Planare Graphen und ihre Dualgraphen auf Zylinderoberflächen

In this thesis, we investigates plane drawings of undirected and directed graphs on cylinder surfaces. In the case of undirected graphs, the vertices are positioned on a line that is parallel to the cylinder’s axis and the edge curves must not intersect this line. We show that a plane drawing is possible if and only if the graph is a double-ended queue (deque) graph, i. e., the vertices of the graph can be processed according to a linear order and the edges correspond to items in the deque inserted and removed at their end vertices. A surprising consequence resulting from these observations is that the deque characterizes planar graphs with a Hamiltonian path. This result extends the known characterization of planar graphs with a Hamiltonian cycle by two stacks. By these insights, we also obtain a new characterization of queue graphs and their duals. We also consider the complexity of deciding whether a graph is a deque graph and prove that it is NP-complete. By introducing a split operation, we obtain the splittable deque and show that it characterizes planarity. For the proof, we devise an algorithm that uses the splittable deque to test whether a rotation system is planar. In the case of directed graphs, we study upward plane drawings where the edge curves follow the direction of the cylinder’s axis (standing upward planarity; SUP) or they wind around the axis (rolling upward planarity; RUP). We characterize RUP graphs by means of their duals and show that RUP and SUP swap their roles when considering a graph and its dual. There is a physical interpretation underlying this characterization: A SUP graph is to its RUP dual graph as electric current passing through a conductor to the magnetic field surrounding the conductor. Whereas testing whether a graph is RUP is NP-hard in general [Bra14], for directed graphs without sources and sink, we develop a linear-time recognition algorithm that is based on our dual graph characterization of RUP graphs.Die Arbeit beschäftigt sich mit planaren Zeichnungen ungerichteter und gerichteter Graphen auf Zylinderoberflächen. Im ungerichteten Fall werden Zeichnungen betrachtet, bei denen die Knoten auf einer Linie parallel zur Zylinderachse positioniert werden und die Kanten diese Linie nicht schneiden dürfen. Es kann gezeigt werden, dass eine planare Zeichnung genau dann möglich ist, wenn die Kanten des Graphen in einer double-ended queue (Deque) verarbeitet werden können. Ebenso lassen sich dadurch Queue, Stack und Doppelstack charakterisieren. Eine überraschende Konsequenz aus diesen Erkenntnissen ist, dass die Deque genau die planaren Graphen mit Hamiltonpfad charakterisiert. Dies erweitert die bereits bekannte Charakterisierung planarer Graphen mit Hamiltonkreis durch den Doppelstack. Im gerichteten Fall müssen die Kantenkurven entweder in Richtung der Zylinderachse verlaufen (SUP-Graphen) oder sich um die Achse herumbewegen (RUP-Graphen). Die Arbeit charakterisiert RUP-Graphen und zeigt, dass RUP und SUP ihre Rollen tauschen, wenn man Graph und Dualgraph betrachtet. Der SUP-Graph verhält sich dabei zum RUP-Graphen wie elektrischer Strom durch einen Leiter zum induzierten Magnetfeld. Ausgehend von dieser Charakterisierung ist es möglich einen Linearzeit-Algorithmus zu entwickeln, der entscheidet ob ein gerichteter Graph ohne Quellen und Senken ein RUP-Graph ist, während der allgemeine Fall NP-hart ist [Bra14]

10091 Abstracts Collection -- Data Structures

From February 28th to March 5th 2010, the Dagstuhl Seminar 10091 "Data Structures" was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. It brought together 45 international researchers to discuss recent developments concerning data structures in terms of research, but also in terms of new technologies that impact how data can be stored, updated, and retrieved. During the seminar a fair number of participants presented their current research and open problems where discussed. This document first briefly describes the seminar topics and then gives the abstracts of the presentations given during the seminar

Enhancing Productivity and Performance Portability of General-Purpose Parallel Programming

