figure 6. A comparison between the node-based algorithm and our
bounding strategy for a balanced tree circuit.
3200 3400 4000
depth = 8
benchmarks, the root-mean-square difference between
the exact distribution and the bounds is 1. 7–4.5% for
the lower bound and 0. 9–6.2% for the upper bound.
Figure 5 illustrates that accurately accounting for edge
delay correlations is crucial when predicting the shape
of the cdf. The edge delay correlation changes with the
ratio of the variance of inter-chip to intra-chip components. For example, the span of the low-correlation case
is smaller than the high-correlation case. Our algorithm
is more accurate than node-based approximation methods when processing balanced timing graphs with equal
path mean delays and delay variances. In Figure 6, we
compare the cumulative distribution functions generated by the node-based algorithm using the procedure
of Visweswariah et al.
17 against the proposed approach.
The cdfs are generated for a balanced tree circuit with
depth of 8. An equal breakdown of total variation into
inter-chip and intra-chip variability terms was used.
Note that the approximate cdf may be noticeably different from the true one, but the bounds accurately contain
the true cdf.
The implementation is very efficient even though the
algorithm runtime is quadratic in the number of deterministically longest paths. The C++ implementation ran
on a single-core machine with a 3.0GHz CPU and 1GB
memory and took less than 4 seconds for the largest circuit in the ISCAS ‘ 85 benchmark suite (3874 nodes). When
evaluated at a fixed number of extracted paths, there is a
close-to-linear growth in algorithm runtime as a function
of circuit size.
We have presented a new statistical timing analysis algorithm for digital integrated circuits. Instead of approximating the cdf of a circuit, we use majorization theory to
compute a tight bound for the delay cdf. The equicoordinate random vectors are used to bound the exact cumulative
100 communIcATIons of The Acm | auGust 2009 | Vol. 52 | no. 8
distribution function, which facilitates the numerical evaluation of the probability.
We thank Ashish K. Singh for his help with the simulations,
and Sachin Sapatnekar for his feedback on the article. The
work has been partially supported by the NSF Career grant
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