Questions tagged [computational-geometry]

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18 votes
0 answers
463 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 ...
Michael Wehar's user avatar
17 votes
1 answer
601 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 ...
Robin Houston's user avatar
12 votes
1 answer
635 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 $X=OY$...
Jarek Duda's user avatar
11 votes
1 answer
249 views

Compute lowest dimensional polytope from a given set of sign vectors

Given a set of hyperplanes determined by the normal vectors $h_1,\dots,h_m \in \mathbf R^d$, its cell types (or sign vectors) are all vectors $t\in\{+,-\}^m$ for which there exists a vector $v\in\...
Holger's user avatar
  • 975
11 votes
1 answer
316 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, ...
Pat Morin's user avatar
  • 571
10 votes
1 answer
171 views

Largest cell in an arrangement

Q. What is the complexity of finding the largest volume bounded cell in an arrangment of $n$ hyperplanes in dimension $d$? I feel I should know this... But I am not finding a definitive reference. ...
Joseph O'Rourke's user avatar
10 votes
1 answer
146 views

Minimum equidecomposable decomposition

Given two polyhedra $P$ and $Q$, $P$ and $Q$ are are equidecomposable if there are finite sets of polyhedra $P_1, \ldots, P_n$ and $Q_1, \ldots, Q_n$ such that $P_i$ and $Q_i$ are congruent for all $i$...
Glencora Borradaile's user avatar
10 votes
0 answers
157 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 ...
a3nm's user avatar
  • 9,232
7 votes
1 answer
180 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 ...
Gabriel Nivasch's user avatar
6 votes
2 answers
571 views

Would a purely topological computational model be useful in decision problems in topology?

If one were to develop a purely topological computational model based upon the equivalence of information in knots and the model would perform transformations of that information. This would be the ...
Joshua Herman's user avatar
6 votes
1 answer
139 views

Maximum number of geometrically disjoint paths - is the complexity known?

Let $G$ be an undirected graph, given with a planar drawing. We do not assume $G$ is a planar graph, we just fix a planar representation of it, such that each vertex is represented by a distinct ...
Andras Farago's user avatar
6 votes
1 answer
313 views

Unclear proof step in Feder and Greene's 1988 paper showing NP-Hardness of approximating k-center problem within a factor of 1.82

I was reading the paper "Optimal Algorithms for Approximate Clustering", Feder and Greene [1988] (https://dl.acm.org/doi/10.1145/62212.62255). Specifically, I was trying to look at the $1.82$...
Agile_Eagle's user avatar
6 votes
1 answer
162 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 ...
Timothy Chu's user avatar
5 votes
2 answers
224 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(...
user36641's user avatar
5 votes
2 answers
251 views

Fast high-dimensional K-nearest neighbors

I'm aware of this question https://stackoverflow.com/questions/4350215/fastest-nearest-neighbor-algorithm But it's not the same question as I'm asking. Because, Octree and its generalization are only ...
AmeerJ's user avatar
  • 679
5 votes
1 answer
91 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 ...
Shaull's user avatar
  • 5,571
5 votes
1 answer
107 views

Finding planes from their points

Given some points $P=\{x_1,\dots,x_m\}$ in a vector space $(Z/2Z)^n$, if $P$ is a union of linear subspaces all of the same dimension $1<d<n$, can we efficiently find these subspaces? (Any ...
Bjørn Kjos-Hanssen's user avatar
5 votes
3 answers
228 views

Locating a point inside a union of simple polygons

I have a set of polygons (convex, concave – non-convex, not self-intersecting) in a plane. There may be intersections between polygons. Polygon is defined by points (cartesian coordinate system). ...
user3102393's user avatar
5 votes
0 answers
93 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}\...
user56067's user avatar
5 votes
0 answers
75 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 ...
Shaull's user avatar
  • 5,571
5 votes
0 answers
70 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 ...
user avatar
5 votes
0 answers
124 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 $X$ ...
Mateus de Oliveira Oliveira's user avatar
5 votes
0 answers
309 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 ...
user avatar
5 votes
0 answers
77 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 ...
Alexey Milovanov's user avatar
4 votes
1 answer
495 views

Does Approx Carathéodory's theorem implies dimensionality reduction

Carathéodory's theorem says that if a point $x$ of $R^d$ lies in the convex hull of a point set $P$, then there is a subset $P′ \subseteq P$ consisting of $d + 1$ or fewer points such that $x$ can be ...
Ram's user avatar
  • 639
4 votes
1 answer
199 views

Configurations from arrangement of lines

Suppose we have an arrangement of $n$ lines in the plane. This partitions the plane into regions. We can label each region by a string in $\{+1, -1\}^n$, where the $i$'th coordinate is +1 if the ...
arnab's user avatar
  • 6,990
4 votes
1 answer
190 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 ...
aelguindy's user avatar
  • 951
4 votes
1 answer
243 views

