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Showing papers in "Discrete and Computational Geometry in 2013"


Journal ArticleDOI
TL;DR: An efficient preprocessing algorithm is introduced to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups through an extension of combinatorial Morse theory from complexes to filtrations.
Abstract: We introduce an efficient preprocessing algorithm to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups. The technique is based on an extension of combinatorial Morse theory from complexes to filtrations.

294 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that if the set of points in the plane has at most two points passing through exactly two points of the plane, then there are at least two lines passing through these points.
Abstract: Let $$P$$ be a set of $$n$$ points in the plane, not all on a line We show that if $$n$$ is large then there are at least $$n/2$$ ordinary lines, that is to say lines passing through exactly two points of $$P$$ This confirms, for large $$n$$ , a conjecture of Dirac and Motzkin In fact we describe the exact extremisers for this problem, as well as all sets having fewer than $$n-C$$ ordinary lines for some absolute constant $$C$$ We also solve, for large $$n$$ , the “orchard-planting problem”, which asks for the maximum number of lines through exactly 3 points of $$P$$ Underlying these results is a structure theorem which states that if $$P$$ has at most $$Kn$$ ordinary lines then all but O(K) points of $$P$$ lie on a cubic curve, if $$n$$ is sufficiently large depending on $$K$$

113 citations


Journal ArticleDOI
TL;DR: For the first time, it is shown how to construct an O(n) size filtered simplicial complex on an n-point metric space such that its persistence diagram is a good approximation to that of the Vietoris–Rips filtration.
Abstract: The Vietoris–Rips filtration is a versatile tool in topological data analysis. It is a sequence of simplicial complexes built on a metric space to add topological structure to an otherwise disconnected set of points. It is widely used because it encodes useful information about the topology of the underlying metric space. This information is of-ten extracted from its so-called persistence diagram. Unfortunately, this filtration is often too large to construct in full. We show how to construct an O(n)-size filtered simplicial complex on an n-point metric space such that its persistence diagram is a good approximation to that of the Vietoris–Rips filtration. This new filtration can be constructed in O(n log n) time. The constant factors in both the size and the run-ning time depend only on the doubling dimension of the metric space and the desired tightness of the approximation. For the first time, this makes it computationally tractable to approximate the persistence di-agram of the Vietoris–Rips filtration across all scales for large data sets. We describe two different sparse filtrations. The first is a zigzag filtration that removes points as the scale increases. The second is a (non-zigzag) filtration that yields the same persistence diagram. Both methods are based on a hierarchical net-tree and yield the same guar-antees.

72 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that for any fixed field and a fixed field with a constant ε ≥ c_d
Abstract: Let $$\Delta _{n-1}$$ denote the $$(n-1)$$ -dimensional simplex. Let $$Y$$ be a random $$d$$ -dimensional subcomplex of $$\Delta _{n-1}$$ obtained by starting with the full $$(d-1)$$ -dimensional skeleton of $$\Delta _{n-1}$$ and then adding each $$d$$ -simplex independently with probability $$p=\frac{c}{n}$$ . We compute an explicit constant $$\gamma _d$$ , with $$\gamma _2 \simeq 2.45$$ , $$\gamma _3 \simeq 3.5$$ , and $$\gamma _d=\Theta (\log d)$$ as $$d \rightarrow \infty $$ , so that for $$c < \gamma _d$$ such a random simplicial complex either collapses to a $$(d-1)$$ -dimensional subcomplex or it contains $$\partial \Delta _{d+1}$$ , the boundary of a $$(d+1)$$ -dimensional simplex. We conjecture this bound to be sharp. In addition, we show that there exists a constant $$\gamma _d< c_d c_d$$ and a fixed field $$\mathbb{F }$$ , asymptotically almost surely $$H_d(Y;\mathbb{F }) e 0$$ .

