BALANCED SUBDIVISIONS WITH BOUNDARY CONDITION OF TWO SETS OF POINTS IN THE PLANE

2010 ◽  
Vol 20 (05) ◽  
pp. 527-541 ◽  
Author(s):  
M. KANO ◽  
MIYUKI UNO
Keyword(s):  
Boundary Condition ◽  
Convex Polygon ◽  
Convex Polygons ◽  
Open Convex ◽  
Disjoint Sets ◽  
Positive Integers

Let R and B be two disjoint sets of red points and blue points in the plane, respectively, such that no three points of R ∪ B are collinear, and let a,b and g be positive integers. We show that if ag ≤ |R| < (a + 1)g and bg ≤ |B| < (b + 1)g, then we can subdivide the plane into g convex polygons so that every open convex polygon contains exactly a red points and b blue points and that the remaining points lie on the boundary of the subdivision. This is a generalization of equitable subdivision of ag red points and bg blue points in the plane.

AN APPROXIMATION ALGORITHM FOR LOCATING MAXIMAL DISKS WITHIN CONVEX POLYGONS

2011 ◽  
Vol 21 (06) ◽  
pp. 661-684
Author(s):  
HIROFUMI AOTA ◽  
TAKURO FUKUNAGA ◽  
HIROSHI NAGAMOCHI
Keyword(s):  
Approximation Algorithm ◽  
Convex Polygon ◽  
Computation Time ◽  
Approximation Ratio ◽  
Convex Polygons ◽  
The Given

This paper considers a problem of locating the given number of disks into a container so that the area covered by the disks is maximized. In the problem, the radii of the disks can be changed arbitrarily unless they overlap outside of the container, and the disks are allowed to overlap with each other. We present an approximation algorithm for this problem assuming that the container is a convex polygon. Our algorithm achieves approximation ratio (0.78 - ϵ) for any small ϵ > 0. Since the computation time of our algorithm depends on the number of corners of the convex polygon exponentially, we also give a heuristic to reduce the number of corners.

scholarly journals SIMULTANEOUS CONTAINMENT OF SEVERAL POLYGONS: ANALYSIS OF THE CONTACT CONFIGURATIONS

1993 ◽  
Vol 03 (04) ◽  
pp. 429-442 ◽  
Author(s):  
OLIVIER DEVILLERS
Keyword(s):  
Convex Polygon ◽  
Main Concern ◽  
Convex Polygons ◽  
Simple Polygons ◽  
General Properties ◽  
Contact Position

The main concern of this paper is the detection of double contact configurations for some polygons moving in translation in a polygonal environment. We first establish some general properties about such configurations and give conditions of existence of double contacts for two or three objects. For three convex polygons moving in a polygonal environment or three simple polygons moving in a rectangle there always exists a double contact. Two examples without possibility of double contacts are given, one with three polygons (not convex) moving in a polygonal environment, and one with four convex polygons moving in a rectangle. We deduce an algorithm detecting a double contact position in time O(n2) (resp. O(n3)) for two (resp. three) convex polygons of constant sizes moving in a non-convex polygon of size n.

Weight-Equitable Subdivision of Red and Blue Points in the Plane

2018 ◽  
Vol 28 (01) ◽  
pp. 39-56 ◽  
Author(s):  
Jude Buot ◽  
Mikio Kano
Keyword(s):  
General Position ◽  
Sufficient Conditions ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Total Weight ◽  
Blue Point ◽  
Disjoint Sets ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Special Case

Let [Formula: see text] and [Formula: see text] be two disjoint sets of red points and blue points, respectively, in the plane in general position. Assign a weight [Formula: see text] to each red point and a weight [Formula: see text] to each blue point, where [Formula: see text] and [Formula: see text] are positive integers. Define the weight of a region in the plane as the sum of the weights of red and blue points in it. We give necessary and sufficient conditions for the existence of a line that bisects the weight of the plane whenever the total weight [Formula: see text] is [Formula: see text], for some integer [Formula: see text]. Moreover, we look closely into the special case where [Formula: see text] and [Formula: see text] since this case is important to generate a weight-equitable subdivision of the plane. Among other results, we show that for any configuration of [Formula: see text] with total weight [Formula: see text], for some integer [Formula: see text] and odd integer [Formula: see text], the plane can be subdivided into [Formula: see text] convex regions of weight [Formula: see text] if and only if [Formula: see text]. Using the proofs of the main result, we also give a polynomial time algorithm in finding a weight-equitable subdivision in the plane.