Binary Trees for Nearest Neighbor Search

Given points $x_1, ..., x_n \in \mathbb{R}^d$, consider a binary decision tree $T$ on $\mathbb{R}^d$ with $L$ leaves, i.e. for a point $y \in \mathbb{R}^d$ at every node of the tree, we check whether $...
Claudio Moneo's user avatar
4 votes
1 answer
75 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 ...
Artemis's user avatar
  • 143
4 votes
1 answer
122 views

Non-Midpoint Segment Splitting in Ruppert's Delaunay Triangulation Refinement Algorithm

Roughly speaking, in Ruppert's Delaunay Triangulation refinement algorithm, so called encroached edges are split until no more encroached edges remain. The algorithm specifies splitting the edges ...
Adi Shavit's user avatar
4 votes
1 answer
116 views

Polytopes convex hull

Assume $P_1$, $P_2$ and $Q$ are axis-parallel polytopes in $\mathbb{R}^d$. Is it true to say that checking $CH(P_1\cup P_2)\neq Q$ is NP-hard? There is a similar result for the general polytopes but I ...
Star's user avatar
  • 253
4 votes
0 answers
87 views

Complexity of detecting general position in the plane?

What is the complexity of detecting whether a given set of points in the plane is in general position? This surely must have been studied, but a quick search turns up nothing. For concreteness, let'...
Neal Young's user avatar
  • 10.1k
4 votes
1 answer
116 views

A counter example for the set mean objective

Let $\mathcal{P} = \{P_1, \cdots,P_n\}$ be a family of finite point sets in $\mathbb{R}^d$, each having at most $m$ points. Consider the following objective function \begin{align} cost(\mathcal{P},c) =...
Sudipta Roy's user avatar
4 votes
0 answers
55 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? ...
Aryeh's user avatar
  • 10.5k
4 votes
0 answers
149 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 $\{0,1\}$-...
FiB's user avatar
  • 141
4 votes
0 answers
305 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 ...
Dibyayan's user avatar
  • 1,006
3 votes
3 answers
440 views

Partitioning a set of 2d polygons into intersection-connected subsets

My question is given a set of 2d polygons how can I find the connected components of polygons according to a criteria based on intersection or proximity of them. In other words I have a set of ...
Juan Besa's user avatar
  • 384
3 votes
1 answer
238 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. ...
orlp's user avatar
  • 835
3 votes
1 answer
127 views

Is minimum knot crossing number elementary recursive?

One result in knot theory is that link crossing number is NP-hard. Another result is that the equivalence problem for knots and links is elementary recursive. So, given that the equivalence problem is ...
Niklas Rosencrantz's user avatar
3 votes
1 answer
167 views

Generalization of Beck's theorem

Beck's theorem is a classical result in discrete geometry which describes about the geometry of points in the plane. The result states that a finite collections of points in the plane fall into one of ...
Ram's user avatar
  • 639
3 votes
1 answer
219 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 ...
ioannis's user avatar
  • 41
3 votes
2 answers
161 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 ...
Chao Xu's user avatar
  • 4,399
3 votes
4 answers
291 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 $R > 0$, ...
Jonas Köhler's user avatar
3 votes
1 answer
58 views

Emptiness of complement of subspace arrangement

Given $k$ affine subspaces in $\{0,1\}^n$, consider the problem of testing whether their union covers all of $\{0,1\}^n$. What's the complexity of this problem? P.S.: It seems that this can be ...
arnab's user avatar
  • 6,990
3 votes
1 answer
478 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 ...
user51847's user avatar
3 votes
2 answers
204 views

Is this covering problem NP-hard?

Given a rectangular region $R$ and a set $D$ of $n$ disks such that the union of all disks in $D$ cover the entire rectangular region $R$, the objective is to find the minimum cardinality set $D'$ ...
Manju's user avatar
  • 31
3 votes
1 answer
164 views

Range searching: what is $\epsilon$?

I am interested in range searching purely as a black box in another problem. One problem in particular is that of shooting 2D rays among 2D segments. It seems that Agarwal and Matousek have a well ...
John's user avatar
  • 235
3 votes
0 answers
41 views

Affine point matching in general dimensions

Fix a positive integer $d$ and consider the $d$-dimensional Euclidean space $\mathbb{R}^d$. Let $S$ and $T$ finite subsets of $\mathbb{R}^d$ of the same size $n$. Under the assumption that $S$ and $T$ ...
rr314's user avatar
  • 131
3 votes
0 answers
83 views

Enforcing general position in $2d$ linear programming

Let $(x_1, y_1), ..., (x_k, y_k)$ be $n$ points in $\Re^2$. For my sake, $k=20$. I am trying to set up a linear program to find a set of $k$ points in the plane $P$ that satisfy some linear ...
user3508551's user avatar
  • 1,143
3 votes
0 answers
27 views

Bound on line with minimum zone complexity in a line arrangement

In an arrangement of $n$ (pseudo)lines, the well known Zone Theorem gives a $O(n)$ bound on the complexity of the zone of any given line (for the purpose of this question, the complexity of the zone ...
Tassle's user avatar
  • 871