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).

Filter by
Sorted by
Tagged with
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? ...
Jeffε's user avatar
  • 23.1k
53 votes
1 answer
2k views

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 $T \in {\cal F}_k$, ...
Gil Kalai's user avatar
  • 6,033
43 votes
3 answers
4k views

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 ...
Kaveh's user avatar
  • 21.6k
37 votes
6 answers
2k views

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 ...
Bob Fraser's user avatar
37 votes
3 answers
2k views

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 ...
James King's user avatar
  • 2,613
28 votes
17 answers
3k views

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 ...
27 votes
1 answer
3k 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 ...
Luca Trevisan's user avatar
26 votes
1 answer
585 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 ...
Gil Kalai's user avatar
  • 6,033
23 votes
5 answers
2k views

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 ...
Simd's user avatar
  • 3,902
23 votes
3 answers
750 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 ...
Ashwinkumar B V's user avatar
22 votes
2 answers
2k views

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 ...
Jeffε's user avatar
  • 23.1k
22 votes
1 answer
1k views

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 ...
Jeffε's user avatar
  • 23.1k
21 votes
1 answer
1k views

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, ...
Erel Segal-Halevi's user avatar
20 votes
6 answers
3k views

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 /...
19 votes
2 answers
2k views

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 ...
ilyaraz's user avatar
  • 1,569
18 votes
1 answer
4k views

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 ...
bigmonachus's user avatar
18 votes
2 answers
891 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....
pantoffski's user avatar
17 votes
3 answers
543 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 ...
Soumitra's user avatar
  • 193
17 votes
2 answers
2k views

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 ...
Alex K.'s user avatar
  • 173
17 votes
1 answer
602 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 ...
Robin Houston's user avatar
17 votes
0 answers
366 views

complexity of checking if a subspace is a Euclidean section of L1

If $X$ is a linear subspace of ${\mathbb R}^n$, $X$ is high-dimensional, and for every $x\in X$ we have $(1-\epsilon) \sqrt n ||x||_2 \leq ||x||_1 \leq \sqrt n ||x||_2$ for some small $\epsilon >...
Luca Trevisan's user avatar
16 votes
2 answers
516 views

Finding the largest set of points of limited diameter

Given points $p_1,\ldots,p_n$ in $\mathbb{R}^{d}$ and a distance $l$ find the largest subset of these points such that the Euclidian distance of no two of them exceeds $l$. What is the complexity of ...
Marcus Ritt's user avatar
  • 1,450
15 votes
2 answers
495 views

Best way to determine the minimum dimension of a structure given only distances between points

I came across this problem in an area of physics quite far removed from computer science, but it seems like the type of question that has been studied in CS, so I thought I'd try my luck asking it ...
Joe Fitzsimons's user avatar
15 votes
1 answer
389 views

Two matrices related by a permutation $B = P A P^T$ - complexity

What is computational complexity of the following problem: given two complex $n\times n$ matrices $A$ and $B$ check if there is a permutation matrix $P$ such that: $$B = P A P^T.$$ If it helps, one ...
Piotr Migdal's user avatar
15 votes
1 answer
803 views

Is the lower bound proof in this paper correct?

In this paper on "Circle Packing for Origami Design Is Hard" by Erik D. Demaine, Sandor P. Fekete, Robert J. Lang, on page 15, figure 13, they claim that the side length of the smallest square that ...
Vinayak Pathak's user avatar
15 votes
1 answer
540 views

Drawing graphs with few "sharp" vertices?

For a planar embedding of a planar graph on a plane with straight edges, define a vertex as a sharp vertex if the maximum angle between two consecutive edges around it is more than 180. Or in other ...
Vinayak Pathak's user avatar
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?
Gil Kalai's user avatar
  • 6,033
14 votes
6 answers
564 views

Planar graph via the intersection of fat thingies?

There is a beautiful theorem of Koebe (see here) that states that any planar graph can be drawn as kissing graph of disks (very romantic...). (Putting it somewhat differently, any planar graph can be ...
Sariel Har-Peled's user avatar
14 votes
2 answers
614 views

The number of triangulations of a set of $n$ planar points: Why so difficult?

