Implicit enumeration of strongly connected components and an application to formal verification

Abstract

V. CONCLUSION We have presented a complete planar clock routing system called ACTD, which is implemented by employing heuristic techniques. Our first algorithm, called CLE routing, is implemented to solve the overlapping of clock net routing and prevent the crossing in the clock net. The second algorithm, called POA routing, reconstructs the clock tree using heuristics, to avoid obstacles with the scheme of changing tapping points. We have validated our techniques experimentally using test circuits. Future work includes extending the CLE routing and POA routing approach to include well-balanced clock trees, since our current implementation constructs a planar clock routing in the presence of obstacles with allowing the path length skew. Extending our present results to the minimum skew goal presents an intriguing direction for future work. We expect that partitioning line adjustment based on the weight of load and tapping point adjustment for a given planar clock tree will help to minimize clock skew.

A Truly Symbolic Linear-Time Algorithm for SCC Decomposition

“…In this section we report our experimental evaluation of the new algorithms Chain and ColoredChain on three classes of benchmarks. We compared their performance to the standard algorithms FwdBwd [30], LockStep [7] (and its recent colored variant [6]) and Skeleton [17]. Our experiments were run on a Linux machine with 2.4GHz CPU speed and 60GB of memory (using 1 core).…”

“…Indeed, Chain simply extends FwdBwd to accept as an argument a set of vertices K, among which to choose a pivot in the current recursive call. It is perhaps surprising that such a simple mechanism has been elusive for decades, as all previous efforts [30,7,17] relied on more elaborate procedures to either reduce or refine the O(n 2 ) time bound. That being said, our new mechanism is somewhat insightful and with a non-trivial complexity analysis.…”

“…The key challenge towards efficient symbolic SCC algorithms is the seeming difficulty to traverse the input graph G in a depth-first fashion, which is the technical underpinning of the O(n)-time explicit SCC algorithms. Nevertheless, a series of improvements have been made in this direction: (i) a variant of FwdBwd was shown in [30] to run in time O(δn), where δ is the diameter of G, and only becomes quadratic when δ = Θ(n), (ii) the LockStep algorithm [7] has complexity O(n log n), while (iii) the Skeleton algorithm with complexity O(n) is provably optimal [11]. Practical improvements based on heuristics have also been proposed [29,16,31].…”

Interconnected Systems Modelling in Food Industry: General Solution Scheme and Stability Conditions for Linear Time-Invariant Systems

“…A minimum number of subsystems is desired in each group because this minimizes the number of equations that must be solved simultaneously. For complex ISM, an algorithm [57,58] is necessary to accomplish the partitioning. No symbolic or other type of manipulation is required.…”