arXiv subjects

Henk Meijer

Publications and source records attributed to Henk Meijer.

18 recordsLinked to original sources

arXiv2024

Unfolding Polycubes with Orthogonally Convex Layers

A polycube is an orthogonal polyhedron composed of unit cubes glued together along entire faces, and homeomorphic to a sphere. A layer of a polycube refers to the portion lying between two horizontal cross-sections spaced one unit apart. We present an unfolding algorithm that flattens any polycube with orthogonally convex layers into a single, non-overlapping planar piece. The algorithm makes cuts only along cube edges-that is, it is an edge unfolding.

cs.CG↗

arXiv2023

New Bounds on the Local and Global Edge-length Ratio of Planar Graphs

The \emph{local edge-length ratio} of a planar straight-line drawing $Γ$ is the largest ratio between the lengths of any pair of edges of $Γ$ that share a common vertex. The \emph{global edge-length ratio} of $Γ$ is the largest ratio between the lengths of any pair of edges of $Γ$. The local (global) edge-length ratio of a planar graph is the infimum over all local (global) edge-length ratios of its planar straight-line drawings. We show that there exist planar graphs with $n$ vertices whose local edge-length ratio is $Ω(\sqrt{n})$. We then show a technique to establish upper bounds on the global (and hence local) edge-length ratio of planar graphs and~apply~it to Halin graphs and to other families of graphs having outerplanarity two.

cs.CG↗

arXiv2019

Packing Trees into 1-planar Graphs

We introduce and study the 1-planar packing problem: Given $k$ graphs with $n$ vertices $G_1, \dots, G_k$, find a 1-planar graph that contains the given graphs as edge-disjoint spanning subgraphs. We mainly focus on the case when each $G_i$ is a tree and $k=3$. We prove that a triple consisting of three caterpillars or of two caterpillars and a path may not admit a 1-planar packing, while two paths and a special type of caterpillar always have one. We then study 1-planar packings with few crossings and prove that three paths (resp. cycles) admit a 1-planar packing with at most seven (resp. fourteen) crossings. We finally show that a quadruple consisting of three paths and a perfect matching with $n \geq 12$ vertices admits a 1-planar packing, while such a packing does not exist if $n \leq 10$.

cs.CG↗

arXiv2019

Line and Plane Cover Numbers Revisited

A measure for the visual complexity of a straight-line crossing-free drawing of a graph is the minimum number of lines needed to cover all vertices. For a given graph $G$, the minimum such number (over all drawings in dimension $d \in \{2,3\}$) is called the \emph{$d$-dimensional weak line cover number} and denoted by $π^1_d(G)$. In 3D, the minimum number of \emph{planes} needed to cover all vertices of~$G$ is denoted by $π^2_3(G)$. When edges are also required to be covered, the corresponding numbers $ρ^1_d(G)$ and $ρ^2_3(G)$ are called the \emph{(strong) line cover number} and the \emph{(strong) plane cover number}. Computing any of these cover numbers -- except $π^1_2(G)$ -- is known to be NP-hard. The complexity of computing $π^1_2(G)$ was posed as an open problem by Chaplick et al. [WADS 2017]. We show that it is NP-hard to decide, for a given planar graph~$G$, whether $π^1_2(G)=2$. We further show that the universal stacked triangulation of depth~$d$, $G_d$, has $π^1_2(G_d)=d+1$. Concerning~3D, we show that any $n$-vertex graph~$G$ with $ρ^2_3(G)=2$ has at most $5n-19$ edges, which is tight.

cs.CG↗

arXiv2018

Polyline Drawings with Topological Constraints

Let $G$ be a simple topological graph and let $Γ$ be a polyline drawing of $G$. We say that $Γ$ \emph{partially preserves the topology} of $G$ if it has the same external boundary, the same rotation system, and the same set of crossings as $G$. Drawing $Γ$ fully preserves the topology of $G$ if the planarization of $G$ and the planarization of $Γ$ have the same planar embedding. We show that if the set of crossing-free edges of $G$ forms a connected spanning subgraph, then $G$ admits a polyline drawing that partially preserves its topology and that has curve complexity at most three (i.e., at most three bends per edge). If, however, the set of crossing-free edges of $G$ is not a connected spanning subgraph, the curve complexity may be $Ω(\sqrt{n})$. Concerning drawings that fully preserve the topology, we show that if $G$ has skewness $k$, it admits one such drawing with curve complexity at most $2k$; for skewness-1 graphs, the curve complexity can be reduced to one, which is a tight bound. We also consider optimal $2$-plane graphs and discuss trade-offs between curve complexity and crossing angle resolution of drawings that fully preserve the topology.