71 citations


Journal ArticleDOI
TL;DR: The problem of moving sensors on a line to form a barrier coverage of a specified segment of the line such that the maximum moving distance of the sensors is minimized is minimized by giving an O(n^2\log n)$$O(n2logn) time algorithm.
Abstract: In this paper, we study the problem of moving $$n$$n sensors on a line to form a barrier coverage of a specified segment of the line such that the maximum moving distance of the sensors is minimized. Previously, it was an open question whether this problem on sensors with arbitrary sensing ranges is solvable in polynomial time. We settle this open question positively by giving an $$O(n^2\log n)$$O(n2logn) time algorithm. For the special case when all sensors have the same-size sensing range, the previously best solution takes $$O(n^2)$$O(n2) time. We present an $$O(n\log n)$$O(nlogn) time algorithm for this case; further, if all sensors are initially located on the coverage segment, our algorithm takes $$O(n)$$O(n) time. Also, we extend our techniques to the cycle version of the problem where the barrier coverage is for a simple cycle and the sensors are allowed to move only along the cycle. For sensors with the same-size sensing range, we solve the cycle version in $$O(n)$$O(n) time, improving the previously best $$O(n^2)$$O(n2) time solution.

70 citations


Journal ArticleDOI
TL;DR: A tropical analogue of the classical double description method, which computes a minimal internal representation (in terms of vertices) of a polyhedron defined externally (by half-spaces or hyperplanes) is developed, showing that this method outperforms the other existing approaches.
Abstract: We establish a characterization of the vertices of a tropical polyhedron defined as the intersection of finitely many half-spaces. We show that a point is a vertex if, and only if, a directed hypergraph, constructed from the subdifferentials of the active constraints at this point, admits a unique strongly connected component that is maximal with respect to the reachability relation (all the other strongly connected components have access to it). This property can be checked in almost linear-time. This allows us to develop a tropical analogue of the classical double description method, which computes a minimal internal representation (in terms of vertices) of a polyhedron defined externally (by half-spaces or hyperplanes). We provide theoretical worst case complexity bounds and report extensive experimental tests performed using the library TPLib, showing that this method outperforms the other existing approaches.

59 citations


Journal ArticleDOI
TL;DR: This work considers the k-power-free points in n-dimensional lattices and explicitly calculates their entropies and diffraction spectra, of particular interest since these sets have holes of unbounded inradius.
Abstract: We consider the $$k$$kth-power-free points in $$n$$n-dimensional lattices and explicitly calculate their entropies and diffraction spectra. This is of particular interest since these sets have holes of unbounded inradius.

56 citations


Journal ArticleDOI
TL;DR: This paper shows that the psd rank of a polytope is at least the dimension of the polytopes plus one, and characterize those polytopes whose pSD rank equals this lower bound.
Abstract: The positive semidefinite (psd) rank of a polytope is the smallest $$k$$k for which the cone of $$k \times k$$k×k real symmetric psd matrices admits an affine slice that projects onto the polytope. In this paper we show that the psd rank of a polytope is at least the dimension of the polytope plus one, and we characterize those polytopes whose psd rank equals this lower bound. We give several classes of polytopes that achieve the minimum possible psd rank including a complete characterization in dimensions two and three.

55 citations


Journal ArticleDOI
TL;DR: This paper analyzes an approximation scheme that keeps the representation linear in the size of the input, while maintaining the guarantees on the inference quality close to those for the exact but costly representation.
Abstract: Distance functions to compact sets play a central role in several areas of computational geometry. Methods that rely on them are robust to the perturbations of the data by the Hausdorff noise, but fail in the presence of outliers. The recently introduced distance to a measure offers a solution by extending the distance function framework to reasoning about the geometry of probability measures, while maintaining theoretical guarantees about the quality of the inferred information. A combinatorial explosion hinders working with distance to a measure as an ordinary power distance function. In this paper, we analyze an approximation scheme that keeps the representation linear in the size of the input, while maintaining the guarantees on the inference quality close to those for the exact but costly representation.

