arXiv subjects

Christian Scheffer

Publications and source records attributed to Christian Scheffer.

30 records · Page 2Linked to original sources

arXiv2019

Packing Disks into Disks with Optimal Worst-Case Density

We provide a tight result for a fundamental problem arising from packing disks into a circular container: The critical density of packing disks in a disk is 0.5. This implies that any set of (not necessarily equal) disks of total area $δ\leq 1/2$ can always be packed into a disk of area 1; on the other hand, for any $\varepsilon>0$ there are sets of disks of area $1/2+\varepsilon$ that cannot be packed. The proof uses a careful manual analysis, complemented by a minor automatic part that is based on interval arithmetic. Beyond the basic mathematical importance, our result is also useful as a blackbox lemma for the analysis of recursive packing algorithms.

cs.CG↗

arXiv2018

CADbots: Algorithmic Aspects of Manipulating Programmable Matter with Finite Automata

We contribute results for a set of fundamental problems in the context of programmable matter by presenting algorithmic methods for evaluating and manipulating a collective of particles by a finite automaton that can neither store significant amounts of data, nor perform complex computations, and is limited to a handful of possible physical operations. We provide a toolbox for carrying out fundamental tasks on a given arrangement of tiles, using the arrangement itself as a storage device, similar to a higher-dimensional Turing machine with geometric properties. Specific results include time- and space-efficient procedures for bounding, counting, copying, reflecting, rotating or scaling a complex given shape.

cs.DS↗

arXiv2018

Conflict-Free Coloring of Planar Graphs

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number chi_CF(G) (the smallest k for which conflict-free k-colorings exist). We provide results both for closed neighborhoods N[v], for which a vertex v is a member of its neighborhood, and for open neighborhoods N(v), for which vertex v is not a member of its neighborhood. For closed neighborhoods, we prove the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. We also give a complete characterization of the computational complexity of conflict-free coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G, but polynomial for outerplanar graphs. Furthermore, deciding whether chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for outerplanar graphs. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs. For open neighborhoods, we show that every planar bipartite graph has a conflict-free coloring with at most four colors; on the other hand, we prove that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite graph has a conflict-free k-coloring. Moreover, we establish that any general} planar graph has a conflict-free coloring with at most eight colors.

cs.DM↗

arXiv2018

Split Packing: Algorithms for Packing Circles with Optimal Worst-Case Density

In the classic circle packing problem, one asks whether a given set of circles can be packed into a given container. Packing problems like this have been shown to be $\mathsf{NP}$-hard. In this paper, we present new sufficient conditions for packing circles into square and triangular containers, using only the sum of the circles' areas: For square containers, it is possible to pack any set of circles with a combined area of up to approximately 53.90% of the square's area. And when the container is a right or obtuse triangle, any set of circles whose combined area does not exceed the triangle's incircle can be packed. These area conditions are tight, in the sense that for any larger areas, there are sets of circles which cannot be packed. Similar results have long been known for squares, but to the best of our knowledge, we give the first results of this type for circular objects. Our proofs are constructive: We describe a versatile, divide-and-conquer-based algorithm for packing circles into various container shapes with optimal worst-case density. It employs an elegant subdivision scheme that recursively splits the circles into two groups and then packs these into subcontainers. We call this algorithm "Split Packing". It can be used as a constant-factor approximation algorithm when looking for the smallest container in which a given set of circles can be packed, due to its polynomial runtime. A browser-based, interactive visualization of the Split Packing approach and other related material can be found at https://morr.cc/split-packing/

cs.CG↗

arXiv2018

Don't Rock the Boat: Algorithms for Balanced Dynamic Loading and Unloading

We consider dynamic loading and unloading problems for heavy geometric objects. The challenge is to maintain balanced configurations at all times: minimize the maximal motion of the overall center of gravity. While this problem has been studied from an algorithmic point of view, previous work only focuses on balancing the final center of gravity; we give a variety of results for computing balanced loading and unloading schemes that minimize the maximal motion of the center of gravity during the entire process. In particular, we consider the one-dimensional case and distinguish between loading and unloading. In the unloading variant, the positions of the intervals are given, and we search for an optimal unloading order of the intervals. We prove that the unloading variant is NP-complete and give a 2.7-approximation algorithm. In the loading variant, we have to compute both the positions of the intervals and their loading order. We give optimal approaches for several variants that model different loading scenarios that may arise, e.g., in the loading of a transport ship with containers.

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

Tilt Assembly: Algorithms for Micro-Factories That Build Objects with Uniform External Forces

We present algorithmic results for the parallel assembly of many micro-scale objects in two and three dimensions from tiny particles, which has been proposed in the context of programmable matter and self-assembly for building high-yield micro-factories. The underlying model has particles moving under the influence of uniform external forces until they hit an obstacle; particles can bond when being forced together with another appropriate particle. Due to the physical and geometric constraints, not all shapes can be built in this manner; this gives rise to the Tilt Assembly Problem (TAP) of deciding constructibility. For simply-connected polyominoes $P$ in 2D consisting of $N$ unit-squares ("tiles"), we prove that TAP can be decided in $O(N\log N)$ time. For the optimization variant MaxTAP (in which the objective is to construct a subshape of maximum possible size), we show polyAPX-hardness: unless P=NP, MaxTAP cannot be approximated within a factor of $Ω(N^{\frac{1}{3}})$; for tree-shaped structures, we give an $O(N^{\frac{1}{2}})$-approximation algorithm. For the efficiency of the assembly process itself, we show that any constructible shape allows pipelined assembly, which produces copies of $P$ in $O(1)$ amortized time, i.e., $N$ copies of $P$ in $O(N)$ time steps. These considerations can be extended to three-dimensional objects: For the class of polycubes $P$ we prove that it is NP-hard to decide whether it is possible to construct a path between two points of $P$; it is also NP-hard to decide constructibility of a polycube $P$. Moreover, it is expAPX-hard to maximize a path from a given start point.

