11947788
Context-Dependent L Forms
The notions of an EIL form and its interpretations are introduced. The resulting families of L systems and languages are studied from a number of different points of view. Some of the main results are: the family of context-free languages is generated by an EIL form (contrary to the earlier results concerning context-free L forms) and, as regards reducibility in the amount of context, EIL forms lie between IL and EIL systems.
39074040
Pure Grammars
7533318
Completeness of Context-Free Grammar Forms
The notions of a grammar form and g-interpretation were first introduced in [2]. However, it quickly became apparent, particularly for EOL forms [5], that (strict) interpretations had a wider interest as well as being better motivated mathematically. The study of parsing of grammar forms detailed in [4], the basic investigation in [8] on linear completeness amongst other results, the study of density in [9, lo] and the connections with graph theory shown in [ 1 l] have affirmed this position. The present paper continues the approach of [8] focussing on the characterization of complete grammar forms, that is grammar forms which generate all’ context-free languages. Some decidability problems will also be discussed. For further morivation and background material we refer the reader to [8, 141, while for all unexplained concepts in language theory, see [ 131. After giving the necessary definitions in the remainder of this section we characterize complete grammar forms in Section 2, 3 and 4. In Section 2 we introduce the central concept of expansion spectrum and in Section 3 the recently proved super normal form theorem [ 121, while in Section 4 the characterization is completed and generators and hierarchies are briefly mentioned. Consider context-free grammars G = (V, C, P, S), where Z is the alphabet of terminals, VZ the alphabet of nonterminals, P is the set of productions and
39838781
Context-Free Grammar Forms with Strict Interpretations
This paper begins a systematic investigation of context-free grammar forms using the mechanism of “strict” interpretations: the interpretations of terminal letters are defined exactly as those of nonterminal letters. Thus, strict interpretations are more closely related to the master grammar than the g-interpretations investigated in the original grammar form paper [A. Cremers and S. Ginsburg, Context-free grammar forms, J. Comput. System Sci. 11 (1975), 86–116]. The main results of this paper concern reducibility, closure properties, generators and hierarchies of language families, as well as the characterization of grammar forms generating (under strict interpretation) the families of regular and linear languages.
21847551
Bounded Delay L Codes
Abstract Three natural notions of bounded delay for L codes are investigated. All interrelations, many of them rather surprising, between ordinary codes of bounded delay and L codes of bounded delay of various types are established.
12582971
Stabbing line segments
An algorithm for the geometric problem of determining a line (called a stabbing line) which intersects each ofn given line segments in the plane is presented. As a matter of fact, the algorithm computes a description of all stabbing lines. A purely geometric fact is proved which infers that this description requiresO(n) space to be specified. Our algorithm computes it inO(n logn) time which is optimal in the worst case.Using the description of the stabbing lines, we are able to decide inO(logn) time whether or not a specified line is a stabbing line. Finally, the problem of maintaining the description of all stabbing lines while inserting and deleting line segments is addressed.
12635352
Planar Strong Visibility
Strong visibility is a generalization of standard visibility, defined with respect to a fixed set of line orientations. We investigate computational properties of this generalized visibility, as well as the related notion of strong convexity, and describe algorithms for the following tasks: 1. Testing the strong visibility of two points in a polygon. 2. Finding the strong convex hull of a point set or polygon. 3. Constructing the strong kernel of a polygon. 4. Identifying the points that are strongly visible from a given point.
125792
Upward Three-Dimensional Grid Drawings of Graphs
A three-dimensional grid drawing of a graph is a placement of the vertices at distinct points with integer coordinates, such that the straight line segments representing the edges do not cross. Our aim is to produce three-dimensional grid drawings with small bounding box volume. Our first main result is that every $n$-vertex graph with bounded degeneracy has a three-dimensional grid drawing with $\mathcal{O}\left({n^{3/2}}\right)$ volume. This is the largest known class of graphs that have such drawings. A three-dimensional grid drawing of a directed acyclic graph (dag) is upward if every arc points up in the $\textsf{z}$-direction. We prove that every dag has an upward three-dimensional grid drawing with $\mathcal{O}\left({n^3}\right)$ volume, which is tight for the complete dag. The previous best upper bound was $\mathcal{O}\left({n^4}\right)$. Our main result concerning upward drawings is that every $c$-colourable dag ($c$ constant) has an upward three-dimensional grid drawing with $\mathcal{O}\left({n^2}\right)$ volume. This result matches the bound in the undirected case, and improves the best known bound from $\mathcal{O}\left({n^3}\right)$ for many classes of dags, including planar, series parallel, and outerplanar. Improved bounds are also obtained for tree dags. We prove a strong relationship between upward three-dimensional grid drawings, upward track layouts, and upward queue layouts. Finally, we study upward three-dimensional grid drawings with bends in the edges.
7767717
On the Chromatic Number of the Visibility Graph of a Set of Points in the Plane
AbstractThe visibility graph V(P) of a point set P \subseteq R2 has vertex set P, such that two points v,w ∈ P are adjacent whenever there is no other point
in P on the line segment between v and w. We study the chromatic number of
V(P). We characterise the 2- and 3-chromatic visibility graphs. It is an open
problem whether the chromatic number of a visibility graph is bounded by its clique
number. Our main result is a super-polynomial lower bound on the chromatic number
(in terms of the clique number).
448131
On the Complexity of the Balanced Vertex Ordering Problem
We consider the problem of finding a balanced ordering of the vertices of a graph. More precisely, we want to minimise the sum, taken over all vertices v, of the difference between the number of neighbours to the left and right of v. This problem, which has applications in graph drawing, was recently introduced by Biedl et al. [Discrete Applied Math. 148:27―48, 2005]. They proved that the problem is solvable in polynomial time for graphs with maximum degree three, but NP-hard for graphs with maximum degree six. One of our main results is to close the gap in these results, by proving NP-hardness for graphs with maximum degree four. Furthermore, we prove that the problem remains NP-hard for planar graphs with maximum degree four and for 5-regular graphs. On the other hand, we introduce a polynomial time algorithm that determines whetherthere is a vertex ordering with total imbalance smaller than a fixed constant, and a polynomial time algorithm that determines whether a given multigraph with even degrees has an 'almost balanced' ordering.
14349700
Normal form algorithms for extended context-free grammars
We investigate the complexity of a variety of normal-form transformations for extended context-free grammars, where by extended we mean that the set of right-hand sides for each nonterminal in such a grammar is a regular set. The study is motivated by the implementation project GraMa which will provide a C++ toolkit for the symbolic manipulation of context-free objects just as Grail does for regular objects. Our results generalize known complexity bounds for context-free grammars but do so in nontrivial ways. Specifically, we introduce a new representation scheme for extended context-free grammars (the symbol-threaded expression forest), a new normal form for these grammars (dot normal form) and new regular expression algorithms. Copyright 2001 Elsevier Science B.V.