53 citations


Journal ArticleDOI
TL;DR: This paper begins the study of two important aspects of the Reeb graph, which can facilitate its broader applications in shape and data analysis and obtains an efficient near-linear time (expected) algorithm for computing the rank of H1(M) from point data.
Abstract: Given a continuous function f:X→ℝ on a topological space X, its level set f −1(a) changes continuously as the real value a changes. Consequently, the connected components in the level sets appear, disappear, split and merge. The Reeb graph of f summarizes this information into a graph structure. Previous work on Reeb graph mainly focused on its efficient computation. In this paper, we initiate the study of two important aspects of the Reeb graph, which can facilitate its broader applications in shape and data analysis. The first one is the approximation of the Reeb graph of a function on a smooth compact manifold M without boundary. The approximation is computed from a set of points P sampled from M. By leveraging a relation between the Reeb graph and the so-called vertical homology group, as well as between cycles in M and in a Rips complex constructed from P, we compute the H 1-homology of the Reeb graph from P. It takes O(nlogn) expected time, where n is the size of the 2-skeleton of the Rips complex. As a by-product, when M is an orientable 2-manifold, we also obtain an efficient near-linear time (expected) algorithm for computing the rank of H 1(M) from point data. The best-known previous algorithm for this problem takes O(n 3) time for point data. The second aspect concerns the definition and computation of the persistent Reeb graph homology for a sequence of Reeb graphs defined on a filtered space. For a piecewise-linear function defined on a filtration of a simplicial complex K, our algorithm computes all persistent H 1-homology for the Reeb graphs in $O(n n_{e}^{3})$ time, where n is the size of the 2-skeleton and n e is the number of edges in K.

52 citations


Journal ArticleDOI
TL;DR: This work disprove a conjecture, posed by two set of authors, that any 4-connected maximal planar graph has a one-legged Hamiltonian cycle, thereby invalidating an attempt to achieve a polygonal complexity 6 in cartograms for this graph class.
Abstract: In a rectilinear dual of a planar graph vertices are represented by simple rectilinear polygons, while edges are represented by side-contact between the corresponding polygons. A rectilinear dual is called a cartogram if the area of each region is equal to a pre-specified weight. The complexity of a cartogram is determined by the maximum number of corners (or sides) required for any polygon. In a series of papers the polygonal complexity of such representations for maximal planar graphs has been reduced from the initial 40 to 34, then to 12 and very recently to the currently best known 10. Here we describe a construction with 8-sided polygons, which is optimal in terms of polygonal complexity as 8-sided polygons are sometimes necessary. Specifically, we show how to compute the combinatorial structure and how to refine it into an area-universal rectangular layout in linear time. The exact cartogram can be computed from the area-universal layout with numerical iteration, or can be approximated with a hill-climbing heuristic. We also describe an alternative construction of cartograms for Hamiltonian maximal planar graphs, which allows us to directly compute the cartograms in linear time. Moreover, we prove that even for Hamiltonian graphs 8-sided rectilinear polygons are necessary, by constructing a non-trivial lower bound example. The complexity of the cartograms can be reduced to 6 if the Hamiltonian path has the extra property that it is one-legged, as in outer-planar graphs. Thus, we have optimal representations (in terms of both polygonal complexity and running time) for Hamiltonian maximal planar and maximal outer-planar graphs. Finally we address the problem of constructing small-complexity cartograms for 4-connected graphs (which are Hamiltonian). We first disprove a conjecture, posed by two set of authors, that any 4-connected maximal planar graph has a one-legged Hamiltonian cycle, thereby invalidating an attempt to achieve a polygonal complexity 6 in cartograms for this graph class. We also prove that it is NP-hard to decide whether a given 4-connected plane graph admits a cartogram with respect to a given weight function.

Journal ArticleDOI
TL;DR: A lower bound for the number of vertex-labeled neighborly polytopes in even dimension d was shown in this article, where it was shown that Gale Sewing can be used to construct many non-realizable neighborly oriented matroids.
Abstract: In this paper we present a new technique to construct neighborly polytopes, and use it to prove a lower bound of $${\big (( r+d ) ^{( \frac{r}{2}+\frac{d}{2} )^{2}}\big )}\big /{\big ({r}^{{(\frac{r}{2})}^{2}} {d}^{{(\frac{d}{2})}^{2}}{\mathrm{e}^{3\frac{r}{2}\frac{d}{2}}}\big )}$$((r+d)(r2+d2)2)/(r(r2)2d(d2)2e3r2d2) for the number of combinatorial types of vertex-labeled neighborly polytopes in even dimension d with $$r+d+1$$r+d+1 vertices. This improves current bounds on the number of combinatorial types of polytopes. The previous best lower bounds for the number of neighborly polytopes were found by Shemer in 1982 using a technique called the Sewing Construction. We provide a new simple proof that sewing works, and generalize it to oriented matroids in two ways: to Extended Sewing and to Gale Sewing. Our lower bound is obtained by estimating the number of polytopes that can be constructed via Gale Sewing. Combining both new techniques, we are also able to construct many non-realizable neighborly oriented matroids.

