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).
267
questions
2
votes
2
answers
131
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
77
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 ...
- 8,133
1
vote
1
answer
219
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 ...
- 37
2
votes
0
answers
38
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 ...
- 1,901
2
votes
2
answers
189
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 ...
- 1,826
6
votes
2
answers
399
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. .
...
- 1,081
12
votes
0
answers
350
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 ...
- 121
13
votes
0
answers
336
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 ...
- 89
4
votes
1
answer
150
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 $\...
- 143
1
vote
1
answer
215
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(...
- 11
0
votes
1
answer
61
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 ...
- 20.5k
4
votes
1
answer
140
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 $p \geq 1$.
...
4
votes
0
answers
94
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\...
- 778
3
votes
0
answers
555
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 ...
- 131
7
votes
1
answer
190
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
151
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 ...
- 1,023
3
votes
1
answer
137
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 ...
- 20.5k
9
votes
1
answer
282
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 $a_1, a_2, \ldots, a_n$ ...
- 3,381
3
votes
0
answers
82
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 ...
- 3,910
3
votes
0
answers
77
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 $ |...
7
votes
1
answer
176
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 ...
- 171
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 ...
- 43
2
votes
0
answers
53
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 ...
- 305
17
votes
1
answer
581
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 ...
- 431
9
votes
0
answers
352
views
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 (...
- 91
2
votes
0
answers
102
views
Minimize the product of distances from n fixed points
Given $n$ fixed points $\{p_i\}_{i=1}^n$ in $D$-dimensional Euclidean space $\mathbb{R}^D$, consider the following optimization problem:
$$
\begin{align}
\text{mininize} \ \ \ &\prod_{i=1}^n \...
- 21
3
votes
1
answer
386
views
What is the proof that visibility graphs can be used to compute the shortest path?
I am trying to understand what the proof is that constructing a visibility graph and searching on can give you the shortest path between two points, avoiding a set of convex polygons. I am trying ...
- 151
4
votes
0
answers
146
views
Computing Minima of the Projection of a Binary Cube
The problem is as follows: I want to compute the minima (with respect to the canonical partial order on vectors "$\leq$") of the linear projection of the extreme points of an $n$-dimensional $\{0,1\}$-...
- 141
11
votes
1
answer
248
views
Smallest axis-aligned box that contains $k$ points
Input: A set of $n$ points in $\mathbb{R}^3$, and an integer $k \le n$.
Output: The smallest volume axis-aligned bounding box that contains at least $k$ of these $n$ points.
I'm wondering if any ...
- 2,183
5
votes
1
answer
158
views
Generalized Priority Queues
I was wondering if there is any literature on the following problem:
Maintain a set $S$ where each element is a function from $\mathbb{R}$ to $\mathbb{R}$ supporting the following operations:
Insert ...
- 423
12
votes
1
answer
339
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 ...
- 1,826
1
vote
0
answers
203
views
Clarification needed on an algorithm for $\epsilon$-net construction for the column space of PSD matrices
I found an algorithm for constructing an $\epsilon$-net for a positive semidefine matrix $A\in[-1,1]^{n\times n}$ which has $rank(A)=d$, described in the paper
The approximate rank of a matrix and ...
- 161
6
votes
1
answer
324
views
Largest rectangle not touching any rock in a square field
Following this mathematical question, I am interested in the algorithmic question:
Given $n$ points in the unit square,
find the largest area of an axis-parallel rectangle in the unit square
...
- 1,826
4
votes
0
answers
109
views
Minkowski decomposition of lattice point cloud
Given two point clouds $A,B\subset\mathbb Z^d$, let $A\oplus B$ be their Minkowski sum, defined as the set $\{ a + b : a\in A, b\in B \}$.
Is there any known result for the following problem?
...
- 4,400
11
votes
1
answer
248
views
Compute lowest dimensional polytope from a given set of sign vectors
Given a set of hyperplanes determined by the normal vectors $h_1,\dots,h_m \in \mathbf R^d$, its cell types (or sign vectors) are all vectors $t\in\{+,-\}^m$ for which there exists a vector $v\in\...
- 975
4
votes
1
answer
112
views
Non-Midpoint Segment Splitting in Ruppert's Delaunay Triangulation Refinement Algorithm
Roughly speaking, in Ruppert's Delaunay Triangulation refinement algorithm, so called encroached edges are split until no more encroached edges remain.
The algorithm specifies splitting the edges ...
- 235
1
vote
1
answer
68
views
Far point queries in high dimensions
Given a set of points $X\subset R^d$ and a number $r\in R$, create a data structure for queries of the form: "given a point $q\in R^d$ return a point $x\in X$ with $\text{dist}(q,x)\ge r$".
This is ...
- 778
-2
votes
1
answer
79
views
arbitrary segment stabbing query for 2d segments
Store a set of 2d segments S in some data structure. For an arbitrary query 2d segment q, answer a yes/no question in sublinear time: whether q intersect with any segment in S?
If the query is a ...
- 101
0
votes
1
answer
3k
views
VC-dimension of triangles in 2D space [closed]
I have been reading in multiple places (e.g. [1], section 4) that the VC-dimension of the class of triangles (in 2D space) is 7.
The issue is that, for the case when 4 points lying on a straight line ...
- 739
3
votes
1
answer
392
views
Sorted intervals query
I'm in search for a data structure which efficiently operates over closed intervals with the following properties:
dynamically add or remove an interval
set, and anytime change, a number ("depth") ...
- 289
4
votes
1
answer
190
views
Algorithm to find all intersections in set of simplices
What is the fastest known algorithm to report all intersecting pairs amongst a collection of $n$ simplices, each with dimension at most $r \leq d$ embedded in $\mathbb{R}^d$ (for small $d$)?
In the ...
- 581
1
vote
0
answers
84
views
k closest points that belong to a set
This is a question from theory community, but I came across this issue in a practical problem. So just have this in mind.
I have a set of real vectors:
$$
S = \lbrace v_1, \dots, v_n \rbrace
$$
$...
- 739
4
votes
0
answers
126
views
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 $r$...
- 581
5
votes
2
answers
230
views
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 ...
- 639
3
votes
0
answers
108
views
$k$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 $k$...
- 191
4
votes
0
answers
171
views
Problems in dynamic algorithms in computational geometry
The publication of Chiang and Tamassia's paper on dynamic algorithms in computational geometry included several algorithms used in solving dynamic computational geometry problems such as:
Dynamic ...
- 313
3
votes
1
answer
781
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 ...
user22369
1
vote
0
answers
104
views
Is there an algorithm that, given a point cloud, infers an optimal wireframe (surface) structure?
I have a point cloud that I would like to convert to a surface, in the form of a wireframe lattice structure.
This means, from a sequence of 3D points (x,y,z), obtaining three 2D matrices X,Y,Z of ...
- 119
5
votes
1
answer
240
views
Number of points on the interior of the convex hull of a random subset
This question is in regards to the following problem:
Suppose you are given a set $S$ of $n$ points in the plane. Let $R$ be a random subset of $S$ of size $r$ with all subsets of size $r$ equally ...
- 235
3
votes
1
answer
127
views
Find all hyperplanes separating unique sets of `k` points
I have a series of n points in d-dimensional continuous space. I want to find a series of hyperplanes such that ...
- 252