Questions tagged [computational-geometry]

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-2
votes
1answer
143 views

How to calculate complexity in a high dimensional space?

Edit: 'Fitness landscape analysis' was mentioned as a relevant measure. If you're going to downvote the post, at least leave a comment what is wrong. For a specific f(), I'm defining a term '...
18
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0answers
374 views

In an $m$ by $n$ Boolean matrix, can you find a square block whose four corners are ones in $O(m \cdot n)$ time?

Decision Problem Input: An $m$ by $n$ Boolean matrix $M$. Decision Question: Does there exist a square block within $M$ such that upper-left corner entry == upper-right corner entry == lower-left ...
0
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0answers
28 views

What is the best way to find circles that contain a given point (in 2D)?

Given $n$ circles all with radius $r$ and one point on a 2D plane, what is the best algorithm to find all circles that contain the given point. The circles and the point can change their positions. ...
0
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1answer
47 views

Find a boundary from set of 3d line segments

I have a set of n 3d line segments ...
5
votes
1answer
80 views

Complexity of computing the union of H-polytopes in three dimensions

Consider a set $P_1,\ldots,P_k$ of polytopes in $\mathbb{R}^3$, each given as an intersection of halfspaces with rational normals (in particular, they are all convex). We are also given a target ...
1
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0answers
34 views

Practical worst-case polylogarithmic dynamic orthogonal range queries?

There are a number of data structures in the literature that solve the dynamic orthogonal range search problem in polylogarithmic time (say, range trees). My understanding is that these structures ...
5
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0answers
84 views

Optimal point placement on integer lattice

What is known about the following point placement problem? For positive integers $N$, $n<N^2$, and $N\times N$ grid $\mathcal{G}$, compute \begin{eqnarray*} \mu_1(N,n)\triangleq\min_{\mathcal{P}\...
-1
votes
1answer
64 views

Example of Delaunay Triangulation where it does not minimize the maximum angle

I know that that the Delaunay triangulation maximizes the minimum angle of triangulation. And it does not minimize the maximum angle. If we consider the set of points in general position(no four ...
0
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0answers
36 views

Delaunay Triangulaition (or Voronoi) for a specific distribution of points

I found a paper about Delaunay triangulation for a set of points that are distributed by the Poisson distribution (https://pdfs.semanticscholar.org/9693/4b7e8e5483893f4874d7ba6afd812bbfe0ba.pdf). The ...
5
votes
0answers
70 views

Computing 3D viewpoint of a set of non-intersecting segments

Consider the following problem: we are given a finite set of bounded line-segments in ${\mathbb R}^3$, and we want to decide whether there exists a point $p\in {\mathbb R}^3$ from which no two ...
2
votes
1answer
114 views

How to choose good diagonals when partitioning an orthogonal polygon into rectangles?

Following this answer on MathOverflow and section 3 of the linked paper by David Eppstein on how to split an orthogonal polygon into rectangles I came to a point where I just fail to understand how to ...
3
votes
1answer
144 views

Convex polygons inclusion relation

I have the following problem which came as a subproblem in some work I was doing and I am completely stuck. Note that I am interested in it only in terms of worst case time complexity (not heuristics ...
1
vote
1answer
76 views

Data structures for embedded simplicial complexes

I am looking for a data structure to encode an $n$-dimensional simplicial complex with an embedding in $\mathbb{R}^{n+1}$. I am aware of combinatorial maps, which generalize rotation systems of planar ...
1
vote
1answer
143 views

The decision procedure of theory of closed real field is in NP-hard?

The decision procedure of theory of closed real field refers to https://en.wikipedia.org/wiki/Decidability_of_first-order_theories_of_the_real_numbers
3
votes
1answer
141 views

Minimal number of hyperplanes needed to separate sets of points from one other set

Let $\mathbb{R}^d$ be our space. We have a single good set of points $g$, and a collection of bad sets of points $B$. We assume that for all $b \in B$ the convex hulls of $g$ and $b$ are disjoint. ...
0
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0answers
47 views

Volume computation of special polytopes

I'm interested in computing the volume of a special class of $\mathcal{H}$-polytopes and the complexity of doing so. I know that in general it is #P-hard to compute the volume of $\mathcal{H}$ -...
4
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0answers
49 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? ...
1
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2answers
100 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
60 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
votes
1answer
155 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
votes
2answers
105 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
70 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 ...
1
vote
0answers
82 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 \...
3
votes
1answer
99 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
76 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
votes
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
votes
0answers
52 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 ...
12
votes
1answer
570 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
vote
1answer
52 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
53 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
votes
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
votes
0answers
40 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
votes
1answer
76 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
votes
2answers
138 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
vote
0answers
103 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 ...
5
votes
0answers
106 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 $X$ ...
6
votes
1answer
149 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 ...
17
votes
1answer
509 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 ...
5
votes
0answers
267 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
68 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
162 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
58 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
56 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
134 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 ...
2
votes
1answer
215 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)?
4
votes
0answers
140 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 $\{0,1\}$-...
3
votes
4answers
250 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
vote
0answers
88 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
votes
0answers
69 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 ...