42 research outputs found
Geometric Separation and Packing Problems
The first part of this thesis investigates combinatorial and algorithmic aspects of geometric separation problems in the plane. In such a setting one is given a set of points and a set of separators such as lines, line segments or disks. The goal is to select a small subset of those separators such that every path between any two points is intersected by at least one separator. We first look at several problems which arise when one is given a set of points and a set of unit disks embedded in the plane and the goal is to separate the points using a small subset of the given disks. Next, we focus on a separation problem involving only one region: Given a region in the plane, bounded by a piecewise linear closed curve, such as a fence, place few guards inside the fenced region such that wherever an intruder cuts through the fence, the closest guard is at most a distance one away. Restricting the separating objects to be lines, we investigate combinatorial aspects which arise when we use them to pairwise separate a set of points in the plane; hereafter we generalize the notion of separability to arbitrary sets and present several enumeration results. Lastly, we investigate a packing problem with a non-convex shape in â3. We show that â3 can be packed at a density of 0.222 with tori of major radius one and minor radius going to zero. Furthermore, we show that the same torus arrangement yields the asymptotically optimal number of pairwise linked tori
Minimum Perimeter-Sum Partitions in the Plane
Let P be a set of n points in the plane. We consider the problem of partitioning P into two subsets P_1 and P_2 such that the sum of the perimeters of CH(P_1) and CH(P_2) is minimized, where CH(P_i) denotes the convex hull of P_i. The problem was first studied by Mitchell and Wynters in 1991 who gave an O(n^2) time algorithm. Despite considerable progress on related problems, no subquadratic time algorithm for this problem was found so far. We present an exact algorithm solving the problem in O(n log^4 n) time and a (1+e)-approximation algorithm running in O(n + 1/e^2 log^4(1/e)) time
Fast Fencing
We consider very natural "fence enclosure" problems studied by Capoyleas,
Rote, and Woeginger and Arkin, Khuller, and Mitchell in the early 90s. Given a
set of points in the plane, we aim at finding a set of closed curves
such that (1) each point is enclosed by a curve and (2) the total length of the
curves is minimized. We consider two main variants. In the first variant, we
pay a unit cost per curve in addition to the total length of the curves. An
equivalent formulation of this version is that we have to enclose unit
disks, paying only the total length of the enclosing curves. In the other
variant, we are allowed to use at most closed curves and pay no cost per
curve.
For the variant with at most closed curves, we present an algorithm that
is polynomial in both and . For the variant with unit cost per curve, or
unit disks, we present a near-linear time algorithm.
Capoyleas, Rote, and Woeginger solved the problem with at most curves in
time. Arkin, Khuller, and Mitchell used this to solve the unit cost
per curve version in exponential time. At the time, they conjectured that the
problem with curves is NP-hard for general . Our polynomial time
algorithm refutes this unless P equals NP
Massively Parallel Computation and Sublinear-Time Algorithms for Embedded Planar Graphs
While algorithms for planar graphs have received a lot of attention, few
papers have focused on the additional power that one gets from assuming an
embedding of the graph is available. While in the classic sequential setting,
this assumption gives no additional power (as a planar graph can be embedded in
linear time), we show that this is far from being the case in other settings.
We assume that the embedding is straight-line, but our methods also generalize
to non-straight-line embeddings. Specifically, we focus on sublinear-time
computation and massively parallel computation (MPC).
Our main technical contribution is a sublinear-time algorithm for computing a
relaxed version of an -division. We then show how this can be used to
estimate Lipschitz additive graph parameters. This includes, for example, the
maximum matching, maximum independent set, or the minimum dominating set. We
also show how this can be used to solve some property testing problems with
respect to the vertex edit distance.
