logical relations or constraints between the
victim and (or) the aggressor nets, aggressor switching combinations
may be restricted to obtain pessimism reduction during coupling-noise
computation on v. A set of aggressor nets to a victim that satisfy given
constraints is termed a feasible aggressor set.
Worst-case analysis considering all aggressors (with overlapping
switching windows) leads to noise overestimation or false noise.
This, in turn, leads to overdesign or causes a designer to spend
unnecessary time in fixing false violations. Pessimism reduction in
noise analysis is consequently a key step for timing and noise closure
for any given design. We thus consider the problem of identifying a
feasible set of aggressor nets that would induce the worst (maximum)
coupling noise on a victim net, under given switching constraints.
This problem is nontrivial; an exhaustive search approach to obtain
the feasible aggressor set that induces maximum coupling noise on
a victim has exponential run-time complexity. The overhead of such
an approach is significant for large coupling clusters, i.e., when a
net has a large number of aggressors. Although aggressors whose
switching windows do not overlap with that of the victim are ignored,
Manuscript received August 4, 2008; revised November 20, 2008 and
January 5, 2009. Current version published June 17, 2009. This work is
a transaction version of the authors’ work in the Proceedings International
Conference on Computer-Aided Design, pp. 790-796, 2008 [1]. This paper was
recommended by Associate Editor Y.-W. Chang.
D. Sinha, A. Rubin, F. Borkam, G. Schaeffer, and S. Abbaspour are with
the IBM Systems and Technology Group, Electronic Design Automation,
Hopewell Junction, NY 12533 USA (e-mail: [email protected]).
C. Visweswariah is with the IBM Thomas J. Watson Research Center,
Yorktown Heights, NY 10598 USA.
Digital Object Identifier 10.1109/TCAD.2009.2018779
1 In
the rest of this paper, the term switching will imply switching in a given
direction.
the problem size can still be large enough to make an exhaustive
search approach impractical in terms of run time. For modern sub90-nm designs, coupling clusters of about 20–50 nets are common,
with larger clusters having more than 150 nets [2]. It is thus important
to have an efficient approach for solving the problem, particularly from
a computer-aided-design-tool perspective that cannot assume limited
(small) cluster sizes. It is equally important to solve this problem
optimally, since a nonoptimal solution can result in an optimistic
timing or noise analysis. As an example, a greedy approach which
iteratively selects an aggressor that induces the worst noise while
satisfying given constraints may be efficient, but cannot guarantee an
optimal solution. Based on the calculated noise from the worst feasible
set, optimizations (e.g., wire spacing, rerouting, and buffering) or signoff timing analysis may be performed on the given chip design.
Multiple approaches toward pessimism reduction based on logical
constraints during coupling-noise analysis have been proposed in
the past [2]–[7]. Most prior work assume that complete functional
information is available and employ satisfiability, a binary decision
diagram, or branch-and-bound-based approaches for feasibility checking. These are often run-time expensive for large designs.
Our contributions to this research area are the following. We present
a novel formulation of the constrained aggressor-set-identification
problem for maximum coupling noise as an integer linear programming (ILP) problem and then show that the problem can be treated as
a linear programming (LP) problem without any loss of accuracy for
most practical cases. Using constrained aggressor groups to represent
simplified logical relations, we propose an efficient approach for solving this problem based on Lagrangian relaxation (LR) that guarantees
the optimal solution. We also formulate this problem considering the
noise susceptibility of the victim’s receiving gate and illustrate the
importance of such a formulation during functional noise analysis.
Prior work does not consider this during feasible aggressor-set identification. An LR-based approach is proposed for solving this problem.
II. P ROBLEM D EFINITION
We consider a victim net v and a set of m coupled aggressor nets
a1 , a2 , . . . , am . Based on available functional information, a set of
n aggressor groups is provided, which we denote as g1 , g2 , . . . , gn .
Each group is fundamentally a collection of nets. A net may belong
to multiple groups. Functional information imposes constraints that, at
most, Ni nets in group gi could switch within a given clock cycle.
