Wojciech Rytter

A Optimal Parallel Algorithm for Dynamic Epression Evaluation and its Applications

We describe a deterministic parallel algorithm to compute algebraic expressions in log n time using n/log(n) processors on a parallel random access machine without write conflicts (P-RAM) with no free preprocessing. The input to our algorithm is a string, given by an array, of the expression. Such a form for the input enables a consecutive numbering of the operands in the expression in log(n) time with n/log(n) processors. This corresponds to a consecutive numbering of leaves in the tree of the expression which further enables a suitable partitioning of the leaves into small segments. We improve the result of Miller and Reif (1985), who described an optimal parallel randomized algorithm. Our algorithm can be used to construct optimal parallel algorithms for the recognition of two nontrivial subclasses of context-free languages: bracket and input-driven languages. These languages are the most complicated context-free languages known to be recognizable in deterministic logarithmic space. This strengthens the result of Matheyses and Fiduccia (1982) who constructed an almost optimal parallel algorithm for Dyck languages, since Dyck languages are a proper subclass of input-driven languages.

On the decidability of some problems about rational subsets of free partially commutative monoids

Let I=A+B be a partially commutative alphabet such that two letters commute if one of them belongs to A and the other one belongs to B. Let M=A* B* denote the free partially commutative monoid generated by I. We consider the following six problems for rational (given by regular expressions) subsets X, Y of M: n nQ1:X ∩ Y= O? nQ2: X ⊆ Y? nDI X = Y? nQ4: X = M? nL25: M — X finite? nX is recognizable? n nIt was proved by Choffrut (see [2]) that all these problems are undecidable if Card A > 1 and Card B > 1, and they are decidable if Card A = Card B = 1 (Card U denotes the cardinality of U). It was conjectured (see [2], p 79) that these problems are decidable in the remaining cases, where Card A = 1 and Card B > 1. In this paper we show that if Card A = 1 and Card B > 1 then the problem 01 is decidable, and problems Q2-06 are undecidable. Our paper is an application of results concerning reversal-bounded nondeterministic multicounter machines and nondeterministic general sequential machines.

On the complexity of parallel parsing of general context-free languages

Let T(n) be the time to recognize context-free languages on a parallel random-access machine without write conflicts (P-RAM) using a polynomial number of processors. We assume that T(n) = Ω(log n). Let P(n) be the time to compute a representation of a parsing tree for strings of length n using a polynomial number of processors. Then we prove P(n) = O(T(n)). n nA related result is a parallel time log n computation of the transitive closure of directed graphs having special structure.

An application of Mehlhorn's algorithm for bracket languages to log(n) space recognition of input-driven languages

The class of input-driven languages (idls, for short) is presently the most complicated subclass of context-free languages known to be recognizable in the deterministic log(n) space. The idls are a generalization of bracket languages; the strings of a bracket language contain an explicit information about the parse tree, while the strings of an idl contain an information about the behaviour of the pushdown store. Therefore, one could expect that there is a log(n) space recognizer of idls which is a natural generalization of the corresponding recognizer of bracket languages. However, the algorithm presented in 1983 by Braunmiahl and Verbeek [2] for log(n) space recognition of idls uses ideas quite different from those of the algorithm presented in 1976 by Mehlhorn [3] for log(n) space recognition of bracket languages. In this paper we show how a log(n) space recognizer for idls can be derived in a natural way from Mehlhorns algorithm. This is an exercise in the efficient elimination of recursion.

Optimal parallel parsing of bracket languages

We prove that the parsing problem for bracket context-free languages can be solved in log n time using n/log n processors on a parallel random access machine without write conflicts (P-RAM). On the way we develop a new general technique for tree compression based on the bracket structure of the tree.

Fast Parallel Algorithms for Optimal Edge-Colouring of some Tree-structured Graphs

We show that every Halin and every outerplanar graph can be optimally edge-coloured in polylog time using a polynomial number of processors on a parallel random access machine without write conflicts (P-RAM). Our algorithms are designed using the divide and conquer technique in a parallel setting.