14560983
Lower bounds for hot-potato permutation routing on trees
In this paper we consider hot-potato packet routing on trees. As a lower bound, for all sufficiently largen we construct a permutation routing problem on an n-node tree for which an oblivious greedy hot-potato algorithm requires at least 2n o n time steps. This lower bound is also valid for the minimumdistance heuristic. Applying the charging argument of Borodin et al. [8] we establish that any greedy hot-potato algorithm routes a permutation on a tree within 2 n 1 steps.
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
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.
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.
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.
30874363
Synchronized EOL Forms Under Uniform Interpretation
— The aim ofthis paper is thejurther study oj EOLjorms under uniform interprétation. We are naturally led to the considération oj synchronized EOLjorms, which become the central thème of this paper. We prove: a ""strongest"" possible normaljorm resuit, the spanning normalJorm theorem; the non-existence oj uni-generatorsjor "reasonable" language amilies, which include thefinite, regular and context-jree j amilies; and a char act erization oj those ynchronized jorms which, under uniform interprétation, yield all regular languages. In closing we remark on three topics which are worthy oj separate investigation. Résumé. — Le but de cet article est de poursuivre l'étude des EOL-jormes sous des interprétations unijormes. Nous sommes conduits naturellement à considérer les EOL-jormes synchronisées qui deviennent le thème central de cet article. Nous établissons : un résultat sur la forme normale la plus « serrée » possible, le théorème sur la forme normale recouvrante, la non-existence d'uni-générateurs pour des jamilles « raisonnables » de langages qui comprennent les langages finis, rationnels et algébriques; et une caractérisation des formes synchronisées qui donnent, par interprétation unijorme, tous les langages rationnels. Pour terminer, nous effleurons trois domaines qui méritent une étude séparée.
3174555
A characterization of Thompson digraphs
A finite-state machine is called a Thompson machine if it can be constructed from an empty-free regular expression using the construction of Thompson as modified by Hopcroft and Ullman. We call the underlying digraph of a Thompson machine a Thompson digraph. We characterize Thompson digraphs and we give an algorithm that generates an equivalent regular expression from a Thompson machine that has size linear in the total number of states and transitions. Although the algorithm is simple, it is novel in that the usual constructions of equivalent regular expressions from finite-state machines produce regular expressions that have size exponential in the size of the given machine, in the worst case. The algorithm provides a tentative first step in the construction of small expressions from finite-state machines.
13899040
The Regularity of Two-Way Nondeterministic Tree Automata Languages
We establish that regularly extended two-way nondeterministic tree automata with unranked alphabets have the same expressive power as regularly extended nondeterministic tree automata with unranked alphabets alphabets. We obtain this results by establishing regularly extended versions of a congruence on trees and of a congruence on, so called, views. Our motivation for the study of these tree models is the Extensible Markup Language (XML), a metalanguage for defining document grammars. Such grammars have regular sets of right-hand sides for their productions and tree automata provide an alternative and useful modeling tool for them. In particular, we believe that they provide a useful computational model for what we call caterpillar expressions.
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.
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.