arXiv subjects

Bettina Speckmann

Publications and source records attributed to Bettina Speckmann.

At least 55 records · Page 3Linked to original sources

arXiv2020

Quantitative Comparison of Time-Dependent Treemaps

Rectangular treemaps are often the method of choice to visualize large hierarchical datasets. Nowadays such datasets are available over time, hence there is a need for (a) treemaps that can handle time-dependent data, and (b) corresponding quality criteria that cover both a treemap's visual quality and its stability over time. In recent years a wide variety of (stable) treemapping algorithms has been proposed, with various advantages and limitations. We aim to provide insights to researchers and practitioners to allow them to make an informed choice when selecting a treemapping algorithm for specific applications and data. To this end, we perform an extensive quantitative evaluation of rectangular treemaps for time-dependent data. As part of this evaluation we propose a novel classification scheme for time-dependent datasets. Specifically, we observe that the performance of treemapping algorithms depends on the characteristics of the datasets used. We identify four potential representative features that characterize time-dependent hierarchical datasets and classify all datasets used in our experiments accordingly. We experimentally test the validity of this classification on more than 2000 datasets, and analyze the relative performance of 14 state-of-the-art rectangular treemapping algorithms across varying features. Finally, we visually summarize our results with respect to both visual quality and stability to aid users in making an informed choice among treemapping algorithms. All datasets, metrics, and algorithms are openly available to facilitate reuse and further comparative studies.

cs.CG↗

arXiv2019

Optimal Morphs of Planar Orthogonal Drawings II

Van Goethem and Verbeek recently showed how to morph between two planar orthogonal drawings $Γ_I$ and $Γ_O$ of a connected graph $G$ while preserving planarity, orthogonality, and the complexity of the drawing during the morph. Necessarily drawings $Γ_I$ and $Γ_O$ must be equivalent, that is, there exists a homeomorphism of the plane that transforms $Γ_I$ into $Γ_O$. Van Goethem and Verbeek use $O(n)$ linear morphs, where $n$ is the maximum complexity of the input drawings. However, if the graph is disconnected their method requires $O(n^{1.5})$ linear morphs. In this paper we present a refined version of their approach that allows us to also morph between two planar orthogonal drawings of a disconnected graph with $O(n)$ linear morphs while preserving planarity, orthogonality, and linear complexity of the intermediate drawings. Van Goethem and Verbeek measure the structural difference between the two drawings in terms of the so-called spirality $s = O(n)$ of $Γ_I$ relative to $Γ_O$ and describe a morph from $Γ_I$ to $Γ_O$ using $O(s)$ linear morphs. We prove that $s+1$ linear morphs are always sufficient to morph between two planar orthogonal drawings, even for disconnected graphs. The resulting morphs are quite natural and visually pleasing.

cs.CG↗

arXiv2019

Non-crossing paths with geographic constraints

A geographic network is a graph whose vertices are restricted to lie in a prescribed region in the plane. In this paper we begin to study the following fundamental problem for geographic networks: can a given geographic network be drawn without crossings? We focus on the seemingly simple setting where each region is a vertical segment, and one wants to connect pairs of segments with a path that lies inside the convex hull of the two segments. We prove that when paths must be drawn as straight line segments, it is NP-complete to determine if a crossing-free solution exists, even if all vertical segments have unit length. In contrast, we show that when paths must be monotone curves, the question can be answered in polynomial time. In the more general case of paths that can have any shape, we show that the problem is polynomial under certain assumptions.