Journal ArticleDOI
TL;DR: A finite collection of computable invariants are provided which answer the basic questions on persistence and in addition encode the topology of the source space and its relevant subspaces with these invariants.
Abstract: We study circle-valued maps and consider the persistence of the homology of their fibers. The outcome is a finite collection of computable invariants which answer the basic questions on persistence and in addition encode the topology of the source space and its relevant subspaces. Unlike persistence of real-valued maps, circle-valued maps enjoy a different class of invariants called Jordan cells in addition to bar codes. We establish a relation between the homology of the source space and of its relevant subspaces with these invariants and provide a new algorithm to compute these invariants from an input matrix that encodes a circle-valued map on an input simplicial complex.

Journal ArticleDOI
TL;DR: It is shown that any planar graph has an even subdivision whose complement is a ray intersection graph, which implies that finding a maximum clique in a segment intersection graph is NP-hard.
Abstract: Ray intersection graphs are intersection graphs of rays, or halflines, in the plane. We show that any planar graph has an even subdivision whose complement is a ray intersection graph. The construction can be done in polynomial time and implies that finding a maximum clique in a segment intersection graph is NP-hard. This solves a 21-year old open problem posed by Kratochvil and Nesetřil (Comment Math Univ Carolinae 31(1):85---93, 1990).

Journal ArticleDOI
TL;DR: This work proves small-deviation estimates for the volume of random convex sets generated by independent random points using (Lebesgue) absolutely continuous probability measures with bounded densities and the class of log-concave measures.
Abstract: We prove small-deviation estimates for the volume of random convex sets. The focus is on convex hulls and Minkowski sums of line segments generated by independent random points. The random models considered include (Lebesgue) absolutely continuous probability measures with bounded densities and the class of log-concave measures.

Journal ArticleDOI
TL;DR: It is proved that equally spaced points maximize the minimum of discrete potentials on the unit circle whenever the potential is of the form k=1n f(d(z,z_k), where k denotes the geodesic distance between z and w on the circle.
Abstract: We prove a conjecture of Ambrus, Ball and Erdelyi that equally spaced points maximize the minimum of discrete potentials on the unit circle whenever the potential is of the form $$\begin{aligned} \sum _{k=1}^n f(d(z,z_k)), \end{aligned}$$źk=1nf(d(z,zk)),where $$f:[0,\pi ]\rightarrow [0,\infty ]$$f:[0,ź]ź[0,ź] is non-increasing and convex and $$d(z,w)$$d(z,w) denotes the geodesic distance between z and w on the circle.

Journal ArticleDOI
TL;DR: The key tool in the proof is a novel subdivision of the free space around n disjoint line segments into at most n+1 convex cells such that the dual graph of the subdivision contains two edge-disjoint spanning trees.
Abstract: We prove that for every set of n pairwise disjoint line segments in the plane in general position, where n is even, there is another set of n segments such that the 2n segments form pairwise disjoint simple polygons in the plane. This settles in the affirmative the Disjoint Compatible Matching Conjecture by Aichholzer et al. (Comput. Geom. 42:617–626, 2009). The key tool in our proof is a novel subdivision of the free space around n disjoint line segments into at most n+1 convex cells such that the dual graph of the subdivision contains two edge-disjoint spanning trees.

Journal ArticleDOI
TL;DR: In this paper, the authors investigated algorithmic ways to classify oriented matroids in terms of realizability, and determined all possible combinatorial types (including degenerate ones) of 3-dimensional configurations of 8 points, 2-dimensional configuration of 9 points, and 5-dimensional polytopes with nine vertices.
Abstract: Enumeration of all combinatorial types of point configurations and polytopes is a fundamental problem in combinatorial geometry. Although many studies have been done, most of them are for 2-dimensional and non-degenerate cases. Finschi and Fukuda (Discrete Comput Geom 27:117–136, 2002) published the first database of oriented matroids including degenerate (i.e., non-uniform) ones and of higher ranks. In this paper, we investigate algorithmic ways to classify them in terms of realizability, although the underlying decision problem of realizability checking is NP-hard. As an application, we determine all possible combinatorial types (including degenerate ones) of 3-dimensional configurations of 8 points, 2-dimensional configurations of 9 points, and 5-dimensional configurations of 9 points. We also determine all possible combinatorial types of 5-polytopes with nine vertices.

