scholarly journals Ordinary hyperspheres and spherical curves

2021 ◽  
Vol 21 (1) ◽  
pp. 15-22
Author(s):  
Aaron Lin ◽  
Konrad Swanepoel
Keyword(s):  
Degenerate Case ◽  
Minimum Number ◽  
Set Of Points ◽  
Orchard Problem

Abstract An ordinary hypersphere of a set of points in real d-space, where no d + 1 points lie on a (d - 2)-sphere or a (d - 2)-flat, is a hypersphere (including the degenerate case of a hyperplane) that contains exactly d + 1 points of the set. Similarly, a (d + 2)-point hypersphere of such a set is one that contains exactly d + 2 points of the set. We find the minimum number of ordinary hyperspheres, solving the d-dimensional spherical analogue of the Dirac–Motzkin conjecture for d ⩾ 3. We also find the maximum number of (d + 2)-point hyperspheres in even dimensions, solving the d-dimensional spherical analogue of the orchard problem for even d ⩾ 4.

APPROXIMATION ALGORITHMS FOR A VARIANT OF DISCRETE PIERCING SET PROBLEM FOR UNIT DISKS

2013 ◽  
Vol 23 (06) ◽  
pp. 461-477 ◽  
Author(s):  
MINATI DE ◽  
GAUTAM K. DAS ◽  
PAZ CARMI ◽  
SUBHAS C. NANDY
Keyword(s):  
Approximation Algorithms ◽  
Simple Algorithm ◽  
Constant Factor ◽  
Performance Ratio ◽  
Approximation Result ◽  
Worst Case ◽  
Approximation Factor ◽  
Minimum Number ◽  
Unit Disks ◽  
Set Of Points

In this paper, we consider constant factor approximation algorithms for a variant of the discrete piercing set problem for unit disks. Here a set of points P is given; the objective is to choose minimum number of points in P to pierce the unit disks centered at all the points in P. We first propose a very simple algorithm that produces 12-approximation result in O(n log n) time. Next, we improve the approximation factor to 4 and then to 3. The worst case running time of these algorithms are O(n8 log n) and O(n15 log n) respectively. Apart from the space required for storing the input, the extra work-space requirement for each of these algorithms is O(1). Finally, we propose a PTAS for the same problem. Given a positive integer k, it can produce a solution with performance ratio [Formula: see text] in nO(k) time.

COVERING A SET OF POINTS WITH A MINIMUM NUMBER OF TURNS

2004 ◽  
Vol 14 (01n02) ◽  
pp. 105-114 ◽  
Author(s):  
MICHAEL J. COLLINS
Keyword(s):  
Euclidean Space ◽  
Lower Bounds ◽  
Piecewise Linear ◽  
Upper And Lower Bounds ◽  
Linear Path ◽  
Change Direction ◽  
Finite Set ◽  
Minimum Number ◽  
Pass Through ◽  
Set Of Points

Given a finite set of points in Euclidean space, we can ask what is the minimum number of times a piecewise-linear path must change direction in order to pass through all of them. We prove some new upper and lower bounds for the rectilinear version of this problem in which all motion is orthogonal to the coordinate axes. We also consider the more general case of arbitrary directions.

scholarly journals Lower Bounds on the Obstacle Number of Graphs

10.37236/2380 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Padmini Mukkamala ◽  
János Pach ◽  
Dömötör Pálvölgyi
Keyword(s):  
Lower Bounds ◽  
Minimum Number ◽  
Obstacle Number ◽  
Set Of Points

Given a graph $G$, an obstacle representation of $G$ is a set of points in the plane representing the vertices of $G$, together with a set of connected obstacles such that two vertices of $G$ are joined by an edge if and only if the corresponding points can be connected by a segment which avoids all obstacles. The obstacle number of $G$ is the minimum number of obstacles in an obstacle representation of $G$. It is shown that there are graphs on $n$ vertices with obstacle number at least $\Omega({n}/{\log n})$.