cs.CG↗

arXiv2018

Coordinated Motion Planning: Reconfiguring a Swarm of Labeled Robots with Bounded Stretch

We present a number of breakthroughs for coordinated motion planning, in which the objective is to reconfigure a swarm of labeled convex objects by a combination of parallel, continuous, collision-free translations into a given target arrangement. Problems of this type can be traced back to the classic work of Schwartz and Sharir (1983), who gave a method for deciding the existence of a coordinated motion for a set of disks between obstacles; their approach is polynomial in the complexity of the obstacles, but exponential in the number of disks. Other previous work has largely focused on {\em sequential} schedules, in which one robot moves at a time. We provide constant-factor approximation algorithms for minimizing the execution time of a coordinated, {\em parallel} motion plan for a swarm of robots in the absence of obstacles, provided some amount of separability. Our algorithm achieves {\em constant stretch factor}: If all robots are at most $d$ units from their respective starting positions, the total duration of the overall schedule is $O(d)$. Extensions include unlabeled robots and different classes of robots. We also prove that finding a plan with minimal execution time is NP-hard, even for a grid arrangement without any stationary obstacles. On the other hand, we show that for densely packed disks that cannot be well separated, a stretch factor $Ω(N^{1/4})$ may be required. On the positive side, we establish a stretch factor of $O(N^{1/2})$ even in this case.

cs.CG↗

arXiv2017

New Results on Edge Partitions of 1-plane Graphs

A $1$-plane graph is a graph embedded in the plane such that each edge is crossed at most once. A NIC-plane graph is a $1$-plane graph such that any two pairs of crossing edges share at most one end-vertex. An edge partition of a $1$-plane graph $G$ is a coloring of the edges of $G$ with two colors, red and blue, such that both the graph induced by the red edges and the graph induced by the blue edges are plane graphs. We prove the following: $(i)$ Every NIC-plane graph admits an edge partition such that the red graph has maximum vertex degree three; this bound on the vertex degree is worst-case optimal. $(ii)$ Deciding whether a $1$-plane graph admits an edge partition such that the red graph has maximum vertex degree two is NP-complete. $(iii)$ Deciding whether a $1$-plane graph admits an edge partition such that the red graph has maximum vertex degree one, and computing one in the positive case, can be done in quadratic time. Applications of these results to graph drawing are also discussed.

cs.DM↗

arXiv2017

Ortho-polygon Visibility Representations of Embedded Graphs

An ortho-polygon visibility representation of an $n$-vertex embedded graph $G$ (OPVR of $G$) is an embedding-preserving drawing of $G$ that maps every vertex to a distinct orthogonal polygon and each edge to a vertical or horizontal visibility between its end-vertices. The vertex complexity of an OPVR of $G$ is the minimum $k$ such that every polygon has at most $k$ reflex corners. We present polynomial time algorithms that test whether $G$ has an OPVR and, if so, compute one of minimum vertex complexity. We argue that the existence and the vertex complexity of an OPVR of $G$ are related to its number of crossings per edge and to its connectivity. More precisely, we prove that if $G$ has at most one crossing per edge (i.e., $G$ is a 1-plane graph), an OPVR of $G$ always exists while this may not be the case if two crossings per edge are allowed. Also, if $G$ is a 3-connected 1-plane graph, we can compute an OPVR of $G$ whose vertex complexity is bounded by a constant in $O(n)$ time. However, if $G$ is a 2-connected 1-plane graph, the vertex complexity of any OPVR of $G$ may be $Ω(n)$. In contrast, we describe a family of 2-connected 1-plane graphs for which an embedding that guarantees constant vertex complexity can be computed in $O(n)$ time. Finally, we present the results of an experimental study on the vertex complexity of ortho-polygon visibility representations of 1-plane graphs.