Seemingly simple aggressor groups allow significant flexibility in
modeling complex functional constraints, including equivalent, logic
exclusivity [2], and hot-/cold-k constraints in global buses where only
k nets of a bus are in a logic high/low state in a clock cycle. A brief
discussion with modeling examples is presented in [1].
Given timing information (e.g., switching windows, input slews,
switching directions, and analysis modes) for the victim and aggressor
nets, we denote the coupling noise induced by a switching aggressor
ai on v as Ψi . Mathematically, the coupling noise on v due to ai is
represented as
Ψi ,
0,
if ai is considered to be switching
otherwise.
(1)
During timing analysis considering coupling, if the switching windows of an aggressor ai and the given victim do not overlap, it is
obvious that noise induced by ai would not affect timing on the
0278-0070/$25.00 © 2009 IEEE
Authorized licensed use limited to: KnowledgeGate from IBM Market Insights. Downloaded on June 25, 2009 at 13:54 from IEEE Xplore. Restrictions apply.
IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 28, NO. 7, JULY 2009
victim, and thus, Ψi = 0. Such an aggressor need not be considered
in the feasible-set-identification problem. For the rest of this paper,
we assume that the aggressors to a victim include only the ones
that have overlapping switching windows with the victim, assuming
that some early timing information is available. The total worst-case
noise induced on the victim when all aggressors switch is assumed
to be given by the superposition of each induced coupling noise
(assuming linear driver models). Mathematically, we represent this as
Ψ1 + Ψ2 + · · · + Ψm .
Using an abstract model Ψi to represent coupling noise facilitates
applying various noise models to our problem formulation. Examples
of noise models that may be employed are presented in [1]. Given
the aforementioned information and constraints, the objective of our
problem is to identify the feasible set of switching aggressors that
would induce maximum total coupling noise on v.
III. P ROBLEM F ORMULATION
In this section, we formulate our problem as an ILP problem. To
denote the set of aggressors that would be considered as switching
(feasible) for coupling-noise computation, we define a set of Boolean
variables as follows (for each i ∈ [1, m]):
Δ
si =
1,
0,
if ai is considered to be switching
otherwise.
Δ
xij =
if aggressor ai ∈ group gj
otherwise.
1,
0,
xij si ≤ Nj .
(3)
(4)
i=1
We thus formulate our primal problem (PP) as follows:
PP :
max
m
si Ψi
i=1
m
s.t.
xij si ≤ Nj
∀j ∈ [1, n]
i=1
0 ≤ si ≤ 1, and integral
∀i ∈ [1, m].
the solution to the subproblem is also the solution to the original
constrained problem is called the Lagrangian dual problem or LDP.
For the Lagrangian subproblem, we choose to relax all constraints
on the maximum number of aggressors that may switch in any
group. Each constraint is multiplied with a nonnegative multiplier λj
(corresponding to aggressor group gj , j ∈ [1, n]) and then added to
the objective function. The Lagrangian subproblem (LRS/λ) is thus
expressed mathematically as follows (after rearranging some terms in
the objective function):
LRS/λ :
max
n
λj Nj + s1
Ψ1 −
j=1
+ s2
The constraint that, at most, Nj aggressors in group gj could switch
within a given clock cycle is expressed as
m
Fig. 1. Algorithm to solve the Lagrangian subproblem.
(2)
From (1) and (2), the effective coupling noise that ai induces on
v is expressed as si Ψi . To denote mathematically if an aggressor
ai belongs to a group gj , we define a set of Boolean coefficients as
follows (for each i ∈ [1, m] and j ∈ [1, n]):
1097
Ψ2 −
n
λj x2j
n
λj x1j
j=1
+ · · · + sm
Ψm −
j=1
s.t. 0 ≤ si ≤ 1, and integral
n
λj xmj
j=1
∀i ∈ [1, m].
