arXiv subjects

Maarten Löffler

Publications and source records attributed to Maarten Löffler.

At least 55 records · Page 3Linked to original sources

arXiv2020

Global Curve Simplification

Due to its many applications, \emph{curve simplification} is a long-studied problem in computational geometry and adjacent disciplines, such as graphics, geographical information science, etc. Given a polygonal curve $P$ with $n$ vertices, the goal is to find another polygonal curve $P'$ with a smaller number of vertices such that $P'$ is sufficiently similar to $P$. Quality guarantees of a simplification are usually given in a \emph{local} sense, bounding the distance between a shortcut and its corresponding section of the curve. In this work, we aim to provide a systematic overview of curve simplification problems under \emph{global} distance measures that bound the distance between $P$ and $P'$. We consider six different curve distance measures: three variants of the \emph{Hausdorff} distance and three variants of the \emph{Fréchet} distance. And we study different restrictions on the choice of vertices for $P'$. We provide polynomial-time algorithms for some variants of the global curve simplification problem and show NP-hardness for other variants. Through this systematic study we observe, for the first time, some surprising patterns, and suggest directions for future research in this important area.

cs.CG↗

arXiv2019

Most vital segment barriers

We study continuous analogues of "vitality" for discrete network flows/paths, and consider problems related to placing segment barriers that have highest impact on a flow/path in a polygonal domain. This extends the graph-theoretic notion of "most vital arcs" for flows/paths to geometric environments. We give hardness results and efficient algorithms for various versions of the problem, (almost) completely separating hard and polynomially-solvable cases.

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↗

arXiv2019

Lombardi Drawings of Knots and Links

Knot and link diagrams are projections of one or more 3-dimensional simple closed curves into $R^2$, such that no more than two points project to the same point in $R^2$. These diagrams are drawings of 4-regular plane multigraphs. Knots are typically smooth curves in $R^3$, so their projections should be smooth curves in $R^2$ with good continuity and large crossing angles: exactly the properties of Lombardi graph drawings (defined by circular-arc edges and perfect angular resolution). We show that several knots do not allow plane Lombardi drawings. On the other hand, we identify a large class of 4-regular plane multigraphs that do have Lombardi drawings. We then study two relaxations of Lombardi drawings and show that every knot admits a plane 2-Lombardi drawing (where edges are composed of two circular arcs). Further, every knot is near-Lombardi, that is, it can be drawn as Lombardi drawing when relaxing the angular resolution requirement by an arbitrary small angular offset $\varepsilon$, while maintaining a $180^\circ$ angle between opposite edges.

cs.CG↗

arXiv2019

Geometric Multicut

We study the following separation problem: Given a collection of colored objects in the plane, compute a shortest "fence" $F$, i.e., a union of curves of minimum total length, that separates every two objects of different colors. Two objects are separated if $F$ contains a simple closed curve that has one object in the interior and the other in the exterior. We refer to the problem as GEOMETRIC $k$-CUT, where $k$ is the number of different colors, as it can be seen as a geometric analogue to the well-studied multicut problem on graphs. We first give an $O(n^4\log^3 n)$-time algorithm that computes an optimal fence for the case where the input consists of polygons of two colors and $n$ corners in total. We then show that the problem is NP-hard for the case of three colors. Finally, we give a $(2-4/3k)$-approximation algorithm.

cs.CG↗

arXiv2018

Convex partial transversals of planar regions

We consider the problem of testing, for a given set of planar regions $\cal R$ and an integer $k$, whether there exists a convex shape whose boundary intersects at least $k$ regions of $\cal R$. We provide a polynomial time algorithm for the case where the regions are disjoint line segments with a constant number of orientations. On the other hand, we show that the problem is NP-hard when the regions are intersecting axis-aligned rectangles or 3-oriented line segments. For several natural intermediate classes of shapes (arbitrary disjoint segments, intersecting 2-oriented segments) the problem remains open.

cs.CG↗

arXiv2018

How to Fit a Tree in a Box

We study compact straight-line embeddings of trees. We show that perfect binary trees can be embedded optimally: a tree with $n$ nodes can be drawn on a $\sqrt n$ by $\sqrt n$ grid. We also show that testing whether a given binary tree has an upward embedding with a given combinatorial embedding in a given grid is NP-hard.

