In the Priority Steiner Tree (PST) problem, we are given an undirected graph $G=(V,E)$ with a source $s in V$ and terminals $T subseteq V setminus {s}$, where each terminal $v in T$ requires a nonnegative priority $P(v)$. The goal is to compute a min
imum weight Steiner tree containing edges of varying rates such that the path from $s$ to each terminal $v$ consists of edges of rate greater than or equal to $P(v)$. The PST problem with $k$ priorities admits a $min{2 ln |T| + 2, krho}$-approximation [Charikar et al., 2004], and is hard to approximate with ratio $c log log n$ for some constant $c$ [Chuzhoy et al., 2008]. In this paper, we first strengthen the analysis provided by [Charikar et al., 2004] for the $(2 ln |T| + 2)$-approximation to show an approximation ratio of $lceil log_2 |T| rceil + 1 le 1.443 ln |T| + 2$, then provide a very simple, parallelizable algorithm which achieves the same approximation ratio. We then consider a more difficult node-weighted version of the PST problem, and provide a $(2 ln |T|+2)$-approximation using extensions of the spider decomposition by [Klein & Ravi, 1995]. This is the first result for the PST problem in node-weighted graphs. Moreover, the approximation ratios for all above algorithms are tight.
Just-in-time (JIT) compilers are used by many modern programming systems in order to improve performance. Bugs in JIT compilers provide exploitable security vulnerabilities and debugging them is difficult as they are large, complex, and dynamic. Curr
ent debugging and visualization tools deal with static code and are not suitable in this domain. We describe a new approach for simplifying the large and complex intermediate representation, generated by a JIT compiler and visualize it with a metro map metaphor to aid developers in debugging.
Set systems are used to model data that naturally arises in many contexts: social networks have communities, musicians have genres, and patients have symptoms. Visualizations that accurately reflect the information in the underlying set system make i
t possible to identify the set elements, the sets themselves, and the relationships between the sets. In static contexts, such as print media or infographics, it is necessary to capture this information without the help of interactions. With this in mind, we consider three different systems for medium-sized set data, LineSets, EulerView, and MetroSets, and report the results of a controlled human-subjects experiment comparing their effectiveness. Specifically, we evaluate the performance, in terms of time and error, on tasks that cover the spectrum of static set-based tasks. We also collect and analyze qualitative data about the three different visualization systems. Our results include statistically significant differences, suggesting that MetroSets performs and scales better.
We consider the construction of a polygon $P$ with $n$ vertices whose turning angles at the vertices are given by a sequence $A=(alpha_0,ldots, alpha_{n-1})$, $alpha_iin (-pi,pi)$, for $iin{0,ldots, n-1}$. The problem of realizing $A$ by a polygon ca
n be seen as that of constructing a straight-line drawing of a graph with prescribed angles at vertices, and hence, it is a special case of the well studied problem of constructing an emph{angle graph}. In 2D, we characterize sequences $A$ for which every generic polygon $Psubset mathbb{R}^2$ realizing $A$ has at least $c$ crossings, for every $cin mathbb{N}$, and describe an efficient algorithm that constructs, for a given sequence $A$, a generic polygon $Psubset mathbb{R}^2$ that realizes $A$ with the minimum number of crossings. In 3D, we describe an efficient algorithm that tests whether a given sequence $A$ can be realized by a (not necessarily generic) polygon $Psubset mathbb{R}^3$, and for every realizable sequence the algorithm finds a realization.
We propose MetroSets, a new, flexible online tool for visualizing set systems using the metro map metaphor. We model a given set system as a hypergraph $mathcal{H} = (V, mathcal{S})$, consisting of a set $V$ of vertices and a set $mathcal{S}$, which
contains subsets of $V$ called hyperedges. Our system then computes a metro map representation of $mathcal{H}$, where each hyperedge $E$ in $mathcal{S}$ corresponds to a metro line and each vertex corresponds to a metro station. Vertices that appear in two or more hyperedges are drawn as interchanges in the metro map, connecting the different sets. MetroSets is based on a modular 4-step pipeline which constructs and optimizes a path-based hypergraph support, which is then drawn and schematized using metro map layout algorithms. We propose and implement multiple algorithms for each step of the MetroSet pipeline and provide a functional prototype with easy-to-use preset configurations. Furthermore, using several real-world datasets, we perform an extensive quantitative evaluation of the impact of different pipeline stages on desirable properties of the generated maps, such as octolinearity, monotonicity, and edge uniformity.