(6)
Given λj (∀j ∈ [1, n]), the aforementioned problem is trivial. Since
si can either be 0 or 1, the objective function canbe maximized
n
by setting si to 1 if its corresponding term (Ψi − j=1 λj xij ) is
positive and setting si to 0 if otherwise. The pseudocode for this
approach to solving the Lagrangian subproblem for a given vector λ =
[λ1 , λ2 , . . . , λn ] is shown in Fig. 1 and has a complexity of O(mn).
It is observed that the optimal solution to LRS/λ will not change
even if the integrality constraint on each si is ignored from (6) and
replaced by a constraint that each si is a real number between 0 and 1
(0 ≤ si ≤ 1 ∀i ∈ [1, m]). This illustrates that the problem can be
treated as a continuous optimization problem. Only in extremely rare
cases that have multiple optimal solutions, treating the problem as an
LP problem may result in non-Boolean solutions for some si terms.
However, these cases can be handled by doing a fast local search for
Boolean values around the vicinity of the nonintegral values obtained
for those si variables.
We employ the subgradient optimization method to solve the
Lagrangian dual problem [8].
(5)
It is obvious that the problem is an ILP problem, and any ILP
solver could be employed to obtain the optimal solution for each si ,
i ∈ [1, m]. In the following section, we describe an efficient approach
for solving this problem optimally.
IV. F EASIBLE A GGRESSOR -S ET I DENTIFICATION
A. LR-Based Solution
In this section, we formulate and solve the problem in (5) using
LR [8]. LR is a widely acclaimed technique for solving constrained
optimization problems. In LR, “troublesome” constraints are “relaxed”
and incorporated into the objective function after multiplying them
with nonnegative constants called Lagrange multipliers, one multiplier
for each constraint. For each fixed vector λ of the Lagrange multiplier
introduced, we obtain an easier subproblem called the LR subproblem
associated with λ or LRS/λ. The problem of finding a λ such that
B. Greedy Approach to Feasible Aggressor-Set Identification
For comparison, we present in brief a greedy heuristic for solving
the feasible aggressor-set-identification problem. A naïve approach
would select aggressors in the feasible set based on a sorted order
of their nonincreasing noise contributions (Ψ), while satisfying group
constraints. We additionally employ the intuition that an aggressor
that belongs to multiple groups has a larger chance of violating a
given constraint than an aggressor that belongs to only a single group.
Based on this, we propose a smarter greedy approach, the pseudocode
of which is shown in Fig. 2. Initially (steps 1–3 in Fig. 2), each
aggressor ai is excluded from the feasible set (si = 0), and the noise
contribution Ψi is scaled down by the number of groups that ai belong
to. Aggressors are then sorted based on their scaled noise contributions
(Ψ∗i ) and pushed into a max-stack S such that the top of the stack
contains the aggressor with the largest Ψ∗ value (step 4). Aggressors
are next iteratively popped from the stack and included in the feasible
Authorized licensed use limited to: KnowledgeGate from IBM Market Insights. Downloaded on June 25, 2009 at 13:54 from IEEE Xplore. Restrictions apply.
1098
IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 28, NO. 7, JULY 2009
TABLE I
CONSTRAINED AGGRESSOR-SET-IDENTIFICATION RESULTS
Fig. 2. Greedy algorithm for feasible aggressor-set identification.
set if they do not violate the given group constraints (steps 5–9). The
feasibility check in step 8 returns a Boolean to indicate if the current
switching vector [s1 , s2 , . . . , sm ] satisfies all group constraints in the
original problem (5).
The complexity of this approach is O(m2 n) since the feasibility
check in step 8 has complexity O(mn) and is done m times within the
while loop in step 5.
C. Experimental Results
A prototype LR solver and a greedy heuristic-based solver for the
feasible aggressor-set-identification problem is implemented using the
C++ programming language in an industrial tool framework. A set of
representative coupling clusters from an industrial test case is used for
study. The results obtained are presented in Table I. In the table, m and
n denote the number of aggressors and aggressor groups, respectively.