cs.DS↗

arXiv2017

Universal Guard Problems

We provide a spectrum of results for the Universal Guard Problem, in which one is to obtain a small set of points ("guards") that are "universal" in their ability to guard any of a set of possible polygonal domains in the plane. We give upper and lower bounds on the number of universal guards that are always sufficient to guard all polygons having a given set of n vertices, or to guard all polygons in a given set of k polygons on an n-point vertex set. Our upper bound proofs include algorithms to construct universal guard sets of the respective cardinalities.

cs.CG↗

arXiv2017

An Efficient Data Structure for Dynamic Two-Dimensional Reconfiguration

In the presence of dynamic insertions and deletions into a partially reconfigurable FPGA, fragmentation is unavoidable. This poses the challenge of developing efficient approaches to dynamic defragmentation and reallocation. One key aspect is to develop efficient algorithms and data structures that exploit the two-dimensional geometry of a chip, instead of just one. We propose a new method for this task, based on the fractal structure of a quadtree, which allows dynamic segmentation of the chip area, along with dynamically adjusting the necessary communication infrastructure. We describe a number of algorithmic aspects, and present different solutions. We also provide a number of basic simulations that indicate that the theoretical worst-case bound may be pessimistic.

cs.DS↗

arXiv2016

New Geometric Algorithms for Fully Connected Staged Self-Assembly

We consider staged self-assembly systems, in which square-shaped tiles can be added to bins in several stages. Within these bins, the tiles may connect to each other, depending on the glue types of their edges. Previous work by Demaine et al. showed that a relatively small number of tile types suffices to produce arbitrary shapes in this model. However, these constructions were only based on a spanning tree of the geometric shape, so they did not produce full connectivity of the underlying grid graph in the case of shapes with holes; designing fully connected assemblies with a polylogarithmic number of stages was left as a major open problem. We resolve this challenge by presenting new systems for staged assembly that produce fully connected polyominoes in O(log^2 n) stages, for various scale factors and temperature τ = 2 as well as τ = 1. Our constructions work even for shapes with holes and uses only a constant number of glues and tiles. Moreover, the underlying approach is more geometric in nature, implying that it promised to be more feasible for shapes with compact geometric description.

cs.DS↗

arXiv2015

Approximating the Integral Fréchet Distance

A pseudo-polynomial time $(1 + \varepsilon)$-approximation algorithm is presented for computing the integral and average Fréchet distance between two given polygonal curves $T_1$ and $T_2$. In particular, the running time is upper-bounded by $\mathcal{O}( ζ^{4}n^4/\varepsilon^{2})$ where $n$ is the complexity of $T_1$ and $T_2$ and $ζ$ is the maximal ratio of the lengths of any pair of segments from $T_1$ and $T_2$. The Fréchet distance captures the minimal cost of a continuous deformation of $T_1$ into $T_2$ and vice versa and defines the cost of a deformation as the maximal distance between two points that are related. The integral Fréchet distance defines the cost of a deformation as the integral of the distances between points that are related. The average Fréchet distance is defined as the integral Fréchet distance divided by the lengths of $T_1$ and $T_2$. Furthermore, we give relations between weighted shortest paths inside a single parameter cell $C$ and the monotone free space axis of $C$. As a result we present a simple construction of weighted shortest paths inside a parameter cell. Additionally, such a shortest path provides an optimal solution for the partial Fréchet similarity of segments for all leash lengths. These two aspects are related to each other and are of independent interest.

cs.CG↗

arXiv2013

Similarity of Polygonal Curves in the Presence of Outliers

The Fréchet distance is a well studied and commonly used measure to capture the similarity of polygonal curves. Unfortunately, it exhibits a high sensitivity to the presence of outliers. Since the presence of outliers is a frequently occurring phenomenon in practice, a robust variant of Fréchet distance is required which absorbs outliers. We study such a variant here. In this modified variant, our objective is to minimize the length of subcurves of two polygonal curves that need to be ignored (MinEx problem), or alternately, maximize the length of subcurves that are preserved (MaxIn problem), to achieve a given Fréchet distance. An exact solution to one problem would imply an exact solution to the other problem. However, we show that these problems are not solvable by radicals over $\mathbb{Q}$ and that the degree of the polynomial equations involved is unbounded in general. This motivates the search for approximate solutions. We present an algorithm, which approximates, for a given input parameter $δ$, optimal solutions for the \MinEx\ and \MaxIn\ problems up to an additive approximation error $δ$ times the length of the input curves. The resulting running time is upper bounded by $\mathcal{O} \left(\frac{n^3}δ \log \left(\frac{n}δ \right)\right)$, where $n$ is the complexity of the input polygonal curves.

cs.CG↗