Journal ArticleDOI
TL;DR: A new technique to investigate crossings in drawings of k-edges is developed and it is proved that the minimum number of crossings determined by a 2-page book drawing of K-n, the complete graph of Hill’s Conjecture.
Abstract: Around 1958, Hill described how to draw the complete graph $$K_n$$ with $$\begin{aligned} Z(n) :=\frac{1}{4}\Big \lfloor \frac{n}{2}\Big \rfloor \Big \lfloor \frac{n-1}{2}\Big \rfloor \Big \lfloor \frac{n-2}{2}\Big \rfloor \Big \lfloor \frac{n-3}{2}\Big \rfloor \end{aligned}$$ crossings, and conjectured that the crossing number $${{\mathrm{cr}}}(K_{n})$$ of $$K_n$$ is exactly $$Z(n)$$ . This is also known as Guy’s conjecture as he later popularized it. Towards the end of the century, substantially different drawings of $$K_{n}$$ with $$Z(n)$$ crossings were found. These drawings are 2-page book drawings, that is, drawings where all the vertices are on a line $$\ell $$ (the spine) and each edge is fully contained in one of the two half-planes (pages) defined by $$\ell $$ . The 2-page crossing number of $$K_{n} $$ , denoted by $$ u _{2}(K_{n})$$ , is the minimum number of crossings determined by a 2-page book drawing of $$K_{n}$$ . Since $${{\mathrm{cr}}}(K_{n}) \le u _{2}(K_{n})$$ and $$ u _{2}(K_{n}) \le Z(n)$$ , a natural step towards Hill’s Conjecture is the weaker conjecture $$ u _{2}(K_{n}) = Z(n)$$ , popularized by Vrt’o. In this paper we develop a new technique to investigate crossings in drawings of $$K_{n}$$ , and use it to prove that $$ u _{2}(K_{n}) = Z(n) $$ . To this end, we extend the inherent geometric definition of $$k$$ -edges for finite sets of points in the plane to topological drawings of $$K_{n}$$ . We also introduce the concept of $${\le }{\le }k$$ -edges as a useful generalization of $${\le }k$$ -edges and extend a powerful theorem that expresses the number of crossings in a rectilinear drawing of $$K_{n}$$ in terms of its number of $${\le }k$$ -edges to the topological setting. Finally, we give a complete characterization of crossing minimal 2-page book drawings of $$K_{n}$$ and show that, up to equivalence, they are unique for $$n$$ even, but that there exist an exponential number of non homeomorphic such drawings for $$n$$ odd.

Journal ArticleDOI
TL;DR: It is shown that, if P is not NP, there is a constant c0>1 such that there is no c0-approximation algorithm for the crossing number, even when restricted to 3-regular graphs.
Abstract: We show that, if $\mathrm{P} ot=\mathrm{NP}$, there is a constant c0>1 such that there is no c0-approximation algorithm for the crossing number, even when restricted to 3-regular graphs.

