Let D(n,r) be a random r-out regular directed multigraph on the set of vertices 1,...,n . In this work, we establish that for every r � 2, there existsr > 0 such that diam(D(n,r)) = (1 + �r + o(1))log r n. Our techniques also allow us to bound some extremal quantities related to the stationary distribution of a simple random walk on D(n,r). In particular, we determine the asymptotic behaviour ofmax andmin, the maximum and the minimum values of the stationary distribution. We show that with
One possibility is to study the average case. This approach can be formalized by considering regular languages defined by random DFAs. In particular, one can ask for an algorithm that with high probability (as the number of the states in the DFA goes to infinity) can learn the regular language recognized by a random DFA. There exists evidence suggesting that such relaxation might not be enough to achieve efficient learning in general: it was recently showed by that generic instances of DFA (as well as decision trees and DNF formulas) are hard to learn from statistical queries when examples can be sampled from an arbitrary distribution @cite_4 . Nevertheless, prior to 's result it was showed that generic decision trees and generic DNF formulas can be efficiently learned when samples are drawn according to the uniform distribution @cite_3 @cite_11 .
@cite_4 @cite_3 @cite_11
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We show that random DNF formulas, random log-depth decision trees and random deterministic finite acceptors cannot be weakly learned with a polynomial number of statistical queries with respect to an arbitrary distribution on examples. We consider three natural models of random logarithmic depth decision trees over Boolean variables. We give an efficient algorithm that for each of these models learns all but an inverse polynomial fraction of such trees using only uniformly distributed random examples from 0,1 n. The learning algorithm constructs a decision tree as its hypothesis. We show that randomly generated c log(n)-DNF formula can be learned exactly in probabilistic polynomial time using randomly generated examples. Our notion of randomly generated is with respect to a uniform distribution. To prove this we extend the concept of well behaved c log(n)-Monotone DNF formulae to c log(n)-DNF formulae, and show that almost every DNF formula is well-behaved, and that there exists a probabilistic polynomial time algorithm that exactly learns all well behaved c log(n)-DNF formula. This is the first algorithm that properly learns (non-monotone) DNF with a polynomial number of terms from random examples alone.