cs.CG↗

arXiv2016

Visibility Representations of Boxes in 2.5 Dimensions

We initiate the study of 2.5D box visibility representations (2.5D-BR) where vertices are mapped to 3D boxes having the bottom face in the plane $z=0$ and edges are unobstructed lines of sight parallel to the $x$- or $y$-axis. We prove that: $(i)$ Every complete bipartite graph admits a 2.5D-BR; $(ii)$ The complete graph $K_n$ admits a 2.5D-BR if and only if $n \leq 19$; $(iii)$ Every graph with pathwidth at most $7$ admits a 2.5D-BR, which can be computed in linear time. We then turn our attention to 2.5D grid box representations (2.5D-GBR) which are 2.5D-BRs such that the bottom face of every box is a unit square at integer coordinates. We show that an $n$-vertex graph that admits a 2.5D-GBR has at most $4n - 6 \sqrt{n}$ edges and this bound is tight. Finally, we prove that deciding whether a given graph $G$ admits a 2.5D-GBR with a given footprint is NP-complete. The footprint of a 2.5D-BR $Γ$ is the set of bottom faces of the boxes in $Γ$.

cs.CG↗

arXiv2008

Minimizing the stabbing number of matchings, trees, and triangulations

The (axis-parallel) stabbing number of a given set of line segments is the maximum number of segments that can be intersected by any one (axis-parallel) line. This paper deals with finding perfect matchings, spanning trees, or triangulations of minimum stabbing number for a given set of points. The complexity of these problems has been a long-standing open question; in fact, it is one of the original 30 outstanding open problems in computational geometry on the list by Demaine, Mitchell, and O'Rourke. The answer we provide is negative for a number of minimum stabbing problems by showing them NP-hard by means of a general proof technique. It implies non-trivial lower bounds on the approximability. On the positive side we propose a cut-based integer programming formulation for minimizing the stabbing number of matchings and spanning trees. We obtain lower bounds (in polynomial time) from the corresponding linear programming relaxations, and show that an optimal fractional solution always contains an edge of at least constant weight. This result constitutes a crucial step towards a constant-factor approximation via an iterated rounding scheme. In computational experiments we demonstrate that our approach allows for actually solving problems with up to several hundred points optimally or near-optimally.

cs.CG↗

arXiv2007

The Distance Geometry of Music

We demonstrate relationships between the classic Euclidean algorithm and many other fields of study, particularly in the context of music and distance geometry. Specifically, we show how the structure of the Euclidean algorithm defines a family of rhythms which encompass over forty timelines (\emph{ostinatos}) from traditional world music. We prove that these \emph{Euclidean rhythms} have the mathematical property that their onset patterns are distributed as evenly as possible: they maximize the sum of the Euclidean distances between all pairs of onsets, viewing onsets as points on a circle. Indeed, Euclidean rhythms are the unique rhythms that maximize this notion of \emph{evenness}. We also show that essentially all Euclidean rhythms are \emph{deep}: each distinct distance between onsets occurs with a unique multiplicity, and these multiplicies form an interval $1,2,...,k-1$. Finally, we characterize all deep rhythms, showing that they form a subclass of generated rhythms, which in turn proves a useful property called shelling. All of our results for musical rhythms apply equally well to musical scales. In addition, many of the problems we explore are interesting in their own right as distance geometry problems on the circle; some of the same problems were explored by Erdős in the plane.

cs.CG↗

arXiv2005

Communication-Aware Processor Allocation for Supercomputers

This paper gives processor-allocation algorithms for minimizing the average number of communication hops between the assigned processors for grid architectures, in the presence of occupied cells. The simpler problem of assigning processors on a free grid has been studied by Karp, McKellar, and Wong who show that the solutions have nontrivial structure; they left open the complexity of the problem. The associated clustering problem is as follows: Given n points in Re^d, find k points that minimize their average pairwise L1 distance. We present a natural approximation algorithm and show that it is a 7/4-approximation for 2D grids. For d-dimensional space, the approximation guarantee is 2-(1/2d), which is tight. We also give a polynomial-time approximation scheme (PTAS) for constant dimension d, and report on experimental results.

cs.DS↗

arXiv2005

An O(n log n)-Time Algorithm for the Restricted Scaffold Assignment

The assignment problem takes as input two finite point sets S and T and establishes a correspondence between points in S and points in T, such that each point in S maps to exactly one point in T, and each point in T maps to at least one point in S. In this paper we show that this problem has an O(n log n)-time solution, provided that the points in S and T are restricted to lie on a line (linear time, if S and T are presorted).