We limit cluster sizes to 40 aggressors in these experiments since larger
clusters (beyond m = 40) are found in less than 5% of the nets in
our test case. The largest cluster size encountered in our test case is
150. In the third column, we indicate the constraints by showing the
maximum number of aggressors Nj that can be considered switching
in each group gj . Larger values of Nj (values closer to m) imply larger
correlation between aggressor groups, i.e., a larger likelihood of an
aggressor belonging to multiple groups. Increased correlation between
groups make the feasible-set-identification problem more difficult to
solve, since, if all aggressor sets were mutually exclusive, a greedy
approach would guarantee an optimal solution.
In each case, the feasible aggressor sets obtained by the proposed
LR and greedy algorithms are tested for optimality using the solution
obtained from an exhaustive search approach. The fourth column in
Table I (|s|) shows the total number of aggressors that are considered
switching (or found feasible) in the optimal solution. In the next set of
three columns, k denotes the number of iterations required in solving
the Lagrangian dual problem. In these columns, we show how allowing
a small error tolerance (shown as %ξ) during the convergence of an
iterative subgradient optimization method for solving the dual problem
may reduce the number of iterations k. Results are presented for error
tolerances of 0%, 2%, and 5%. As an example, we observe that, for
the case of m = 15 and n = 4, k reduces from 101 to 15 and 6 after
allowing a 2% and 5% error tolerance, respectively. The proposed
LR approach yields the optimal solution for all cases with ξ = 0%.
However, for positive ξ, LR yields suboptimal solutions in some
cases. The greedy approach also fails to yield the optimal solution in
some cases. To quantify the error in a computed suboptimal feasible
aggressor set with total coupling noise Ψ and given the noise Ψopt
from the optimal feasible aggressor set (evaluated using exhaustive
search), we define an error metric as
Δ Ψopt − Ψ
%Error in calculated noise =
× 100.
(7)
Ψopt
The aforementioned error metric is shown for the LR approach with
ξ = 2% and = 5% in columns eight and nine and for the greedy
approach (Gr.) in column ten of Table I, respectively. For the LR
approach with ξ = 0%, the obtained solution is optimal for all cases.
It is observed from the table that, for ξ = 2%, we obtain suboptimal
results in three of the 12 cases, with at most 1.6% of error (for the case
with m = 15 and n = 3). Similarly, the LR approach with ξ = 5%
and the greedy approach yield suboptimal solutions in six and three of
the 12 cases, with errors in computed noise being, at most, 4.7% and
4.2%, respectively. A solution obtained using the greedy approach or
the LR approach with a positive error tolerance (ξ) may be suboptimal,
but it satisfies all group constraints. Since an obtained aggressor set is
always feasible, the noise induced by the obtained set on a victim is at
most the noise induced on the victim by the optimal feasible set (the
optimal set induces the maximum noise by definition). This explains
why the error metric in Table I is always positive.
It is observed from the table that the number of required iterations
for the LR approach is reasonable. For a qualitative run-time comparison, we restate that the complexity of the LR, greedy, and exhaustive
search approaches are O(kmn), O(m2 n), and O(2m ), respectively.
In practice, the run times of the LR and greedy approaches are found
to be comparable, while the run time of the exhaustive search approach
is found slower by orders of magnitude than the others for m ≥ 11.
Although run times for the LR and greedy approaches are found
comparable, the LR approach (even with positive error tolerance)
is preferred since the former provides a way to quantify an upper
bound on the error and thereby facilitates a tradeoff between run time
and error. The simplicity of the greedy approach makes it a good
candidate for feasible-set identification possibly in an early design
flow. However, the LR approach would be desirable in a sign-off
flow to avoid potential optimistic coupling analysis, resulting from an
approximate (e.g., greedy) approach.
Individual coupling delay pushout due to each aggressor are used
as Ψi in these experiments. The group constraints employed in these
experiments are a combination of designer asserted ones and those
obtained from a proprietary functional analysis tool.
In Table II, we present experimental results when the proposed LR
approach (with ξ = 0%) is employed for aggressor filtering on a set
of five 65-nm high-performance microprocessor units (subdesigns).
