Dual-Eulerian Graphs with Applications to VLSI Design

A Dual-Eulerian graph is a plane multigraph G that contains an edge list which is simultaneously an Euler tour in G and an Euler tour in the dual of G. Dual-Eulerian tours play an important role in optimizing CMOS layouts of Boolean functions. When circuits are represented by undirected multigraphs the layout area of the circuit can be optimized through finding the minimum number of disjoint dual trails that cover the graph. This paper presents an implementation of a polynomial time algorithm for determining whether or not a plane multigraph is Dual-Eulerian and for finding the Dual-Eulerian trail if it exists

Minimum Makespan Multi-vehicle Dial-a-Ride

Dial a ride problems consist of a metric space (denoting travel time between vertices) and a set of m objects represented as source-destination pairs, where each object requires to be moved from its source to destination vertex. We consider the multi-vehicle Dial a ride problem, with each vehicle having capacity k and its own depot-vertex, where the objective is to minimize the maximum completion time (makespan) of the vehicles. We study the "preemptive" version of the problem, where an object may be left at intermediate vertices and transported by more than one vehicle, while being moved from source to destination. Our main results are an O(log^3 n)-approximation algorithm for preemptive multi-vehicle Dial a ride, and an improved O(log t)-approximation for its special case when there is no capacity constraint. We also show that the approximation ratios improve by a log-factor when the underlying metric is induced by a fixed-minor-free graph.Comment: 22 pages, 1 figure. Preliminary version appeared in ESA 200

An Overview of Rendering from Volume Data --- including Surface and Volume Rendering

Volume rendering is a title often ambiguously used in science. One meaning often quoted is: `to render any three volume dimensional data set'; however, within this categorisation `surface rendering'' is contained. Surface rendering is a technique for visualising a geometric representation of a surface from a three dimensional volume data set. A more correct definition of Volume Rendering would only incorporate the direct visualisation of volumes, without the use of intermediate surface geometry representations. Hence we state: `Volume Rendering is the Direct Visualisation of any three dimensional Volume data set; without the use of an intermediate geometric representation for isosurfaces'; `Surface Rendering is the Visualisation of a surface, from a geometric approximation of an isosurface, within a Volume data set'; where an isosurface is a surface formed from a cross connection of data points, within a volume, of equal value or density. This paper is an overview of both Surface Rendering and Volume Rendering techniques. Surface Rendering mainly consists of contouring lines over data points and triangulations between contours. Volume rendering methods consist of ray casting techniques that allow the ray to be cast from the viewing plane into the object and the transparency, opacity and colour calculated for each cell; the rays are often cast until an opaque object is `hit' or the ray exits the volume

Embedding Digraphs on Orientable Surfaces

AbstractWe consider a notion of embedding digraphs on orientable surfaces, applicable to digraphs in which the indegree equals the outdegree for every vertex, i.e., Eulerian digraphs. This idea has been considered before in the context of compatible Euler tours or orthogonal A-trails by Andersen and by Bouchet. This prior work has mostly been limited to embeddings of Eulerian digraphs on predetermined surfaces and to digraphs with underlying graphs of maximum degree at most 4. In this paper, a foundation is laid for the study of all Eulerian digraph embeddings. Results are proved which are analogous to those fundamental to the theory of undirected graph embeddings, such as Duke's theorem [5], and an infinite family of digraphs which demonstrates that the genus range for an embeddable digraph can be any nonnegative integer given. We show that it is possible to have genus range equal to one, with arbitrarily large minimum genus, unlike in the undirected case. The difference between the minimum genera of a digraph and its underlying graph is considered, as is the difference between the maximum genera. We say that a digraph is upper-embeddable if it can be embedded with two or three regions and prove that every regular tournament is upper-embeddable

Counting and sampling problems on Eulerian graphs

In this thesis we consider two sets of combinatorial structures defined on an Eulerian graph: the Eulerian orientations and Euler tours. We are interested in the computational problems of counting (computing the number of elements in the set) and sampling (generating a random element of the set). Specifically, we are interested in the question of when there exists an efficient algorithm for counting or sampling the elements of either set. The Eulerian orientations of a number of classes of planar lattices are of practical significance as they correspond to configurations of certain models studied in statistical physics. In 1992 Mihail and Winkler showed that counting Eulerian orientations of a general Eulerian graph is #P-complete and demonstrated that the problem of sampling an Eulerian orientation can be reduced to the tractable problem of sampling a perfect matching of a bipartite graph. We present a proof that this problem remains #Pcomplete when the input is restricted to being a planar graph, and analyse a natural algorithm for generating random Eulerian orientations of one of the afore-mentioned planar lattices. Moreover, we make some progress towards classifying the range of planar graphs on which this algorithm is rapidly mixing by exhibiting an infinite class of planar graphs for which the algorithm will always take an exponential amount of time to converge. The problem of counting the Euler tours of undirected graphs has proven to be less amenable to analysis than that of Eulerian orientations. Although it has been known for many years that the number of Euler tours of any directed graph can be computed in polynomial time, until recently very little was known about the complexity of counting Euler tours of an undirected graph. Brightwell and Winkler showed that this problem is #P-complete in 2005 and, apart from a few very simple examples, e.g., series-parellel graphs, there are no known tractable cases, nor are there any good reasons to believe the problem to be intractable. Moreover, despite several unsuccessful attempts, there has been no progress made on the question of approximability. Indeed, this problem was considered to be one of the more difficult open problems in approximate counting since long before the complexity of exact counting was resolved. By considering a randomised input model, we are able to show that a very simple algorithm can sample or approximately count the Euler tours of almost every d-in/d-out directed graph in expected polynomial time. Then, we present some partial results towards showing that this algorithm can be used to sample or approximately count the Euler tours of almost every 2d-regular graph in expected polynomial time. We also provide some empirical evidence to support the unproven conjecture required to obtain this result. As a sideresult of this work, we obtain an asymptotic characterisation of the distribution of the number of Eulerian orientations of a random 2d-regular graph