On Upward Drawings of Trees on a Given Grid

Computing a minimum-area planar straight-line drawing of a graph is known to be NP-hard for planar graphs, even when restricted to outerplanar graphs. However, the complexity question is open for trees. Only a few hardness results are known for straight-line drawings of trees under various restrictions such as edge length or slope constraints. On the other hand, there exist polynomial-time algorithms for computing minimum-width (resp., minimum-height) upward drawings of trees, where the height (resp., width) is unbounded. In this paper we take a major step in understanding the complexity of the area minimization problem for strictly-upward drawings of trees, which is one of the most common styles for drawing rooted trees. We prove that given a rooted tree $T$ and a $W\times H$ grid, it is NP-hard to decide whether $T$ admits a strictly-upward (unordered) drawing in the given grid.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Drawing Binary Tanglegrams: An Experimental Evaluation

A binary tanglegram is a pair of binary trees whose leaf sets are in one-to-one correspondence; matching leaves are connected by inter-tree edges. For applications, for example in phylogenetics or software engineering, it is required that the individual trees are drawn crossing-free. A natural optimization problem, denoted tanglegram layout problem, is thus to minimize the number of crossings between inter-tree edges. The tanglegram layout problem is NP-hard and is currently considered both in application domains and theory. In this paper we present an experimental comparison of a recursive algorithm of Buchin et al., our variant of their algorithm, the algorithm hierarchy sort of Holten and van Wijk, and an integer quadratic program that yields optimal solutions.Comment: see http://www.siam.org/proceedings/alenex/2009/alx09_011_nollenburgm.pd

Stress-Minimizing Orthogonal Layout of Data Flow Diagrams with Ports

We present a fundamentally different approach to orthogonal layout of data flow diagrams with ports. This is based on extending constrained stress majorization to cater for ports and flow layout. Because we are minimizing stress we are able to better display global structure, as measured by several criteria such as stress, edge-length variance, and aspect ratio. Compared to the layered approach, our layouts tend to exhibit symmetries, and eliminate inter-layer whitespace, making the diagrams more compact

Exploring atmospheric radon with airborne gamma-ray spectroscopy

$^{222}$Rn is a noble radioactive gas produced along the $^{238}$U decay chain, which is present in the majority of soils and rocks. As $^{222}$Rn is the most relevant source of natural background radiation, understanding its distribution in the environment is of great concern for investigating the health impacts of low-level radioactivity and for supporting regulation of human exposure to ionizing radiation in modern society. At the same time, $^{222}$Rn is a widespread atmospheric tracer whose spatial distribution is generally used as a proxy for climate and pollution studies. Airborne gamma-ray spectroscopy (AGRS) always treated $^{222}$Rn as a source of background since it affects the indirect estimate of equivalent $^{238}$U concentration. In this work the AGRS method is used for the first time for quantifying the presence of $^{222}$Rn in the atmosphere and assessing its vertical profile. High statistics radiometric data acquired during an offshore survey are fitted as a superposition of a constant component due to the experimental setup background radioactivity plus a height dependent contribution due to cosmic radiation and atmospheric $^{222}$Rn. The refined statistical analysis provides not only a conclusive evidence of AGRS $^{222}$Rn detection but also a (0.96 $\pm$ 0.07) Bq/m$^{3}$ $^{222}$Rn concentration and a (1318 $\pm$ 22) m atmospheric layer depth fully compatible with literature data.Comment: 17 pages, 8 figures, 2 table

An SDP approach to multi-level crossing minimization

We present an approach based on semidefinite programs (SDP) to tackle the multi-level crossing minimization prob- lem. Thereby, we are given a layered graph (i.e., the graph´s vertices are assigned to multiple parallel levels) and ask for an ordering of the nodes on their levels such that, when draw- ing the graph with straight lines, the resulting number of crossings is minimized. Solving this step is crucial in the probably most widely used graph drawing scheme, the so- called Sugiyama framework. The problem has received a lot of attention both in the field of heuristics and exact methods. For a long time, integer linear programming (ILP) approaches were the only exact algorithms applicable at least to small graphs. Recently, SDP formulations for the special case of two levels were proposed and dominated the ILP for dense instances. In this paper, we present a new SDP formulation for the general multi-level version that, for two-levels, is even stronger than the aforementioned specialized SDP. As a side- product, we also obtain an SDP-based heuristic which in practice always gives (near-)optimal solutions. We conduct a large set of experiments, both on random- ized and on real-world instances, and compare our approach to a state-of-the-art ILP-based branch-and-cut implementa- tion. The SDP clearly dominates for denser graphs, while the ILP approach is usually faster for sparse instances. However, even for such sparse graphs, the SDP solves more instances to optimality than the ILP. In fact, there is no single instance the ILP solved, which the SDP did not. Overall, our experi- ments reveal that for sparse graphs, one should usually try to find an optimal solution with the ILP first. If this approach does not solve the instance to optimality within reasonable time, the SDP still has a good chance to do so. Being able to solve larger real-world instances than reported before, we are also able to evaluate heuristics for this problem. In this paper we do so for the traditional barycenter-heuristic (showing that it leaves a large gap to the true optimum) and the state-of-the-art upward-planarization method (showing that it is usually close to the optimum)

Simultaneous Drawing of Layered Trees

We study the crossing-minimization problem in a layered graph drawing of planar-embedded rooted trees whose leaves have a given total order on the first layer, which adheres to the embedding of each individual tree. The task is then to permute the vertices on the other layers (respecting the given tree embeddings) in order to minimize the number of crossings. While this problem is known to be NP-hard for multiple trees even on just two layers, we describe a dynamic program running in polynomial time for the restricted case of two trees. If there are more than two trees, we restrict the number of layers to three, which allows for a reduction to a shortest-path problem. This way, we achieve XP-time in the number of trees.Comment: Appears in Proc. 18th International Conference and Workshops on Algorithms and Computation 2024 (WALCOM'24

Turbulent bubbly flow in pipe under gravity and microgravity conditions

Experiments on vertical turbulent flow with millimetric bubbles, under three gravity conditions, upward, downward and microgravity flows (1g, -1g and 0g), have been performed to understand the influence of gravity upon the flow structure and the phase distribution. The mean and fluctuating phase velocities, shear stress, turbulence production, gas fraction and bubble size have been measured or determined. The results for 0g flow obtained during parabolic flights are taken as reference for buoyant 1g and -1g flows. Three buoyancy numbers are introduced to understand and quantify the effects of gravity with respect to friction. We show that the kinematic structure of the liquid is similar to single-phase flow for 0g flow whereas it deviates in 1g and -1g buoyant flows. The present results confirm the existence of a two-layer structure for buoyant flows with a nearly homogeneous core and a wall layer similar to the single-phase inertial layer whose thickness seems to result from a friction–gravity balance. The distributions of phase velocity, shear stress and turbulence are discussed in the light of various existing physical models. This leads to a dimensionless correlation that quantifies the wall shear stress increase due to buoyancy. The turbulent dispersion, the lift and the nonlinear effects of added mass are taken into account in a simplified model for the phase distribution. Its analytical solution gives a qualitative description of the gas fraction distribution in the wall layer