On Indecomposable Graphs

1967 ◽  
Vol 19 ◽  
pp. 800-809 ◽  
Author(s):  
Frank Harary ◽  
Michael D. Plummer
Keyword(s):  
Covering Number ◽  
Minimum Cover ◽  
Minimum Number ◽  
Set Of Points

A set of points M of a graph G is a point cover if each line of G is incident with at least one point of M. A minimum cover (abbreviated m.c.) for G is a point cover with a minimum number of points. The point covering number α(G) is the number of points in any minimum cover of G. Let [V1, V2, … , Vr], r > 1 be a partition of V(G), the set of points of G. Let Gi be the subgraph of G spanned by Vi for i = 1, 2, … , r.

scholarly journals Traversing a Set of Points with a Minimum Number of Turns

2008 ◽  
Vol 41 (4) ◽  
pp. 513-532 ◽  
Author(s):  
Sergey Bereg ◽  
Prosenjit Bose ◽  
Adrian Dumitrescu ◽  
Ferran Hurtado ◽  
Pavel Valtr
Keyword(s):  
Minimum Number ◽  
Set Of Points

FITTING A STEP FUNCTION TO A POINT SET WITH OUTLIERS BASED ON SIMPLICIAL THICKNESS DATA STRUCTURES

2012 ◽  
Vol 22 (03) ◽  
pp. 215-241
Author(s):  
DANNY Z. CHEN ◽  
HAITAO WANG
Keyword(s):  
Data Structures ◽  
Dimensional Space ◽  
Piecewise Linear ◽  
Approximation Error ◽  
Step Function ◽  
Dynamic Data Structures ◽  
Dynamic Data ◽  
Minimum Number ◽  
Point Set ◽  
Set Of Points

Given a real ∊ > 0, an integer g ≥ 0 and a set of points in the plane, we study the problem of computing a piecewise linear functional curve with minimum number of line segments to approximate all points after removing g outliers such that the approximation error is at most ∊. We give an improved algorithm over the previous work. The algorithm is based on two dynamic data structures developed in this paper for the simplicial thickness queries, which are of independent interest. For a set S of simplices in the d-dimensional space ℝd(d ≥ 2 is a constant), the simplicial thickness of a point p is defined as the number of simplices in S that contain p. Given a set P of n points in ℝd, we develop two dynamic data structures to support the following operations. (1) Simplex insertion: Insert a simplex into S. (2) Simplex deletion: Delete a simplex from S. (3) Simplicial thickness query: Given a query simplex σ, compute the minimum simplicial thickness among all points in σ∩P. The first data structure is constructed in O(n1+δ) time, for any constant δ > 0, and can support each operation in O(n1-1/d) time; the second one is con-structed in O(n log n) time and can support each operation in O(n1-1/d( log n)O(1)) time. Both data structures use O(n). space. These data structures may find other applications as well.

scholarly journals ON THE HULL NUMBERS OF TORUS LINKS

2006 ◽  
Vol 15 (05) ◽  
pp. 589-600
Author(s):  
IVAN IZMESTIEV
Keyword(s):  
Plane Passing ◽  
Minimum Number ◽  
Set Of Points ◽  
Torus Links

The nth hull of a union of curves in ℝ3 is the set of points with the property: Any plane passing through the point intersects the curves at least 2n times. The hull number of a link L is defined as the minimum number of non-empty hulls a representative of L can have. We show that the hull numbers of torus links is smaller than expected, but still large. In particular, for a torus link of type (p, p) hull number equals [Formula: see text], and in general for the type (p, q) with q > p it is greater than or equal to [Formula: see text].

Extremal Point and Edge Sets in n-Graphs

1969 ◽  
Vol 21 ◽  
pp. 1069-1075
Author(s):  
N. Sauer
Keyword(s):  
Finite Graph ◽  
Extremal Point ◽  
Minimal Cover ◽  
Covering Problems ◽  
Graph Covering ◽  
Minimum Number ◽  
Image Position ◽  
Set Of Points

A set of points (edges) of a graph is independent if no two distinct members of the set are adjacent. Gallai (1) observed that, if A0 (B0) is the minimum number of points (edges) of a finite graph covering all the edges (points) and A1 (B1) is the maximum number of independent points (edges), then:holds, where m is the number of points of the graph.The concepts of independence and covering are generalized in various ways for n-graphs. In this paper we establish certain connections between the corresponding extreme numbers analogous to the above result of Gallai.Ray-Chaudhuri considered (2) independence and covering problems in n-graphs and determined algorithms for finding the minimal cover and some associated numbers.

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