normal distribution based MAX-operation for very large dimensions of the skew- normal random vector X = (X1, . . . , Xn)T ∼ S Nn(µ,Σ, λ). The goal is to test that the asymptotic behaviour of the implementation matches the theoretical worst-case runtime complexity of the algorithm for the skew-normal distribution based MAX- operation. For this, the skew-normal distribution based MAX-operation was applied to approximate the distribution of the random vector
with a (n−1)-dimensional skew-normal distribution. Except for the very first ap- proximation step, the inverse Cholesky factor L−1is obtained by updating the inverse
Cholesky factor of the previous application of the skew-normal distribution based MAX-operation. To represent this typical scenario, it is assumed that the inverse Cholesky factor L−1is available, and hence, the runtime of the initial computation of
L−1 with LLT = Σ was ignored. As a consequence, the runtime appears to be lower
than infig. 7.11.
The runtime of the algorithm insection 6.5.3is measured for very large dimensions n∈ {2000, 4000, 6000, 8000, 10000, 12000, 14000, 16000}.
Therefore, even the smallest size (n=2000) is large enough for the high order terms to dominate all other terms. The result of these measurements consists of 8 data points (ni, ti)for 1≤i≤8, as shown infig. 7.15. In order to predict the runtime t(n)with the power-law function
t(n) =anb, (7.23)
linear regression is applied to the set of data points (log(ni), log(ti)). The resulting coefficients are a ≈1.20176×10−6 and b≈2, which confirms the theoretical quadratic worst case runtime complexity O(n2)of the proposed algorithm for the skew-normal distribution based MAX-operation.
1.20176 × 10-6n2.00058 2000 4000 6000 8000 10 000 16 000 4 8 16 32 64 128 256
Dimension n of multivariate skew-normal distribution
R
un
time
[ms
]
q Figure 7.15 —Log-log plot of the runtime of the proposed algorithm for the skew- normal distribution based MAX-operation for a huge number of random variables
C
h
a
p
t
8
Conclusions
Tremendous advances in semiconductor process technology have created new delay test challenges for digital integrated circuits. The complexity of state-of-the-art manu- facturing processes does not only exacerbate the problem of process variability, it also makes today’s integrated circuits more prone to defects such as resistive shorts and opens. The resulting large delay variations severely degrade the quality and reliability of all delay tests. A delay test might detect a delay fault of fixed size in only a subset of all manufactured circuits, which can potentially result in a large number of test escapes. Statistical timing analysis is an integral component of any variation-aware delay test generation method and is required to analyse and predict the effectiveness of delay tests in a population of circuits which are functionally identical but have varying timing properties. Efficient statistical timing analysis of large circuits is a well known hard problem. Under the impact of delay variations, a path is sensitized by a particular test vector-pair only with some probability. Furthermore, the event that any of the sensitized paths has a path delay fault is also described by a probability. Recent variation-aware delay test generation methods must therefore be guided by the probability that at least one of the sensitized paths has a path delay fault. One of the most challenging problems of statistical timing analysis is the efficient computation of the statistical MAX-operation. The normal distribution based MAX-operation [Clark61] approximates the skewed distribution of the result with a normal distribution, leading to large approximation errors.
8.1 Contributions of this Work
This work targets key statistical timing analysis problems, which arise in delay test applications for innovative technology nodes. Novel and efficient statistical timing
analysis algorithms for path and small delay fault testing applications have been presented. In addition, accurate skew-normal distribution based SUM and MAX- operations are proposed, which provide the foundation for the efficient statistical delay fault simulation.
Probabilistic sensitization analysis is proposed to guide the delay test generation process into generating path delay fault tests, which are more tolerant towards delay variations. The analysis not only provides the location of the gate which blocks the propagation of the transition along the target path, but it also identifies those paths that are responsible for the invalidation of the delay test. Further important quality characteristics of the given test vector-pair can be efficiently computed by a Monte-Carlo simulation of a small subcircuit, which is constructed by the proposed analysis.
For the detection of small delay faults, two efficient algorithms for the computation of the target paths delay fault probability are proposed. The non-incremental algorithm provides high accuracy but may become inefficient if the delay test parameters are frequently modified. To minimize the computational cost after delay test parameter modifications, an efficient incremental algorithm has also been presented, which is suitable for the inner loop of automatic test pattern generation methods. Compared to extensive Monte-Carlo simulations, the experimental results show a very large speedup with only a small approximation error, which is mainly due to the impact of delay variations on the sensitization of the critical paths.
To minimize the error of block-based statistical timing analysis, the more accurate skew- normal distribution based SUM and MAX-operations are introduced. Compared to the normal distribution based MAX-operation [Clark61], the proposed MAX-operation is defined on the far more flexible skew-normal distribution, which allows the accurate approximation of the result with another skew-normal distribution. Although the worst case runtime complexity of the proposed algorithm is quadratic in the number of random variables, the runtime remains very small even for several hundreds of random variables. The superior accuracy and low runtime makes the skew-normal distribution based MAX-operation ideal for block-based statistical timing analysis and many other statistical timing analysis problems.
The experimental results demonstrate the high efficiency of the proposed algorithms.