cs.CG↗

arXiv2019

Preprocessing Ambiguous Imprecise Points

Let ${R} = \{R_1, R_2, ..., R_n\}$ be a set of regions and let $ X = \{x_1, x_2, ..., x_n\}$ be an (unknown) point set with $x_i \in R_i$. Region $R_i$ represents the uncertainty region of $x_i$. We consider the following question: how fast can we establish order if we are allowed to preprocess the regions in $R$? The preprocessing model of uncertainty uses two consecutive phases: a preprocessing phase which has access only to ${R}$ followed by a reconstruction phase during which a desired structure on $X$ is computed. Recent results in this model parametrize the reconstruction time by the ply of ${R}$, which is the maximum overlap between the regions in ${R}$. We introduce the ambiguity $A({R})$ as a more fine-grained measure of the degree of overlap in ${R}$. We show how to preprocess a set of $d$-dimensional disks in $O(n \log n)$ time such that we can sort $X$ (if $d=1$) and reconstruct a quadtree on $X$ (if $d\geq 1$ but constant) in $O(A({R}))$ time. If $A({R})$ is sub-linear, then reporting the result dominates the running time of the reconstruction phase. However, we can still return a suitable data structure representing the result in $O(A({R}))$ time. In one dimension, ${R}$ is a set of intervals and the ambiguity is linked to interval entropy, which in turn relates to the well-studied problem of sorting under partial information. The number of comparisons necessary to find the linear order underlying a poset $P$ is lower-bounded by the graph entropy of $P$. We show that if $P$ is an interval order, then the ambiguity provides a constant-factor approximation of the graph entropy. This gives a lower bound of $Ω(A({R}))$ in all dimensions for the reconstruction phase (sorting or any proximity structure), independent of any preprocessing; hence our result is tight.

cs.CG↗

arXiv2018

SETH Says: Weak Fréchet Distance is Faster, but only if it is Continuous and in One Dimension

We show by reduction from the Orthogonal Vectors problem that algorithms with strongly subquadratic running time cannot approximate the Fréchet distance between curves better than a factor $3$ unless SETH fails. We show that similar reductions cannot achieve a lower bound with a factor better than $3$. Our lower bound holds for the continuous, the discrete, and the weak discrete Fréchet distance even for curves in one dimension. Interestingly, the continuous weak Fréchet distance behaves differently. Our lower bound still holds for curves in two dimensions and higher. However, for curves in one dimension, we provide an exact algorithm to compute the weak Fréchet distance in linear time.

cs.CG↗

arXiv2018

Agglomerative Clustering of Growing Squares

We study an agglomerative clustering problem motivated by interactive glyphs in geo-visualization. Consider a set of disjoint square glyphs on an interactive map. When the user zooms out, the glyphs grow in size relative to the map, possibly with different speeds. When two glyphs intersect, we wish to replace them by a new glyph that captures the information of the intersecting glyphs. We present a fully dynamic kinetic data structure that maintains a set of $n$ disjoint growing squares. Our data structure uses $O(n (\log n \log\log n)^2)$ space, supports queries in worst case $O(\log^3 n)$ time, and updates in $O(\log^7 n)$ amortized time. This leads to an $O(nα(n)\log^7 n)$ time algorithm to solve the agglomerative clustering problem. This is a significant improvement over the current best $O(n^2)$ time algorithms.

cs.DS↗

arXiv2017

Packing Plane Spanning Trees and Paths in Complete Geometric Graphs

We consider the following question: How many edge-disjoint plane spanning trees are contained in a complete geometric graph $GK_n$ on any set $S$ of $n$ points in general position in the plane? We show that this number is in $Ω(\sqrt{n})$. Further, we consider variants of this problem by bounding the diameter and the degree of the trees (in particular considering spanning paths).

cs.CG↗

arXiv2016

Colored Spanning Graphs for Set Visualization

We study an algorithmic problem that is motivated by ink minimization for sparse set visualizations. Our input is a set of points in the plane which are either blue, red, or purple. Blue points belong exclusively to the blue set, red points belong exclusively to the red set, and purple points belong to both sets. A \emph{red-blue-purple spanning graph} (RBP spanning graph) is a set of edges connecting the points such that the subgraph induced by the red and purple points is connected, and the subgraph induced by the blue and purple points is connected. We study the geometric properties of minimum RBP spanning graphs and the algorithmic problems associated with computing them. Specifically, we show that the general problem can be solved in polynomial time using matroid techniques. In addition, we discuss more efficient algorithms for the case in which points are located on a line or a circle, and also describe a fast $(\frac 12ρ+1)$-approximation algorithm, where $ρ$ is the Steiner ratio.

