# Questions tagged [computational-geometry]

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### 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 ...
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### 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 ...
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 $X=OY$...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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). ...
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}\...
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### 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 ...
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### 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 ...
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 $X$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $R > 0$, ...
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### 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 ...
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### 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 ...
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### 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 $D'$ ...
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### 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 ...
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### 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 $T$ ...
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### 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 ...
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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 ...