Scattered one-counter languges have rank less than ω2\omega^2

A linear ordering is called context-free if it is the lexicographic ordering of some context-free language and is called scattered if it has no dense subordering. Each scattered ordering has an associated ordinal, called its rank. It is known that scattered context-free (regular, resp.) orderings have rank less than $\omega^\omega$ ($\omega$, resp). In this paper we confirm the conjecture that one-counter languages have rank less than $\omega^2$

DFS is unsparsable and lookahead can help in maximal matching

In this paper we study two problems in the context of fully dynamic graph algorithms that is, when we have to handle updates (insertions and removals of edges), and answer queries regarding the current graph, preferably with a better time bound than that when running a classical algorithm from scratch each time a query arrives. In the first part we show that there are dense (directed) graphs having no nontrivial strong certificates for maintaining a depth-first search tree, hence the so-called sparsification technique cannot be applied effectively to this problem. In the second part, we show that a maximal matching can be maintained in an (undirected) graph with a deterministic amortized update cost of O(log m) (where m is the all-time maximum number of the edges), provided that a lookahead of length m is available, i.e. we can “take a peek” at the next m update operations in advance

On the Order Type of Scattered Context-Free Orderings

We show that if a context-free grammar generates a language whose lexicographic ordering is well-ordered of type less than $\omega^2$, then its order type is effectively computable.Comment: In Proceedings GandALF 2019, arXiv:1909.05979. arXiv admin note: text overlap with arXiv:1907.1157

Regular expressions for muller context-free languages

Muller context-free languages (MCFLs) are languages of countable words, that is, labeled countable linear orders, generated by Muller context-free grammars. Equivalently, they are the frontier languages of (nondeterministic Muller-)regular languages of infinite trees. In this article we survey the known results regarding MCFLs, and show that a language is an MCFL if and only if it can be generated by a so-called µη-regular expression

Lookahead can help in maximal matching

In this paper we study a problems in the context of fully dynamic graph algorithms that is, when we have to handle updates (insertions and removals of edges), and answer queries regarding the current graph, preferably in a better time bound than running a classical algorithm from scratch each time a query arrives. We show that a maximal matching can be maintained in an (undirected) graph with a deterministic amortized update cost of O(log m) (where m is the all-time maximum number of the edges), provided that a lookahead of length m is available, i.e. we can “peek” the next m update operations in advance