arXiv subjects

Maarten Löffler

Publications and source records attributed to Maarten Löffler.

At least 73 records · Page 4Linked to original sources

arXiv2016

Single-Player and Two-Player Buttons & Scissors Games

We study the computational complexity of the Buttons \& Scissors game and obtain sharp thresholds with respect to several parameters. Specifically we show that the game is NP-complete for $C = 2$ colors but polytime solvable for $C = 1$. Similarly the game is NP-complete if every color is used by at most $F = 4$ buttons but polytime solvable for $F \leq 3$. We also consider restrictions on the board size, cut directions, and cut sizes. Finally, we introduce several natural two-player versions of the game and show that they are PSPACE-complete.

cs.CC↗

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

On the complexity of minimum-link path problems

We revisit the minimum-link path problem: Given a polyhedral domain and two points in it, connect the points by a polygonal path with minimum number of edges. We consider settings where the vertices and/or the edges of the path are restricted to lie on the boundary of the domain, or can be in its interior. Our results include bit complexity bounds, a novel general hardness construction, and a polynomial-time approximation scheme. We fully characterize the situation in 2D, and provide first results in dimensions 3 and higher for several variants of the problem. Concretely, our results resolve several open problems. We prove that computing the minimum-link diffuse reflection path, motivated by ray tracing in computer graphics, is NP-hard, even for two-dimensional polygonal domains with holes. This has remained an open problem [1] despite a large body of work on the topic. We also resolve the open problem from [2] mentioned in the handbook [3] (see Chapter 27.5, Open problem 3) and The Open Problems Project [4] (see Problem 22): "What is the complexity of the minimum-link path problem in 3-space?" Our results imply that the problem is NP-hard even on terrains (and hence, due to discreteness of the answer, there is no FPTAS unless P=NP), but admits a PTAS.

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

The Flip Diameter of Rectangulations and Convex Subdivisions

We study the configuration space of rectangulations and convex subdivisions of $n$ points in the plane. It is shown that a sequence of $O(n\log n)$ elementary flip and rotate operations can transform any rectangulation to any other rectangulation on the same set of $n$ points. This bound is the best possible for some point sets, while $Θ(n)$ operations are sufficient and necessary for others. Some of our bounds generalize to convex subdivisions of $n$ points in the plane.

cs.DM↗

arXiv2015

Counting Carambolas

We give upper and lower bounds on the maximum and minimum number of geometric configurations of various kinds present (as subgraphs) in a triangulation of $n$ points in the plane. Configurations of interest include \emph{convex polygons}, \emph{star-shaped polygons} and \emph{monotone paths}. We also consider related problems for \emph{directed} planar straight-line graphs.

math.CO↗

arXiv2015

Mixed Map Labeling

Point feature map labeling is a geometric problem, in which a set of input points must be labeled with a set of disjoint rectangles (the bounding boxes of the label texts). Typically, labeling models either use internal labels, which must touch their feature point, or external (boundary) labels, which are placed on one of the four sides of the input points' bounding box and which are connected to their feature points by crossing-free leader lines. In this paper we study polynomial-time algorithms for maximizing the number of internal labels in a mixed labeling model that combines internal and external labels. The model requires that all leaders are parallel to a given orientation $θ\in [0,2π)$, whose value influences the geometric properties and hence the running times of our algorithms.

cs.CG↗

arXiv2014

Region-based approximation of probability distributions (for visibility between imprecise points among obstacles)

Let $p$ and $q$ be two imprecise points, given as probability density functions on $\mathbb R^2$, and let $\cal R$ be a set of $n$ line segments (obstacles) in $\mathbb R^2$. We study the problem of approximating the probability that $p$ and $q$ can see each other; that is, that the segment connecting $p$ and $q$ does not cross any segment of $\cal R$. To solve this problem, we approximate each density function by a weighted set of polygons; a novel approach to dealing with probability density functions in computational geometry.

cs.CG↗

arXiv2014

Terrain visibility with multiple viewpoints

We study the problem of visibility in polyhedral terrains in the presence of multiple viewpoints. We consider a triangulated terrain with $m>1$ viewpoints (or guards) located on the terrain surface. A point on the terrain is considered \emph{visible} if it has an unobstructed line of sight to at least one viewpoint. We study several natural and fundamental visibility structures: (1) the visibility map, which is a partition of the terrain into visible and invisible regions; (2) the \emph{colored} visibility map, which is a partition of the terrain into regions whose points have exactly the same visible viewpoints; and (3) the Voronoi visibility map, which is a partition of the terrain into regions whose points have the same closest visible viewpoint. We study the complexity of each structure for both 1.5D and 2.5D terrains, and provide efficient algorithms to construct them. Our algorithm for the visibility map in 2.5D terrains improves on the only existing algorithm in this setting. To the best of our knowledge, the other structures have not been studied before.

cs.CG↗

arXiv2014

Unions of Onions: Preprocessing Imprecise Points for Fast Onion Decomposition

Let $\mathcal{D}$ be a set of $n$ pairwise disjoint unit disks in the plane. We describe how to build a data structure for $\mathcal{D}$ so that for any point set $P$ containing exactly one point from each disk, we can quickly find the onion decomposition (convex layers) of $P$. Our data structure can be built in $O(n \log n)$ time and has linear size. Given $P$, we can find its onion decomposition in $O(n \log k)$ time, where $k$ is the number of layers. We also provide a matching lower bound. Our solution is based on a recursive space decomposition, combined with a fast algorithm to compute the union of two disjoint onion