The term aggressor filtering is used to denote that aggressors not
included in the obtained feasible set of a victim are disregarded during
subsequent coupling-aware timing analysis. The filtering is performed
after an initial timing analysis of the design with grounded coupling
capacitances. For each design, we show the number of timing points
(input and output pins of gates, also includes internal pins for complex
gates), the number of timing arcs (arcs between timing points), number
of globally defined aggressor groups, and the number of aggressors in
the largest aggressor group in columns 2–5 of the table, respectively.
Authorized licensed use limited to: KnowledgeGate from IBM Market Insights. Downloaded on June 25, 2009 at 13:54 from IEEE Xplore. Restrictions apply.
IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 28, NO. 7, JULY 2009
1099
TABLE II
AGGRESSOR FILTERING RESULTS FOR INDUSTRIAL TEST CASES
The sixth column shows the percentage improvement in the worstslack (for setup, hold, or clock pulsewidth tests) due to couplingdelay pessimism relief from aggressor filtering. The next two columns
present run times for aggressor filtering using the proposed LR-based
algorithm (indicated as LR) and an exhaustive search-based feasible
set solver (indicated as EXH), respectively. The run-time gain (as a
ratio) of the proposed algorithm over the exhaustive search approach
is shown. Some of the designs have large coupling clusters (up to about
150 nets in a cluster). To avoid impractical run times of the exhaustive
search algorithm on such clusters, the EXH algorithm is forced to fail
for clusters that have larger than 35 nets. As a result, this approach
is unsuccessful for a number of nets in three of the five designs. The
corresponding numbers are indicated in the last column of the table.
The designs contain 14.4 K–126.2 K timing points and 8–8833
aggressor groups. In the design Ckt1, only eight aggressor groups are
defined for some global buses. As expected, run times for aggressor
filtering with the LR and EXH approaches are comparable, and slack
improvements are negligible. However, with the growing number of
aggressor groups and the size of each aggressor group, the run-time
overhead of the EXH approach in comparison to the LR approach
grows significantly (up to 800 times). It should be noted that the run
times of the EXH algorithm are actually underestimated since some
nets (that are part of large coupling clusters) are not filtered. For
the design Ckt5, the EXH algorithm took an additional 25 h to filter
the remaining 143 nets, while the LR approach filters all nets within
a minute. Worst-slack improvements (includes setup, hold, or clock
pulsewidth tests) between 3% and 36% are observed for the designs,
indicating the importance of feasible aggressor-set identification.
V. C ONSIDERING N OISE R EJECTION C URVE
During functional noise verification, the noise injected on a quiet
(nonswitching) victim is required to be less than the threshold that
causes a potential false switching at the output of the victim’s receiving
gate. The injected noise is often referred to as a noise bump or glitch.
It is required that the noise bump height (peak of the glitch) be less
than a threshold, which, in turn, is a function of the noise bump width.
This function is termed the noise rejection curve of a gate [9] and
represents the susceptibility of the gate to noise. Mathematically, the
noise rejection curve of a gate denotes the maximum height of a noise
bump that the gate can tolerate without failure as a function of the
width of the noise bump. The width of a noise bump is traditionally
defined as the ratio of the area to the peak of a noise bump. For a
given victim net v, we denote the noise rejection curve of its receiving
gate as N RC(w), where w denotes the width of the noise bump at the
input of this gate and define δ to be the amount of noise that exceeds
the gate’s noise susceptibility. We formulate the feasible aggressor-setidentification problem in the following section to maximize δ under
given aggressor group constraints.
bump induced on v due to ai as pi and wi , respectively. Various noise
models may be employed to obtain the noise bump peak and height.
For example, the 2π noise model proposed in [10] presents analytical
forms for the noise bump peak and width.
One commonly employed approach (in industrial tools with proven
good accuracy) to compute the cumulative noise bump peak and
width given the peaks and widths of the individual noise bumps is
the following. The individual noise bumps are aligned so that the
cumulative noise bump peak pT on v is equal to the sum of the
noise bump peaks induced by each aggressor [6]. The area of the cumulative noise bump is set to the sum of all the individual noise bump
areas to compute the cumulative noise bump width wT . Only those
aggressors whose switching windows overlap with that of the victim
are considered in the aforementioned approach.
