52806192
Minimising the Number of Bends and Volume in 3-Dimensional
Orthogonal Graph Drawings with a Diagonal Vertex Layout
Abstract 
A 3-dimensional orthogonal drawing of a graph with maximum
degree at most 6, positions the vertices at grid-points in the 3-dimensional
orthogonal grid, and routes edges along grid-lines such that edge routes only
intersect at common end-vertices. Minimising the number of bends and the volume
of 3-dimensional orthogonal drawings are established criteria for measuring the
aesthetic quality of a given drawing. In this paper we present two algorithms
for producing 3-dimensional orthogonal graph drawings with the vertices
positioned along the main diagonal of a cube, so-called diagonal
drawings. This vertex-layout strategy was introduced in the 3-BENDS
algorithm of Eades et al. [Discrete Applied Math.
103:55–87, 2000]. We show that minimising the number of bends in a
diagonal drawing of a given graph is NP-hard. Our first algorithm minimises the
total number of bends for a fixed ordering of the vertices along the diagonal
in linear time. Using two heuristics for determining this vertex-ordering we
obtain upper bounds on the number of bends. Our second algorithm, which is a
variation of the above-mentioned 3-BENDS algorithm, produces 3-bend
drawings with n3+o(n3) volume, which is the best known upper bound for
the volume of 3-dimensional orthogonal graph drawings with at most three bends per
edge. 


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.
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.
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.
122961893
Lower Bounds for the Number of Bends in Three-Dimensional Orthogonal Graph Drawings
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.
5598249
MSW Spaces
1926264
L Codes and Number Systems
Abstract The notion of an L code is introduced as a generalization of the customary notion of a code. Attention is focused on unary L codes. where the target alphabet consists of one letter only. Unary L codes can be identified with such number systems where the representation is unique. Surprisingly many unary morphisms are L codes. Furthermore, even for non-L-codes there is only one ‘exceptional length’ We also give a complete classification for the binary case.
29528938
Caterpillars: A Context Specification Technique
195884052
Token Coherence: decoupling performance and correctness
Many future shared memory multiprocessor servers will both target commercial workloads and use highly integrated "glueless" designs. Implementing low latency cache coherence in these systems is difficult, because traditional approaches either add indirection for common cache to cache misses (directory protocols) or require a totally ordered interconnect (traditional snooping protocols). Unfortunately, totally ordered interconnects are difficult to implement in glueless designs. An ideal coherence protocol would avoid indirections and interconnect ordering; however, such an approach introduces numerous protocol races that are difficult to resolve. We propose a new coherence framework to enable such protocols by separating performance from correctness. A performance protocol can optimize for the common case (i.e., absence of races) and rely on the underlying correctness substrate to resolve races, provide safety, and prevent starvation. We call the combination Token Coherence, since it explicitly exchanges and counts tokens to control coherence permissions. We develop TokenB, a specific Token Coherence performance protocol that allows a glueless multiprocessor to both exploit a low latency unordered interconnect (like directory protocols) and avoid indirection (like snooping protocols). Simulations using commercial workloads show that our new protocol can significantly outperform traditional snooping and directory protocols.
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