In the second part of our paper, we show an MPC algorithm that computes an
-division of the input graph. We show how this can be used to solve various
classical graph problems with space per machine of for
some , and while performing rounds. This includes for
example approximate shortest paths or the minimum spanning tree. Our results
also imply an improved MPC algorithm for Euclidean minimum spanning tree
Genuinely multipartite entangled states and orthogonal arrays
A pure quantum state of N subsystems with d levels each is called
k-multipartite maximally entangled state, written k-uniform, if all its
reductions to k qudits are maximally mixed. These states form a natural
generalization of N-qudits GHZ states which belong to the class 1-uniform
states. We establish a link between the combinatorial notion of orthogonal
arrays and k-uniform states and prove the existence of several new classes of
such states for N-qudit systems. In particular, known Hadamard matrices allow
us to explicitly construct 2-uniform states for an arbitrary number of N>5
qubits. We show that finding a different class of 2-uniform states would imply
the Hadamard conjecture, so the full classification of 2-uniform states seems
to be currently out of reach. Additionally, single vectors of another class of
2-uniform states are one-to-one related to maximal sets of mutually unbiased
bases. Furthermore, we establish links between existence of k-uniform states,
classical and quantum error correction codes and provide a novel graph
representation for such states.Comment: 24 pages, 7 figures. Comments are very welcome
Local-Global Results on Discrete Structures
Local-global arguments, or those which glean global insights from local information, are central ideas in many areas of mathematics and computer science. For instance, in computer science a greedy algorithm makes locally optimal choices that are guaranteed to be consistent with a globally optimal solution. On the mathematical end, global information on Riemannian manifolds is often implied by (local) curvature lower bounds. Discrete notions of graph curvature have recently emerged, allowing ideas pioneered in Riemannian geometry to be extended to the discrete setting. Bakry- Ămery curvature has been one such successful notion of curvature. In this thesis we use combinatorial implications of Bakry- Ămery curvature on graphs to prove a sort of local discrepancy inequality. This then allows us to derive a number of results regarding the local structure of graphs, dependent only on a curvature lower bound. For instance, it turns out that a curvature lower bound implies a nontrivial lower bound on graph connectivity. We also use these results to consider the curvature of strongly regular graphs, a well studied and important class of graphs. In this regard, we give a partial solution to an open conjecture: all SRGs satisfy the curvature condition CD(â, 2). Finally we transition to consider a facility location problem motivated by using Unmanned Aerial Vehicles (UAVs) to guard a border. Here, we find a greedy algorithm, acting on local geometric information, which finds a near optimal placement of base stations for the guarding of UAVs
Bridging the gap between general probabilistic theories and the device-independent framework for nonlocality and contextuality
Characterizing quantum correlations in terms of information-theoretic
principles is a popular chapter of quantum foundations. Traditionally, the
principles adopted for this scope have been expressed in terms of conditional
probability distributions, specifying the probability that a black box produces
a certain output upon receiving a certain input. This framework is known as
"device-independent". Another major chapter of quantum foundations is the
information-theoretic characterization of quantum theory, with its sets of
states and measurements, and with its allowed dynamics. The different
frameworks adopted for this scope are known under the umbrella term "general
probabilistic theories". With only a few exceptions, the two programmes on
characterizing quantum correlations and characterizing quantum theory have so
far proceeded on separate tracks, each one developing its own methods and its
own agenda. This paper aims at bridging the gap, by comparing the two
frameworks and illustrating how the two programmes can benefit each other.Comment: 61 pages, no figures, published versio
Novel Split-Based Approaches to Computing Phylogenetic Diversity and Planar Split Networks
EThOS - Electronic Theses Online ServiceGBUnited Kingdo
Graph Theory
Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem
Grouping Uncertain Oriented Projective Geometric Entities with Application to Automatic Building Reconstruction
The fully automatic reconstruction of 3d scenes from a set of 2d images has always been a key issue in photogrammetry and computer vision and has not been solved satisfactory so far. Most of the current approaches match features between the images based on radiometric cues followed by a reconstruction using the image geometry. The motivation for this work is the conjecture that in the presence of highly redundant data it should be possible to recover the scene structure by grouping together geometric primitives in a bottom-up manner. Oriented projective geometry will be used throughout this work, which allows to represent geometric primitives, such as points, lines and planes in 2d and 3d space as well as projective cameras, together with their uncertainty. The first major contribution of the work is the use of uncertain oriented projective geometry, rather than uncertain projective geometry, that enables the representation of more complex compound entities, such as line segments and polygons in 2d and 3d space as well as 2d edgels and 3d facets. Within the uncertain oriented projective framework a procedure
is developed, which allows to test pairwise relations between the various uncertain oriented projective entities. Again, the novelty lies in the possibility to check relations between the novel compound entities.
