On the Properties of a Tree-Structured Server Process

Let X0 be a nonnegative integer-valued random variable and let an independent copy of X0 be assigned to each leaf of a binary tree of depth k. If X0 and X0′ are adjacent leaves, let X1=(X0−1)++(X0′−1)+ be assigned to the parent node. In general, if Xj and Xj′ are assigned to adjacent nodes at level j = 0,⋯, k − 1, then Xj and Xj′ are, in turn, independent and the value assigned to their parent node is then Xj+1=(Xj−1)++(Xj′−1)+. We ask what is the behavior of Xk as k→∞. We give sufficient conditions for Xk→∞ and for Xk→0 and ask whether these are the only nontrivial possibilities. The problem is of interest because it asks for the asymptotics of a nonlinear transform which has an expansive term (the + in the sense of addition) and a contractive term (the + in the sense of positive part)

An Algorithmic Version of the Blow-up Lemma

Recently we have developed a new method in graph theory based on the Regularity Lemma. The method is applied to find certain spanning subgraphs in dense graphs. The other main general tool of the method, beside the Regularity Lemma, is the so-called Blow-up Lemma ([24]). This lemma helps to find bounded degree spanning subgraphs in "-regular graphs. Our original proof of the lemma is not algorithmic, it applies probabilistic methods. In this paper we provide an algorithmic version of the Blow-up Lemma. The desired subgraph, for an n-vertex graph, can be found in time O(nM (n)), where M (n) = O(n2:376) is the time needed to multiply two n by n matrices with 0,1 entries over the integers. We show that the algorithm can be parallelized and implemented in NC^5