Journal ArticleDOI
TL;DR: It is proved that for every graph G with n vertices, m edges and no isolated vertices the number of weak isomorphism classes of simple topological graphs that realize G is at most 2, and at least 2 for graphs with m>(6+\varepsilon )n$$m>(6-ε)n.
Abstract: A simple topological graph $$T=(V(T), E(T))$$ is a drawing of a graph in the plane where every two edges have at most one common point (an endpoint or a crossing) and no three edges pass through a single crossing. Topological graphs $$G$$ and $$H$$ are isomorphic if $$H$$ can be obtained from $$G$$ by a homeomorphism of the sphere, and weakly isomorphic if $$G$$ and $$H$$ have the same set of pairs of crossing edges. We generalize results of Pach and Toth and the author’s previous results on counting different drawings of a graph under both notions of isomorphism. We prove that for every graph $$G$$ with $$n$$ vertices, $$m$$ edges and no isolated vertices the number of weak isomorphism classes of simple topological graphs that realize $$G$$ is at most $$2^{O(n^2\log (m/n))}$$ , and at most $$2^{O(mn^{1/2}\log n)}$$ if $$m\le n^{3/2}$$ . As a consequence we obtain a new upper bound $$2^{O(n^{3/2}\log n)}$$ on the number of intersection graphs of $$n$$ pseudosegments. We improve the upper bound on the number of weak isomorphism classes of simple complete topological graphs with $$n$$ vertices to $$2^{n^2\cdot \alpha (n)^{O(1)}}$$ , using an upper bound on the size of a set of permutations with bounded VC-dimension recently proved by Cibulka and the author. We show that the number of isomorphism classes of simple topological graphs that realize $$G$$ is at most $$2^{m^2+O(mn)}$$ and at least $$2^{\Omega (m^2)}$$ for graphs with $$m>(6+\varepsilon )n$$ .

Journal ArticleDOI
TL;DR: In this paper, a general construction for intersection graphs of geometric objects in the plane is presented, where the chromatic number of a graph may be arbitrarily large compared to its clique number.
Abstract: Several classical constructions illustrate the fact that the chromatic number of a graph may be arbitrarily large compared to its clique number. However, until very recently no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set $$X$$X in $$\mathbb{R }^2$$R2 that is not an axis-aligned rectangle and for any positive integer $$k$$k produces a family $$\mathcal{F }$$F of sets, each obtained by an independent horizontal and vertical scaling and translation of $$X$$X, such that no three sets in $$\mathcal{F }$$F pairwise intersect and $$\chi (\mathcal{F })>k$$ź(F)>k. This provides a negative answer to a question of Gyarfas and Lehel for L-shapes. With extra conditions we also show how to construct a triangle-free family of homothetic (uniformly scaled) copies of a set with arbitrarily large chromatic number. This applies to many common shapes, like circles, square boundaries or equilateral L-shapes. Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-line coloring of intervals on the line.

Journal ArticleDOI
TL;DR: A deterministic algorithm is described that finds a Tverberg partition of size n/4(d+1)^3 \rceil in time and can compute an approximate TVerberg point (and hence also an approximate centerpoint) in linear time.
Abstract: Let $$P \subseteq \mathbb{R }^d$$P⊆Rd be a $$d$$d-dimensional $$n$$n-point set. A Tverberg partition is a partition of $$P$$P into $$r$$r sets $$P_1, \dots , P_r$$P1,ź,Pr such that the convex hulls $$\hbox {conv}(P_1), \dots , \hbox {conv}(P_r)$$conv(P1),ź,conv(Pr) have non-empty intersection. A point in $$\bigcap _{i=1}^{r} \hbox {conv}(P_i)$$źi=1rconv(Pi) is called a Tverberg point of depth $$r$$r for $$P$$P. A classic result by Tverberg shows that there always exists a Tverberg partition of size $$\lceil n/(d+1) \rceil $$źn/(d+1)ź, but it is not known how to find such a partition in polynomial time. Therefore, approximate solutions are of interest. We describe a deterministic algorithm that finds a Tverberg partition of size $$\lceil n/4(d+1)^3 \rceil $$źn/4(d+1)3ź in time $$d^{O(\log d)} n$$dO(logd)n. This means that for every fixed dimension we can compute an approximate Tverberg point (and hence also an approximate centerpoint) in linear time. Our algorithm is obtained by combining a novel lifting approach with a recent result by Miller and Sheehy (Comput Geom Theory Appl 43(8):647---654, 2010).

Journal ArticleDOI
TL;DR: In this paper, the geodesic diameter of polygonal domains with holes and corners is computed in worst-case time, in which the distance between two interior points can be computed in linear time.
Abstract: This paper studies the geodesic diameter of polygonal domains having $$h$$ holes and $$n$$ corners. For simple polygons (i.e., $$h=0$$ ), the geodesic diameter is determined by a pair of corners of a given polygon and can be computed in linear time, as shown by Hershberger and Suri. For general polygonal domains with $$h \ge 1$$ , however, no algorithm for computing the geodesic diameter was known prior to this paper. In this paper, we present the first algorithms that compute the geodesic diameter of a given polygonal domain in worst-case time $$O(n^{7.73})$$ or $$O(n^7 (\log n + h))$$ . The main difficulty unlike the simple polygon case relies on the following observation revealed in this paper: two interior points can determine the geodesic diameter and in that case there exist at least five distinct shortest paths between the two.