After hearing Emo Welzl speak on the subject this summer, I know the number of of triangulations of a set of $n$ points in the plane is somewhere between about $\Omega(8.48^n)$ and $O(30^n)$. ...
Joseph O'Rourke's user avatar
14 votes
3 answers
399 views

Separation of a preprocessed polyhedron and a plane

I have serious trouble understanding one step in the paper of Dobkin and Kirkpatrick about the separation of polyhedra. I am trying to understand this version: http://www.cs.princeton.edu/~dpd/Papers/...
domotorp's user avatar
  • 14k
14 votes
1 answer
353 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,...
Sariel Har-Peled's user avatar
14 votes
1 answer
885 views

Computing the Löwner-John ellipsoid of a polyhedron

The Löwner-John ellipsoid of a convex set $C$ is the minimum-volume ellipsoid (MVE) that encloses it. The ellipsoid can be computed using Khachiyan's method, and there are a number of approximations ...
Suresh Venkat's user avatar
14 votes
2 answers
421 views

Data structure for updates on intervals and querying number of zeros

I am looking for a data structure that would maintain an integer table $t$ of size $n$, and allowing the following operations in time $O(\log n)$. $\text{increase}(a,b)$, which increases $t[a],t[a+1],...
Christoph Dürr's user avatar
14 votes
0 answers
383 views

Applications of fat shattering dimension in computational geometry

The fat shattering dimension generalizes the notion of VC-dimension to handle function classes where the range is $(0,1)$, instead of $\{0,1\}$. Fat shattering dimension plays the same role as VC-...
Suresh Venkat's user avatar
13 votes
2 answers
507 views

Learning triangles in the plane

I assigned my students the problem of finding a triangle consistent with a collection of $m$ points in $\mathbb{R}^2$, labeled with $\pm1$. (A triangle $T$ is consistent with the labeled sample if $T$ ...
Aryeh's user avatar
  • 10.5k
13 votes
2 answers
991 views

Testing whether a set of n points in the plane form a convex n polygon in o(nlogn) time

Assume that you are given a set of n points in the plane and you want to check whether they form a convex n polygon, i.e., if they all lie on the convex hull. I was wondering if anyone knows how to do ...
Babis Tsourakakis 's user avatar
13 votes
2 answers
657 views

Estimating VC-Dimension

What is known about the following problem? Given a collection $C$ of functions $f:\{0,1\}^n\rightarrow\{0,1\}$, find a largest subcollection $S \subseteq C$ subject to the constraint that VC-...
Aaron Roth's user avatar
  • 9,870
13 votes
1 answer
463 views

Problem NP-complete for Euclidean geometry but in P for Non-Euclidean geometry?

Are there any problems that are NP-complete when using Euclidean geometry but are well-defined and solvable in polynomial time for some non-euclidean geometry?
Sorin Istrail's user avatar
13 votes
0 answers
362 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 ...
asdfqwer's user avatar
13 votes
0 answers
411 views

Exact nearest neighbor in $d$-dimensional Euclidean space

Suppose that we have $n$ points in $d$-dimensional Euclidean space $\mathbb{R}^d$. We wish to solve the standard exact nearest neighbor problem: build a data structure such that on any query $q\in \...
Lawrence Cayton's user avatar
12 votes
6 answers
900 views

What is a really good problem to get your hands dirty in computational-geometry?

Computational geometry is an area I find pretty interesting, and I'd like to devote about a month or two to a project that will introduce me to this and help me learn key concepts. What is a good way ...
user avatar
12 votes
2 answers
642 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 ...
Bob Fraser's user avatar
12 votes
5 answers
452 views

Motivation for volume estimation

What are some concrete and compelling applications for estimating the volume of convex polyhedra of the sort considered in the more recent papers on random walk methods? These papers on volume ...
user avatar
12 votes
3 answers
350 views

Complexity of Localization in Wireless Networks

Let distinct points $1 ... n$ sit in $\mathbb{R}^2$. We say points $i$ and $j$ are neighbors if $|i-j| < 3 \pmod{n-2}$, meaning each point is neighbors with points with indexes within $2$, ...
Lev Reyzin's user avatar
  • 12k
12 votes
2 answers
3k views

Cover a Concave Polygon with a minimum number of rectangles

I am trying to cover a simple concave polygon with a minimum rectangles. My rectangles can be any length, but they have maximum widths, and the polygon will never have an acute angle. I thought about ...
12 votes
1 answer
350 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 ...
Erel Segal-Halevi's user avatar
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 ...
Andres Fuentes's user avatar
11 votes
4 answers
491 views

Dimensionality reduction with slack?

The Johnson-Lindenstrauss lemma says roughly that for any collection $S$ of $n$ points in $\mathbb{R}^d$, there exists a map $f:\mathbb{R}^d \rightarrow \mathbb{R}^k$ where $k = O(\log n/\epsilon^2)$ ...
Aaron Roth's user avatar
  • 9,870
11 votes
1 answer
864 views

Finding a dual of a graph

According to the book Topological Graph Theory by Gross and Tucker, given a cellular embedding of a graph on a surface (by 'surface' I mean here a sphere with some $n\geq 0$ handles, and below $S_n$ ...
a06e's user avatar
  • 669
11 votes
1 answer
260 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 ...
GMB's user avatar
  • 2,403

1
2 3 4 5 6