Discrete & Computational Geometry最新文献

We study the cone of completely positive (cp) matrices for the first interesting case (n = 5). This is a semialgebraic set for which the polynomial equalities and inequlities that define its boundary can be derived. We characterize the different loci of this boundary and we examine the two open sets with cp-rank 5 or 6. A numerical algorithm is presented that is fast and able to compute the cp-factorization even for matrices in the boundary. With our results, many new example cases can be produced and several insightful numerical experiments are performed that illustrate the difficulty of the cp-factorization problem.

Abstract

Almost 50 years ago Erdős and Purdy asked the following question: Given n points in the plane, how many triangles can be approximate congruent to equilateral triangles? They pointed out that by dividing the points evenly into three small clusters built around the three vertices of a fixed equilateral triangle, one gets at least (leftlfloor frac{n}{3} rightrfloor cdot leftlfloor frac{n+1}{3} rightrfloor cdot leftlfloor frac{n+2}{3} rightrfloor ) such approximate copies. In this paper we provide a matching upper bound and thereby answer their question. More generally, for every triangle T we determine the maximum number of approximate congruent triangles to T in a point set of size n. Parts of our proof are based on hypergraph Turán theory: for each point set in the plane and a triangle T, we construct a 3-uniform hypergraph (mathcal {H}=mathcal {H}(T)) , which contains no hypergraph as a subgraph from a family of forbidden hypergraphs (mathcal {F}=mathcal {F}(T)) . Our upper bound on the number of edges of (mathcal {H}) will determine the maximum number of triangles that are approximate congruent to T.

We investigate how the complexity of Euclidean TSP for point sets P inside the strip ((-infty ,+infty )times [0,delta ]) depends on the strip width (delta ). We obtain two main results.

• For the case where the points have distinct integer x-coordinates, we prove that a shortest bitonic tour (which can be computed in (O(nlog ^2 n)) time using an existing algorithm) is guaranteed to be a shortest tour overall when (delta leqslant 2sqrt{2}), a bound which is best possible.

• We present an algorithm that is fixed-parameter tractable with respect to (delta ). Our algorithm has running time (2^{O(sqrt{delta })} n + O(delta ^2 n^2)) for sparse point sets, where each (1times delta ) rectangle inside the strip contains O(1) points. For random point sets, where the points are chosen uniformly at random from the rectangle ([0,n]times [0,delta ]), it has an expected running time of (2^{O(sqrt{delta })} n). These results generalise to point sets P inside a hypercylinder of width (delta ). In this case, the factors (2^{O(sqrt{delta })}) become (2^{O(delta ^{1-1/d})}).

An arc in (mathbb F_q^2) is a set (P subset mathbb F_q^2) such that no three points of P are collinear. We use the method of hypergraph containers to prove several counting results for arcs. Let ({mathcal {A}}(q)) denote the family of all arcs in (mathbb F_q^2). Our main result is the bound

This matches, up to the factor hidden in the o(1) notation, the trivial lower bound that comes from considering all subsets of an arc of size q. We also give upper bounds for the number of arcs of a fixed (large) size. Let (k ge q^{2/3}(log q)^3), and let ({mathcal {A}}(q,k)) denote the family of all arcs in (mathbb F_q^2) with cardinality k. We prove that

This result improves a bound of Roche-Newton and Warren [12]. A nearly matching lower bound

follows by considering all subsets of size k of an arc of size q.

Using persistent homology to guide optimization has emerged as a novel application of topological data analysis. Existing methods treat persistence calculation as a black box and backpropagate gradients only onto the simplices involved in particular pairs. We show how the cycles and chains used in the persistence calculation can be used to prescribe gradients to larger subsets of the domain. In particular, we show that in a special case, which serves as a building block for general losses, the problem can be solved exactly in linear time. This relies on another contribution of this paper, which eliminates the need to examine a factorial number of permutations of simplices with the same value. We present empirical experiments that show the practical benefits of our algorithm: the number of steps required for the optimization is reduced by an order of magnitude.

Consider a set V of voters, represented by a multiset in a metric space (X, d). The voters have to reach a decision—a point in X. A choice (pin X) is called a (beta )-plurality point for V, if for any other choice (qin X) it holds that (|{vin Vmid beta cdot d(p,v)le d(q,v)}| ge frac{|V|}{2}). In other words, at least half of the voters “prefer” p over q, when an extra factor of (beta ) is taken in favor of p. For (beta =1), this is equivalent to Condorcet winner, which rarely exists. The concept of (beta )-plurality was suggested by Aronov, de Berg, Gudmundsson, and Horton [TALG 2021] as a relaxation of the Condorcet criterion. Let (beta ^*_{(X,d)}=sup {beta mid text{ every } text{ finite } text{ multiset } V{ in}X{ admitsa}beta text{-plurality } text{ point }}). The parameter (beta ^*) determines the amount of relaxation required in order to reach a stable decision. Aronov et al. showed that for the Euclidean plane (beta ^*_{({mathbb {R}}^2,Vert cdot Vert _2)}=frac{sqrt{3}}{2}), and more generally, for d-dimensional Euclidean space, (frac{1}{sqrt{d}}le beta ^*_{({mathbb {R}}^d,Vert cdot Vert _2)}le frac{sqrt{3}}{2}). In this paper, we show that (0.557le beta ^*_{({mathbb {R}}^d,Vert cdot Vert _2)}) for any dimension d (notice that (frac{1}{sqrt{d}}<0.557) for any (dge 4)). In addition, we prove that for every metric space (X, d) it holds that (sqrt{2}-1le beta ^*_{(X,d)}), and show that there exists a metric space for which (beta ^*_{(X,d)}le frac{1}{2}).