Do algorithms for drawing graphs pass the Turing Test? That is, are their outputs indistinguishable from graphs drawn by humans? We address this question through a human-centred experiment, focusing on `small graphs, of a size for which it would be r
easonable for someone to choose to draw the graph manually. Overall, we find that hand-drawn layouts can be distinguished from those generated by graph drawing algorithms, although this is not always the case for graphs drawn by force-directed or multi-dimensional scaling algorithms, making these good candidates for Turing Test success. We show that, in general, hand-drawn graphs are judged to be of higher quality than automatically generated ones, although this result varies with graph size and algorithm.
Given a graph $G = (V,E)$ and a subset $T subseteq V$ of terminals, a emph{Steiner tree} of $G$ is a tree that spans $T$. In the vertex-weighted Steiner tree (VST) problem, each vertex is assigned a non-negative weight, and the goal is to compute a m
inimum weight Steiner tree of $G$. We study a natural generalization of the VST problem motivated by multi-level graph construction, the emph{vertex-weighted grade-of-service Steiner tree problem} (V-GSST), which can be stated as follows: given a graph $G$ and terminals $T$, where each terminal $v in T$ requires a facility of a minimum grade of service $R(v)in {1,2,ldotsell}$, compute a Steiner tree $G$ by installing facilities on a subset of vertices, such that any two vertices requiring a certain grade of service are connected by a path in $G$ with the minimum grade of service or better. Facilities of higher grade are more costly than facilities of lower grade. Multi-level variants such as this one can be useful in network design problems where vertices may require facilities of varying priority. While similar problems have been studied in the edge-weighted case, they have not been studied as well in the more general vertex-weighted case. We first describe a simple heuristic for the V-GSST problem whose approximation ratio depends on $ell$, the number of grades of service. We then generalize the greedy algorithm of [Klein & Ravi, 1995] to show that the V-GSST problem admits a $(2 ln |T|)$-approximation, where $T$ is the set of terminals requiring some facility. This result is surprising, as it shows that the (seemingly harder) multi-grade problem can be approximated as well as the VST problem, and that the approximation ratio does not depend on the number of grades of service.
A emph{Stick graph} is an intersection graph of axis-aligned segments such that the left end-points of the horizontal segments and the bottom end-points of the vertical segments lie on a `ground line, a line with slope $-1$. It is an open question to
decide in polynomial time whether a given bipartite graph $G$ with bipartition $Acup B$ has a Stick representation where the vertices in $A$ and $B$ correspond to horizontal and vertical segments, respectively. We prove that $G$ has a Stick representation if and only if there are orderings of $A$ and $B$ such that $G$s bipartite adjacency matrix with rows $A$ and columns $B$ excludes three small `forbidden submatrices. This is similar to characterizations for other classes of bipartite intersection graphs. We present an algorithm to test whether given orderings of $A$ and $B$ permit a Stick representation respecting those orderings, and to find such a representation if it exists. The algorithm runs in time linear in the size of the adjacency matrix. For the case when only the ordering of $A$ is given, we present an $O(|A|^3|B|^3)$-time algorithm. When neither ordering is given, we present some partial results about graphs that are, or are not, Stick representable.
Symmetry is an important factor in human perception in general, as well as in the visualization of graphs in particular. There are three main types of symmetry: reflective, translational, and rotational. We report the results of a human subjects expe
riment to determine what types of symmetries are more salient in drawings of graphs. We found statistically significant evidence that vertical reflective symmetry is the most dominant (when selecting among vertical reflective, horizontal reflective, and translational). We also found statistically significant evidence that rotational symmetry is affected by the number of radial axes (the more, the better), with a notable exception at four axes.
Timeslices are often used to draw and visualize dynamic graphs. While timeslices are a natural way to think about dynamic graphs, they are routinely imposed on continuous data. Often, it is unclear how many timeslices to select: too few timeslices ca
n miss temporal features such as causality or even graph structure while too many timeslices slows the drawing computation. We present a model for dynamic graphs which is not based on timeslices, and a dynamic graph drawing algorithm, DynNoSlice, to draw graphs in this model. In our evaluation, we demonstrate the advantages of this approach over timeslicing on continuous data sets.