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

Dynamic Planar Point Location with Sub-Logarithmic Local Updates

We study planar point location in a collection of disjoint fat regions, and investigate the complexity of \emph {local updates}: replacing any region by a different region that is "similar" to the original region. (i.e., the size differs by at most a constant factor, and distance between the two regions is a constant times that size). We show that it is possible to create a linear size data structure that allows for insertions, deletions, and queries in logarithmic time, and allows for local updates in sub-logarithmic time on a pointer machine.

cs.CG↗

arXiv2012

Flow Computations on Imprecise Terrains

We study the computation of the flow of water on imprecise terrains. We consider two approaches to modeling flow on a terrain: one where water flows across the surface of a polyhedral terrain in the direction of steepest descent, and one where water only flows along the edges of a predefined graph, for example a grid or a triangulation. In both cases each vertex has an imprecise elevation, given by an interval of possible values, while its (x,y)-coordinates are fixed. For the first model, we show that the problem of deciding whether one vertex may be contained in the watershed of another is NP-hard. In contrast, for the second model we give a simple O(n log n) time algorithm to compute the minimal and the maximal watershed of a vertex, where n is the number of edges of the graph. On a grid model, we can compute the same in O(n) time.

cs.CG↗

arXiv2012

How Many Potatoes are in a Mesh?

We consider the combinatorial question of how many convex polygons can be made by using the edges taken from a fixed triangulation of n vertices. For general triangulations, there can be exponentially many: we show a construction that has Omega(1.5028^n) convex polygons, and prove an O(1.62^n) upper bound in the worst case. If the triangulation is fat (every triangle has its angles lower-bounded by a constant delta>0), then there can be only polynomially many. We also consider the problem of counting convex outerplanar polygons (i.e., they contain no vertices of the triangulation in their interiors) in the same triangulations. In this setting, we get the same exponential bounds in general triangulations, and lower polynomial bounds in fat triangulations. If the triangulation is furthermore compact (the ratio between the longest and shortest distance between any two vertices is bounded), the bounds drop further to Theta (n^2) for general convex outerplanar polygons, and Theta (n) for fat convex outerplanar polygons.

cs.DM↗

arXiv2012

Triangulating the Square and Squaring the Triangle: Quadtrees and Delaunay Triangulations are Equivalent

We show that Delaunay triangulations and compressed quadtrees are equivalent structures. More precisely, we give two algorithms: the first computes a compressed quadtree for a planar point set, given the Delaunay triangulation; the second finds the Delaunay triangulation, given a compressed quadtree. Both algorithms run in deterministic linear time on a pointer machine. Our work builds on and extends previous results by Krznaric and Levcopolous and Buchin and Mulzer. Our main tool for the second algorithm is the well-separated pair decomposition(WSPD), a structure that has been used previously to find Euclidean minimum spanning trees in higher dimensions (Eppstein). We show that knowing the WSPD (and a quadtree) suffices to compute a planar Euclidean minimum spanning tree (EMST) in linear time. With the EMST at hand, we can find the Delaunay triangulation in linear time. As a corollary, we obtain deterministic versions of many previous algorithms related to Delaunay triangulations, such as splitting planar Delaunay triangulations, preprocessing imprecise points for faster Delaunay computation, and transdichotomous Delaunay triangulations.

cs.CG↗

arXiv2012

Geometric Computations on Indecisive and Uncertain Points

We study computing geometric problems on uncertain points. An uncertain point is a point that does not have a fixed location, but rather is described by a probability distribution. When these probability distributions are restricted to a finite number of locations, the points are called indecisive points. In particular, we focus on geometric shape-fitting problems and on building compact distributions to describe how the solutions to these problems vary with respect to the uncertainty in the points. Our main results are: (1) a simple and efficient randomized approximation algorithm for calculating the distribution of any statistic on uncertain data sets; (2) a polynomial, deterministic and exact algorithm for computing the distribution of answers for any LP-type problem on an indecisive point set; and (3) the development of shape inclusion probability (SIP) functions which captures the ambient distribution of shapes fit to uncertain or indecisive point sets and are admissible to the two algorithmic constructions.

cs.CG↗

arXiv2011

Category-Based Routing in Social Networks: Membership Dimension and the Small-World Phenomenon (Full)

A classic experiment by Milgram shows that individuals can route messages along short paths in social networks, given only simple categorical information about recipients (such as "he is a prominent lawyer in Boston" or "she is a Freshman sociology major at Harvard"). That is, these networks have very short paths between pairs of nodes (the so-called small-world phenomenon); moreover, participants are able to route messages along these paths even though each person is only aware of a small part of the network topology. Some sociologists conjecture that participants in such scenarios use a greedy routing strategy in which they forward messages to acquaintances that have more categories in common with the recipient than they do, and similar strategies have recently been proposed for routing messages in dynamic ad-hoc networks of mobile devices. In this paper, we introduce a network property called membership dimension, which characterizes the cognitive load required to maintain relationships between participants and categories in a social network. We show that any connected network has a system of categories that will support greedy routing, but that these categories can be made to have small membership dimension if and only if the underlying network exhibits the small-world phenomenon.

cs.SI↗

arXiv2011

Planar and Poly-Arc Lombardi Drawings

In Lombardi drawings of graphs, edges are represented as circular arcs, and the edges incident on vertices have perfect angular resolution. However, not every graph has a Lombardi drawing, and not every planar graph has a planar Lombardi drawing. We introduce k-Lombardi drawings, in which each edge may be drawn with k circular arcs, noting that every graph has a smooth 2-Lombardi drawing. We show that every planar graph has a smooth planar 3-Lombardi drawing and further investigate topics connecting planarity and Lombardi drawings.

cs.CG↗