Optimization convex hull algorithm

  • What is convex hull optimization?

    The convex hull of a set of points in S is the boundary of the smallest convex region that contain all the points of S inside it or on its boundary..

  • What is the algorithm for computing the convex hull?

    Graham's Algorithm
    Graham's scan algorithm is a method of computing the convex hull of a finite set of points in the plane with time complexity O ( n log ⁡ n ) O(n \\log n) O(nlogn)..

  • Chan's algorithm starts by shattering the input points into n/h arbitrary subsets, each of size h, and computing the convex hull of each subset using (say) Graham's scan.
    This much of the algorithm requires O((n/h) \xb7 hlogh) = O(nlogh) time.
    Shattering the points and computing subhulls in O(nlogh) time.
  • The Convex Hull Trick is a technique used to efficiently determine which member of a set of linear functions attains an extremal value for a given value of the independent variable.
    It can be used to optimize dynamic programming problems with certain conditions.
  • The definition of convex hull is as follows: A set Y is said to be convex if for any points a, b ∈ Y, every point on the straight-line segment joining them is also in Y.
    The convex hull of a set of points X in Euclidean space is the smallest convex set containing X.
Feb 17, 2020Convex hull trick (CHT) . Introduction . This post on Codeforces explained how CHT works thorough. I'll focus on when to use CHT here.Convex hull trick (CHT)IntroductionExample problemsCF311B - Cats Transport
It is based on the efficient convex hull algorithm by Selim Akl and G. T. Toussaint, 1978. The idea is to quickly exclude many points that would not be part of  Optimal output-sensitive AlgorithmsOn-line and dynamic convex

Planar case

Consider the general case when the input to the algorithm is a finite unordered set of points on a Cartesian plane. An important special c…

Higher dimensions

A number of algorithms are known for the three-dimensional case, as well as for arbitrary dimensions. Chan's algorithm is used for dimensi…

See also

• Orthogonal convex hull

Further reading

• Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2…

External links

• Weisstein, Eric W. "Convex Hull". MathWorld.

What are convex hull problems?

Convex hull problems arise as one of the most important subproblems in many problems of computational geometry, optimization, etc

Algorithms have been reported for points in two-, three-, and even higher-dimensional Euclidean space

What happens if a convex hull is a finite set of points?

If Sis a finite set of points, then the extreme points Eof the convex hull of Sform the unique convexly independent subset of Swith convex hull equal to the convex hull K(S) of S

Proof

Which hull algorithm is the fastest based on a convex hull?

Its idea is based on the quicksort algorithm and as the quicksort is frequently the fastest among sorting algorithms, the quickhull algorithm tends to be the fastest among the convex hull algorithms for points


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