Questions tagged [computational-geometry]

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4
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0answers
46 views

$\ell_\infty$ partially enclosing ball problem

Suppose I have $n$ points in $\mathbb{R}^d$ endowed with the $\ell_\infty$ metric, and I wish to find a minimum-diameter ball that contains some $k$ of these points. What is known about this problem? ...
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0answers
26 views

Two question regarding coreset construictions

I have two questions regarding coreset construction of clustering problem In A Unified Framework for Approximating and Clustering Data, a very general framework is given to construct coresets for ...
0
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0answers
55 views

Linear time algorithm for projective clustering

There is a lot of work in clustering of high dimensional data. In case of k-means, it is shown here that one can get an $(1+\epsilon)$-approximation in linear time, yielding a PTAS, by random sampling....
1
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1answer
68 views

Embedding a n-tree into a b-dimensional space

Given a (directed) n-tree $T=(N,E,r)$ rooted in $r\in N$, I want to represent each node $n\in N$ at most as a $m$-dimensional vector $v_n\in \mathbb{R}^m$ (From the current Yuri's reply, m cannot be $...
1
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0answers
48 views

Mapping of entire balls using Locality Sensitive Hashing (LSH)

LSH functions are useful for approximate nearest neighbor search. They are usually defined, for distance metric $d$ and $c>1$ as follows: A family of hash functions is $(r, cr, p_1, p_2)$-LSH ...
7
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1answer
129 views

Reference request: Shortest homotopic curve via vertex releases

Let $C$ be a piecewise-linear path (or closed curve) in the plane, in the presence of polygonal obstacles. We would like to find the shortest path (or curve) homotopic to $C$. (A path $D$ is homotopic ...
0
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0answers
21 views

Does optimal fitting flat must pass through the mean of the point set?

I am confused about a statement made in the paper Linear Time Algorithm for Projective Clustering, section 5.1, second paragraph, second line. Project clustering is a natural generalization of k-...
3
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2answers
95 views

Data structure for radial orderings of points on the plane

Assume points are always in general position. For a set of $n$ points $S$ on the plane, a radial ordering with respect to $x\in S$ is a total ordering of the elements in $S-x$. Consider shooting an ...
3
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0answers
67 views

Rearranging angles of a convex polyline to make it closed

Let {$\alpha_1, \alpha_2, ... ,\alpha_n$} be a string of n positive reals summing up to 2$\pi$. We inductively construct the following 2D polyline, denoting with $R[\alpha]$ the clockwise rotation by ...
3
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1answer
92 views

Select circle with given radius that contains most points

Given some points on a coordinate system and some radius r, I need to place a circle with radius r somewhere on the coordinate system such that that circle includes the most points. I tried solving ...
1
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0answers
71 views

LSH Probabilistic guarantees

A family $H$ is $(r,cr,p_1,p_2)$-sensitive if for all $x,y \in \mathbb{R}^d$ we have: $\lVert x-y\rVert <r\quad \Rightarrow\quad \Pr[h(x)=h(y)] \geq p_1$, and $\lVert x-y\rVert > cr \quad \...
1
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0answers
73 views

Points of a finite set wihtin a ball

I am looking for data-structures to store efficiently a set of points $E$ in an euclidean space of dimension $d$. In particular, I would like to be able to solve the problem of finding all the point ...
2
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1answer
140 views

A least sized partition of a set under a distance metric

What is the worst case complexity of an algorithm to find a least partition of a set under a distance metric, described as follows: Input: A set $S=\{s_1,\ldots,s_n\}$, where the elements $s_i$ are ...
5
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0answers
51 views

Inapproximability Results for APX-hard Geometric Optimization Problems

A lot of Geometric Optimization problems are NP-hard and APX-hard to approximate (No PTAS unless P=NP). One example of the geometric set cover problem can be found here. However, it is not easy to ...
17
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1answer
431 views

How not to compute the smallest circle enclosing a finite set of circles

Suppose we have a finite set $L$ of disks in $\mathbb{R}^2$, and we wish to compute the smallest disk $D$ for which $\bigcup L\subseteq D$. A standard way to do this is to use the algorithm of ...
12
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1answer
460 views

Complexity of testing if two sets of $m$ points in $\mathbb{R}^n$ differ only by rotation?