Journal ArticleDOI
TL;DR: It is shown that if P is a convex multiple tiler in R3, with a discrete multiset of translation vectors, thenΛ has to be a finite union of translated lattices, unless P belongs to a special class of zonotopes, and this conclusion is indeed true for R3.
Abstract: We study multiple tilings of 3-dimensional Euclidean space by a convex body. In a multiple tiling, a convex body $$P$$P is translated with a discrete multiset $$\Lambda $$ź in such a way that each point of $${\mathbb {R}}^d$$Rd gets covered exactly $$k$$k times, except perhaps the translated copies of the boundary of $$P$$P. It is known that all possible multiple tilers in $${\mathbb {R}}^3$$R3 are zonotopes. In $${\mathbb {R}}^2$$R2 it was known by the work of Kolountzakis (Discrete Comput Geom 23(4):537---553, 2000) that, unless $$P$$P is a parallelogram, the multiset of translation vectors $$\Lambda $$ź must be a finite union of translated lattices (also known as quasi periodic sets). In that work (Kolountzakis, Discrete Comput Geom 23(4):537---553, 2000) the author asked whether the same quasi-periodic structure on the translation vectors would be true in $${\mathbb {R}}^3$$R3. Here we prove that this conclusion is indeed true for $${\mathbb {R}}^3$$R3. Namely, we show that if $$P$$P is a convex multiple tiler in $${\mathbb {R}}^3$$R3, with a discrete multiset $$\Lambda $$ź of translation vectors, then $$\Lambda $$ź has to be a finite union of translated lattices, unless $$P$$P belongs to a special class of zonotopes. This exceptional class consists of two-flat zonotopes $$P$$P, defined by the Minkowski sum of two 2-dimensional symmetric polygons in $${\mathbb {R}}^3$$R3, one of which may degenerate into a single line segment. It turns out that rational two-flat zonotopes admit a multiple tiling with an aperiodic (nonquasi-periodic) set of translation vectors $$\Lambda $$ź. We note that it may be quite difficult to offer a visualization of these 3-dimensional non-quasi-periodic tilings, and that we discovered them by using Fourier methods.

Journal ArticleDOI
TL;DR: In this article, a new series of radii called core-radii is introduced to deal with the containment problem under homothetics, which has the minimal enclosing ball (MEB) problem as a prominent representative.
Abstract: This paper deals with the containment problem under homothetics which has the minimal enclosing ball (MEB) problem as a prominent representative. We connect the problem to results in classic convex geometry and introduce a new series of radii, which we call core-radii. For the MEB problem, these radii have already been considered from a different point of view and sharp inequalities between them are known. In this paper sharp inequalities between core-radii for general containment under homothetics are obtained. Moreover, the presented inequalities are used to derive sharp upper bounds on the size of core-sets for containment under homothetics. In the MEB case, this yields a tight (dimension-independent) bound for the size of such core-sets. In the general case, we show that there are core-sets of size linear in the dimension and that this bound stays sharp even if the container is required to be symmetric.

Journal ArticleDOI
TL;DR: It is shown that linear repetitivity is equivalent to positivity of weights combined with a certain balancedness of the shape of return patterns.
Abstract: We consider Delone sets with finite local complexity. We characterize the validity of a subadditive ergodic theorem by uniform positivity of certain weights. The latter can be considered to be an averaged version of linear repetitivity. In this context, we show that linear repetitivity is equivalent to positivity of weights combined with a certain balancedness of the shape of return patterns.