cs.CG↗

arXiv2016

Grouping Time-varying Data for Interactive Exploration

We present algorithms and data structures that support the interactive analysis of the grouping structure of one-, two-, or higher-dimensional time-varying data while varying all defining parameters. Grouping structures characterise important patterns in the temporal evaluation of sets of time-varying data. We follow Buchin et al. [JoCG 2015] who define groups using three parameters: group-size, group-duration, and inter-entity distance. We give upper and lower bounds on the number of maximal groups over all parameter values, and show how to compute them efficiently. Furthermore, we describe data structures that can report changes in the set of maximal groups in an output-sensitive manner. Our results hold in $\mathbb{R}^d$ for fixed $d$.

cs.CG↗

arXiv2016

Distance-Sensitive Planar Point Location

Let $\mathcal{S}$ be a connected planar polygonal subdivision with $n$ edges that we want to preprocess for point-location queries, and where we are given the probability $γ_i$ that the query point lies in a polygon $P_i$ of $\mathcal{S}$. We show how to preprocess $\mathcal{S}$ such that the query time for a point~$p\in P_i$ depends on~$γ_i$ and, in addition, on the distance from $p$ to the boundary of~$P_i$---the further away from the boundary, the faster the query. More precisely, we show that a point-location query can be answered in time $O\left(\min \left(\log n, 1 + \log \frac{\mathrm{area}(P_i)}{γ_i Δ_{p}^2}\right)\right)$, where $Δ_{p}$ is the shortest Euclidean distance of the query point~$p$ to the boundary of $P_i$. Our structure uses $O(n)$ space and $O(n \log n)$ preprocessing time. It is based on a decomposition of the regions of $\mathcal{S}$ into convex quadrilaterals and triangles with the following property: for any point $p\in P_i$, the quadrilateral or triangle containing~$p$ has area $Ω(Δ_{p}^2)$. For the special case where $\mathcal{S}$ is a subdivision of the unit square and $γ_i=\mathrm{area}(P_i)$, we present a simpler solution that achieves a query time of $O\left(\min \left(\log n, \log \frac{1}{Δ_{p}^2}\right)\right)$. The latter solution can be extended to convex subdivisions in three dimensions.

cs.CG↗

arXiv2016

Convex-Arc Drawings of Pseudolines

A weak pseudoline arrangement is a topological generalization of a line arrangement, consisting of curves topologically equivalent to lines that cross each other at most once. We consider arrangements that are outerplanar---each crossing is incident to an unbounded face---and simple---each crossing point is the crossing of only two curves. We show that these arrangements can be represented by chords of a circle, by convex polygonal chains with only two bends, or by hyperbolic lines. Simple but non-outerplanar arrangements (non-weak) can be represented by convex polygonal chains or convex smooth curves of linear complexity.

cs.CG↗

arXiv2015

Computing the Similarity Between Moving Curves

In this paper we study similarity measures for moving curves which can, for example, model changing coastlines or retreating glacier termini. Points on a moving curve have two parameters, namely the position along the curve as well as time. We therefore focus on similarity measures for surfaces, specifically the Fréchet distance between surfaces. While the Fréchet distance between surfaces is not even known to be computable, we show for variants arising in the context of moving curves that they are polynomial-time solvable or NP-complete depending on the restrictions imposed on how the moving curves are matched. We achieve the polynomial-time solutions by a novel approach for computing a surface in the so-called free-space diagram based on max-flow min-cut duality.

cs.CG↗

arXiv2013

Computing the Fréchet distance with shortcuts is NP-hard

We study the shortcut Fréchet distance, a natural variant of the Fréchet distance, that allows us to take shortcuts from and to any point along one of the curves. The classic Fréchet distance is a bottle-neck distance measure and hence quite sensitive to outliers. The shortcut Fréchet distance allows us to cut across outliers and hence produces significantly more meaningful results when dealing with real world data. Driemel and Har-Peled recently described approximation algorithms for the restricted case where shortcuts have to start and end at input vertices. We show that, in the general case, the problem of computing the shortcut Fréchet distance is NP-hard. This is the first hardness result for a variant of the Fréchet distance between two polygonal curves in the plane. We also present two algorithms for the decision problem: a 3-approximation algorithm for the general case and an exact algorithm for the vertex-restricted case. Both algorithms run in O(n^3 log n) time.