cs.CG↗

arXiv2018

Approximating $(k,\ell)$-center clustering for curves

The Euclidean $k$-center problem is a classical problem that has been extensively studied in computer science. Given a set $\mathcal{G}$ of $n$ points in Euclidean space, the problem is to determine a set $\mathcal{C}$ of $k$ centers (not necessarily part of $\mathcal{G}$) such that the maximum distance between a point in $\mathcal{G}$ and its nearest neighbor in $\mathcal{C}$ is minimized. In this paper we study the corresponding $(k,\ell)$-center problem for polygonal curves under the Fréchet distance, that is, given a set $\mathcal{G}$ of $n$ polygonal curves in $\mathbb{R}^d$, each of complexity $m$, determine a set $\mathcal{C}$ of $k$ polygonal curves in $\mathbb{R}^d$, each of complexity $\ell$, such that the maximum Fréchet distance of a curve in $\mathcal{G}$ to its closest curve in $\mathcal{C}$ is minimized. In this paper, we substantially extend and improve the known approximation bounds for curves in dimension $2$ and higher. We show that, if $\ell$ is part of the input, then there is no polynomial-time approximation scheme unless $\mathsf{P}=\mathsf{NP}$. Our constructions yield different bounds for one and two-dimensional curves and the discrete and continuous Fréchet distance. In the case of the discrete Fréchet distance on two-dimensional curves, we show hardness of approximation within a factor close to $2.598$. This result also holds when $k=1$, and the $\mathsf{NP}$-hardness extends to the case that $\ell=\infty$, i.e., for the problem of computing the minimum-enclosing ball under the Fréchet distance. Finally, we observe that a careful adaptation of Gonzalez' algorithm in combination with a curve simplification yields a $3$-approximation in any dimension, provided that an optimal simplification can be computed exactly. We conclude that our approximation bounds are close to being tight.

cs.CG↗

arXiv2018

Largest and Smallest Area Triangles on Imprecise Points

Assume we are given a set of parallel line segments in the plane, and we wish to place a point on each line segment such that the resulting point set maximizes or minimizes the area of the largest or smallest triangle in the set. We analyze the complexity of the four resulting computational problems, and we show that three of them admit polynomial-time algorithms, while the fourth is NP-hard. Specifically, we show that maximizing the largest triangle can be done in $O(n^2)$ time (or in $O(n \log n)$ time for unit segments); minimizing the largest triangle can be done in $O(n^2 \log n)$ time; maximizing the smallest triangle is NP-hard; but minimizing the smallest triangle can be done in $O(n^2)$ time. We also discuss to what extent our results can be generalized to polygons with $k>3$ sides.

cs.CG↗

arXiv2018

On Optimal Polyline Simplification using the Hausdorff and Fréchet Distance

We revisit the classical polygonal line simplification problem and study it using the Hausdorff distance and Fréchet distance. Interestingly, no previous authors studied line simplification under these measures in its pure form, namely: for a given $\varepsilon$ > 0, choose a minimum size subsequence of the vertices of the input such that the Hausdorff or Fréchet distance between the input and output polylines is at most $\varepsilon$. We analyze how the well-known Douglas-Peucker and Imai-Iri simplification algorithms perform compared to the optimum possible, also in the situation where the algorithms are given a considerably larger error threshold than $\varepsilon$. Furthermore, we show that computing an optimal simplification using the undirected Hausdorff distance is NP-hard. The same holds when using the directed Hausdorff distance from the input to the output polyline, whereas the reverse can be computed in polynomial time. Finally, to compute the optimal simplification from a polygonal line consisting of $n$ vertices under the Fréchet distance, we give an $O(kn^5)$ time algorithm that requires $O(kn^2)$ space, where $k$ is the output complexity of the simplification.

cs.CG↗

arXiv2018

Dynamic smooth compressed quadtrees (Fullversion)

We introduce dynamic smooth (a.k.a. balanced) compressed quadtrees with worst-case constant time updates in constant dimensions. We distinguish two versions of the problem. First, we show that quadtrees as a space-division data structure can be made smooth and dynamic subject to split and merge operations on the quadtree cells. Second, we show that quadtrees used to store a set of points in $\mathbb{R}^d$ can be made smooth and dynamic subject to insertions and deletions of points. The second version uses the first but must additionally deal with compression and alignment of quadtree components. In both cases our updates take $2^{\mathcal{O}(d\log d )}$ time, except for the point location part in the second version which has a lower bound of $Θ(\log n)$---but if a pointer (finger) to the correct quadtree cell is given, the rest of the updates take worst-case constant time. Our result implies that several classic and recent results (ranging from ray tracing to planar point location) in computational geometry which use quadtrees can deal with arbitrary point sets on a real RAM pointer machine.

