5762022
Orthogonal Drawings with Few Layers
In this paper, we study 3-dimensional orthogonal graph drawings. Motivated by the fact that only a limited number of layers is possible in VLSI technology, and also noting that a small number of layers is easier to parse for humans, we study drawings where one dimension is restricted to be very small. We give algorithms to obtain point-drawings with 3 layers and 4 bends per edge, and algorithms to obtain box-drawings with 2 layers and 2 bends per edge. Several other related results are included as well. Our constructions have optimal volume, which we prove by providing lower bounds.
45354174
On the Parameterized Complexity of Layered Graph Drawing
We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight line-segments between vertices on adjacent layers. We prove that graphs admitting crossing-free h-layer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a linear-time algorithm to decide if a graph has a crossing-free h-layer drawing (for fixed h). This algorithm is extended to solve a large number of related problems, including allowing at most k crossings, or removing at most r edges to leave a crossing-free drawing (for fixed k or r). If the number of crossings or deleted edges is a non-fixed parameter then these problems are NP-complete. For each setting, we can also permit downward drawings of directed graphs and drawings in which edges may span multiple layers, in which case the total span or the maximum span of edges can be minimized. In contrast to the so-called Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers.
341729
Lower Bounds for the Number of Bends in Three-Dimensional Orthogonal Graph Drawings
In this paper we present the first non-trivial lower bounds for the total number of bends in 3-D orthogonal drawings of maximum degree six graphs. In particular, we prove lower bounds for the number of bends in 3-D orthogonal drawings of complete simple graphs and multigraphs, which are tight in most cases. These result are used as the basis for the construction of infinite classes of c-connected simple graphs and multigraphs (2 ≤ c ≤ 6) of maximum degree Δ (3 ≤ Δ ≤ 6) with lower bounds on the total number of bends for all members of the class. We also present lower bounds for the number of bends in general position 3-D orthogonal graph drawings. These results have significant ramifications for the '2-bends' problem, which is one of the most important open problems in the field.
32158792
An Algorithm for Three-Dimensional Orthogonal Graph Drawing
In this paper we present an algorithm for 3-dimensional orthogonal graph drawing based on the movement of vertices from an initial layout along the main diagonal of a cube. For an n-vertex m-edge graph with maximum degree six, the algorithm produces drawings with bounding box volume at most 2:37n3 and with a total of 7m/3 bends, using no more than 4 bends per edge route. For maximum degree five graphs the bounding box has volume n3 and each edge route has two bends. These results establish new bounds for 3-dimensional orthogonal graph drawing algorithms and improve on some existing bounds.
123081730
Simultaneous diagonal flips in plane triangulations
Simultaneous diagonal flips in plane triangulations are investigated. It is proved that every triangulation with at least six vertices has a simultaneous flip into a 4-connected triangulation, and that it can be computed in linear time. It follows that every triangulation has a simultaneous flip into a Hamiltonian triangulation. This result is used to prove that for any two n-vertex triangulations, there exists a sequence of O(log n) simultaneous flips to transform one into the other. The total number of edges flipped in this sequence is O(n). The maximum size of a simultaneous flip is then studied. It is proved that every triangulation has a simultaneous flip of at least 1/3(n − 2) edges. On the other hand, every simultaneous flip has at most n − 2 edges, and there exist triangulations with a maximum simultaneous flip of 6/7 (n − 2) edges.
122961893
Lower Bounds for the Number of Bends in Three-Dimensional Orthogonal Graph Drawings
23304788
On non context-free grammar forms
A grammar formF defines via a so-calledinterpretation mechanism, a family of languages(F). In this paper we establish that for many grammar formsF, (the family of context-free languages) implies(F)=RE (the family of recursively enumerable sets). We conjecture that this is also true in general. Because of this, it seems necessary to use restricted interpretations for non context-free grammar forms, a form giving then rise to a family We compare the obvious alternatives for restricting interpretations and focus attention on two promising alternatives,Q(F) andst(F) and their combinationQ, st(F). Using st-interpretations, surprising families can be generated and strong normal form results can be obtained. Closure results and decidability results are also given. UnderQ, st-interpretation, it is possible to characterize a number of well-known families of languages betweenCF andRE, including the families of EOL, ETOL, matrix and scattered context languages.
29528938
Caterpillars: A Context Specification Technique
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.
12975927
Uniform Interpretations of L Forms
This paper investigates interpretations of L forms in which the substitution defining the interpretation is uniform in terminal letters. Such uniform interpretations have several advantages over the ordinary ones, both from a mathematical and biological point of view. A number of different aspects of uniform interpretations are studied. It is felt that the results will shed new light on the theory of L forms and, in general, on the theory of L systems.
12937575
Application-specific protocols for user-level shared memory
Recent distributed shared memory (DSM) systems and proposed shared-memory machines have implemented some or all of their cache coherence protocols in software. One way to exploit the flexibility of this software is to tailor a coherence protocol to match an application's communication patterns and memory semantics. This paper presents evidence that this approach can lead to large performance improvements. It shows that application-specific protocols substantially improved the performance of three application programs-appbt, em3d, and barnes-over carefully tuned transparent shared memory implementations. The speed-ups were obtained on Blizzard, a fine-grained DSM system running on a 32-node Thinking Machines CM-5.<<ETX>>