arXiv subjects

Michael Kaufmann

Publications and source records attributed to Michael Kaufmann.

At least 55 records · Page 3Linked to original sources

arXiv2019

Addressing Algorithmic Bottlenecks in Elastic Machine Learning with Chicle

Distributed machine learning training is one of the most common and important workloads running on data centers today, but it is rarely executed alone. Instead, to reduce costs, computing resources are consolidated and shared by different applications. In this scenario, elasticity and proper load balancing are vital to maximize efficiency, fairness, and utilization. Currently, most distributed training frameworks do not support the aforementioned properties. A few exceptions that do support elasticity, imitate generic distributed frameworks and use micro-tasks. In this paper we illustrate that micro-tasks are problematic for machine learning applications, because they require a high degree of parallelism which hinders the convergence of distributed training at a pure algorithmic level (i.e., ignoring overheads and scalability limitations). To address this, we propose Chicle, a new elastic distributed training framework which exploits the nature of machine learning algorithms to implement elasticity and load balancing without micro-tasks. We use Chicle to train deep neural network as well as generalized linear models, and show that Chicle achieves performance competitive with state of the art rigid frameworks, while efficiently enabling elastic execution and dynamic load balancing.

cs.LG↗

arXiv2019

Efficient Generation of Different Topological Representations of Graphs Beyond-Planarity

Beyond-planarity focuses on combinatorial properties of classes of non-planar graphs that allow for representations satisfying certain local geometric or topological constraints on their edge crossings. Beside the study of a specific graph class for its maximum edge density, another parameter that is often considered in the literature is the size of the largest complete or complete bipartite graph belonging to it. Overcoming the limitations of standard combinatorial arguments, we present a technique to systematically generate all non-isomorphic topological representations of complete and complete bipartite graphs, taking into account the constraints of the specific class. As a proof of concept, we apply our technique to various beyond-planarity classes and achieve new tight bounds for the aforementioned parameter.

cs.DS↗

arXiv2019

The QuaSEFE Problem

We initiate the study of Simultaneous Graph Embedding with Fixed Edges in the beyond planarity framework. In the QuaSEFE problem, we allow edge crossings, as long as each graph individually is drawn quasiplanar, that is, no three edges pairwise cross. We show that a triple consisting of two planar graphs and a tree admit a QuaSEFE. This result also implies that a pair consisting of a 1-planar graph and a planar graph admits a QuaSEFE. We show several other positive results for triples of planar graphs, in which certain structural properties for their common subgraphs are fulfilled. For the case in which simplicity is also required, we give a triple consisting of two quasiplanar graphs and a star that does not admit a QuaSEFE. Moreover, in contrast to the planar SEFE problem, we show that it is not always possible to obtain a QuaSEFE for two matchings if the quasiplanar drawing of one matching is fixed.

cs.DS↗

arXiv2018

Elastic CoCoA: Scaling In to Improve Convergence

In this paper we experimentally analyze the convergence behavior of CoCoA and show, that the number of workers required to achieve the highest convergence rate at any point in time, changes over the course of the training. Based on this observation, we build Chicle, an elastic framework that dynamically adjusts the number of workers based on feedback from the training algorithm, in order to select the number of workers that results in the highest convergence rate. In our evaluation of 6 datasets, we show that Chicle is able to accelerate the time-to-accuracy by a factor of up to 5.96x compared to the best static setting, while being robust enough to find an optimal or near-optimal setting automatically in most cases.

cs.LG↗

arXiv2018

Queue Layouts of Planar 3-Trees

A queue layout of a graph G consists of a linear order of the vertices of G and a partition of the edges of G into queues, so that no two independent edges of the same queue are nested. The queue number of G is the minimum number of queues required by any queue layout of G. In this paper, we continue the study of the queue number of planar 3-trees. As opposed to general planar graphs, whose queue number is not known to be bounded by a constant, the queue number of planar 3-trees has been shown to be at most seven. In this work, we improve the upper bound to five. We also show that there exist planar 3-trees, whose queue number is at least four; this is the first example of a planar graph with queue number greater than three.

cs.DS↗

arXiv2018

1-Fan-Bundle-Planar Drawings of Graphs

Edge bundling is an important concept heavily used for graph visualization purposes. To enable the comparison with other established near-planarity models in graph drawing, we formulate a new edge-bundling model which is inspired by the recently introduced fan-planar graphs. In particular, we restrict the bundling to the end segments of the edges. Similarly to 1-planarity, we call our model 1-fan-bundle-planarity, as we allow at most one crossing per bundle. For the two variants where we allow either one or, more naturally, both end segments of each edge to be part of bundles, we present edge density results and consider various recognition questions, not only for general graphs, but also for the outer and 2-layer variants. We conclude with a series of challenging questions.

cs.CG↗

arXiv2018

Orthogonal and Smooth Orthogonal Layouts of 1-Planar Graphs with Low Edge Complexity