Imagine we have two size $m$ sets of points $X,Y\subset \mathbb{R}^n$. What is (time) complexity of testing if they differ only by rotation?: there exists rotation matrix $OO^T=O^TO=I$ such that $X=OY$...
1
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1answer
49 views

Inclusion probability of irregularly shaped polygon

I have two shapes, one is a circle, let's call it Circle A, and another irregularly shaped Polygon B. A will always have a greater area than B. Both of these polygons exist inside an area S. My task ...
2
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0answers
51 views

Congruency check for set of points in 3D using inertia tensor

You're given two set of points $A, B\subset \mathbb R^3:|A|=|B|=n$. You have to check if those sets are congruent, i.e. there exist some mapping $\sigma : A \to B$ and combination of translation and ...
2
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0answers
48 views

Given an algebraic variaties of n multivarieties polynomial equations, is there any algorithm to decide whether there is n-cube inscribing to it?

Given an algebraic variaties of n multivarieties polynomial equations, is there any algorithm to decide whether there is n-cube inscribing to it? And if there is, what is the computational ...
2
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0answers
35 views

Lower bound on light spanners in Euclidean space reference

It is well-known that Euclidean space of dimension $d$ has a $(1+\epsilon)$-spanner of weight at most $\epsilon^{-O(d)}\cdot w(MST)$ (see Chapter 14 of Geometric Spanner Network book by Narashimhan ...
-1
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1answer
61 views

given a set of $n$ points in $d$-dimensional space and the basis vectors of some subspace, how to find all the points on that space?

given a set $A$ of $n$ points with integer coordinates in $\mathbb{R}^d$, and $k<d$ basis vectors of a subspace $K$ of $\mathbb{R}^d$, is there an efficient algorithm that returns all points from $...
-1
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2answers
122 views

How to continue this algorithm? [closed]

I want to create an algorithm to fill a fixed-size big rectangle (W,H) with the maximum number of fixed-size smaller rectangles (w,h) (I can rotate the small rectangles 90º). I have thought about ...
1
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0answers
80 views

Hausdorff Distance and Convex Hull

Given two sets of points A and B, both in $R^d$, is there a relation between the convex hulls of A and B, i.e. conv(A) and conv(B), w.r.t. the Hausdorff distance between A and B? In other words, does ...
6
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1answer
142 views

Embedding points in Euclidean space into a box

If I give you a set of points in Euclidean space, is there a criterion to determine whether there exists a (potentially higher-dimensional) rectangular prism / box that has these points as their ...
5
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0answers
100 views

Structures obtained by gluing simplices

I'm looking for the correct name of geometric structures obtained as follows. 2-structures: A collection $X$ of triangles is a $2$-structure. If $X$ is a $2$-structure and $Y$ is obtained from $X$ ...
2
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2answers
303 views

Coreset and VC dimension

I am trying to understand the notion of $\epsilon$-coreset and its relation with sampling bounds of a range space having a finite VC-dimension. Although both of them give an $\epsilon$-approximation ...
0
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1answer
2k views

VC-dimension of triangles in 2D space [closed]

I have been reading in multiple places (e.g. [1], section 4) that the VC-dimension of the class of triangles (in 2D space) is 7. The issue is that, for the case when 4 points lying on a straight line ...
5
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0answers
213 views

Total time complexity of convex hull problem

The convex hull problem is to compute the facets of the convex hull of finitely many given points in $\mathbb{R}^d.$ By cone polarity it is equivalent to computing the vertices and rays of a ...
4
votes
1answer
65 views

Reference needed for lower bound on number of guards in three-dimensional art gallery guarding

During my research (writing my master's thesis) I've stumbled across the book Art Gallery Theorems and Algorithms by O’Rourke. In chapter 10 (pdf available on the above site), section 10.2.2. he shows ...
4
votes
1answer
159 views

Complexity of counting maximum number of co-linear points in Euclidean plane

The problem: given a set of points in the Euclidean plane, find the maximum number of co-linear points. I already know that the problem can be solved in quadratic time using hashing or projective ...
2
votes
1answer
55 views

Finding a cell in an arrangement of simplices

My question is n-dimensional, but I will begin by dropping the problem down to two dimensions for clarity's sake. It regards defining what is a solution by defining one or more data points that are ...
1
vote
1answer
54 views

Finding sets of heavily intersecting objects, while minimizing their size [closed]

Assume I have some array $a$ of length $n$. If I place the elements contained in the array $a$ into a $\sqrt{n} \times \sqrt{n}$ matrix, then every row "intersects" with every column. That is, I ...
10
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0answers
120 views

Which convex polytopes have volumes of polynomial bit-length?

