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# Questions tagged [convex-hull]

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### Algorithms for unordered vertices of the convex hull

For the sake of this question a "non-ordering" "set of vertices of convex hull" algorithm produces the collection of all points on the convex hull of its input without producing ...
16 views

### Algorithm for surface mesh of convex decomposition

I have an object mesh. This is loaded into a physics simulator, which does a convex decomposition. Now I want to perform operations outside the simulator that require a mesh representation. I could ...
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### On a set of points uniformly distributed in a circle, on average, what would be better? Graham's scan or Jarvis's march

I got this question in an exam and my approach was to use probability. Jarvis's march is better than Graham's scan if h < log(n) So I calculated the probability of a point being on the circle ...
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### Detailed exposition for proof of Localization Lemma in paper "Random Walks in a Convex Body and an Improved Volume Algorithm"

I've begun reading the paper "Random Walks in a Convex Body and an Improved Volume Algorithm" by Lovász-Simonovits ('93). Although the paper for the most part is pretty self-contained and ...
119 views

### Computing an initial ellipsoid for solving a convex program

Suppose we want to find a vector $x\in R^n$ that satisfies the constraints $g_i(x)\leq 0$, for $i\in 1,\ldots, m$, where all $g_i$ are convex functions. The functions can be given by an oracle access: ...
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### Quickhull vs T.M. Chan vs Avis and Fukuda

From Discrete lizard's answer and Handbook of Computational Geometry (Third edition, 2018) Section 26, Chazelle's solution of high dimensional convex hull achieved worst case optimal. T.M. Chan's ...
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### From the boundary description to the lattice description of the David Avis & Komei Fukuda's convex hull algorithm

This is a continued question from: The updated convex hull algorithms in 2023? From Handbook of Computational Geometry (Third edition, 2018) section 26.3, Seidel mentioned the boundary and the lattice ...
86 views

### The updated convex hull algorithms in 2023?

I'm studying the convex hull algorithms in the high dimensions. There were two papers by Bernard Chazelle and T.M. Chan from the 90s, to have achieved the at then the state of the art complexity. ...
19 views

### Implementation of algorithm to enumerate all vertices of a convex polyhedron defined as linear inequalities?

I am looking for an implementation for any of the methods to enumerate all vertices of a convex polyhedron defined by $Ax \leq b$ I have found some papers that talk about the problem, for example this ...
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### Collinear convex hull

Is there an algorithm where I provide a list of non-overlapping squares, for example: ...
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### Graham's scan algorithm including all colinear points

I'm solving the Erect the Fence problem on leetcode. My approach is to use Graham's scan algorithm with the following steps: Find the leftmost point p0 Sort the points according to their angle ...
26 views

### Topologizing boundaries in 3D space

I have a set of closed curves (not convex, not planar) in 3D. The goal is to produce A mesh (any will do) that is manifold and contains the closed curves as boundaries/holes. For example like this ...
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### Compute upper convex hull of a function surface

I have a sampled function sruface $$z = f(x, y)$$ represented as a grayscale raster image. How to find the upper convex hull of that surface. That is I am looking for the minimal $$g(x, y)$$ such ...
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### Why isn't there a computational "Carpenter's Algorithm" for Planar Convex Hull?

Planar Convex Hull is a problem where you have $N$ points in a Cartesian plane, and you want to find the smallest convex polygon that encloses the points. QuickHull seems to be the best available ...
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1 vote
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### What is the optimal algorithm for merging an arbitrary number of convex hulls?

Preparata & Shamos in "Convex Hulls: Basic Algorithms" (1985), give a linear algorithm for merging two convex hulls in $O(n+m)$ time, where $n$ and $m$ are the numbers of vertices in ...
173 views

### The optimal complexity of intersecting a line with a convex hull of a set of points in 2d

The problem: in 2d, given a line and an unordered set of $N$ points with real coordinates, find the intersection between the line and the convex hull of the points. Clearly, one can explicitly ...
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1 vote
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### Computing convex hull by triangle point inclusion

In my computational geometry book they present the following $O(n^4)$ algorithm for computing the convex hull of a pointset $P \subset \mathbb{R}^2$ assuming general position on the points in $P$: For ...
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### Shamos algorithm , cannot understand the Area part

I wanted to find shortest time algorithm for finding the diameter of a convex hull, so I found Shamos algorithm on wikipedia: ...
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### How can vector angle comparison between lattice points be done without using floating-points? (Convex Hull)

Let's say I have a point $(x_0, y_0)$, and some other points $(x_1, y_1), (x_2, y_2) ... (x_n, y_n)$, such that all of them are lattice points; all have integer coordinates. Let's further assume that ...
1k views

### Difference between convex hull algorithms

I was wondering what are the main differences in terms of efficiency of convex hull algorithms? Brute force algorithm is inefficient due to iterating over every three vertices in our set and its time ...
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1 vote
2k views

### Upper and lower tangent line to convex hull from a point

Is it possible to find an upper and lower tangent line to a convex hull in $log(n)$ time where $n$ is number of points on a convex hull? I have just done it in linear time where I checked for upper ...
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258 views

### Efficient algorithm to compute the diameter of a convex set?

Is there a polynomial algorithm that can compute the diameter (the distance between the furthest points) of a convex set? It is possible to do it efficiently for a set of points, but imagine that the ...
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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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### 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 ...
156 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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### Why are basic feasible solutions the same as vertices geometrically?

The first line on the Wikipedia page for basic feasible solutions reads, ...
56 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 ...
53 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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### 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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### 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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### 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 ...
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### 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 ...
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561 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 ...
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### 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 ...
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### 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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### 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 $O(n)$?
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