Transcript Document

Convex Hull Problem
Presented By Erion Lin
Outline
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Convex Hull Problem
Voronoi Diagram
Fermat Point
Outline
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Convex Hull Problem
Voronoi Diagram
Fermat Point
Convex Hull Problem
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Definitions
Algorithm
Definitions
Concave Polygon
Convex Polygon
A Convex Hull
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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
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Input：A set S of planar points
Output：A Convex hull of S
Initial Points
An algorithm to Construct a Convex Hull(Cont’d)
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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)
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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)
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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)
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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
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T(n)=2T(n/2) + O(n) = O(nlogn)
Outline
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Convex Hull Problem
Voronoi Diagram
Fermat Point
Voronoi Diagram
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Definitions
Algorithms
Applications
Definitions
Definitions
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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)
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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)
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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
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Input: A Set S on n planar points.
Output: The Voronoi Diagram of S
Initial Points
An algorithm to Construct Voronoi Diagrams(Cont’d)
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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)
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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)
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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
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T(n)=2T(n/2) + O(n) = O(nlogn)
An algorithm to Merge Two Voronoi Diagrams
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Input: (a)SL and SR where SL and SR are
divided by a perpendicular line L.
(b)VD(SL) and VD(SR).
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Output: VD(S) where S  SL SR
Initial Voronoi Diagrams
An algorithm to Merge Two Voronoi Diagrams(Cont’d)
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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 Pa Pb and Pc Pd 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,SG
y=b,
 Px Py ,
Pa Pb
Pc Pd
and HP  
An algorithm to Merge Two Voronoi Diagrams(Cont’d)
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Step 3.
Find the perpendicular bisector of SG. Denote it by BS.
Let HP  HP BS. If SG  Pc Pd , 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 Px Py or
Py Pz for some z. If this ray is the perpendicular bisector
of Py Pz , then let SG  Px Pz ; otherwise, let SG  Pz Py.
Go to Step 3.
Voronoi Diagram after Step 3、4
An algorithm to Merge Two Voronoi Diagrams(Cont’d)
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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 S  SL SR
The Resulting Voronoi Diagram
Applications
Constructing a Convex Hull from a Voronoi Diagram
Time Complexity Analysis
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Preprocessing Time：O(nlogn)
Constructing Time：O(n)
Euclidean Nearest Neighbor Searching Problem
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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
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Preprocessing Time：O(nlogn)
Searching Time：O(logn)
Euclidean All Nearest Neighbor Problem
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Definition：
The Euclidean all nearest neighbor problem is to
find a nearest neighbor of every Pi.
Euclidean All Nearest Neighbor Problem (Cont’d)
PN
 PM
and PL
 PN
i
i
i
i .
This means that
PP
 2 PN
 2 PM
 PP
i k  2 PL
i
i
i
i j
Euclidean All Nearest Neighbor Problem(Cont’d)
Time Complexity Analysis
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O(nlogn)
Outline
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Convex Hull Problem
Voronoi Diagram
Fermat Point
Fermat Point
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To find the shortest distance between A、B、C
Fermat Point (Cont’d)
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Prove：
Fermat Point (Cont’d)
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