Kevin Verbeek

Flow Map Layout via Spiral Trees

Flow maps are thematic maps that visualize the movement of objects, such as people or goods, between geographic regions. One or more sources are connected to several targets by lines whose thickness corresponds to the amount of flow between a source and a target. Good flow maps reduce visual clutter by merging (bundling) lines smoothly and by avoiding self-intersections. Most flow maps are still drawn by hand and only few automated methods exist. Some of the known algorithms do not support edge-bundling and those that do, cannot guarantee crossing-free flows. We present a new algorithmic method that uses edge-bundling and computes crossing-free flows of high visual quality. Our method is based on so-called spiral trees, a novel type of Steiner tree which uses logarithmic spirals. Spiral trees naturally induce a clustering on the targets and smoothly bundle lines. Our flows can also avoid obstacles, such as map features, region outlines, or even the targets. We demonstrate our approach with extensive experiments.

On the Most Likely Convex Hull of Uncertain Points

Consider a set of points in d dimensions where the existence or the location of each point is determined by a probability distribution. The convex hull of this set is a random variable distributed over exponentially many choices. We are interested in finding the most likely convex hull, namely, the one with the maximum probability of occurrence. We investigate this problem under two natural models of uncertainty: the point (also called the tuple) model where each point (site) has a fixed position s i but only exists with some probability π i , for 0 < π i ≤ 1, and the multipoint model where each point has multiple possible locations or it may not appear at all. We show that the most likely hull under the point model can be computed in O(n 3) time for n points in d = 2 dimensions, but it is NP–hard for d ≥ 3 dimensions. On the other hand, we show that the problem is NP–hard under the multipoint model even for d = 2 dimensions. We also present hardness results for approximating the probability of the most likely hull. While we focus on the most likely hull for concreteness, our results hold for other natural definitions of a probabilistic hull.

Metric Embedding, Hyperbolic Space, and Social Networks

We consider the problem of embedding an undirected graph into hyperbolic space with minimum distortion. A fundamental problem in its own right, it has also drawn a great deal of interest from applied communities interested in empirical analysis of large-scale graphs. In this paper, we establish a connection between distortion and quasi-cyclicity of graphs, and use it to derive lower and upper bounds on metric distortion. Two particularly simple and natural graphs with large quasi-cyclicity are n-node cycles and n × n square lattices, and our lower bound shows that any hyperbolic-space embedding of these graphs incurs a multiplicative distortion of at least Ω(n/log n). This is in sharp contrast to Euclidean space, where both of these graphs can be embedded with only constant multiplicative distortion. We also establish a relation between quasi-cyclicity and Δ-hyperbolicity of a graph as a way to prove upper bounds on the distortion. Using this relation, we show that graphs with small quasi-cyclicity can be embedded into hyperbolic space with only constant additive distortion. Finally, we also present an efficient (linear-time) randomized algorithm for embedding a graph with small quasi-cyclicity into hyperbolic space, so that with high probability at least a (1 − &epsis;) fraction of the node-pairs has only constant additive distortion. Our results also give a plausible theoretical explanation for why social networks have been observed to embed well into hyperbolic space: they tend to have small quasi-cyclicity.

On the Most Likely Voronoi Diagramand Nearest Neighbor Searching

We consider the problem of nearest-neighbor searching among a set of stochastic sites, where a stochastic site is a tuple \((s_i, \pi _i)\) consisting of a point \(s_i\) in a \(d\)-dimensional space and a probability \(\pi _i\) determining its existence. The problem is interesting and non-trivial even in \(1\)-dimension, where the Most Likely Voronoi Diagram (LVD) is shown to have worst-case complexity \(\Omega (n^2)\). We then show that under more natural and less adversarial conditions, the size of the \(1\)-dimensional LVD is significantly smaller: (1) \(\Theta (k n)\) if the input has only \(k\) distinct probability values, (2) \(O(n \log n)\) on average, and (3) \(O(n \sqrt{n})\) under smoothed analysis. We also present an alternative approach to the most likely nearest neighbor (LNN) search using Pareto sets, which gives a linear-space data structure and sub-linear query time in 1D for average and smoothed analysis models, as well as worst-case with a bounded number of distinct probabilities. Using the Pareto-set approach, we can also reduce the multi-dimensional LNN search to a sequence of nearest neighbor and spherical range queries.

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.

Geometric simultaneous embeddings of a graph and a matching

The geometric simultaneous embedding problem asks whether two planar graphs on the same set of vertices in the plane can be drawn using straight lines, such that each graph is plane. Geometric simultaneous embedding is a current topic in graph drawing and positive and negative results are known for various classes of graphs. So far only connected graphs have been considered. In this paper we present the first results for the setting where one of the graphs is a matching. In particular, we show that there exists a planar graph and a matching which do not admit a geometric simultaneous embedding. This generalizes the same result for a planar graph and a path. On the positive side, we describe algorithms that compute a geometric simultaneous embedding of a matching and a wheel, outerpath, or tree. Our proof for a matching and a tree sheds new light on a major open question: do a tree and a path always admit a geometric simultaneous embedding? Our drawing algorithms minimize the number of orientations used to draw the edges of the matching. Specifically, when embedding a matching and a tree, we can draw all matching edges horizontally. When embedding a matching and a wheel or an outerpath, we use only two orientations.

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.

A framework for algorithm stability and its application to kinetic Euclidean MSTs

We say that an algorithm is stable if small changes in the input result in small changes in the output. This kind of algorithm stability is particularly relevant when analyzing and visualizing time-varying data. Stability in general plays an important role in a wide variety of areas, such as numerical analysis, machine learning, and topology, but is poorly understood in the context of (combinatorial) algorithms.

Tight bounds for conflict-free chromatic guarding of orthogonal art galleries

Abstract The chromatic art gallery problem asks for the minimum number of “colors” t so that a collection of point guards, each assigned one of the t colors, can see the entire polygon subject to some conditions on the colors visible to each point. In this paper, we explore this problem for orthogonal polygons using orthogonal visibility—two points p and q are mutually visible if the smallest axis-aligned rectangle containing them lies within the polygon. Our main result establishes that for a conflict-free guarding of an orthogonal n-gon, in which at least one of the colors seen by every point is unique, the number of colors is in the worst case Θ ( log ⁡ log ⁡ n ) . By contrast, the best known upper bound for orthogonal polygons under standard (non-orthogonal) visibility is O ( log ⁡ n ) colors. We also show that the number of colors needed for strong guarding of simple orthogonal polygons, where all the colors visible to a point are unique, is, again in the worst case, Θ ( log ⁡ n ) . Finally, our techniques also help us establish the first non-trivial lower bound of Ω ( log ⁡ log ⁡ n / log ⁡ log ⁡ log ⁡ n ) for conflict-free guarding under standard visibility. To this end we introduce and utilize a novel discrete combinatorial structure called multicolor tableau.

Mapping Polygons to the Grid with Small Hausdorff and Fréchet Distance

We show how to represent a simple polygon P by a (pixel-based) grid polygon Q that is simple and whose Hausdorff or Frechet distance to P is small. For any simple polygon P, a grid polygon exists with constant Hausdorff distance between their boundaries and their interiors. Moreover, we show that with a realistic input assumption we can also realize constant Frechet distance between the boundaries. We present algorithms accompanying these constructions, heuristics to improve their output while keeping the distance bounds, and experiments to assess the output.