Cologne Excellence Cluster on Cellular Stress Responses in Aging Associated Diseases
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Optimal Pants Decompositions and Shortest Homotopic Cycles on an Orientable Surface
No full textA pants decomposition of a compact orientable surface M is a set of disjoint simple cycles which cuts M into pairs of pants, i.e., spheres with three boundaries. Assuming M is a polyhedral surface, with weighted vertex-edge graph G, we consider combinatorial pants decompositions: the cycles are closed walks in G that may overlap but do not cross.
We give an algorithm which, given a pants decomposition, computes a homotopic pants decomposition in which each cycle is a shortest cycle in its homotopy class. In particular, the resulting decomposition is optimal (as short as possible among all homotopic pants decompositions), and any optimal pants decomposition is made of shortest homotopic cycles. Our algorithm is polynomial in the complexity of the input and in the longest-to-shortest edge ratio of G. The same algorithm can be applied, given a simple cycle C, to compute a shortest cycle homotopic to C which is itself simple
HexGraph: Applying Graph Drawing Algorithms to the Game of Hex
No full textHex [1] is a two player board game which is traditionally played on a rhombic hexagonal pattern (See Figure (1)). Players are assigned a colour and make moves by putting a token of their colour onto an empty field on the board. The first player to connect the two borders of the board in his colour by a path of his tokens on the board wins the game. Alternatively, Hex is played on an undirected, tricoloured (Red, Blue, Unclaimed) graph G [2]. The fields are represented by nodes and adjacent fields on the board are connected by an edge. The four borders of the board are represented by one node of equivalent colour each (See Figure(1))
Drawing Power Law Graphs
No full textWe present methods for drawing graphs that arise in various information networks. It has been noted that many realistic graphs have a power law degree distribution and exhibit the small world phenomenon. Our methods are influenced by recent developments in the modeling of such graphs
An Efficient Implementation of Sugiyama's Algorithm for Layered Graph Drawing
No full textSugiyama's algorithmic framework for layered graph drawing is commonly used in practical software. The extensive use of dummy vertices to break long edges between non-adjacent layers often leads to unsatisfactorial performance. The worst-case running-time of Sugiyamarsquos approach is O(|V||E|\log|E|) requiring O(|V||E|) memory, which makes it unusable for the visualization of large graphs. By a conceptually simple new technique we are able to keep the number of dummy vertices and edges linear in the size of the graph and hence reduce the worst-case time complexity of Sugiyamarsquos approach by an order of magnitude to O((|V|+|E|)\log|E|) requiring O(|V|+|E|) space.
This work has partially been supported by the DFG-grant Ka512/8-2. It has been performed when the first author was with the Universität Tübingen
Algorithms for Drawing Media
No full textWe describe algorithms for drawing media, systems of states, tokens and actions that have state transition graphs in the form of partial cubes. Our algorithms are based on two principles: embedding the state transition graph in a low-dimensional integer lattice and projecting the lattice onto the plane, or drawing the medium as a planar graph with centrally symmetric faces
Confluent Layered Drawings
No full textWe combine the idea of confluent drawings with Sugiyama style drawings, in order to reduce the edge crossings in the resultant drawings. Furthermore, it is easier to understand the structures of graphs from the mixed style drawings. The basic idea is to cover a layered graph by complete bipartite subgraphs (bicliques), then replace bicliques with tree-like structures. The biclique cover problem is reduced to a special edge coloring problem and solved by heuristic coloring algorithms. Our method can be extended to obtain multi-depth confluent layered drawings.
Work by the first author is supported by NSF grant CCR-9912338. Work by the second and the third author is supported by NSF grants CCR-0098068, CCR-0225642, and DUE-0231467
Drawing Large Graphs with a Potential-Field-Based Multilevel Algorithm (Extended Abstract)
No full textForce-directed graph drawing algorithms are widely used for drawing general graphs. However, these methods do not guarantee a sub-quadratic running time in general. We present a new force-directed method that is based on a combination of an efficient multilevel scheme and a strategy for approximating the repulsive forces in the system by rapidly evaluating potential fields. Given a graph G=(V,E), the asymptotic worst case running time of this method is O(|V|\log|V|+|E|) with linear memory requirements. In practice, the algorithm generates nice drawings of graphs containing 100000 nodes in less than 5 minutes. Furthermore, it clearly visualizes even the structures of those graphs that turned out to be challenging for some other methods
A More Practical Algorithm for Drawing Binary Trees in Linear Area with Arbitrary Aspect Ratio
No full textTrees are usually drawn using planar straight-line drawings. [1] presented an algorithm for constructing a planar straight-line grid drawing of an n-node binary tree with area O(n) and any pre-specified aspect ratio in the range [n^{-\alpha}, n^{\alpha}], where 0 /le /alpha < 1 is any constant, in time. Unfortunately, the algorithm of [1] is not suitable for practical use. The main problem is that the constant hidden in the "Oh" notation for area is quite large (e.g., it can be as large as 3900).
In this paper, we have made several practical improvements to the algorithm, which make it suitable for practical use. We have also conducted experiments on this newer version of the algorithm for randomly-generated and complete binary trees with up to 50,000, and 65,535 nodes, respectively. Our experiments show that it constructs area-efficient drawings in practice, with area at most 10 times and 8 times the number of nodes for randomly-generated and complete binary trees, respectively.
Research supported by NSF CAREER Award IIS-9985136, NSF CISE Research Infrastructure Award No. 0101244, and Mark Diamond Research Grant No. 13-Summer-2003 from GSA of The State University of New York
GraphEx: An Improved Graph Translation Service
No full textThe Internet-based translation service GraphEX automatically converting between different graph formats, making it easier for users to take advantage of the full variety of graph drawing tools. GraphEx builds on the prototype translation component of the Graph Drawing Server [2], improving the handling of information mismatch problems and adding support for user-defined formats
Intersection-Free Morphing of Planar Graphs
No full textGiven two different drawings of a planar graph we consider the problem of morphing one drawing into the other. We designed and implemented an algorithm for intersection-free morphing of planar graphs. Our algorithm uses a combination of different techniques to achieve smooth transformations: rigid morphing, compatible triangulations, as well as morphing based on interpolation of the convex representations of the graphs. Our algorithm can morph between drawings with straight-line segments, bends, and curves. Our system is implemented in Java and available as an applet at http://gmorph.cs.arizona.edu