Questions tagged [convex-hull]
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minimum number of points a convex hull must have
Quick question:
Say for example there are 10 colinear points. my question is does a convex hull have to be a convex polygon? or can it be a line as well according to the formal definition of the ...
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12 views
Integer Linear (ILP / MILP) Formulation for Collision Avoidance of Convex Polytopes / Polyhedra
I am looking for a possibility to avoid the collision of two convex polytopes using (mixed integer) linear programming. I know how I can detect a collision (Akgunduz, A., Banerjee, P., and Mehrotra, S....
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15 views
minimum moves to be made to get out of convex hull
Given a convex hull in $XY$ plane. and we have $n$ points sitting inside it. In one move we can move all the points to right by $1$ or move all the points to up by $1$. what are the minimum number of ...
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1answer
72 views
Concentric convex hulls
Given N points in a 2D plane, if we start at a given point and start including points in a set ordered by their distance from the starting point. After including every point, we check if there is a ...
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17 views
Why are basic feasible solutions the same as vertices geometrically?
The first line on the Wikipedia page for basic feasible solutions reads, ...
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1answer
40 views
Finidng edges of convexhull from rectangles
Green Boxes = rectangles
red dots = edge points
red lines = to be generated from convex hull algorithm
I have a problem with creating the convex hull algorithm.
I want to select or collect all the ...
4
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0answers
45 views
Minimal set of inequalities including good points but excluding bad points
Suppose I have a collection of good convex sets and bad convex sets in $\mathbb{R}^d$ (where $d$ can be big). Each convex set is defined by a series of closed ranges in each dimension $d$ - a ...
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39 views
Distance from high dimensional convex hull to target point T
I have a set S of high dimensional points in Euclidean space, with convex hull H (not known); and some target point T in that space not in or on H.
Rather than worry about calculating both H and the ...
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47 views
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, ...
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2answers
690 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
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1answer
42 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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55 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\...
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1answer
637 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
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1answer
294 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
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1answer
532 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
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1answer
109 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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16 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
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1answer
66 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 $\...
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1answer
192 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 ...
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1answer
153 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 ...
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1answer
497 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 ...
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178 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
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1answer
717 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 ...
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1answer
75 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
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1answer
120 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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628 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
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1answer
903 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
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1answer
172 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
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0answers
97 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
150 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
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1answer
331 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 ...
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42 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 ...
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1answer
217 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
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1answer
93 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 ...
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1answer
380 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, \...