Simple drawings are drawings of graphs in which the edges are Jordan arcs and each pair of edges share at most one point (a proper crossing or a common endpoint). A simple drawing is c-monotone if there is a point O such that each ray emanating from O crosses each edge of the drawing at most once. We introduce a special kind of c-monotone drawings that we call generalized twisted drawings. A c-monotone drawing is generalized twisted if there is a ray emanating from O that crosses all the edges of the drawing. Via this class of drawings, we show that every simple drawing of the complete graph with n vertices contains (Omega (n^{frac{1}{2}})) pairwise disjoint edges and a plane cycle (and hence path) of length (Omega (frac{log n }{log log n})). Both results improve over best previously published lower bounds. On the way we show several structural results and properties of generalized twisted and c-monotone drawings, some of which we believe to be of independent interest. For example, we show that a drawing D is c-monotone if there exists a point O such that no edge of D is crossed more than once by any ray that emanates from O and passes through a vertex of D.

Let (S subseteq mathbb {R}^2) be a set of n sites in the plane, so that every site (s in S) has an associated radius (r_s > 0). Let (mathcal {D}(S)) be the disk intersection graph defined by S, i.e., the graph with vertex set S and an edge between two distinct sites (s, t in S) if and only if the disks with centers s, t and radii (r_s), (r_t) intersect. Our goal is to design data structures that maintain the connectivity structure of (mathcal {D}(S)) as sites are inserted and/or deleted in S. First, we consider unit disk graphs, i.e., we fix (r_s = 1), for all sites (s in S). For this case, we describe a data structure that has (O(log ^2 n)) amortized update time and (O(log n/log log n)) query time. Second, we look at disk graphs with bounded radius ratio (Psi ), i.e., for all (s in S), we have (1 le r_s le Psi ), for a parameter (Psi ) that is known in advance. Here, we not only investigate the fully dynamic case, but also the incremental and the decremental scenario, where only insertions or only deletions of sites are allowed. In the fully dynamic case, we achieve amortized expected update time (O(Psi log ^{4} n)) and query time (O(log n/log log n)). This improves the currently best update time by a factor of (Psi ). In the incremental case, we achieve logarithmic dependency on (Psi ), with a data structure that has (O(alpha (n))) amortized query time and (O(log Psi log ^{4} n)) amortized expected update time, where (alpha (n)) denotes the inverse Ackermann function. For the decremental setting, we first develop an efficient decremental disk revealing data structure: given two sets R and B of disks in the plane, we can delete disks from B, and upon each deletion, we receive a list of all disks in R that no longer intersect the union of B. Using this data structure, we get decremental data structures with a query time of (O(log n/log log n)) that supports deletions in (O(nlog Psi log ^{4} n)) overall expected time for disk graphs with bounded radius ratio (Psi ) and (O(nlog ^{5} n)) overall expected time for disk graphs with arbitrary radii, assuming that the deletion sequence is oblivious of the internal random choices of the data structures.

We investigate the rigidity properties of rod configurations. Rod configurations are realizations of rank two incidence geometries as points (joints) and straight lines (rods) in the Euclidean plane, such that the lines move as rigid bodies, connected at the points. Note that not all incidence geometries have such realizations. We show that under the assumptions that the rod configuration exists and is sufficiently generic, its infinitesimal rigidity is equivalent to the infinitesimal rigidity of generic frameworks of the graph defined by replacing each rod by a cone over its point set. To put this into context, the molecular conjecture states that the infinitesimal rigidity of rod configurations realizing 2-regular hypergraphs is determined by the rigidity of generic body and hinge frameworks realizing the same hypergraph. This conjecture was proven by Jackson and Jordán in the plane, and by Katoh and Tanigawa in arbitrary dimension. Whiteley proved a version of the molecular conjecture for hypergraphs of arbitrary degree that have realizations as independent body and joint frameworks. Our result extends his result to hypergraphs that do not necessarily have realizations as independent body and joint frameworks, under the assumptions listed above.

Let (n in mathbb {N}) and (k in mathbb {N}_0). Given a set P of n points in the plane, a pair ({p,q}) of points in P is called k-deep, if there are at least k points from P strictly on each side of the line spanned by p and q. A k-deep clique is a subset of P with all its pairs k-deep. We show that if P is in general position (i.e., no three points on a line), there is a k-deep clique of size at least ( max {1,lfloor frac{n}{k+1} rfloor }); this is tight, for example in convex position. A k-deep clique in any set P of n points cannot have size exceeding (n-lceil frac{3k}{2} rceil ); this is tight for (k le frac{n}{3}). Moreover, for (k le lfloor frac{n}{2} rfloor - 1), a k-deep clique cannot have size exceeding (2sqrt{n(lfloor frac{n}{2} rfloor -k)}); this is tight within a constant factor. We also pay special attention to ((frac{n}{2}-1))-deep cliques (for n even), which are called halving cliques. These have been considered in the literature by Khovanova and Yang, 2012, and they play a role in the latter bound above. Every set P in general position with a halving clique Q of size m must have at least (lfloor frac{(m-1)(m+3)}{2}rfloor ) points. If Q is in convex position, the set P must have size at least (m(m-1)). This is tight, i.e., there are sets (Q_m) of m points in convex position which can be extended to a set of (m(m-1)) points where (Q_m) is a halving clique. Interestingly, this is not the case for all sets Q in convex position (even if parallel connecting lines among point pairs in Q are excluded).