Given logical constraints (as aggressor groups) and employing
notations introduced in Section III, the cumulative noise bump peak
pT on v is mathematically represented as follows:
pT =
m
si pi .
(8)
i=1
Based on the definition of a noise bump width in the preceding
section, the cumulative noise bump width wT is computed by matching
the area as follows:
m
spw
i=1 i i i
.
wT = m
i=1
The amount of noise that exceeds the victim’s receiving gate’s
threshold (denoted as δ in the preceding section) is thus given by
pT − N RC(wT ) =
m
si pi − N RC
m
si pi wi
i=1
.
m
i=1
i=1
Given a set of aggressor nets {a1 , a2 , . . . , am } that induce noise
on a victim net v, we denote the peak and the width of the noise
si pi
(10)
Noise rejection curves are specified as lookup tables and also
commonly modeled mathematically as
N RC(w) = k1 +
k2
w
(11)
for some constants k1 and k2 . Our new primal problem (N PP)
considering noise rejection curve is thus formulated mathematically
as follows (after ignoring the constant k1 from the objective function):
N PP :
max
m
m
si pi
si pi − k2 mi=1
i=1
i=1
m
s.t.
xij si ≤ Nj
si pi wi
∀j ∈ [1, n]
i=1
0 ≤ si ≤ 1, and integral
A. Problem Formulation Considering Noise Rejection Curve
(9)
si pi
∀i ∈ [1, m].
(12)
It is observed that ignoring the noise rejection curve makes the
objective function identical to that of our original problem in (5), by
setting Ψi = pi .
Authorized licensed use limited to: KnowledgeGate from IBM Market Insights. Downloaded on June 25, 2009 at 13:54 from IEEE Xplore. Restrictions apply.
1100
IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL. 28, NO. 7, JULY 2009
TABLE III
CONSTRAINED AGGRESSOR-SET-IDENTIFICATION RESULTS
Fig. 3. Algorithm to solve the Lagrangian subproblem.
B. LR-Based Solution
In this section, we present in brief an LR-based approach toward
solving the feasible aggressor-set-identification problem considering
noise rejection curve. For the Lagrangian subproblem, we choose
to relax all constraints on the maximum number of aggressors that
may switch in any group as described in Section IV-A. The new
Lagrangian subproblem associated with a given vector of multipliers
λ is expressed mathematically as follows (after rearranging the terms
in the objective function):
max
n
λj Nj + s1
j=1
+ s2
k2 p1 −
p1 −
λj x1j
A
n
k2 p2 −
λj x2j
A
n
p2 −
+ · · · + sm
j=1
j=1
k2 pm −
pm −
λj xmj
A
n
j=1
s.t. 0 ≤ si ≤ 1, and integral
m
∀i ∈ [1, m]
(13)
where A = I=1 si pi wi . It should be noted that A is not a constant
and the objective function in (13) is nonposynomial. We thus propose
the following heuristic for solving this problem. Each si is initialized
to
Next, each si is iteratively set to 0 if the term (pi − (k2 pi /A) −
1.
n
λ x ) is negative until convergence. Since this term is
j=1 j ij
monotonically nondecreasing with increasing si , convergence is
guaranteed. The pseudocode of this approach is shown in Fig. 3. We
employ subgradient optimization for solving the dual problem.
C. Experimental Results
In Table III, we present results when the proposed algorithm is
employed to identify feasible aggressor sets while considering noise
rejection curves for a set of coupling clusters. The table format and
legend is similar to Table I. Comparisons to solutions from exhaustive
simulations (for m ≤ 30) show nonoptimal solutions obtained by the
proposed heuristic for only three of the test cases; the error, as defined
in (7) is shown as %Error in the last column of Table III. Errors in
calculated coupling noise are found to be less than 1% on the average.