scholarly journals Intersecting Families in a Subset of Boolean Lattices

10.37236/724 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Jun Wang ◽  
Huajun Zhang
Keyword(s):  
Intersecting Family ◽  
Intersecting Families ◽  
Boolean Lattices ◽  
Disjoint Sets ◽  
Positive Integers

Let $n, r$ and $\ell$ be distinct positive integers with $r < \ell\leq n/2$, and let $X_1$ and $X_2$ be two disjoint sets with the same size $n$. Define $$\mathcal{F}=\left\{A\in \binom{X}{r+\ell}: \mbox{$|A\cap X_1|=r$ or $\ell$}\right\},$$ where $X=X_1\cup X_2$. In this paper, we prove that if $\mathcal{S}$ is an intersecting family in $\mathcal{F}$, then $|\mathcal{S}|\leq \binom{n-1}{r-1}\binom{n}{\ell}+\binom{n-1}{\ell-1}\binom{n}{r}$, and equality holds if and only if $\mathcal{S}=\{A\in\mathcal{F}: a\in A\}$ for some $a\in X$.

scholarly journals Mining Convex Polygon Patterns with Formal Concept Analysis

2017 ◽  
Author(s):  
Aimene Belfodil ◽  
Sergei O. Kuznetsov ◽  
Céline Robardet ◽  
Mehdi Kaytoue
Keyword(s):  
Spatial Data ◽  
Formal Concept Analysis ◽  
Convex Polygon ◽  
Pattern Mining ◽  
Concept Analysis ◽  
Formal Concept ◽  
Arbitrary Form ◽  
Convex Polygons ◽  
Trade Off ◽  
Good Trade

Pattern mining is an important task in AI for eliciting hypotheses from the data. When it comes to spatial data, the geo-coordinates are often considered independently as two different attributes. Consequently, rectangular patterns are searched for. Such an arbitrary form is not able to capture interesting regions in general. We thus introduce convex polygons, a good trade-off for capturing high density areas in any pattern mining task. Our contribution is threefold: (i) We formally introduce such patterns in Formal Concept Analysis (FCA), (ii) we give all the basic bricks for mining polygons with exhaustive search and pattern sampling, and (iii) we design several algorithms that we compare experimentally.

EXACT SOLUTION OF THE ROW-CONVEX POLYGON GENERATING FUNCTION FOR THE CHECKERBOARD LATTICE

1991 ◽  
Vol 05 (15) ◽  
pp. 2551-2562 ◽  
Author(s):  
W.J. TZENG ◽  
K.Y. LIN
Keyword(s):  
Exact Solution ◽  
Generating Function ◽  
Convex Polygon ◽  
Square Lattice ◽  
Honeycomb Lattice ◽  
Rectangular Lattice ◽  
Convex Polygons ◽  
Recursion Relations ◽  
Special Cases ◽  
Special Case

We have studied the row-convex polygons on a general checkerboard lattice and derived the recursion relations for the four-variable generating function. Exact solution of the row-convex polygon generating function is obtained for a special case of the checkerboard lattice. Our result includes the square lattice, rectangular lattice and isotropic honeycomb lattice as special cases.

ON THE ROW-CONVEX POLYGON GENERATING FUNCTION FOR THE CHECKERBOARD LATTICE

1991 ◽  
Vol 05 (20) ◽  
pp. 3275-3285 ◽  
Author(s):  
K.Y. LIN ◽  
W.J. TZENG
Keyword(s):  
Exact Solution ◽  
Generating Function ◽  
Convex Polygon ◽  
Square Lattice ◽  
Rectangular Lattice ◽  
Convex Polygons ◽  
Special Cases

Exact solution for the most general four-variable generating function of the number of row-convex polygons on the checkerboard lattice is derived. Previous results for the square lattice, rectangular lattice, and honeycomb latticc are special cases of our solution.

scholarly journals A Note on the Minimum Size of a Point Set Containing Three Nonintersecting Empty Convex Polygons

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 447
Author(s):  
Qing Yang ◽  
Zengtai You ◽  
Xinshang You
Keyword(s):  
Pairwise Disjoint ◽  
Convex Polygon ◽  
Minimum Size ◽  
Convex Polygons ◽  
Planar Point ◽  
Empty Convex ◽  
Point Set

Let P be a planar point set with no three points collinear, k points of P be a k-hole of P if the k points are the vertices of a convex polygon without points of P. This article proves 13 is the smallest integer such that any planar points set containing at least 13 points with no three points collinear, contains a 3-hole, a 4-hole and a 5-hole which are pairwise disjoint.

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