Recognizing and Drawing IC-planar Graphs

IC-planar graphs are those graphs that admit a drawing where no two crossed edges share an end-vertex and each edge is crossed at most once. They are a proper subfamily of the 1-planar graphs. Given an embedded IC-planar graph $G$ with $n$ vertices, we present an $O(n)$-time algorithm that computes a straight-line drawing of $G$ in quadratic area, and an $O(n^3)$-time algorithm that computes a straight-line drawing of $G$ with right-angle crossings in exponential area. Both these area requirements are worst-case optimal. We also show that it is NP-complete to test IC-planarity both in the general case and in the case in which a rotation system is fixed for the input graph. Furthermore, we describe a polynomial-time algorithm to test whether a set of matching edges can be added to a triangulated planar graph such that the resulting graph is IC-planar

Axis-Parallel Right Angle Crossing Graphs

A RAC graph is one admitting a RAC drawing, that is, a polyline drawing in which each crossing occurs at a right angle. Originally motivated by psychological studies on readability of graph layouts, RAC graphs form one of the most prominent graph classes in beyond planarity. In this work, we study a subclass of RAC graphs, called axis-parallel RAC (or apRAC, for short), that restricts the crossings to pairs of axis-parallel edge-segments. apRAC drawings combine the readability of planar drawings with the clarity of (non-planar) orthogonal drawings. We consider these graphs both with and without bends. Our contribution is as follows: (i) We study inclusion relationships between apRAC and traditional RAC graphs. (ii) We establish bounds on the edge density of apRAC graphs. (iii) We show that every graph with maximum degree 8 is 2-bend apRAC and give a linear time drawing algorithm. Some of our results on apRAC graphs also improve the state of the art for general RAC graphs. We conclude our work with a list of open questions and a discussion of a natural generalization of the apRAC model

Axis-Parallel Right Angle Crossing Graphs

A RAC graph is one admitting a RAC drawing, that is, a polyline drawing in which each crossing occurs at a right angle. Originally motivated by psychological studies on readability of graph layouts, RAC graphs form one of the most prominent graph classes in beyond planarity. In this work, we study a subclass of RAC graphs, called axis-parallel RAC (or apRAC, for short), that restricts the crossings to pairs of axis-parallel edge-segments. apRAC drawings combine the readability of planar drawings with the clarity of (non-planar) orthogonal drawings. We consider these graphs both with and without bends. Our contribution is as follows: (i) We study inclusion relationships between apRAC and traditional RAC graphs. (ii) We establish bounds on the edge density of apRAC graphs. (iii) We show that every graph with maximum degree 8 is 2-bend apRAC and give a linear time drawing algorithm. Some of our results on apRAC graphs also improve the state of the art for general RAC graphs. We conclude our work with a list of open questions and a discussion of a natural generalization of the apRAC model

Fixed-Parameter Algorithms for Computing RAC Drawings of Graphs

In a right-angle crossing (RAC) drawing of a graph, each edge is represented as a polyline and edge crossings must occur at an angle of exactly $90^\circ$, where the number of bends on such polylines is typically restricted in some way. While structural and topological properties of RAC drawings have been the focus of extensive research, little was known about the boundaries of tractability for computing such drawings. In this paper, we initiate the study of RAC drawings from the viewpoint of parameterized complexity. In particular, we establish that computing a RAC drawing of an input graph $G$ with at most $b$ bends (or determining that none exists) is fixed-parameter tractable parameterized by either the feedback edge number of $G$, or $b$ plus the vertex cover number of $G$.Comment: Accepted at GD 202

Exploring algorithms to score control points in metrogaine events

Metrogaining is an urban outdoor navigational sport that uses a street map to which scored control points have been added. The objective is to collect maximum score points within a set time by visiting a subset of the scored control points. There is currently no metrogaining scoring standard, only guidelines on how to allocate scores. Accordingly, scoring approaches were explored to create new score sets by using scoring algorithms based on a simple relationship between the score of, and the number of visits to a control point. A spread model, which was developed to evaluate the score sets, generated a range of routes by solving a range of orienteering problems, which belongs to the class of NP-hard combinatorial optimisation problems. From these generated routes, the control point visit frequencies of each control point were determined. Using the visit frequencies, test statistics were subsequently adapted to test the goodness of scoring for each score set. The ndings indicate that the score-visits relationship is not a simple one, as the number of visits to a control point is not only dependent on its score, but also on the scores of the surrounding control points. As a result, the scoring algorithms explored were unable to cope with the complex scoring process uncovered.Decision SciencesM. Sc. (Operations Research

