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.
29528938
Caterpillars: A Context Specification Technique
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.
205885836
On Predecessors of Finite Languages
We continue the study of finite grammatical families by characterizing when a family has a predecessor. This characterization implies that (i) the existence of predecessors for finite families is decidable and (ii) the collection of context-free grammatical families, each family of which contains all regular languages, is not maximal dense.
12322077
Bounded Degree Book Embeddings and Three-Dimensional Orthogonal Graph Drawing
A book embedding of a graph consists of a linear ordering of the vertices along a line in 3-space (the spine), and an assignment of edges to half-planes with the spine as boundary (the pages), so that edges assigned to the same page can be drawn on that page without crossings. Given a graph G = (V,E), let f : V → ℕ be a function such that 1 ≤ f(υ) ≤ deg(υ). We present a Las Vegas algorithm which produces a book embedding of G with \( O(\sqrt {|E| \cdot \max _\upsilon \left\lceil {\deg (\upsilon )/f(\upsilon )} \right\rceil } ) \) pages, such that at most f(v) edges incident to a vertex v are on a single page. This algorithm generalises existing results for book embeddings. We apply this algorithm to produce 3-D orthogonal drawings with one bend per edge and O(∣V ∣3/2∣E∣) volume, and single-row drawings with two bends per edge and the same volume. In the produced drawings each edge is entirely contained in some Z-plane; such drawings are without so-called cross-cuts, and are particularly appropriate for applications in multilayer VLSI. Using a different approach, we achieve two bends per edge with O(∣V ∣∣E∣) volume but with cross-cuts. These results establish improved bounds for the volume of 3-D orthogonal graph drawings.
2392498
Partitions of Complete Geometric Graphs into Plane Trees
Consider the following open problem: does every complete geometric graph K2n have a partition of its edge set into n plane spanning trees? We approach this problem from three directions. First, we study the case of convex geometric graphs. It is well known that the complete convex graph K2n has a partition into n plane spanning trees. We characterise all such partitions. Second, we give a sufficient condition, which generalises the convex case, for a complete geometric graph to have a partition into plane spanning trees. Finally, we consider a relaxation of the problem in which the trees of the partition are not necessarily spanning. We prove that every complete geometric graph Kn can be partitioned into at most $n-\sqrt{n/12}$ plane trees.
5331533
Balanced vertex-orderings of graphs
In this paper we consider the problem of determining a balanced ordering of the vertices of a graph; that is, the neighbors of each vertex v are as evenly distributed to the left and fight of v as possible. This problem, which has applications in graph drawing for example, is shown to be NP- hard, and remains NP- hard for bipartite simple graphs with maximum degree six. We then describe and analyze a number of methods for determining a balanced vertex-ordering, obtaining optimal orderings for directed acyclic graphs, trees, and graphs with maximum degree three. For undirected graphs, we obtain a 13/8-approximation algorithm. Finally we consider the problem of determining a balanced vertex-ordering of a bipartite graph with a fixed ordering of one bipartition. When only the imbalances of the fixed vertices count, this problem is shown to be NP-hard. On the other hand, we describe an optimal linear time algorithm when the final imbalances of all vertices count. We obtain a linear time algorithm to compute an optimal vertex-ordering of a bipartite graph with one bipartition of constant size.
9788732
On generators and generative capacity of EOL forms
SummaryIn this paper we study the generative capacity of EOL forms from two different points of view. On the one hand, we consider the generative capacity of special EOL forms which one could call ‘linear like’ and ‘context free like’, establishing the existence of a rich variety of non-regular sub-EOL language families. On the other hand, we propose the notion of a ‘generator’ L of a language family ℒ We mean by this that any synchronized EOL system generating L generates — if understood as an EOL form — all languages of ℒ. We characterize the generators of the family of regular languages, and prove that other well known language families do not have generators.