cs.CG↗

arXiv2013

Strict Confluent Drawing

We define strict confluent drawing, a form of confluent drawing in which the existence of an edge is indicated by the presence of a smooth path through a system of arcs and junctions (without crossings), and in which such a path, if it exists, must be unique. We prove that it is NP-complete to determine whether a given graph has a strict confluent drawing but polynomial to determine whether it has an outerplanar strict confluent drawing with a fixed vertex ordering (a drawing within a disk, with the vertices placed in a given order on the boundary).

cs.CG↗

arXiv2013

Trajectory Grouping Structures

The collective motion of a set of moving entities like people, birds, or other animals, is characterized by groups arising, merging, splitting, and ending. Given the trajectories of these entities, we define and model a structure that captures all of such changes using the Reeb graph, a concept from topology. The trajectory grouping structure has three natural parameters that allow more global views of the data in group size, group duration, and entity inter-distance. We prove complexity bounds on the maximum number of maximal groups that can be present, and give algorithms to compute the grouping structure efficiently. We also study how the trajectory grouping structure can be made robust, that is, how brief interruptions of groups can be disregarded in the global structure, adding a notion of persistence to the structure. Furthermore, we showcase the results of experiments using data generated by the NetLogo flocking model and from the Starkey project. The Starkey data describe the movement of elk, deer, and cattle. Although there is no ground truth for the grouping structure in this data, the experiments show that the trajectory grouping structure is plausible and has the desired effects when changing the essential parameters. Our research provides the first complete study of trajectory group evolvement, including combinatorial, algorithmic, and experimental results.

cs.CG↗

arXiv2012

Flip Graphs of Degree-Bounded (Pseudo-)Triangulations

We study flip graphs of triangulations whose maximum vertex degree is bounded by a constant $k$. In particular, we consider triangulations of sets of $n$ points in convex position in the plane and prove that their flip graph is connected if and only if $k > 6$; the diameter of the flip graph is $O(n^2)$. We also show that, for general point sets, flip graphs of pointed pseudo-triangulations can be disconnected for $k \leq 9$, and flip graphs of triangulations can be disconnected for any $k$. Additionally, we consider a relaxed version of the original problem. We allow the violation of the degree bound $k$ by a small constant. Any two triangulations with maximum degree at most $k$ of a convex point set are connected in the flip graph by a path of length $O(n \log n)$, where every intermediate triangulation has maximum degree at most $k+4$.

math.CO↗

arXiv2012

Locally Correct Frechet Matchings

The Frechet distance is a metric to compare two curves, which is based on monotonous matchings between these curves. We call a matching that results in the Frechet distance a Frechet matching. There are often many different Frechet matchings and not all of these capture the similarity between the curves well. We propose to restrict the set of Frechet matchings to "natural" matchings and to this end introduce locally correct Frechet matchings. We prove that at least one such matching exists for two polygonal curves and give an O(N^3 log N) algorithm to compute it, where N is the total number of edges in both curves. We also present an O(N^2) algorithm to compute a locally correct discrete Frechet matching.

cs.CG↗

arXiv2011

Angle-Restricted Steiner Arborescences for Flow Map Layout

We introduce a new variant of the geometric Steiner arborescence problem, motivated by the layout of flow maps. Flow maps show the movement of objects between places. They reduce visual clutter by bundling lines smoothly and avoiding self-intersections. To capture these properties, our angle-restricted Steiner arborescences, or flux trees, connect several targets to a source with a tree of minimal length whose arcs obey a certain restriction on the angle they form with the source. We study the properties of optimal flux trees and show that they are planar and consist of logarithmic spirals and straight lines. Flux trees have the shallow-light property. We show that computing optimal flux trees is NP-hard. Hence we consider a variant of flux trees which uses only logarithmic spirals. Spiral trees approximate flux trees within a factor depending on the angle restriction. Computing optimal spiral trees remains NP-hard, but we present an efficient 2-approximation, which can be extended to avoid "positive monotone" obstacles.

cs.CG↗