# 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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### 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 ...
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### Covering a set of intervals, continued

Peter Taylor and Tsuyoshi Ito solved a previous question that I posted: Covering a set of intervals I have a slight variation on that question that I'd also like to ask. I'm not sure what the ...
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### 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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### Triangle arrangement problem

Suppose you are given an undirected graph $G$, with each vertex representing an equilateral triangle with sides of unit length. Does there exist an arrangement of these triangles in two dimensions (...
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### Efficient Enumeration and Parameterization of Graphs Deriving from Delaunay Tesselations in 3D

This is a repost from a question on Computational Science. No proper answer was found and it was suggested that TCS might be able to answer. So here it goes: Is there an algorithm that enumerates the ...
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### Minimum length cuts needed to remove holes in a polygon

Suppose I'm given a connected polygon in the plane with holes. I can "remove" a hole by drawing a straight line from the boundary of a hole to another boundary (either of another hole, or the boundary ...
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### Is there any known result for 1-median problem with negative weights in Euclidean space?

Given a set of $n$ points $v_1,\ldots,v_n$ in $R^d$, find a point $p$ such that $\sum_{i=1}^n w_i d(p,v_i)$ is minimized, where $w_i$ is the weight of $v_i$, which may be positive or negative. ...
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### Lower bound for the maximal vectors problem

I am studying the (worst-case) complexity $C(n,d)$ required to solve the maximal vector problem: given a finite set $V$ of $n$ $d$-dimensional vectors, compute the set of undominated (a.k.a. skyline, ...
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### Examples of simple charging schemes

I am looking for a simple example of a charging scheme. That is, one that will take 1-2 minutes to explain in a talk. I know of one such example, though it concerns a geometric question, while I hope ...
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### Cheapest dissection of a grid polygon into rectangles with cost

My problem: Dissect a grid polygon into rectangles. (A grid polygon is a rectilinear polygon all of whose vertices have integer coordinates.) The rectangles must be taken from a predefined set (which ...
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### Can we reduce dimensions before applying high dimensional approximate nearest neighbor algorithms?

Andoni and Indyk presented a paper in FOCS 2006: Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions. In this paper they present an algorithm for the $c$-approximate ...
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### Practical algorithm for testing whether an edge is Delaunay

I have a set of vertices $V\subset\mathbb R^3$ and a set of edges $S=\{(a,b)|a,b \in V\}$. I want to know whether an edge in the set $S$ is Delaunay against the vertices in $V$. My assumed ...
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### Dynamic 2-dimensional orthogonal range reporting in external memory and linear space

Orthogonal 2-dimensional range reporting is the problem of storing a set of values from $U \times V$, where $U$ and $V$ are totally ordered universes, subject to queries of the form "Return all stored ...
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### Voronoi partitioning on the 2nd and 3rd closest: how many pieces?

Where order-$k$ Voronoi diagrams partition according to the $k$ closest sites (de Berg et al. p. 169), consider partitioning on the 2nd and 3rd closest — call these [2,3]-Voronoi partitions. ...
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### Strongly edge-guarding a 3d triangulation

Let $T$ be a planar triangulation. It is known that one can guard the faces of $T$ using at most $\lfloor n/3 \rfloor$ edge-guards (Worst-case-optimal algorithms for guarding planar graphs and ...
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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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### Computing the convex hull of several polyhedra

Let $K_1,..., K_m$ be a set of $m$ polyhedra, with $K_i\subseteq \mathbf{R}^n$, for all $i\in [m]$, and each is described by a set of $poly(n)$ linear inequalities. How easy is it to compute an ...
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### Fixed orientation metrics

I am currently working on some computational geometry problems with non-euclidean metrics, and have some trouble on a fact that sounds easy enough, at least intuitively. The fact is as follows: Given ...
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### References for algorithms to compute approximating polytopes for arbitrary convex sets

There is a vast theoretical literature on approximating convex, compact bodies in $d$-dimensional space $\Bbb R^d$ by convex polytopes. One of the main results in this area is that under some mild ...
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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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### 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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### Finding the largest set of points of limited diameter (2)

Problem: Given a set of points $S = \{(x_1, y_1), (x_2, y_2),\cdots,(x_n, y_n)\}$ in $\mathbb{R}^2$ and a distance threshold $\tau$, find a subset of $S$ such that (1) the Euclidean distance between ...
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### 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 ...
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