Questions tagged [convex-hull]
The convex-hull tag has no usage guidance.
27
questions
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Convex hull in a discrete space
I know some algorithms which compute the convex hull in a continuous space. Are there efficient algorithms to compute it in a discrete domain?
For example in 3D discrete space, given the blue points, ...
5
votes
2answers
155 views
If a convex optimization problem can be NP-Hard, in what sense are convex problems easier than non-convex problems?
Being new to the OR and Optimization world, I've always assumed that a problem being convex meant that it can be solved in polynomial time.
Now I am learning that a convex optimization problem can ...
2
votes
1answer
27 views
Determine image of hypercube under linear map
Let $A$ be an $3\times N$ matrix (where $N$ is large) with nonnegative real entries. I'd like an algorithm for determining when a vector $v\in\Bbb R^3$ can be written as $Aw$ for some vector $w\in\Bbb ...
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vote
0answers
34 views
Game Theory: Using a convex hull algorithm to map out Pareto outcomes
I have started studying the Pareto efficiency notion in Game theory. The definition I am familiar with is this:
Strategy profile $\mathbf{s}$ Pareto dominates strategy $\mathbf{s}'$ if for all $i\in\...
2
votes
1answer
363 views
3D gift wrapping algorithm: how to find the first face in the convex hull?
I am implementing the gift wrapping algorithm to find the convex hull of a set of points in the 3D space.
However, all the articles I have read seem to omit the description of the first step of the ...
3
votes
1answer
202 views
Is binary-search really required in Chan's convex hull algorithm?
I have a little doubt about Chan's algorithm.
From Wikipedia's description we see that the second phase of an algorithm works with $K = \mathrm{ceil}(\frac{n}{m})$ subsets $Q_i$. The goal of the ...
2
votes
1answer
355 views
Minimum distance between two convex hulls maximized
I want to implement a program that splits a set $S$ of $n$ points in the plane into two sets such that the distance of the convex hulls of the two sets is maximized.
It should be done in $O(n^3)$.
I ...
3
votes
1answer
71 views
Why is the graph inside Graham Scan always planar
One of the ways to prove that Graham Scan constructs convex hull in linear time is using planarity of the graph obtained by running the algorithm. This graph is always planar, so according to Euler's ...
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0answers
13 views
Convex Hull for known output size in O(n)
Given a set $S$ of $n$ planar points, we know that the $|CH(S)| = 17$. How can I create an algorithm that computes $CH(S)$ in $O(n)$ time? Why is it really $O(n)$?
2
votes
1answer
60 views
Lower bound for point set triangulation
How can I show that the lower bound for getting a triangulation from a point set in the plane is $\Omega(n \log n)$?
I know that the lower bound for finding the convex hull for a point set is also $\...
1
vote
1answer
139 views
How to prove that non-antipodal vertices cannot be a diameter of a convex polygon?
I am learning Shamos's rotating calipers algorithm for finding the diameter of a convex polygon in his Ph.D. thesis; Page 78.
It reads
Consult Figure 3.23 and notice that parallel lines of support ...
1
vote
1answer
130 views
Convex hull of fixed size
I have heard that the quickhull algorithm can be modified if the size of the convex hull (the number of points it consists of) is known beforehand, in which case it will run in linear time. What ...
6
votes
1answer
440 views
How to find the supremum over all the “good” (interior) polytopes for a given set of 3D points?
Let $S \subset \mathbf{R}^3$ be a set of points in 3D and let $O=(x_0,y_0,z_0)$ be the origin/point of reference.
We consider a convex polytope $P$ good / interior if:
$P$ is wholly contained ...
1
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0answers
133 views
Nested Convex Hulls Algorithm
The convex hull of a point set is a well understood problem and nice optimal solutions are known in the case of a finite point set and a simple polygon.
For a convex polygon, the hull is the polygon ...
2
votes
1answer
547 views
The use of binary search when determining whether a point lies inside a given convex hull
In an answer to the problem of determining whether or not a point lies inside a given convex hull, a thesis is mentioned, which says :
For repeated queries with preprocessing allowed, we develop a ...
1
vote
1answer
64 views
How to convert two conflicting objective functions into a single objective function
I have two objective functions say f1 where I have to minimize (X+Y) and another function f2 where I have to maximize (A-B). The two functions are conflicting.
I need to convert them into a ...
2
votes
1answer
97 views
Internal tangent intersection of two point sets in linear time
I need to find the intersection of the internal tangents of two point sets $V_a, V_b$ in $\mathbb{R}^2$, defined via their convex hulls. We can assume that the sets are disjoint and linearly separable,...
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0answers
515 views
Finding the Convex Layers of a given set of points
Definition of convex layers can be found at wikipedia.
I was trying to understand this algorithm , which works in O(n log n) time, which is optimal.
In the paper, the author has described two ...
4
votes
1answer
697 views
Convex-hull of a star shaped polygon in O(n)
I'm trying to develop an algorithm to find a convex hull of a star shaped polygon in $O(n)$ time. The specific problem, which is also described here, is as follows:
A polygon $P$ is star-shaped if ...
5
votes
1answer
148 views
Optimization over convex combinations in a circle
Consider the following situation: given a triangle $ABC$ inscribed in a circle, define $f$ as the product
$$f(P) = d(P, A) \; d(P,B) \; d(P,C)$$
where $P$ is a point on the circle and $d$ are ...
6
votes
0answers
93 views
$3$-dimensional convex hull using only a desired number of planes
I would like to find the convex polytope with the smallest volume that envelops (contains) all the points of a given 3D point cloud and that can be constructed from only $k$ planes. This is similar to ...
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1answer
122 views
Convex Hull Problem proof [closed]
In a given set of points, Prove that the two farthest points are the vertices of the convex hull,
How can i get the accurate proof, so that the question can be explained in the class
3
votes
1answer
235 views
Convex hull algorithm in $O(\min(mn, n\log n))$
I am looking for an algorithm to compute the convex hull of a set of $n$ points $P$. The hull should contains $m$ points. This algorithm should work in time $O(\min(mn,n \log n))$.
My first guess was ...
1
vote
0answers
39 views
Does Optimal Substructure implies Convexity and vice versa?
In undergraduate CS, Dynamic Programming problems are often related to Overlapping Optimal Substructure (https://en.wikipedia.org/wiki/Optimal_substructure).
Dynamic Programming is also often used in ...
2
votes
1answer
142 views
About the Max-Cut SDP
The Max-Cut optimization problem on a graph $G=(V,E)$ can be written as the question of wanting to maximize the function $\frac{1}{4} \sum_{(i,j) \in E } (x_i -x_j)^2$ under the constraint $x_i^2 = 1, ...
3
votes
1answer
89 views
How to use convex hull for this problem
Problem:
You are to collect a total of $N$ litres of red and $M$ litres of blue liquid. For doing job $i$ for time $t$, you get $a_i t$ litres of red and $b_i t$ litres of blue liquid. $t$ need not ...
0
votes
1answer
373 views
Understanding a few intricacies related to two naive algorithms to compute the convex hull of a set of points
I already understood how the well-known algorithms like Graham's scan, Quickhull, etc. work, but I have difficulties in understanding 2 naive algorithms to compute the convex hull:
Let $S= \{p_1, \...
