# 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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### Super Mario Galaxy problem

Suppose Mario is walking on the surface of a planet. If he starts walking from a known location, in a fixed direction, for a predetermined distance, how quickly can we determine where he will stop? ...
1answer
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### A combinatorial version for the polynomial Hirsch conjecture

Consider $t$ disjoint families of subsets of {1,2,…,n}, ${\cal F}_1,{\cal F_2},\dots {\cal F_t}$ . Suppose that (*) For every $i \lt j \lt k$ and every $R \in {\cal F}_i$, and $T \in {\cal F}_k$, ...
3answers
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### What are the reasons that researchers in computational geometry prefer the BSS/real-RAM model?

Background The computation over real numbers are more complicated than computation over natural numbers, since real numbers are infinite objects and there are uncountably many real numbers, therefore ...
6answers
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### Geometric problems that are NP-complete in $R^3$ but tractable in $R^2$?

A number of geometric problems are easy when considered in $R^1$, but are NP-complete in $R^d$ for $d\geq2$ (including one of my favourite problems, unit disk cover). Does anyone know of a problem ...
3answers
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### Parameterized complexity of Hitting Set in finite VC-dimension

I'm interested in the parameterized complexity of what I'll call the d-Dimensional Hitting Set problem: given a range space (i.e. a set system / hypergraph) S = (X,R) having VC-dimension at most d and ...
17answers
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### Examples where insight from geometry was useful for solving something completely non-geometric

One of the nice things about having evolved in a universe with three spatial dimensions is that we have developed problem solving skills pertaining to objects in space. Thus, for example, we can think ...
1answer
2k views

### Isometric embedding of L2 into L1

It is known that given an $n$-point subset of $\ell_2^d$ (that is, given $n$ points in ${\mathbb R}^d$ with Euclidean distance) it is possible to embed them isometrically in $\ell^{n\choose 2}_1$. Is ...
1answer
534 views

### Approximately sampling from convex polyhedrons with quantum computers

Quantum computers are very good for sampling distributions that we don't know how to sample using classical computers. For example if $f$ is a Boolean function (from $\{-1,1\}^n$ to $\{-1,1\}$) that ...
5answers
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### Packing rectangles into convex polygons but without rotations

I am interested in the problem of packing identical copies of (2 dimensional) rectangles into a convex (2 dimensional) polygon without overlaps. In my problem you are not allowed to rotate the ...
3answers
724 views

### Convex Body with minimum expected l2 norm

Consider a convex body $K$ centered at origin and symmetric (i.e. if $x\in K$ then $-x\in K$). I desire to find a different convex body $L$ such that $K\subseteq L$ and the following measure is ...
2answers
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### Detecting two kinds of almost-simple polygons

I'm interested in the complexity of deciding whether a given non-simple polygon is almost simple, in either of two different formal senses: weakly simple or non-self-crossing. Since these terms are ...
1answer
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### Complexity of computing shortest paths in the plane with polygonal obstacles

Suppose we are given several disjoint simple polygons in the plane, and two points $s$ and $t$ outside every polygon. The Euclidean shortest path problem is to compute the Euclidean shortest path ...
1answer
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### Maximum disjoint set: what is the actual approximation factor of the greedy algorithm?

Consider the problem of finding a maximum disjoint set - a maximum set of non-overlapping geometric shapes, from a given collection of candidates. This is an NP-complete problem, but in many cases, ...
6answers
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### Network / Social network analysis visualization tools?

I was using Jung ( http://jung.sourceforge.net/ ) to visualize page rank and found it a little slow and difficult to scale it beyond 100 nodes. I was wondering what other tools people use for network /...
2answers
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### A data structure for minimum dot product queries

Consider $\mathbb{R}^n$ equipped with the standard dot product $\langle \cdot, \cdot \rangle$ and $m$ vectors there: $v_1, v_2, \ldots, v_m$. We want to build a data structure that allows queries of ...
1answer
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### The quad-edge data structure (Delaunay/Voronoi)

2 questions for the computational geometers or algebraists: I am just beginning to dive into computational geometry and I am loving it =) I am attempting to read the famous article by Guibas and ...
2answers
731 views

### Find a largest cube contained in the union of cuboids

I have a lot of cuboids in 3D space, each has a starting point at (x,y,z) and has size of (Lx,Ly,Lz). I wonder how to find a largest cube in this 3D space that is contained in the union of the cuboids....
3answers
487 views

### Is there a constant factor approximation algorithm for 2D rectangle coloring problem?

The problem we consider here is the extension of the well-known interval coloring problem. Instead of intervals we consider rectangles having sides parallel to axes. The objective is to color the ...
2answers
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### Sorting by Euclidean distance

$S$ is a set of points on a plane. A random point $x \notin S$ is given on the same plane. The task is to sort all $y \in S$ by Euclidean distance between $x$ and $y$. A no-brain approach is to ...
1answer
545 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 ...
0answers
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0answers
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### 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 ...
0answers
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2answers
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### Art gallery variants with pairwise visibility?

The traditional art gallery problem sets up a region and guards with some notion of visibility, and asks for the minimum number of guards that need to be placed to see the entire region. Has anyone ...