The second major contribution of the work is the development of a data structure, specifically
designed to enable performing the tests between large numbers of entities in an efficient manner. Being able to efficiently test relations between the geometric entities, a framework for grouping those entities together is developed. Various different grouping methods are discussed. The third major contribution of this work is the development of a novel grouping method that by analyzing the entropy change incurred by incrementally adding observations into an estimation is able to balance efficiency against robustness in order to achieve better grouping results. Finally the applicability of the proposed representations, tests and grouping methods for the task of purely geometry based building reconstruction from oriented aerial images is demonstrated. lt will be shown that in the presence of highly redundant datasets it is possible
to achieve reasonable reconstruction results by grouping together geometric primitives.Gruppierung unsicherer orientierter projektiver geometrischer Elemente mit Anwendung in der automatischen GebÀuderekonstruktion
Die vollautomatische Rekonstruktion von 3D Szenen aus einer Menge von 2D Bildern war immer ein Hauptanliegen in der Photogrammetrie und Computer Vision und wurde bisher noch nicht zufriedenstellend gelöst. Die meisten aktuellen AnsĂ€tze ordnen Merkmale zwischen den Bildern basierend auf radiometrischen Eigenschaften zu. Daran schlieĂt sich dann eine Rekonstruktion auf der Basis der Bildgeometrie an. Die Motivation fĂŒr diese Arbeit ist die These, dass es möglich sein sollte, die Struktur einer Szene durch Gruppierung geometrischer Primitive zu rekonstruieren, falls die Eingabedaten genĂŒgend redundant sind. Orientierte projektive Geometrie wird in dieser Arbeit zur ReprĂ€sentation geometrischer Primitive, wie Punkten, Linien und Ebenen in 2D und 3D sowie projektiver Kameras, zusammen mit ihrer Unsicherheit verwendet. Der erste Hauptbeitrag dieser Arbeit ist die Verwendung unsicherer orientierter projektiver Geometrie, anstatt von unsicherer projektiver Geometrie, welche die ReprĂ€sentation von komplexeren zusammengesetzten Objekten, wie Liniensegmenten und Polygonen in 2D und 3D sowie 2D Edgels und 3D Facetten, ermöglicht. Innerhalb dieser unsicheren orientierten projektiven ReprĂ€sentation wird ein Verfahren zum Testen paarweiser Relationen zwischen den verschiedenen unsicheren orientierten projektiven geometrischen Elementen entwickelt. Dabei liegt die Neuheit wieder in der Möglichkeit, Relationen zwischen den neuen zusammengesetzten Elementen zu prĂŒfen. Der zweite Hauptbeitrag dieser Arbeit ist die Entwicklung einer Datenstruktur, welche speziell auf die effiziente PrĂŒfung von solchen Relationen zwischen vielen Elementen ausgelegt ist. Die Möglichkeit zur effizienten PrĂŒfung von Relationen zwischen den geometrischen Elementen erlaubt nun die Entwicklung eines Systems zur Gruppierung dieser Elemente. Verschiedene Gruppierungsmethoden werden vorgestellt. Der dritte Hauptbeitrag dieser Arbeit ist die Entwicklung einer neuen Gruppierungsmethode, die durch die Analyse der Ănderung der Entropie beim HinzufĂŒgen von Beobachtungen in die SchĂ€tzung Effizienz und Robustheit gegeneinander ausbalanciert und dadurch bessere Gruppierungsergebnisse erzielt. Zum Schluss wird die Anwendbarkeit der vorgeschlagenen ReprĂ€sentationen, Tests und Gruppierungsmethoden fĂŒr die ausschlieĂlich geometriebasierte GebĂ€uderekonstruktion aus orientierten Luftbildern demonstriert. Es wird gezeigt, dass unter der Annahme von hoch redundanten DatensĂ€tzen vernĂŒnftige Rekonstruktionsergebnisse durch Gruppierung von geometrischen Primitiven erzielbar sind