The complexity of circuit value and network stability

Abstract

E r n s t W. M a y r Ashok S u b r a m a n i a n C o m p u t e r Science D e p a r t m e n t Stanford UniversityS t a n f o r d , C A 94305-2140Fachbereich Informatik (20) J . W. G o e t h e Universitat F r a n k f u r t a m Main 11West G e r m a n y U S A AbstractWe develop a method for non-trivially restricting fanout in a circuit. We study the complexity of the Circuit Value problem and a new problem, Network Stability, when fanout is limited. This leads to new classes of problems within P. We conjecture that the new classes are different from P a n d incomparable to n/C. One of these classes, CC, contains several natural complete problems, includingCircuit Value for comparator circuits, Lex-first Maximal Matching, and problems related to Stable Marriage and Stable Roommates. When fanout is appropriately limited, we get positive results: a parallel algorithm for Circuit Value that runs in time about the square root of t h e number of gates, a linear-time sequential algorithm for Network Stability, and logspace reductions between Circuit Value and Network Stability.

On the parallel complexity of hierarchical clustering and CC‐complete problems

“…On the other hand, no fast parallel algorithms are known for them, with the partial exception of √ n(log n) O(1) -time algorithms for the CCVP and some related problems. Such algorithms were independently discovered by Danny Soroker (unpublished) and by Mayr and Subramanian [18]. Richard Anderson (also unpublished) has improved the algorithms so as to use only √ n processors.…”

Matching and Vertex Packing: How “hard” are They?

“…(This will be the last class in our lattice diagram to be discussed.) This class is discussed at length in [193,192,1941 with perhaps the most succinct treatment in [221.…”

“…(This will be the last class in our lattice diagram to be discussed.) This class is discussed at length in [193,192,1941 with perhaps the most succinct treatment in [221.First we must define the circuit value problem--or CV: "Given a Boolean circuit with n inputs and a single output, together with a binary input value for each input gate, is the output = 1? "…”

Design and Implementation of Parallel Algorithms

“…The paper Subramanian [1989] shows that this problem may be viewed as a problem about fanout-limited circuits with feedback. Consequences include putting this problem into the class CC (one of the classes defined in Mayr and Subramanian [1989]), thus giving evidence that this problem has intermediate difficulty of paralleization.…”