Journal ArticleDOI
TL;DR: The abstract tracing strategy is applied to two different classes of normal curves: abstract curves represented by normal coordinates, which record the number of intersections with each edge of the surface triangulation, and simple geodesics, represented by a starting point and direction in the local coordinate system of some triangle.
Abstract: A simple path or cycle in a triangulated surface is normal if it intersects any triangle in a finite set of arcs, each crossing from one edge of the triangle to another. A normal curve is a finite set of disjoint normal paths and normal cycles. We describe an algorithm to “trace” a normal curve in $$O(\min \{ X, n^2\log X \})$$ time, where $$n$$ is the complexity of the surface triangulation and $$X$$ is the number of times the curve crosses edges of the triangulation. In particular, our algorithm runs in polynomial time even when the number of crossings is exponential in $$n$$ . Our tracing algorithm computes a new cellular decomposition of the surface with complexity $$O(n)$$ ; the traced curve appears in the 1-skeleton of the new decomposition as a set of simple disjoint paths and cycles. We apply our abstract tracing strategy to two different classes of normal curves: abstract curves represented by normal coordinates, which record the number of intersections with each edge of the surface triangulation, and simple geodesics, represented by a starting point and direction in the local coordinate system of some triangle. Our normal-coordinate algorithms are competitive with and conceptually simpler than earlier algorithms by Schaefer et al. (Proceedings of 8th International Conference Computing and Combinatorics. Lecture Notes in Computer Science, vol. 2387, pp. 370–380. Springer, Berlin 2002; Proceedings of 20th Canadian Conference on Computational Geometry, pp. 111–114, 2008) and by Agol et al. (Trans Am Math Soc 358(9): 3821–3850, 2006).

Journal ArticleDOI
TL;DR: This paper characterises when the rth secant variety to X is an irreducible component of the algebraic boundary of the convex hull of the real points of X, a real curve embedded into an even-dimensional affine space.
Abstract: Let $$X\subset \mathbb{A }^{2r}$$XźA2r be a real curve embedded into an even-dimensional affine space. We characterise when the $$r$$rth secant variety to $$X$$X is an irreducible component of the algebraic boundary of the convex hull of the real points $$X(\mathbb{R })$$X(R) of $$X$$X. This fact is then applied to $$4$$4-dimensional $$\mathrm{SO}(2)$$SO(2)-orbitopes and to the so called Barvinok---Novik orbitopes to study when they are basic closed semi-algebraic sets. In the case of $$4$$4-dimensional $$\mathrm{SO}(2)$$SO(2)-orbitopes, we find all irreducible components of their algebraic boundary.

Journal ArticleDOI
TL;DR: This work shows how to compute the tight value z_I$$zI for any inequality that is redundant for an associahedron but facet-defining for the classical permutahedron and a combinatorial interpretation of the values.
Abstract: Realisations of associahedra with linear non-isomorphic normal fans can be obtained by alteration of the right-hand sides of the facet-defining inequalities from a classical permutahedron. These polytopes can be expressed as Minkowski sums and differences of dilated faces of a standard simplex as described by Ardila et al. (Discret Comput Geom, 43:841---854, 2010). The coefficients $$y_I$$yI of such a Minkowski decomposition can be computed by Mobius inversion if tight right-hand sides $$z_I$$zI are known not just for the facet-defining inequalities of the associahedron but also for all inequalities of the permutahedron that are redundant for the associahedron. We show for certain families of these associahedra: (1) How to compute the tight value $$z_I$$zI for any inequality that is redundant for an associahedron but facet-defining for the classical permutahedron. More precisely, each value $$z_I$$zI is described in terms of tight values $$z_J$$zJ of facet-defining inequalities of the corresponding associahedron determined by combinatorial properties of $$I$$I. (2) The computation of the values $$y_I$$yI of Ardila, Benedetti & Doker can be significantly simplified and depends on at most four values $$z_{a(I)},\,z_{b(I)},\,z_{c(I)}$$za(I),zb(I),zc(I) and $$z_{d(I)}$$zd(I). (3) The four indices $$a(I),\,b(I),\,c(I)$$a(I),b(I),c(I) and $$d(I)$$d(I) are determined by the geometry of the normal fan of the associahedron and are described combinatorially. (4) A combinatorial interpretation of the values $$y_I$$yI using a labeled $$n$$n-gon. This result is inspired from similar interpretations for vertex coordinates originally described by Loday and well-known interpretations for the $$z_I$$zI-values of facet-defining inequalities.