Konstantin Skodinis

Computing Optimal Linear Layouts of Trees in Linear Time

We present a linear time algorithm which, given a tree, computes a linear layout optimal with respect to vertex separation. As a consequence optimal edge search strategies, optimal node search strategies, and optimal interval augmentations can be computed also in O(n) for trees. This improves the running time of former algorithms from O(n log n) to O(n) and answers two related open questions raised in [7] and [15].

Construction of linear tree-layouts which are optimal with respect to vertex separation in linear time

We present a linear time algorithm which, given a tree, computes a linear layout optimal with respect to vertex separation. As a consequence optimal edge search strategies, optimal node search strategies, and optimal interval augmentations can be computed also in O(n) for trees. This improves the running time of former algorithms from O(n log n) to O(n) and answers the open questions raised in [Ellis et al., Inform. Comput. 113 (1994) 50-79, Megiddo, J. Assoc. Comput. Mach. 35 (1) (1988) 18-44].

Finite graph automata for linear and boundary graph languages

Graph grammars can be regarded as a generalization of context-free grammars from strings to graphs. Over the past 30 years a rich theory of graph grammars and their languages has been developed. However, there are no graph automata. There is no duality between generative and recognizing devices, as it is known for the Chomsky hierarchy of formal languages.Here we introduce graph automata as devices for the recognition of sets of undirected node labelled graphs. A graph automaton consists of a finite state control, a finite set of instructions, and a collection of heads or guards. It reads an input graph in a systematic way and performs a graph search directed by the instructions. As our main results we show that finite graph automata recognize exactly the set of graph languages generated by linear NCE graph grammars and that alternating finite graph automata recognize exactly the languages of boundary graph grammars. Finally, we generalize some automata theoretic properties from string to graph automata, integrate the connectivity of graphs into graph automata, and explain why graph automata cannot be generalized to deal with dynamic edge relabellings and eNCE graph languages.

Emptiness problems of eNCE graph languages

We consider the complexity of the emptiness problem for various classes of graph languages defined by eNCE (edge label neighborhood controlled embedding) graph grammars. In particular, we show that the emptiness problem is undecidable for general eNCE graph grammars, DEXPTIME-complete for confluent and boundary eNCE graph grammars, PSPACE-complete for linear eNCE graph grammars, NL-complete for deterministic confluent, deterministic boundary, and deterministic linear eNCE graph grammars. The exponential time algorithm for deciding emptiness of confluent eNCE graph grammars is based on an exponential time transformation of a confluent eNCE graph grammar into a nonblocking confluent eNCE graph grammar generating the same language.

The Bounded Degree Problem for eNCE Graph Grammars

The complexity of the bounded degree problem is analyzed for graph languages generated by eNCE graph grammars. In particular, the bounded degree problem is shown to be undecidable for eNCE graph grammars, DEXPTIME-complete for confluent/boundary eNCE graph grammars, PSPACE-complete for linear eNCE graph grammars, and P-complete for non-blocking eNCE graph grammars. In our main theorem we show that the bounded degree problem is NL-complete for reduced non-blocking eNCE graph grammars. Many of the shown results carry over to other types of graph grammars.

Graph Automata for Linear Graph Languages

We introduce graph automata as devices for the recognition of linear graph languages. A graph automaton is the canonical extension of a finite state automaton recognizing a set of connected labeled graphs. It consists of a finite state control and a collection of heads, which search the input graph. In a move the graph automaton reads a new subgraph, checks some consistency conditions, changes states and moves some of its heads beyond the read subgraph. It proceeds such that the set of currently visited edges is an edge-separator between the visited and the yet undiscovered part of the input graph. Hence, the graph automaton realizes a graph searching strategy. Our main result states that finite graph automata recognize exactly the set of graph languages generated by connected linear NCE graph grammars.

Efficient Analysis of Graphs with Small Minimal Separators

Abstract We consider the class C* of graphs whose minimal separators have a fixed bounded size. We give an O(nm)-time algorithm computing an optimal tree-decomposition of every graph in C* with n vertices and m edges. Furthermore we make evident that many NP-complete problems are solvable in polynomial time when restricted to this class. Both claims hold although C* contains graphs of arbitrarily large tree-width.

Neighborhood-Preserving Node Replacements

We introduce a general normal form for eNCE graph grammars and use it for a characterization of the class of all neighborhood-preserving eNCE graph languages in terms of a weak notion of order-independency.

The Bounded Tree-Width Problem of Context-Free Graph Languages

We show that the following (equivalent) problems are P-complete: 1. Does a given confluent NCE graph grammar only generate graphs of bounded tree-width? and 2. is the graph language generated by a given confluent NCE graph grammar an HR language?

The Bounded Degree Problem for Non-Obstructing eNCE Graph Grammars

A graph grammar is called non-obstructing if each graph G derivable from the axiom can derive a terminal graph. In this paper, the bounded degree problem for non-obstructing eNCE graph grammars is proved to be in the complexity class NL.