Drawing graphs is an important problem that combines flavors of computational geometry and graph theory. Applications can be found in a variety of areas including circuit layout, network management, software engineering, and graphics. The main contributions of this paper can be summarized as follows: ffl We devise a model for dynamic graph algorithms, based on performing queries and updates on an implicit representation of the drawing, and we show its applications. ffl We present several efficient dynamic drawing algorithms for trees, series-parallel digraphs, planar st-digraphs, and planar graphs. These algorithms adopt a variety of representations (e.g., straight-line, polyline, visibility), and update the drawing in a smooth way.

structure prediction with pseudoknots

Channel assignment problems in the time, frequency and code domains have hitherto been studied separately. Exploiting the similarity of constraints that characterize assignments within and across these domains, we introduce the first unified framework for the study of assignment problems. Our framework identifies eleven atomic constraints underlying most current and potential assignment problems, and characterizes a problem as a combination of these constraints. Based on this framework, we present a unified algorithm for efficient (T/F/C)DMA channel assignments to network nodes or to inter-nodal links in a (multihop) wireless network. The algorithm is parametrized to allow for use as three different variants - RAND, MNF, and PMNF. We provide comprehensive theoretical analysis characterizing the worst-case performance of our algorithm for several classes of problems. In particular, we show that the assignments produced by the PMNF variant are proportional to the thickness of the network...

In this paper, we study the protein threading problem, which was proposed for finding a folded 3D protein structure from an amino acid sequence. Since this problem was already proved to be NP-hard by Lathrop, we study polynomial time approximation algorithms. First we show that the protein threading problem is MAX SNP-hard. Next we show that the protein threading problem can be approximated within a factor 4 for a special case in which a graph representing interaction between residues (amino acids) is planar. This case corresponds to a fi-sheet substructure, which appears in most protein structures. 1 Introduction The protein folding problem is, given an amino acid sequence (a string), to find its correctly folded 3D protein structure. It is one of the most important computational problems in molecular biology. Although this problem can be defined as a minimization problem, it is too hard to be solved directly. Recently, an indirect approach called inverse folding was proposed [2, 4,...

There is much current interest among researchers to find algorithms that will draw graphs in three dimensions. It is well known that every 3-connected planar graph can be represented as a strictly convex polyhedron. However, no practical algorithms exist to draw a general 3-connected planar graph as a convex polyhedron. In this paper we review the concept of a stressed graph and how it relates to convex polyhedra; we present a practical algorithm that uses stressed graphs to draw 3-connected planar graphs as strictly convex polyhedra; and show some examples. Key words: graph, stressed graph, convex polyhedron, reciprocal polyhedron 1 Introduction It is well known that 3-connected planar graphs can be drawn as convex polyhedra. However, no practical algorithms exist to draw general 3-connected planar graphs as convex polyhedra. The two-dimensional (2D) drawing in Figure 1 is 3-connected and planar, and the corresponding polyhedron is drawn in Figure 2 as three different views. The 2D ...

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We study a variety of geometric network optimization problems on a set of points, in which we are given a resource bound, B, on the total length of the network, and our objective is to maximize the number of points visited (or the total "value" of points visited). In particular, we resolve the well-publicized open problem on the approximability of the rooted "orienteering problem" for the case in which the sites are given as points in the plane and the network required is a cycle. We obtain a 2-approximation for this problem. We also obtain approximation algorithms for variants of this problem in which the network required is a tree (3-approximation) or a path (2-approximation). No prior approximation bounds were known for any of these problems. We also obtain improved approximation algorithms for geometric instances of the unrooted orienteering problem, where we obtain a 2-approximation for both the cycle and tree versions of the problem on points in the plane, as well as a ...

We develop a linear time algorithm for the following problem: Given a graph G and a hierarchical clustering of the vertices, such that all clusters induce connected subgraphs, determine whether G can be embedded into the plane, such that no cluster has a hole. This is an improvement to the O(n 2 )-algorithm of Q.W. Feng et al. [6] and the algorithm of Lengauer [12].

We propose a new method for constructing all-hexahedral finite element meshes. The core of our method is to build up a compatible combinatorial cell complex of hexahedra for a solid body which is topologically a ball and for which a quadrilateral surface mesh of a certain structure is prescribed. The step-wise creation of the hex complex is guided by the cycle structure of the combinatorial dual of the surface mesh. Our method transforms the graph of the surface mesh iteratively by changing the dual cycle structure until we get the surface mesh of a single hexahedron. Starting with a single hexahedron and reversing the order of the graph transformations, each transformation step can be interpreted as adding one or more hexahedra to the so far created hex complex. Given an arbitrary solid body, we first decompose it into simpler subdomains equivalent to topological balls by adding virtual 2-manifolds. Second, we determine a compatible quadrilateral surface mesh for all created...

Abstract. Graphs are widely used for information visualization purposes, since they provide a natural and intuitive representation of complex abstract structures. The automatic generation of drawings of graphs has applications a variety of fields such as software engineering, database systems, and graphical user interfaces. In this paper, we survey algorithmic techniques for graph drawing that support the expression and satisfaction of user-defined constraints. 1.