Convex Hull Problem

Presented By Erion Lin

Outline

Convex Hull Problem Voronoi Diagram Fermat Point

Outline

Convex Hull Problem Voronoi DiagramVoronoi Diagram Fermat PointFermat Point

Convex Hull Problem

Definitions Algorithm

Definitions

Concave Polygon

Convex Polygon

A Convex Hull

The convex hull of a set of planar points is defined as the smallest convex polygon containing all of the points.

Algorithm

An algorithm to Construct a Convex Hull

Input ： A set S of planar points Output ： A Convex hull of S

Initial Points

An algorithm to Construct a Convex Hull(Cont’d)

Step 1.

If S contains no more than five points, use exhaustive searching to find the convex hull and return.

Step 2.

Find a median line perpendicular to the x-axis which divides S into SL and SR.

An algorithm to Construct a Convex Hull(Cont’d)

Step 3.

Recursively construct convex hulls for SL and SR. Denote these convex hulls by Hull by Hull(SL) and Hull(SR) respectively.

Convex Hulls after Step 3

An algorithm to Construct a Convex Hull(Cont’d)

Step 4.

Find an interior point P of SL. Find the vertices v1 and v2 of Hull(SR) which divide the vertices of Hull(SR) into two sequences of vertices which have increasing polar angles with respect to P. Without loss of generality, let us assume that v1 has y-value greater than v2. Then form three sequences as follows ：

An algorithm to Construct a Convex Hull(Cont’d)

(a) Sequence 1 ： all of the convex hull vertices

of Hull(SL) in counterclockwise direction.

(b) Sequence 2 ： the convex hull vertices of Hull(SR) from v2 to v1 in counterclockwise direction.

(c) Sequence 3 ： the convex hull vertices of Hull (SR) from v2 to v1 in clockwise direction

Graham Scan

An algorithm to Construct a Convex Hull(Cont’d)

Merge these three sequences and conduct the Graham scan. Eliminate the points which are reflexive and the remaining points from the convex hull.

Graham Scan(Cont’d)

Convex Hull

Algorithm Analysis

T(n)=2T(n/2) + O(n) = O(nlogn)

Outline

Convex Hull ProblemConvex Hull Problem Voronoi Diagram Fermat PointFermat Point

Voronoi Diagram

Definitions Algorithms Applications

Definitions

Definitions

The Voronoi diagram is, as the minimal spanning tree, a very interesting data structure, and it can be used to store important information about nearest neighbor of points on a plane.

A Voronoi Diagram for Two Points

A Voronoi Diagram for Three Points

A Voronoi Diagram for Sex Points

Definitions(Cont’d)

The Voronoi polygon associated with Pi is a convex polygon region having no more than n-1 sides, defined by V(i)=H(Pi, Pj)

A Voronoi Polygon

Definitions(Cont’d)

The Delaunay triangulation is a line segment connecting Pi and Pj if and only if the Voronoi polygons of Pi and Pj share the same edge.

A Delaunay Triangulation for Six Points

Algorithms

An algorithm to Construct Voronoi Diagrams

Input: A Set S on n planar points. Output: The Voronoi Diagram of S

Initial Points

An algorithm to Construct Voronoi Diagrams(Cont’d)

Step 1.

If S contains only one point, return. Step 2.

Find a median line L perpendicular to the x-axis which divides S into SL and SR such that SL(SR) lies to the left(right) of L and the sizes of SL and SR are equal.

An algorithm to Construct Voronoi Diagrams(Cont’d)

Step 3.

Construct Voronoi diagrams of SL and SR recursively. Denote these Voronoi diagrams by VD(SL) and VD(SR).

An algorithm to Construct Voronoi Diagrams(Cont’d)

Step 4.

Construct a dividing piece-wise linear hyperplane HP which is the locus of points simultaneously closest to a point in SL and a point SR.

Discard all segments of VD(SL) which lie to the right of HP and all segments of VD(SR) that lie to the left of HP. The resulting graph is the Voronoi diagram of S.

Voronoi Diagrams Step 4-1

Voronoi Diagrams Step 4-2

Voronoi Diagrams

Algorithm Analysis

T(n)=2T(n/2) + O(n) = O(nlogn)

An algorithm to Merge Two Voronoi Diagrams

Input: (a)SL and SR where SL and SR are divided by a perpendicular line L.

(b)VD(SL) and VD(SR). Output: VD(S) where L RS S S

Initial Voronoi Diagrams

An algorithm to Merge Two Voronoi Diagrams(Cont’d)

Step 1.

Find the convex hulls of SL and SR. Let them be denoted as Hull(SL) and Hull(SR) respectively.

Step 2.

Find segments and which join Hull(SL) and Hull (SR) into a convex hull (Pa and Pc belong to SL and Pb and Pd belong to SR.) Assume that

lies above .Let x=a, y=b,

a bP P c dP P

a bP Pc dP P ,x ySG P P

and HP

An algorithm to Merge Two Voronoi Diagrams(Cont’d)

Step 3.

Find the perpendicular bisector of SG. Denote it by BS. Let . If , Go to Step5; otherwise, Go to Step4.

Step 4.

Let BS first intersect with a ray from VD(SL) or VD(SR). This ray must be a perpendicular bisector of either or for some z. If this ray is the perpendicular bisector of , then let ; otherwise, let .

Go to Step 3.

HP HP BS c dSG P P

x yP P

y zP P

y zP Px zSG P P z ySG P P

Voronoi Diagram after Step 3 、 4

An algorithm to Merge Two Voronoi Diagrams(Cont’d)

Step 5.

Discard the edges of VD(SL) which extend to the right of HP and discard the edges of VD(SR) which extend to the left of HP. The resulting graph is the Voronoi diagram of L RS S S

The Resulting Voronoi Diagram

Applications

Constructing a Convex Hull from a Voronoi Diagram

Time Complexity Analysis

Preprocessing Time ： O(nlogn) Constructing Time ： O(n)

Euclidean Nearest Neighbor Searching Problem

Definition ：We are given a set of n planar points: P1, P2, …Pn and a testing point P. Our problem is to find a nearest neighbor of P among Pi.

Euclidean Nearest Neighbor Searching Problem(Cont’d)

Time Complexity Analysis

Preprocessing Time ： O(nlogn) Searching Time ： O(logn)

Euclidean All Nearest Neighbor Problem

Definition ：The Euclidean all nearest neighbor problem is to find a nearest neighbor of every Pi.

Euclidean All Nearest Neighbor Problem (Cont’d)

and .

This means that

2 2 2

i i i i

i i i i ik j

PN PM PL PN

PP PL PN PM PP

Euclidean All Nearest Neighbor Problem(Cont’d)

Time Complexity Analysis

O(nlogn)

Outline

Convex Hull ProblemConvex Hull Problem Voronoi DiagramVoronoi Diagram Fermat Point

Fermat Point

To find the shortest distance between A 、 B 、 C

Fermat Point (Cont’d)

Prove ：

                                                                                                                                                                              

Fermat Point (Cont’d)

                                        

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