Instantly Decodable Network Coding: From Point to Multi-Point to Device-to-Device Communications

The network coding paradigm enhances transmission efficiency by combining information flows and has drawn significant attention in information theory, networking, communications and data storage. Instantly decodable network coding (IDNC), a subclass of network coding, has demonstrated its ability to improve the quality of service of time critical applications thanks to its attractive properties, namely the throughput enhancement, delay reduction, simple XOR-based encoding and decoding, and small coefficient overhead. Nonetheless, for point to multi-point (PMP) networks, IDNC cannot guarantee the decoding of a specific new packet at individual devices in each transmission. Furthermore, for device-to-device (D2D) networks, the transmitting devices may possess only a subset of packets, which can be used to form coded packets. These challenges require the optimization of IDNC algorithms to be suitable for different application requirements and network configurations. In this thesis, we first study a scalable live video broadcast over a wireless PMP network, where the devices receive video packets from a base station. Such layered live video has a hard deadline and imposes a decoding order on the video layers. We design two prioritized IDNC algorithms that provide a high level of priority to the most important video layer before considering additional video layers in coding decisions. These prioritized algorithms are shown to increase the number of decoded video layers at the devices compared to the existing network coding schemes. We then study video distribution over a partially connected D2D network, where a group of devices cooperate with each other to recover their missing video content. We introduce a cooperation aware IDNC graph that defines all feasible coding and transmission conflictfree decisions. Using this graph, we propose an IDNC solution that avoids coding and transmission conflicts, and meets the hard deadline for high importance video packets. It is demonstrated that the proposed solution delivers an improved video quality to the devices compared to the video and cooperation oblivious coding schemes. We also consider a heterogeneous network wherein devices use two wireless interfaces to receive packets from the base station and another device concurrently. For such network, we are interested in applications with reliable in-order packet delivery requirements. We represent all feasible coding opportunities and conflict-free transmissions using a dual interface IDNC graph. We select a maximal independent set over the graph by considering dual interfaces of individual devices, in-order delivery requirements of packets and lossy channel conditions. This graph based solution is shown to reduce the in-order delivery delay compared to the existing network coding schemes. Finally, we consider a D2D network with a group of devices experiencing heterogeneous channel capacities. For such cooperative scenarios, we address the problem of minimizing the completion time required for recovering all missing packets at the devices using IDNC and physical layer rate adaptation. Our proposed IDNC algorithm balances between the adopted transmission rate and the number of targeted devices that can successfully receive the transmitted packet. We show that the proposed rate aware IDNC algorithm reduces the completion time compared to the rate oblivious coding scheme

Algorithms for visualization of graph-based structures

Buildings today are built to maintain a healthy indoor environment and an efficient energy usage which is probably why damages caused by dampness has increased since the 1960’s. A study between year 2008 and 2010 showed that 26 percent of the 110 000 examined houses had damages and flaws caused by dampness that could prove to be harmful later on. This means that one out of four bathrooms risk the chance to develop damages by dampness. Approximately 2 percent of the houses had already developed water damages. It is here where the problems appear. A house or a building that is damaged by water of dampness need time to dry out before any renovation can take place. This means that damaged parts must be removed and allowed to dry out, this takes a long time to do and the costs are high and at the same time it can cause inconvenience to the residents. Here is where the Air Gap Method enters the picture. The meaning with the method is to drain and dry out the moisture without the need to perform a larger renovation. The Air Gap Method is a so called "forgiving"-system that is if water damages occur the consequences will be small. The Air Gap method means that an air gap is created in the walls, ceiling and the floor where a heating cable in the gap heats up the air and creates an air movement. The point is to create a stack effect in the gap that with the help of the air movement transports the damp air through an opening by the ceiling. The aim of this thesis is to examine if it’s necessary with the heating cable in the air gap and if there is a specific drying out pattern of the water damaged bathroom floor. The possibility of mould growth will also be examined. The study showed that the damped floor did dry out even without a heating cable, but as one of the studies showed signs of mould growth it is shown that the risk for mould growth is higher without a heating cable. There was a seven days difference in the drying out time between the studies with and without the heating cable; this difference can be decisive for mould growth which is why the heating cable is recommended. The Air Gap method is quite easy to apply in houses with light frame constructions simply by using a smaller dimension on the studs to create the air gap in the floor and walls. The method can also be applied in apartment buildings with a concrete frame by using the room-in- room principal. When renovating existing bathrooms it’s easier to use prefabricated elements to create the air gap in the floor and walls. ~