cs.CG↗

arXiv2004

The one-round Voronoi game replayed

We consider the one-round Voronoi game, where player one (``White'', called ``Wilma'') places a set of n points in a rectangular area of aspect ratio r <=1, followed by the second player (``Black'', called ``Barney''), who places the same number of points. Each player wins the fraction of the board closest to one of his points, and the goal is to win more than half of the total area. This problem has been studied by Cheong et al., who showed that for large enough $n$ and r=1, Barney has a strategy that guarantees a fraction of 1/2+a, for some small fixed a. We resolve a number of open problems raised by that paper. In particular, we give a precise characterization of the outcome of the game for optimal play: We show that Barney has a winning strategy for n>2 and r>sqrt{2}/n, and for n=2 and r>sqrt{3}/2. Wilma wins in all remaining cases, i.e., for n>=3 and r<=sqrt{2}/n, for n=2 and r<=sqrt{3}/2, and for n=1. We also discuss complexity aspects of the game on more general boards, by proving that for a polygon with holes, it is NP-hard to maximize the area Barney can win against a given set of points by Wilma.

cs.CG↗

arXiv2003

Maximum dispersion and geometric maximum weight cliques

We consider a facility location problem, where the objective is to ``disperse'' a number of facilities, i.e., select a given number k of locations from a discrete set of n candidates, such that the average distance between selected locations is maximized. In particular, we present algorithmic results for the case where vertices are represented by points in d-dimensional space, and edge weights correspond to rectilinear distances. Problems of this type have been considered before, with the best result being an approximation algorithm with performance ratio 2. For the case where k is fixed, we establish a linear-time algorithm that finds an optimal solution. For the case where k is part of the input, we present a polynomial-time approximation scheme.

cs.DS↗

arXiv2002

Solving a "Hard" Problem to Approximate an "Easy" One: Heuristics for Maximum Matchings and Maximum Traveling Salesman Problems

We consider geometric instances of the Maximum Weighted Matching Problem (MWMP) and the Maximum Traveling Salesman Problem (MTSP) with up to 3,000,000 vertices. Making use of a geometric duality relationship between MWMP, MTSP, and the Fermat-Weber-Problem (FWP), we develop a heuristic approach that yields in near-linear time solutions as well as upper bounds. Using various computational tools, we get solutions within considerably less than 1% of the optimum. An interesting feature of our approach is that, even though an FWP is hard to compute in theory and Edmonds' algorithm for maximum weighted matching yields a polynomial solution for the MWMP, the practical behavior is just the opposite, and we can solve the FWP with high accuracy in order to find a good heuristic solution for the MWMP.

cs.DS↗

arXiv2002

Long Proteins with Unique Optimal Foldings in the H-P Model

It is widely accepted that (1) the natural or folded state of proteins is a global energy minimum, and (2) in most cases proteins fold to a unique state determined by their amino acid sequence. The H-P (hydrophobic-hydrophilic) model is a simple combinatorial model designed to answer qualitative questions about the protein folding process. In this paper we consider a problem suggested by Brian Hayes in 1998: what proteins in the two-dimensional H-P model have unique optimal (minimum energy) foldings? In particular, we prove that there are closed chains of monomers (amino acids) with this property for all (even) lengths; and that there are open monomer chains with this property for all lengths divisible by four.

cs.CG↗

arXiv2000

Flipturning polygons

A flipturn is an operation that transforms a nonconvex simple polygon into another simple polygon, by rotating a concavity 180 degrees around the midpoint of its bounding convex hull edge. Joss and Shannon proved in 1973 that a sequence of flipturns eventually transforms any simple polygon into a convex polygon. This paper describes several new results about such flipturn sequences. We show that any orthogonal polygon is convexified after at most n-5 arbitrary flipturns, or at most 5(n-4)/6 well-chosen flipturns, improving the previously best upper bound of (n-1)!/2. We also show that any simple polygon can be convexified by at most n^2-4n+1 flipturns, generalizing earlier results of Ahn et al. These bounds depend critically on how degenerate cases are handled; we carefully explore several possibilities. We describe how to maintain both a simple polygon and its convex hull in O(log^4 n) time per flipturn, using a data structure of size O(n). We show that although flipturn sequences for the same polygon can have very different lengths, the shape and position of the final convex polygon is the same for all sequences and can be computed in O(n log n) time. Finally, we demonstrate that finding the longest convexifying flipturn sequence of a simple polygon is NP-hard.

cs.CG↗