13 research outputs found
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 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 (12¿+1)-approximation algorithm, where ¿ is the Steiner ratio.Peer ReviewedPostprint (author's final draft
Colored Non-Crossing Euclidean Steiner Forest
Given a set of -colored points in the plane, we consider the problem of
finding trees such that each tree connects all points of one color class,
no two trees cross, and the total edge length of the trees is minimized. For
, this is the well-known Euclidean Steiner tree problem. For general ,
a -approximation algorithm is known, where is the
Steiner ratio.
We present a PTAS for , a -approximation algorithm
for , and two approximation algorithms for general~, with ratios
and
On Embeddability of Buses in Point Sets
Set membership of points in the plane can be visualized by connecting
corresponding points via graphical features, like paths, trees, polygons,
ellipses. In this paper we study the \emph{bus embeddability problem} (BEP):
given a set of colored points we ask whether there exists a planar realization
with one horizontal straight-line segment per color, called bus, such that all
points with the same color are connected with vertical line segments to their
bus. We present an ILP and an FPT algorithm for the general problem. For
restricted versions of this problem, such as when the relative order of buses
is predefined, or when a bus must be placed above all its points, we provide
efficient algorithms. We show that another restricted version of the problem
can be solved using 2-stack pushall sorting. On the negative side we prove the
NP-completeness of a special case of BEP.Comment: 19 pages, 9 figures, conference version at GD 201
Short Plane Supports for Spatial Hypergraphs
A graph is a support of a hypergraph if every hyperedge
induces a connected subgraph in . Supports are used for certain types of
hypergraph visualizations. In this paper we consider visualizing spatial
hypergraphs, where each vertex has a fixed location in the plane. This is the
case, e.g., when modeling set systems of geospatial locations as hypergraphs.
By applying established aesthetic quality criteria we are interested in finding
supports that yield plane straight-line drawings with minimum total edge length
on the input point set . We first show, from a theoretical point of view,
that the problem is NP-hard already under rather mild conditions as well as a
negative approximability results. Therefore, the main focus of the paper lies
on practical heuristic algorithms as well as an exact, ILP-based approach for
computing short plane supports. We report results from computational
experiments that investigate the effect of requiring planarity and acyclicity
on the resulting support length. Further, we evaluate the performance and
trade-offs between solution quality and speed of several heuristics relative to
each other and compared to optimal solutions.Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018
Gap-ETH-Tight Approximation Schemes for Red-Green-Blue Separation and Bicolored Noncrossing Euclidean Travelling Salesman Tours
In this paper, we study problems of connecting classes of points via
noncrossing structures. Given a set of colored terminal points, we want to find
a graph for each color that connects all terminals of its color with the
restriction that no two graphs cross each other. We consider these problems
both on the Euclidean plane and in planar graphs.
On the algorithmic side, we give a Gap-ETH-tight EPTAS for the two-colored
traveling salesman problem as well as for the red-blue-green separation problem
(in which we want to separate terminals of three colors with two noncrossing
polygons of minimum length), both on the Euclidean plane. This improves the
work of Arora and Chang (ICALP 2003) who gave a slower PTAS for the simpler
red-blue separation problem. For the case of unweighted plane graphs, we also
show a PTAS for the two-colored traveling salesman problem. All these results
are based on our new patching procedure that might be of independent interest.
On the negative side, we show that the problem of connecting terminal pairs
with noncrossing paths is NP-hard on the Euclidean plane, and that the problem
of finding two noncrossing spanning trees is NP-hard in plane graphs.Comment: 36 pages, 15 figures (colored
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 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 -approximation algorithm, where ρ is the Steiner ratio
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 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 is NP-hard. Hence we give an (12ρ+1) -approximation, where ρ is the Steiner ratio. We also present efficient exact solutions if the points are located on a line or a circle. Finally we consider extensions to more than two sets
Colored spanning graphs for set visualization
\u3cp\u3eWe 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 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 ([Formula presented]ρ+1)-approximation algorithm, where ρ is the Steiner ratio.\u3c/p\u3