cs.CG↗

arXiv2017

Maximum-Area Quadrilateral in a Convex Polygon, Revisited

In this note we show by example that the algorithm presented in 1979 by Dobkin and Snyder for finding the largest-area k-gon that is inscribed in a convex polygon fails to find the optimal solution for k=4. This question, posed by Keikha et al. where they showed that the Dobkin Snyder algorithm fails for k=3.

cs.CG↗

arXiv2017

Maximum-Area Triangle in a Convex Polygon, Revisited

We revisit the following problem: Given a convex polygon $P$, find the largest-area inscribed triangle. We show by example that the linear-time algorithm presented in 1979 by Dobkin and Snyder for solving this problem fails. We then proceed to show that with a small adaptation, their approach does lead to a quadratic-time algorithm. We also present a more involved $O(n\log n)$ time divide-and-conquer algorithm. Also we show by example that the algorithm presented in 1979 by Dobkin and Snyder for finding the largest-area $k$-gon that is inscribed in a convex polygon fails to find the optimal solution for $k=4$. Finally, we discuss the implications of our discoveries on the literature.

cs.CG↗

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↗

arXiv2017

On the Complexity of Barrier Resilience for Fat Regions

In the \emph {barrier resilience} problem (introduced by Kumar {\em et al.}, Wireless Networks 2007), we are given a collection of regions of the plane, acting as obstacles, and we would like to remove the minimum number of regions so that two fixed points can be connected without crossing any region. In this paper, we show that the problem is NP-hard when the collection only contains fat regions with bounded ply $Δ$ (even when they are axis-aligned rectangles of aspect ratio $1 : (1 + \varepsilon)$). We also show that the problem is fixed-parameter tractable (FPT) for unit disks and for similarly-sized $β$-fat regions with bounded ply $Δ$ and $O(1)$ pairwise boundary intersections. Furthermore, we can use our FPT algorithm to construct an $(1+\varepsilon)$-approximation algorithm that runs in $O(2^{f(Δ, \varepsilon,β)}n^5)$ time, where $f\in O(\frac{Δ^4β^8}{\varepsilon^4}\log(βΔ/\varepsilon))$.

cs.CC↗

arXiv2017

Space-Efficient Hidden Surface Removal

We propose a space-efficient algorithm for hidden surface removal that combines one of the fastest previous algorithms for that problem with techniques based on bit manipulation. Such techniques had been successfully used in other settings, for example to reduce working space for several graph algorithms. However, bit manipulation is not usually employed in geometric algorithms because the standard model of computation (the real RAM) does not support it. For this reason, we first revisit our model of computation to have a reasonable theoretical framework. Under this framework we show how the use of a bit representation for the union of triangles, in combination with rank-select data structures, allows us to implicitly compute the union of $n$ triangles with roughly $O(1)$ bits per union boundary vertex. This results in an algorithm that uses at most as much space as the previous one, and depending on the input, can give a reduction of up to a factor $Θ(\log n)$, while maintaining the running time.

cs.CG↗

arXiv2016

Multi-Colored Spanning Graphs

We study a problem proposed by Hurtado et al. (2016) motivated by sparse set visualization. Given $n$ points in the plane, each labeled with one or more primary colors, a \emph{colored spanning graph} (CSG) is a graph such that for each primary color, the vertices of that color induce a connected subgraph. The \textsc{Min-CSG} problem asks for the minimum sum of edge lengths in a colored spanning graph. We show that the problem is NP-hard for $k$ primary colors when $k\ge 3$ and provide a $(2-\frac{1}{3+2\varrho})$-approximation algorithm for $k=3$ that runs in polynomial time, where $\varrho$ is the Steiner ratio. Further, we give a $O(n)$ time algorithm in the special case that the input points are collinear and $k$ is constant.

cs.CG↗

arXiv2016

Critical Placements of a Square or Circle amidst Trajectories for Junction Detection

Motivated by automated junction recognition in tracking data, we study a problem of placing a square or disc of fixed size in an arrangement of lines or line segments in the plane. We let distances among the intersection points of the lines and line segments with the square or circle define a clustering, and study the complexity of \emph{critical} placements for this clustering. Here critical means that arbitrarily small movements of the placement change the clustering. A parameter $\varepsilon$ defines the granularity of the clustering. Without any assumptions on $\varepsilon$, the critical placements have a trivial $O(n^4)$ upper bound. When the square or circle has unit size and $0 < \varepsilon < 1$ is given, we show a refined $O(n^2/\varepsilon^2)$ bound, which is tight in the worst case. We use our combinatorial bounds to design efficient algorithms to compute junctions. As a proof of concept for our algorithms we have a prototype implementation that showcases their application in a basic visualization of a set of $n$ trajectories and their $k$ most important junctions.

cs.CG↗