# Questions tagged [cg.comp-geom]

Computational Geometry is the study of geometric problems from a computational perspective. Examples of problems include: computation of geometric objects such as convex hulls, dimensionality reduction, shortest path problems in metric spaces, or finding a small subset of points that approximates some measure of the whole set (i.e. a coreset).

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### Complexity of polygon intersection test

In the book "Spatial Databases: With Application to GIS (The Morgan Kaufmann Series in Data Management Systems)", there's a section on simple polygon intersection detection , and it mentions ...
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1 vote
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### Diameter queries for stream of points

Given an online stream of $k$ points $x_1, x_2,\ldots,x_k$ with $x_i \in \mathbb{R}^2$. By online we mean that when $x_i$ arrives we have no knowledge of points $x_j$ for $j > i$. Denote by $S_i$ ...
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### Partial linear spaces and partial projective planes

In 1948, De Bruijn and Erdős [1] proved that a finite linear space on $n$ points has at least $n$ lines, with equality occurring if and only if the space is either a near-pencil (all points but one ...
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### Status of certain problems in knot theory

I found it somewhat difficult to understand the status of certain problems from knot theory. Is it correct to say that it's been neither proved nor disproved that any of the following problems are NP-...
1 vote
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### partitioning points in the plane into two clusters to minimize maximum cluster diameter

What is a fast algorithm for the following problem? input: a set of $n$ pairs of points in the Euclidean plane output: a partition of the points into two clusters so that, for each given pair, the ...
• 10.1k
1 vote
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### Is this a novel technique for determining whether or not two rotated rectangles collide?

I was trying to determine whether or not two rectangles rotated around their centers were colliding and randomly thought to try the following algorithm: Rotate both rectangles by the negative rotation ...
54 views

### Efficiency of building orthogonal range search structures?

I've been reading up on data structures for 2D range searching. I've noticed that, in many of the papers I've read, there's close attention paid to the query cost and the space usage required, but ...
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### Partitioning a connected polygon into connected pieces of equal area

Armaselu and Daescu (TCS, 2015) present algorithms that, given a convex polygon $P$ and an integer $m$ (which must be a power of $2$), return a partition of $P$ into $m$ convex polygons with the same ...
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### Complexity of Unknotting problems

The complexity of the Unknotting problem is known to be in $\mathrm{NP} \cap\mathrm{co\text-NP}$, see references: The Computational Complexity of Knot Problems. Knottedness is in NP, modulo GRH. . ...
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### NP complete problem help

I'm currently trying to find a reduction to this problem: Given a set S of n points (in the plane) in general position, is there a set of at least k triangles (formed using only points in S as ...
359 views

### NP-hardness for one-dimensional facility location problem with entrance fee for each customer [closed]

We have $n$ customers, $(x_1, \dots, x_n)$, sorted on the read line. For convenience, we also use $x_i$ to denote its coordinate on the line. We need to locate $m$ facilities on the real line. We note ...
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### Time Complexity for Nearest Neighbor Searches in kd-trees

Nearest neighbor searches in kd-trees run in logarithmic time, as shown by Friedman et al. However, I have some difficulty to fully understand the proof. In order to calculate the average number of ...
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### Document references describing weaknesses for cutting planes and algebraic proof system?

Here, Fortnow says (section 4.3): Since then complexity theorists have shown similar weaknesses in a number of other proof systems including cutting planes, algebraic proof systems based on ...
157 views

### Optimal bee swarm plots: NP-hard?

Bee swarm plots are a way of visualizing one-dimensional data sets, similar to box plots. The idea is that if there's not too many points (e.g. <300) we can just plot them along the $x$-axis with ...
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### Complexity of existence of simple polygonalization with prescribed area?

This is a followup on my previous question. Fekete proved the NP-completeness of deciding the existence of simple polygonalization with minimum (or maximum) enclosed area (simple polygonalization is ...
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### Proof for Upper Bound of Sum of Square Roots Problem

In [1], Garey et al. identify what would later be known as the Sum of Square Roots Problem in the course of working out the NP-completeness of Euclidean TSP. Given integers $a_1, a_2, \ldots, a_n$ ...
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### Data structure to report points in the intersection of two circles

The circular range reporting is defined as follow: preprocess $n$ points in the plane so that the points inside a query circle, of any radius, can be reported quickly. This was solved beautifully ...
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### Bichromatic all nearest neighbors

Given two finite sets of points, $R, B \subseteq \mathbb{R}^d$, compute a map $f : R \to B$ where: $f(r) = \text{argmin}_{b \in B} |r - b|$ That is $f(r)$ is the closest point in $B$ to the point $r$...
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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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### $k$th closest pair of points

The problem of finding the pair of closest points in a set of points has been a classical example of divide and conquer technique for some decades now. But I could not find any results on finding $k$...
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Consider the convex hull problem in $\Re^d$: Input: a list of $n$ points $S$ in $\Re^d$, Output: the vertices of the convex hull of $S$. What is the best lower bound on the time complexity of ...