A convex polytope is described as an intersection of halfspaces, given as inequalities between linear combinations of variables with rational coefficients. The volume computation problem for convex ...
4
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0answers
136 views

Computing Minima of the Projection of a Binary Cube

The problem is as follows: I want to compute the minima (with respect to the canonical partial order on vectors "$\leq$") of the linear projection of the extreme points of an $n$-dimensional $\{0,1\}$-...
2
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1answer
196 views

Intersection graphs of squares and rectangles

Is it known if the class of intersection graphs of rectangles is equal to the class of intersection graphs of squares (not necessarily unit)?
3
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4answers
243 views

Find the maximum subset contained by a ball of radius R

I am searching for the name of / literature to the algorithmic problem as follows: Given a metric space $(M,d)$, a finite Subset $X = \{ x_1, \dots, x_n \} \subset M$ and a fixed Radius $R > 0$, ...
1
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0answers
84 views

Complexity of Maximum Independent Set (or Vertex Cover) on disk packing graphs

I'm interested in complexity results for Maximum Independent Set (or Vertex Cover) problem over the class of disk packing graphs. Having a set of disks we build a graph that has its vertices at the ...
0
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0answers
68 views

Trade-off between number of spheres and wasted space in covering a 3d object by spheres

Consider the following optimization problem: Input: a 3-dimensional "object" $O$. Output: a covering of $O$ by a list of $k$ spheres $S_1, \ldots, S_k$ (given by their centers and radii) minimizing ...
2
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0answers
123 views

Weighted $l_1$ distance

So there are many well known algorithms for approximate nearest neighbor on the $\ell_1$ distance. My question is, what about the weighted version of the problem (where the weights are specified along ...
5
votes
2answers
207 views

Voronoi Diagram of Lines

Let $S$ be a set of lines (or line segments) in $\mathbb{R}^3$. Consider the Voronoi diagram $VD(S)$. The best lower bound on the complexity of $VD(S)$ is $\Omega(n^2)$ and the best upper bound is $O(...
2
votes
1answer
277 views

maximizing inner product

Given two lists $L,L'\subseteq\mathbb{R}^d$ of $n$ vectors each, how fast can we compute for all $p\in L$ the vector of $L'$ that maximizes the inner product with $p$, i.e., $\arg\max_{p'\in L'} \...
5
votes
0answers
71 views

Finding of dimension of algebraic varieties

I have found that the problem of finding of dimension of algebraic varieties over $\mathbb{C}$ is $NP$-complete (https://pdfs.semanticscholar.org/a947/463a29ee512b89823176f6e8c9f9b2bb1a5e.pdf). Are ...
0
votes
1answer
114 views

Lowerbound for the minimum distance between points in a “nice” triangulation

Suppose we have a unit square $S$ that contains $n$ points. Assume we always have a point at each of the four corners. No we triangulate $S$ by adding non-intersecting segments between the points. ...
4
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0answers
183 views

Voronoi diagram in presence of polygonal obstacle

Suppose there is a set of convex polygons ($\mathbb{P}$) on the plane. For each convex polygon $P_i$ there is one "facility" $f_i$ placed on the boundary of $P_i$. The distance between a point $p \in ...
2
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0answers
513 views

Shortest non-crossing geometric paths

I have a plane graph $G$ and a set of $k$ vertex pairs $\{s_1,t_1\}, \dots, \{s_k, t_k\}$. The goal is to find $k$ non-crossing paths connecting the pairs of terminals $s_i$ with $t_i$ in the graph so ...
0
votes
1answer
80 views

Algorithms to classify the geometrical relationship of two 3D-geometries

I have multiple 3D volumes which are represented by their boundaries (set of polygons each represented as a list of 3D-coordinates). I am now looking for algorithms to decide if volume A is inside ...
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0answers
38 views

Current review on polygon partition problems

Mark Keil(1) provides an extensive survey of polygon partitioning and polygon covering alrogithms. This survey was written in 2000. Is there a more recent survey on this topic?
2
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0answers
171 views

Inclusion of polytopes

Consider the following two system of linear (in)eqaulities: $S = Ax \leq b;\; Cx = e$ $T = Dx \leq d;\; Gx = g$ How can I check if $S\cap \neg T=\emptyset$ where $\neg T$ is the complement of the ...
11
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1answer
275 views

Implementation of partition trees?

Have partition trees ever been implemented? Here, I'm talking about the partition trees from computational geometry. The earliest (near-)optimal versions of which were due to Matousek and others, ...
3
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0answers
99 views

Equivalence of weighted Minkowski sums

Given $n$ polytopes $P_1, \cdots, P_n$, each $P_i$ is given as the V-representation, i.e., a set of $m$ points as its set of vertices. Furthermore, consider a variant of the Minkowski sum (somehow ...