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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2 votes
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66 views

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 ...
3 votes
0 answers
127 views

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 ...
2 votes
1 answer
177 views

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 ...
15 votes
1 answer
404 views

Triangulating a Planar Polygon

Are there by now simpler algorithms/proofs for triangulating a planar polygon in linear time? What is a good resource on the state of the art of this famous problem?
1 vote
0 answers
50 views

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 $S_i$ ...
14 votes
1 answer
352 views

Reference to lower bound on separator in a grid?

It is easy to verify that given the d dimensional grid of the integer points $\{1,\ldots,n\}^d$, with the regular adjacency, one can find a separator of size $n^{d-1}$ (just pick any middle hyperplane,...
2 votes
0 answers
65 views

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 ...
7 votes
4 answers
667 views

Small $\epsilon$-nets for points and half-planes without VC dimension

I have recently learned the proof of Haussler and Welzl of the following theorem. Theorem. Suppose we have a set system $\mathcal{F} \subseteq 2^X$, where $X$ is a finite set. Suppose $\mathcal{F}$ ...
12 votes
2 answers
641 views

Which problems in computational geometry or graph theory are believed to be $\Omega(n^3)$?

This is intended as a follow up question to Robin Kothari's previous post on polynomial time hardness results. Specifically, I'm interested in seeing some hardness proofs for problems that are ...
0 votes
3 answers
1k views

What is the problem in "closest pair problem" if all points share the same x-coordinate

The closest pair of points problem deals with the task to find a pair of points with the global minimum distance. There is a problem, when all points share the same x-coordinate, or at least a large ...
6 votes
1 answer
2k views

VC dimension of intersection of half-spaces

Define $$l_i(x) := \text{sgn} \left( w_i^\top x - b_i \right)$$ for $i=1,...,n$, where $x \in \mathbb{R}^d$. Then define the classifier $$ g(x) := \max \{ l_1(x),..., l_n(x) \}$$ which represents ...
1 vote
1 answer
134 views

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 ...
2 votes
2 answers
308 views

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
1 answer
267 views

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 ...
2 votes
0 answers
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 ...
2 votes
2 answers
281 views

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 ...
6 votes
2 answers
458 views

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. . ...
12 votes
0 answers
356 views

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 ...
3 votes
1 answer
146 views

(eps,delta)-approx with VC-Dimension 1?

I have a domain $X$ and a set system $R$ on $X$, such that the sets in $R$ are one included in the other, that is, for any $A,B\in R$, either $A\subseteq B$ or $B\subseteq A$. The sets are not all ...
13 votes
0 answers
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 ...
4 votes
1 answer
164 views

Minimizing $L_2$ norm of a vector with two distinct entries

Let $d\in\mathbb N$ and denote $V=\bigcup_{a,b\in\mathbb R}\{a,b\}^d$, the set of all vectors with two distinct values. Given a vector $x\in \mathbb R^d$, I want to compute some $v^*\in V$ such that $\...
1 vote
1 answer
245 views

VC-dimension of infinite set of triangle wave

I am searching for the VC-dimension of the following: What is the VC-dimension of the infinite set of triangle wave functions with amplitude 1 and period parameter p on points on the line? 2πarcsin⁡(...
0 votes
1 answer
82 views

How hard is deciding the existence of a polygonization with prescribed perimeter?

Polygonization problem of a set of points in the Euclidean plane (2D lattice) is to find a simple polygon that passes through all points. Deciding the existence of a polygonization with minimum (or ...
8 votes
3 answers
6k views

VC dimension of polynomials (in one variable) of degree d

Linear functions in one variable have VC dimension =3 and I remember reading somewhere that the VC for polynomials of degree $d$ is $(d^2 + 3d + 2)/2$. I am searching for ideas that can prove the ...
4 votes
1 answer
159 views

Citation for isometric embeddability of $\ell_2$ into $\ell_p^\binom{n}{2}$ for $p \geq 1$?