The results show a reasonable number of iterations incurred during
the optimization process. The experimental setup used is similar to
that described in Section IV-C. In these experiments, k1 = 0.55 V and
k2 = 0.007 V · nS.
VI. C ONCLUSION
In this paper, we present approaches for constrained aggressorset identification that induces maximum coupling noise. An efficient
approach based on LR is proposed, which guarantees the optimal
solution and can be easily extended to simultaneously consider all
nets in a path for coupling delay pessimism reduction. The proposed
approach is readily extensible to also model temporal constraints, for
example, when the switching windows of two aggressors of a victim
overlap with that of the victim, but do not overlap with each other. We
introduce the idea and importance of considering the noise rejection
curve while formulating the feasible aggressor-set-identification problem and present a heuristic to solve the problem that has less than 1%
error on the average.
The proposed approaches highlight their efficiency for large problem sizes (coupling cluster sizes). An engineering perspective to
consider is to use an exhaustive search for feasible-set identification
when the cluster size is small and use the proposed LR approach for
larger clusters (m > 11).
We do not limit the proposed approaches to any particular noise
model. Since Ψi is used as a score to select the final set of aggressors,
accurate noise analysis tools including SPICE may be employed
during final timing or noise analysis while considering the obtained
set of aggressors. The proposed approach cannot guarantee optimality
if the noise models used to obtain the feasible set and to compute the
final noise are different. However, if the noise model used to obtain
the feasible set has high fidelity, this approach may be used to obtain
significant pessimism reduction in functional noise or coupling aware
timing analysis.
R EFERENCES
[1] D. Sinha, S. Abbaspour, G. Schaeffer, A. Rubin, and F. Borkam,
“Constrained aggressor set selection for maximum coupling noise,” in
Proc. Int. Conf. Comput.-Aided Des., 2008, pp. 790–796.
[2] R. Li, A. Shey, and M. Laudes, “Incorporating logic exclusivity (LE)
constraints in noise analysis using gain guided backtracking method,” in
Proc. Int. Conf. Comput.-Aided Des., 2008, pp. 783–789.
[3] P. Chen and K. Keutzer, “Towards true crosstalk noise analysis,” in Proc.
Int. Conf. Comput.-Aided Des., 1999, pp. 132–137.
[4] T. Xiao and M. Marek-Sadowska, “Functional correlation analysis in
crosstalk induced critical paths identification,” in Proc. Des. Autom. Conf.,
2001, pp. 653–656.
[5] D. Chai, A. Kondratyev, Y. Ran, K. Tseng, Y. Watanabe, and
M. Marek-Sadowska, “Temporofunctional crosstalk noise analysis,” in
Proc. Des. Autom. Conf., 2003, pp. 860–863.
[6] A. Glebov, S. Gavrilov, R. Soloviev, V. Zolotov, M. Becer, C. Oh, and
R. Panda, “Delay noise pessimism reduction by logic correlations,” in
Proc. Int. Conf. Comput.-Aided Des., 2004, pp. 160–167.
[7] K. Tseng and M. Horowitz, “False coupling exploration in timing analysis,” IEEE Trans. Comput.-Aided Des., vol. 24, no. 11, pp. 1795–1805,
2005.
[8] C. Chen, C. Chu, and D. F. Wong, “Fast and exact simultaneous gate and
wire sizing by Lagrangian Relaxation,” IEEE Trans. Comput.-Aided Des.,
pp. 1014–1025, Jul. 1999.
[9] A. Korshak, “Noise-rejection model based on charge-transfer equation for
digital CMOS circuits,” IEEE Trans. Comput.-Aided Des., vol. 23, no. 10,
pp. 1460–1465, 2004.
[10] J. Cong, D. Z. Pang, and P. V. Srinivas, “Improved crosstalk modeling
for noise constrained interconnect optimization,” in ACM Intl. Workshop
Timing Issues Specification Synthesis Digital Systems, 2000.
Authorized licensed use limited to: KnowledgeGate from IBM Market Insights. Downloaded on June 25, 2009 at 13:54 from IEEE Xplore. Restrictions apply.