While orthogonal drawings have a long history, smooth orthogonal drawings have been introduced only recently. So far, only planar drawings or drawings with an arbitrary number of crossings per edge have been studied. Recently, a lot of research effort in graph drawing has been directed towards the study of beyond-planar graphs such as 1-planar graphs, which admit a drawing where each edge is crossed at most once. In this paper, we consider graphs with a fixed embedding. For 1-planar graphs, we present algorithms that yield orthogonal drawings with optimal curve complexity and smooth orthogonal drawings with small curve complexity. For the subclass of outer-1-planar graphs, which can be drawn such that all vertices lie on the outer face, we achieve optimal curve complexity for both, orthogonal and smooth orthogonal drawings.

cs.DS↗

arXiv2018

On RAC Drawings of Graphs with one Bend per Edge

A k-bend right-angle-crossing drawing or (k-bend RAC drawing}, for short) of a graph is a polyline drawing where each edge has at most k bends and the angles formed at the crossing points of the edges are 90 degrees. Accordingly, a graph that admits a k-bend RAC drawing is referred to as k-bend right-angle-crossing graph (or k-bend RAC, for short). In this paper, we continue the study of the maximum edge-density of 1-bend RAC graphs. We show that an n-vertex 1-bend RAC graph cannot have more than $5.5n-O(1)$ edges. We also demonstrate that there exist infinitely many n-vertex 1-bend RAC graphs with exactly $5n-O(1)$ edges. Our results improve both the previously known best upper bound of $6.5n-O(1)$ edges and the corresponding lower bound of $4.5n-O(\sqrt{n})$ edges by Arikushi et al. (Comput. Geom. 45(4), 169--177 (2012)).

cs.DS↗

arXiv2018

A Heuristic Approach towards Drawings of Graphs with High Crossing Resolution

The crossing resolution of a non-planar drawing of a graph is the value of the minimum angle formed by any pair of crossing edges. Recent experiments have shown that the larger the crossing resolution is, the easier it is to read and interpret a drawing of a graph. However, maximizing the crossing resolution turns out to be an NP-hard problem in general and only heuristic algorithms are known that are mainly based on appropriately adjusting force-directed algorithms. In this paper, we propose a new heuristic algorithm for the crossing resolution maximization problem and we experimentally compare it against the known approaches from the literature. Our experimental evaluation indicates that the new heuristic produces drawings with better crossing resolution, but this comes at the cost of slightly higher aspect ratio, especially when the input graph is large.

cs.DS↗

arXiv2018

On Dispersable Book Embeddings

In a dispersable book embedding, the vertices of a given graph $G$ must be ordered along a line l, called spine, and the edges of G must be drawn at different half-planes bounded by l, called pages of the book, such that: (i) no two edges of the same page cross, and (ii) the graphs induced by the edges of each page are 1-regular. The minimum number of pages needed by any dispersable book embedding of $G$ is referred to as the dispersable book thickness $dbt(G)$ of $G$. Graph $G$ is called dispersable if $dbt(G) = Δ(G)$ holds (note that $Δ(G) \leq dbt(G)$ always holds). Back in 1979, Bernhart and Kainen conjectured that any $k$-regular bipartite graph $G$ is dispersable, i.e., $dbt(G)=k$. In this paper, we disprove this conjecture for the cases $k=3$ (with a computer-aided proof), and $k=4$ (with a purely combinatorial proof). In particular, we show that the Gray graph, which is 3-regular and bipartite, has dispersable book thickness four, while the Folkman graph, which is 4-regular and bipartite, has dispersable book thickness five. On the positive side, we prove that 3-connected 3-regular bipartite planar graphs are dispersable, and conjecture that this property holds, even if 3-connectivity is relaxed.

cs.DM↗

arXiv2017

Beyond-Planarity: Density Results for Bipartite Graphs

Beyond-planarity focuses on the study of geometric and topological graphs that are in some sense nearly-planar. Here, planarity is relaxed by allowing edge crossings, but only with respect to some local forbidden crossing configurations. Early research dates back to the 1960s (e.g., Avital and Hanani 1966) for extremal problems on geometric graphs, but is also related to graph drawing problems where visual clutter by edge crossings should be minimized (e.g., Huang et al. 2008) that could negatively affect the readability of the drawing. Different types of forbidden crossing configurations give rise to different families of nearly-planar graphs. Most of the literature focuses on Turán-type problems, which ask for the maximum number of edges a nearly-planar graph can have. Here, we study this problem for bipartite topological graphs, considering several types of nearly-planar graphs, i.e., 1-planar, 2-planar, fan-planar, and RAC graphs. We prove bounds on the number of edges that are tight up to small additive constants; some of them are surprising and not along the lines of the known results for non-bipartite graphs. Our findings lead to an improvement of the leading constant of the well-known Crossing Lemma for bipartite graphs, as well as to a number of interesting research questions on topological graphs.