I need to use the following well-known result in my paper: Let $X$ be a set of $n$ points in $\mathbb{R}^d$. Then $(X,\ell_2^d)$ embeds isometrically in $\ell_p^\binom{n}{2}$ for all $p \geq 1$. ...
26 votes
1 answer
581 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 ...
4 votes
0 answers
103 views

Can we make Tensor Sketch any faster?

For all constants $\epsilon,\delta>0$, let $k=\epsilon^{-2}\log1/\delta$. We know there exists a linear transformation $M : \mathbb R^{k^2}\to \mathbb R^{\tilde O(k)}$, such that for all $x\in\...
11 votes
2 answers
3k views

Covering a simple polygon with circles

Suppose I have a simple polygon $S$ and an integer $k$. What are some existing approaches for finding the smallest radius $r$ such that I can cover $S$ with $k$ circles of radius $r$? How about if $...
3 votes
1 answer
1k views

Time complexity of d-dimensional convex hull

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 ...
3 votes
0 answers
628 views

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 ...
7 votes
1 answer
205 views

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 ...
9 votes
0 answers
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 ...
1 vote
3 answers
2k views

Sorting array of distances by proximity to each other

I was playing with geolocation on maps and stumbled on an interesting problem: I retrieve data from the db ordered by increasing distance from a user input, like a postcode or street, which makes ...
3 votes
1 answer
144 views

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 ...
9 votes
1 answer
331 views

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 $a_1, a_2, \ldots, a_n$ ...
7 votes
2 answers
287 views

Paths in a weighted line arrangement

This is a followup to my followup to David's question. This question admittedly leaves the original motivation far behind, but it might provide some useful intuition. Suppose we are given a set of n ...
3 votes
0 answers
94 views

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 ...
3 votes
0 answers
80 views

Embed graph in $\ell_2$ space so that edge and non-edge distances are separated by a constant factor

Suppose I have an undirected unweighted graph $G = (V,E)$. Is there a way to compute points $x_v \in \mathbb{R}^d$ for each vertex $v \in V$ such that $||x_v - x_u|| = 1$ whenever $(u,v) \in E$ and $ |...
10 votes
1 answer
252 views

Places where the order of points along a simple polygon passing through them is useful

We know that finding the convex hull of $n$ points on a plane has a lower bound of $\Omega(n\log n)$ on its running time. However, if the points are given in the order in which they occur along some ...
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 ...
5 votes
0 answers
353 views

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 ...
5 votes
1 answer
342 views

Do combinatorial discrepancy upper bounds lead to smaller $\epsilon$-nets (as with $\epsilon$-samples)?

An $\epsilon$-sample (or $\epsilon$-approximation) of a family of subsets $\mathcal{S}$ of a ground set $X$ is a subset $P \subseteq X$ which preserves relative sizes of sets up to $\epsilon$. I.e., ...
6 votes
3 answers
320 views

Linear Time Maximum Clearance Computation on a Grid Graph?

I have a uniform NxN grid with a non-empty subset of vertices marked as obstacles. My goal is to compute, for each non-obstacle vertex, the "maximum clearance" from the obstacle set. In other words, ...
17 votes
3 answers
542 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 ...
12 votes
1 answer
348 views

Partitioning a rectangle without harming inner rectangles

$C$ is an axis-parallel rectangle. $C_1,\dots,C_n$ are pairwise-interior-disjoint axis-parallel rectangles such that $C_1\cup\dots\cup C_n \subsetneq C$, like this: A rectangle-preserving partition ...
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 ...
2 votes
0 answers
66 views

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 ...
9 votes
1 answer
433 views

VC dimension of Voronoi cells in R^d?

Suppose I have $k$ points in $\mathbb{R}^d$. These induce a Voronoi diagram. If I assign to each of the $k$ points a $\pm$ label, these induce a binary function on $\mathbb{R}^d$. Question: what is ...
2 votes
0 answers
56 views

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 ...
145 votes
2 answers
19k views

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? ...

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