10 research outputs found
Algorithms for Incremental Planar Graph Drawing and Two-page Book Embeddings
Subject of this work are two problems related to ordering the vertices
of planar graphs. The first one is concerned with the properties of
vertex-orderings that serve as a basis for incremental drawing algorithms.
Such a drawing algorithm usually extends a drawing by adding the vertices
step-by-step as provided by the ordering. In the field of graph drawing
several orderings are in use for this purpose. Some of them, however,
lack certain properties that are desirable or required for classic
incremental drawing methods. We narrow down these properties, and
introduce the bitonic st-ordering, an ordering which combines the
features only available when using canonical orderings with the flexibility
of st-orderings. The additional property of being bitonic enables an
st-ordering to be used in algorithms that usually require a canonical
ordering.
With this in mind, we describe a linear-time algorithm that computes
such an ordering for every biconnected planar graph. Unlike canonical
orderings, st-orderings extend to directed graphs, in particular planar
st-graphs. Being able to compute bitonic st-orderings for planar st-graphs
is of particular interest for upward planar drawing algorithms, since
traditional incremental algorithms for undirected planar graphs might
be adapted to directed graphs. Based on this observation, we give a
full characterization of the class of planar st-graphs that admit such
an ordering. This includes a linear-time algorithm for recognition
and ordering. Furthermore, we show that by splitting specific edges of
an instance that is not part of this class, one is able to transform
it into one for which then such an ordering exists. To do so, we describe
a linear-time algorithm for finding the smallest set of edges to split.
We show that for a planar st-graph G=(V,E), |V|â3 edge splits
are sufficient and every edge is split at most once. This immediately
translates to the number of bends required for upward planar poly-line
drawings. More specifically, we show that every planar st-graph admits
an upward planar poly-line drawing in quadratic area with at most |V|â3
bends in total and at most one bend per edge. Moreover, the drawing
can be obtained in linear time.
The second part is concerned with embedding planar graphs with maximum
degree three and four into books. Besides providing a simplified
incremental linear-time algorithm for embedding triconnected 3-planar
graphs into a book of two pages, we describe a linear-time algorithm
to compute a subhamiltonian cycle in a triconnected 4-planar graph
Planar L-Drawings of Directed Graphs
We study planar drawings of directed graphs in the L-drawing standard. We
provide necessary conditions for the existence of these drawings and show that
testing for the existence of a planar L-drawing is an NP-complete problem.
Motivated by this result, we focus on upward-planar L-drawings. We show that
directed st-graphs admitting an upward- (resp. upward-rightward-) planar
L-drawing are exactly those admitting a bitonic (resp. monotonically
increasing) st-ordering. We give a linear-time algorithm that computes a
bitonic (resp. monotonically increasing) st-ordering of a planar st-graph or
reports that there exists none.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Upward planar drawings with two slopes
In an upward planar 2-slope drawing of a digraph, edges are drawn as
straight-line segments in the upward direction without crossings using only two
different slopes. We investigate whether a given upward planar digraph admits
such a drawing and, if so, how to construct it. For the fixed embedding
scenario, we give a simple characterisation and a linear-time construction by
adopting algorithms from orthogonal drawings. For the variable embedding
scenario, we describe a linear-time algorithm for single-source digraphs, a
quartic-time algorithm for series-parallel digraphs, and a fixed-parameter
tractable algorithm for general digraphs. For the latter two classes, we make
use of SPQR-trees and the notion of upward spirality. As an application of this
drawing style, we show how to draw an upward planar phylogenetic network with
two slopes such that all leaves lie on a horizontal line
Efficient Algorithms for a Mesh-Connected Computer with Additional Global Bandwidth
This thesis shows that adding additional global bandwidths to a mesh-connected computer can greatly improve the performance. The goal of this project is to design algorithms for mesh-connected computers augmented with limited global bandwidth, so that we can further enhance our understanding of the parallel/serial nature of the problems on evolving parallel architectures. We do this by first solving several problems associated with fundamental data movement, then summarize ways to resolve different situations one may observe in data movement in parallel computing. This can help us to understand whether the problem is easily parallelizable on different parallel models. We give efficient algorithms to solve several fundamental problems, which include sorting, counting, fast Fourier transform, finding a minimum spanning tree, finding a convex hull, etc. We show that adding a small amount of global bandwidth makes a practical design that combines aspects of mesh and fully connected models to achieve the benefits of each. Most of the algorithms are optimal. For future work, we believe that algorithms with peak-power constrains can make our model well adapted to the recent architectures in high performance computing.PHDComputer Science & EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/150001/1/anyujie_1.pd
Bitonic st-orderings for Upward Planar Graphs
Canonical orderings serve as the basis for many incremental planar drawing algorithms. All these techniques, however, have in common that they are limited to undirected graphs. While st-orderings do extend to directed graphs, especially planar st-graphs, they do not offer the same properties as canonical orderings. In this work we extend the so called bitonic st-orderings to directed graphs. We fully characterize planar st-graphs that admit such an ordering and provide a linear-time algorithm for recognition and ordering. If for a graph no bitonic st-ordering exists, we show how to find in linear time a minimum set of edges to split such that the resulting graph admits one. With this new technique we are able to draw every upward planar graph on n vertices by using at most one bend per edge, at most nâ3 bends in total and within quadratic area
Collection of abstracts of the 24th European Workshop on Computational Geometry
International audienceThe 24th European Workshop on Computational Geomety (EuroCG'08) was held at INRIA Nancy - Grand Est & LORIA on March 18-20, 2008. The present collection of abstracts contains the 63 scientific contributions as well as three invited talks presented at the workshop
Scheduling in Transactional Memory Systems: Models, Algorithms, and Evaluations
Transactional memory provides an alternative synchronization mechanism that removes many limitations of traditional lock-based synchronization so that concurrent program writing is easier than lock-based code in modern multicore architectures. The fundamental module in a transactional memory system is the transaction which represents a sequence of read and write operations that are performed atomically to a set of shared resources; transactions may conflict if they access the same shared resources. A transaction scheduling algorithm is used to handle these transaction conflicts and schedule appropriately the transactions. In this dissertation, we study transaction scheduling problem in several systems that differ through the variation of the intra-core communication cost in accessing shared resources. Symmetric communication costs imply tightly-coupled systems, asymmetric communication costs imply large-scale distributed systems, and partially asymmetric communication costs imply non-uniform memory access systems. We made several theoretical contributions providing tight, near-tight, and/or impossibility results on three different performance evaluation metrics: execution time, communication cost, and load, for any transaction scheduling algorithm. We then complement these theoretical results by experimental evaluations, whenever possible, showing their benefits in practical scenarios. To the best of our knowledge, the contributions of this dissertation are either the first of their kind or significant improvements over the best previously known results
The Fifth NASA Symposium on VLSI Design
The fifth annual NASA Symposium on VLSI Design had 13 sessions including Radiation Effects, Architectures, Mixed Signal, Design Techniques, Fault Testing, Synthesis, Signal Processing, and other Featured Presentations. The symposium provides insights into developments in VLSI and digital systems which can be used to increase data systems performance. The presentations share insights into next generation advances that will serve as a basis for future VLSI design