This work focuses on compiler and run-time techniques for improving the productivity and the performance portability of general-purpose parallel programming. More specifically, we focus on shared-memory task-parallel languages, where the programmer explicitly exposes parallelism in the form of short tasks that may outnumber the cores by orders of magnitude. The compiler, the run-time, and the platform (henceforth the system) are responsible for harnessing this unpredictable amount of parallelism, which can vary from none to excessive, towards efficient execution. The challenge arises from the aspiration to support fine-grained irregular computations and nested parallelism. This work is even more ambitious by also aspiring to lay the foundations to efficiently support declarative code, where the programmer exposes all available parallelism, using high-level language constructs such as parallel loops, reducers or futures. The appeal of declarative code is twofold for general-purpose programming: it is often easier for the programmer who does not have to worry about the granularity of the exposed parallelism, and it achieves better performance portability by avoiding overfitting to a small range of platforms and inputs for which the programmer is coarsening. Furthermore, PRAM algorithms, an important class of parallel algorithms, naturally lend themselves to declarative programming, so supporting it is a necessary condition for capitalizing on the wealth of the PRAM theory. Unfortunately, declarative codes often expose such an overwhelming number of fine-grained tasks that existing systems fail to deliver performance. Our contributions can be partitioned into three components. First, we tackle the issue of coarsening, which declarative code leaves to the system. We identify two goals of coarsening and advocate tackling them separately, using static compiler transformations for one and dynamic run-time approaches for the other. Additionally, we present evidence that the current practice of burdening the programmer with coarsening either leads to codes with poor performance-portability, or to a significantly increased programming effort. This is a ``show-stopper'' for general-purpose programming. To compare the performance portability among approaches, we define an experimental framework and two metrics, and we demonstrate that our approaches are preferable. We close the chapter on coarsening by presenting compiler transformations that automatically coarsen some types of very fine-grained codes. Second, we propose Lazy Scheduling, an innovative run-time scheduling technique that infers the platform load at run-time, using information already maintained. Based on the inferred load, Lazy Scheduling adapts the amount of available parallelism it exposes for parallel execution and, thus, saves parallelism overheads that existing approaches pay. We implement Lazy Scheduling and present experimental results on four different platforms. The results show that Lazy Scheduling is vastly superior for declarative codes and competitive, if not better, for coarsened codes. Moreover, Lazy Scheduling is also superior in terms of performance-portability, supporting our thesis that it is possible to achieve reasonable efficiency and performance portability with declarative codes. Finally, we also implement Lazy Scheduling on XMT, an experimental manycore platform developed at the University of Maryland, which was designed to support codes derived from PRAM algorithms. On XMT, we manage to harness the existing hardware support for scheduling flat parallelism to compose it with Lazy Scheduling, which supports nested parallelism. In the resulting hybrid scheduler, the hardware and software work in synergy to overcome each other's weaknesses. We show the performance composability of the hardware and software schedulers, both in an abstract cost model and experimentally, as the hybrid always performs better than the software scheduler alone. Furthermore, the cost model is validated by using it to predict if it is preferable to execute a code sequentially, with outer parallelism, or with nested parallelism, depending on the input, the available hardware parallelism and the calling context of the parallel code

Optimization with pattern-avoiding input

Permutation pattern-avoidance is a central concept of both enumerative and extremal combinatorics. In this paper we study the effect of permutation pattern-avoidance on the complexity of optimization problems. In the context of the dynamic optimality conjecture (Sleator, Tarjan, STOC 1983), Chalermsook, Goswami, Kozma, Mehlhorn, and Saranurak (FOCS 2015) conjectured that the amortized access cost of an optimal binary search tree (BST) is $O(1)$ whenever the access sequence avoids some fixed pattern. They showed a bound of $2^{\alpha{(n)}^{O(1)}}$, which was recently improved to $2^{\alpha{(n)}(1+o(1))}$ by Chalermsook, Pettie, and Yingchareonthawornchai (2023); here $n$ is the BST size and $\alpha(\cdot)$ the inverse-Ackermann function. In this paper we resolve the conjecture, showing a tight $O(1)$ bound. This indicates a barrier to dynamic optimality: any candidate online BST (e.g., splay trees or greedy trees) must match this optimum, but current analysis techniques only give superconstant bounds. More broadly, we argue that the easiness of pattern-avoiding input is a general phenomenon, not limited to BSTs or even to data structures. To illustrate this, we show that when the input avoids an arbitrary, fixed, a priori unknown pattern, one can efficiently compute a $k$-server solution of $n$ requests from a unit interval, with total cost $n^{O(1/\log k)}$, in contrast to the worst-case $\Theta(n/k)$ bound; and a traveling salesman tour of $n$ points from a unit box, of length $O(\log{n})$, in contrast to the worst-case $\Theta(\sqrt{n})$ bound; similar results hold for the euclidean minimum spanning tree, Steiner tree, and nearest-neighbor graphs. We show both results to be tight. Our techniques build on the Marcus-Tardos proof of the Stanley-Wilf conjecture, and on the recently emerging concept of twin-width; we believe our techniques to be more generally applicable

On The Growth Of Permutation Classes

We study aspects of the enumeration of permutation classes, sets of permutations closed downwards under the subpermutation order. First, we consider monotone grid classes of permutations. We present procedures for calculating the generating function of any class whose matrix has dimensions m × 1 for some m, and of acyclic and unicyclic classes of gridded permutations. We show that almost all large permutations in a grid class have the same shape, and determine this limit shape. We prove that the growth rate of a grid class is given by the square of the spectral radius of an associated graph and deduce some facts relating to the set of grid class growth rates. In the process, we establish a new result concerning tours on graphs. We also prove a similar result relating the growth rate of a geometric grid class to the matching polynomial of a graph, and determine the effect of edge subdivision on the matching polynomial. We characterise the growth rates of geometric grid classes in terms of the spectral radii of trees. We then investigate the set of growth rates of permutation classes and establish a new upper bound on the value above which every real number is the growth rate of some permutation class. In the process, we prove new results concerning expansions of real numbers in non-integer bases in which the digits are drawn from sets of allowed values. Finally, we introduce a new enumeration technique, based on associating a graph with each permutation, and determine the generating functions for some previously unenumerated classes. We conclude by using this approach to provide an improved lower bound on the growth rate of the class of permutations avoiding the pattern 1324. In the process, we prove that, asymptotically, patterns in Łukasiewicz paths exhibit a concentrated Gaussian distribution