# Questions tagged [computational-geometry]

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### Complexity of Maximum Independent Set (or Vertex Cover) on disk packing graphs

I'm interested in complexity results for Maximum Independent Set (or Vertex Cover) problem over the class of disk packing graphs. Having a set of disks we build a graph that has its vertices at the ...
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### Trade-off between number of spheres and wasted space in covering a 3d object by spheres

Consider the following optimization problem: Input: a 3-dimensional "object" $O$. Output: a covering of $O$ by a list of $k$ spheres $S_1, \ldots, S_k$ (given by their centers and radii) minimizing ...
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### Weighted $l_1$ distance

So there are many well known algorithms for approximate nearest neighbor on the $\ell_1$ distance. My question is, what about the weighted version of the problem (where the weights are specified along ...
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### Polytopes convex hull

Assume $P_1$, $P_2$ and $Q$ are axis-parallel polytopes in $\mathbb{R}^d$. Is it true to say that checking $CH(P_1\cup P_2)\neq Q$ is NP-hard? There is a similar result for the general polytopes but I ...
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### Is this covering problem NP-hard?

Given a rectangular region $R$ and a set $D$ of $n$ disks such that the union of all disks in $D$ cover the entire rectangular region $R$, the objective is to find the minimum cardinality set $D'$ ...
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### H-representation of convex hull

Consider a set of polytopes $P_j\;\;j=1,2,\dots,r$ with the same structure as follows: $P_j=\Big\{(x_{j1},\dots, x_{jt})\Big| \sum_{i=1}^t x_{ji}=1, x_{ji}\in [a_{ji},b_{ji}]\subseteq [0,1]\Big\}$ ...
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### centralized deterministic Spanner construction with low degree and low stretch

Does there exist a centralized deterministic spanner construction with low degree and low stretch both independent of the graph diameter (no log D factor), but can be dependent on the number of nodes. ...
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### Finding intersections of numerically implemented 1-dimensional curves on a 2-dimensional plane

Question summary: what are the known efficient algorithms to find the intersections of 1-dimensional curves living on a 2-dimensional plane? Detail: I have a set of 1-dimensional curves on a 2-...
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### VC-dimension of triangles in 2D space [closed]

I have been reading in multiple places (e.g. , 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 ...
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### Largest cell in an arrangement

Q. What is the complexity of finding the largest volume bounded cell in an arrangment of $n$ hyperplanes in dimension $d$? I feel I should know this... But I am not finding a definitive reference. ...
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### Maximum number of geometrically disjoint paths - is the complexity known?

Let $G$ be an undirected graph, given with a planar drawing. We do not assume $G$ is a planar graph, we just fix a planar representation of it, such that each vertex is represented by a distinct ...
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### Emptiness of complement of subspace arrangement

Given $k$ affine subspaces in $\{0,1\}^n$, consider the problem of testing whether their union covers all of $\{0,1\}^n$. What's the complexity of this problem? P.S.: It seems that this can be ...
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### Generalization of Beck's theorem

Beck's theorem is a classical result in discrete geometry which describes about the geometry of points in the plane. The result states that a finite collections of points in the plane fall into one of ...
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### 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 ...
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### Extending Delaunay graphs in d-space

I am new to computational geometry so pardon me for the lack of formalism. I am currently experimenting with an algorithm of mine in which I need to extend recursively a Delaunay graph in $d$-space. ...
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### Weighted furthest point voronoi diagrams

I found that Weighted nearest neighbor voronoi diagrams are widely studied and there are optimal algorithms for that. But I could not find anything on Weighted furthest point voronoi diagrams !! But ...
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### Implementation of partition trees?

Have partition trees ever been implemented? Here, I'm talking about the partition trees from computational geometry. The earliest (near-)optimal versions of which were due to Matousek and others, ...
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### Approximating the value of k in $k$-mean clustering problem

Consider a set of $n$ points in $R^d$ which are covered by some finitely many (say $k$) unit balls. Can we approximate the value of $k$ by querying only sublinear many points. More precisely, by ...
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### Finding a point outside of each of a set of polygons in a bounded space

I know there are algorithms for finding a point inside a simple polygon. Given a set of polygons inside a rectangle (think a bunch of polygons on a computer screen), is there an efficient algorithm ...
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### Clustering in sublinear time/query

Given a set of $n$ points in $R^d$, the goal is to cover them with (finitely many) unit balls such that following conditions satisfy: 1) Minimizing the number of balls that are required to cover all ...
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### A continuous center point of a convex spherical polygon

In discrete geometry, the center point $c$ of a discrete set $S$ of $n$ points in the plane is such that any half plane containing $c$ contains (roughly) $n/3$ points of $S$. (Such a center point ...
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### Minimum equidecomposable decomposition

Given two polyhedra $P$ and $Q$, $P$ and $Q$ are are equidecomposable if there are finite sets of polyhedra $P_1, \ldots, P_n$ and $Q_1, \ldots, Q_n$ such that $P_i$ and $Q_i$ are congruent for all $i$...
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### maximum weighted 2D coverage problem by a rectangle

Let $P=\{p_1,\ldots,p_n\}$ be a set of $n$ points in a 2D plane, that is $p_i\in \mathbb{R}^2$, $\forall i=1,\ldots,n$. Each point, $p_i$, is associated with a weight, $w_i \geq 0$. Imagine a axis-...
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### Would a purely topological computational model be useful in decision problems in topology?

If one were to develop a purely topological computational model based upon the equivalence of information in knots and the model would perform transformations of that information. This would be the ...
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### Algorithms to classify the geometrical relationship of two 3D-geometries

I have multiple 3D volumes which are represented by their boundaries (set of polygons each represented as a list of 3D-coordinates). I am now looking for algorithms to decide if volume A is inside ...
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### Locating a point inside a union of simple polygons

I have a set of polygons (convex, concave – non-convex, not self-intersecting) in a plane. There may be intersections between polygons. Polygon is defined by points (cartesian coordinate system). ...
194 views

### Eulerian Triangulations

Hi i am looking for algorithms to decide whether a planar pointset has a eulerian triangulation i.e. a triangulation that makes every vertex of even degree. I cam across this page http://cs.anu.edu....
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### Worst-case optimal Delaunay algorithm based on spatial sort and walking?

Buchin  showed that under "reasonable" assumptions, (polynomial point spread) incremental construction of Delaunay triangulations using the biased randomized ordering of Amenta et al.  with a ...
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### Rectangular constraints in Delaunay Triangulation without edges within

I'm using a triangulation library to compute the Constrained Delaunay Triangulation of a set of rectangles within some large boundary. The algorithm returns all the edges, but also adds edges inside ...
Given a set of points $p_1, p_2, ..., p_n$ and a collection of sets from $n$ points $S_1, S_2,...,S_m$. The pairwise distance between any three points $(p_1, p_2, p_3)$ satisfies triangle inequality \$...