cs.DM↗

arXiv2017

On Vertex- and Empty-Ply Proximity Drawings

We initiate the study of the vertex-ply of straight-line drawings, as a relaxation of the recently introduced ply number. Consider the disks centered at each vertex with radius equal to half the length of the longest edge incident to the vertex. The vertex-ply of a drawing is determined by the vertex covered by the maximum number of disks. The main motivation for considering this relaxation is to relate the concept of ply to proximity drawings. In fact, if we interpret the set of disks as proximity regions, a drawing with vertex-ply number 1 can be seen as a weak proximity drawing, which we call empty-ply drawing. We show non-trivial relationships between the ply number and the vertex-ply number. Then, we focus on empty-ply drawings, proving some properties and studying what classes of graphs admit such drawings. Finally, we prove a lower bound on the ply and the vertex-ply of planar drawings.

cs.DS↗

arXiv2017

On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings

We study two variants of the well-known orthogonal drawing model: (i) the smooth orthogonal, and (ii) the octilinear. Both models form an extension of the orthogonal, by supporting one additional type of edge segments (circular arcs and diagonal segments, respectively). For planar graphs of max-degree 4, we analyze relationships between the graph classes that can be drawn bendless in the two models and we also prove NP-hardness for a restricted version of the bendless drawing problem for both models. For planar graphs of higher degree, we present an algorithm that produces bi-monotone smooth orthogonal drawings with at most two segments per edge, which also guarantees a linear number of edges with exactly one segment.

cs.DS↗

arXiv2017

An Interactive Tool to Explore and Improve the Ply Number of Drawings

Given a straight-line drawing $Γ$ of a graph $G=(V,E)$, for every vertex $v$ the ply disk $D_v$ is defined as a disk centered at $v$ where the radius of the disk is half the length of the longest edge incident to $v$. The ply number of a given drawing is defined as the maximum number of overlapping disks at some point in $\mathbb{R}^2$. Here we present a tool to explore and evaluate the ply number for graphs with instant visual feedback for the user. We evaluate our methods in comparison to an existing ply computation by De Luca et al. [WALCOM'17]. We are able to reduce the computation time from seconds to milliseconds for given drawings and thereby contribute to further research on the ply topic by providing an efficient tool to examine graphs extensively by user interaction as well as some automatic features to reduce the ply number.

cs.DS↗

arXiv2017

3D Visibility Representations of 1-planar Graphs

We prove that every 1-planar graph G has a z-parallel visibility representation, i.e., a 3D visibility representation in which the vertices are isothetic disjoint rectangles parallel to the xy-plane, and the edges are unobstructed z-parallel visibilities between pairs of rectangles. In addition, the constructed representation is such that there is a plane that intersects all the rectangles, and this intersection defines a bar 1-visibility representation of G.

cs.CG↗

arXiv2017

On Optimal 2- and 3-Planar Graphs

A graph is $k$-planar if it can be drawn in the plane such that no edge is crossed more than $k$ times. While for $k=1$, optimal $1$-planar graphs, i.e., those with $n$ vertices and exactly $4n-8$ edges, have been completely characterized, this has not been the case for $k \geq 2$. For $k=2,3$ and $4$, upper bounds on the edge density have been developed for the case of simple graphs by Pach and Tóth, Pach et al. and Ackerman, which have been used to improve the well-known "Crossing Lemma". Recently, we proved that these bounds also apply to non-simple $2$- and $3$-planar graphs without homotopic parallel edges and self-loops. In this paper, we completely characterize optimal $2$- and $3$-planar graphs, i.e., those that achieve the aforementioned upper bounds. We prove that they have a remarkably simple regular structure, although they might be non-simple. The new characterization allows us to develop notable insights concerning new inclusion relationships with other graph classes.

cs.CG↗

arXiv2016

Low Ply Drawings of Trees

We consider the recently introduced model of \emph{low ply graph drawing}, in which the ply-disks of the vertices do not have many common overlaps, which results in a good distribution of the vertices in the plane. The \emph{ply-disk} of a vertex in a straight-line drawing is the disk centered at it whose radius is half the length of its longest incident edge. The largest number of ply-disks having a common overlap is called the \emph{ply-number} of the drawing. We focus on trees. We first consider drawings of trees with constant ply-number, proving that they may require exponential area, even for stars, and that they may not even exist for bounded-degree trees. Then, we turn our attention to drawings with logarithmic ply-number and show that trees with maximum degree $6$ always admit such drawings in polynomial area.

cs.DS↗

arXiv2016

On the Density of non-Simple 3-Planar Graphs

A \emph{$k$-planar graph} is a graph that can be drawn in the plane such that every edge is crossed at most $k$ times. For $k \leq 4$, Pach and Tóth proved a bound of $(k+3)(n-2)$ on the total number of edges of a $k$-planar graph, which is tight for $k=1,2$. For $k=3$, the bound of $6n-12$ has been improved to $\frac{11}{2}n-11$ and has been shown to be optimal up to an additive constant for simple graphs. In this paper, we prove that the bound of $\frac{11}{2}n-11$ edges also holds for non-simple $3$-planar graphs that admit drawings in which non-homotopic parallel edges and self-loops are allowed. Based on this result, a characterization of \emph{optimal $3$-planar graphs} (that is, $3$-planar graphs with $n$ vertices and exactly $\frac{11}{2}n-11$ edges) might be possible, as to the best of our knowledge the densest known simple $3$-planar is not known to be optimal.

cs.CG↗