Convex Hull Problem I

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Convex Hull Problem I

• S. Y. Shin
• Computer Science Div.
• KAIST

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Contents

• Problem Definition
• Background
• 2.1 Historical Review
• 2.2 Examples
• Mathematical Preliminaries
• Lower bound in time complexity
• Algorithm

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1. Problem Definition
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1. Problem Definition (cont.)
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2. Background
2.1 Historical Review

• R. Graham, IPL, (1972) O(n logn)
• F. Preparata S. Hong, CACM, (1977)
• Divide and Conquer
• O(n logn)
• Extanded to 3D.

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2.1 Historical Review (cont.)

• J. Sklansky, IEEE Trans. Comput, (1972)
• M. Shamos, (1975)
• O(n) algorithm for finding HULL(P), where P is a
  polygon.
• A. Bykat, IPL, (1978) O(n)
• Counter-examples to Sklanskys algorithm

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2.1 Historical Review (cont.)

• D. McCallum D. Avis, IPL, (1979) O(n)
• using two stacks
• very complicated
• D. Lee, Intl J. of Computer and Information
  Science, (1983)
• O(n)
• Lobe(pocket)
• Forward and Backward Tests
• R. Graham F. Yao, J. of Algorithms, (1983) O(n)

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2.1 Historical Review (cont.)
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2.2 Example
Farthest distance problem
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2.2 Example (cont.)
Observation (Farthest distance problem)
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2.2 Example (cont.)
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2.2 Example (cont.)
Intersection problem

• Given two nonconvex polygons, A and B, find

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2.2 Example (cont.)
Observation (Intersection problem)
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2.2 Example (cont.)
Observation (Intersection problem)
Can you give a counter-example?
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2.2 Example (cont.)
Counter example (Intersection problem)
An algorithm for finding the convex hull of a
polygon can be used as a preprocessing step to
possibly avoid an expensive step for computing
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2.2 Example (cont.)
Optimization
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2.2 Example (cont.)
Optimization
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3. Mathematical Preliminaries
Defn Convex set
A set S is said to be convex if and only if the
line segment joining every pairs of points lies
entirely within S.
Notes How can you algebraically express an
interior point x of a convex polygon?
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3. Mathematical Preliminaries (cont.)
algebraic expression
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3. Mathematical Preliminaries (cont.)
Convex Hull
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3. Mathematical Preliminaries (cont.)
Simple Curve
Defn Simple Curve (Jordan Curve)
A sequence of connected line is said to be a
simple curve if they satisfy the following
conditions (1) No pair of line segments
intersects except at their endpoints (2)
No more than two line segments intersects
at an endpoint.
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3. Mathematical Preliminaries (cont.)
Example (Simple Curve)
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3. Mathematical Preliminaries (cont.)
Polygon
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3. Mathematical Preliminaries (cont.)
Polygon
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3. Mathematical Preliminaries (cont.)
Presenting a Simple Polygon
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3. Mathematical Preliminaries (cont.)
Characterization
minimality
The convex hull of S is the intersection of all
convex sets containing S.
convexity
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3. Mathematical Preliminaries (cont.)
Characterization (cont.)
A sequence of vertices in the order appearing on
the boundary of HULL(S)
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3. Mathematical Preliminaries (cont.)
Characterization (cont.)
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3. Mathematical Preliminaries (cont.)
Characterizing the solution
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3. Mathematical Preliminaries (cont.)
Check of Convexity
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3. Mathematical Preliminaries (cont.)
Check of Convexity (cont.)
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3. Mathematical Preliminaries (cont.)
Cross product
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3. Mathematical Preliminaries (cont.)
Cross product (cont.)
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4. Lower bound in Time Complexity
P1 Given a set S of n points in ,
construct its convex hull.
P2 Given a set S of n points in ,
identify all extreme points in S.
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4. Lower bound in Time Complexity (cont.)
well,
In fact,
(next lecture)
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4. Lower bound in Time Complexity (cont.)
Simple Polygon
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4. Lower bound in Time Complexity (cont.)
Simple Polygon (cont.)
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4. Lower bound in Time Complexity (cont.)
Simple Polygon (cont.)
well,
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4. Lower bound in Time Complexity (cont.)
For Simple Curve Construction
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4. Lower bound in Time Complexity (cont.)
For Simple Curve Construction (cont.)
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4. Lower bound in Time Complexity (cont.)
For Simple Curve Construction (cont.)
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5. Algorithms
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5. Algorithms (cont.)
Unfortunately, such a device is not
available. Constructive but not computationally
feasible
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5. Algorithms (cont.)
algorithm
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5. Algorithms (cont.)
algorithm (cont.)
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5. Algorithms (cont.)
Grahams algorithm
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5. Algorithms (cont.)
Grahams algorithm (cont.)
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5. Algorithms (cont.)
Grahams algorithm (cont.)
Algorithm
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5. Algorithms (cont.)
Grahams algorithm (cont.)
VPTR
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5. Algorithms (cont.)
Grahams algorithm (cont.)
Time complexity

• Choose an internal point Sort and
  Construct a polygonFind convex hull

Optimal
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5. Algorithms (cont.)
Any problems with Grahams algorithm?
Akl and Toussaint, efficient convex hull
algorithm for pattern recognition applications,
Proc. 4th Intl joint Conf. on Pattern
Recognition, Kyoto, Japan, 483-487, (1978).
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5. Algorithms (cont.)
Akl and Toussaints Algorithm
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5. Algorithms (cont.)
Akl and Toussaints Algorithm
Akl and Toussaints algorithm is specialization
of Grahams algorithm!!
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5. Algorithms (cont.)
Jarvis March

• Choose an extreme point How?
• Choose next extreme point in sequences

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5. Algorithms (cont.)
Jarvis March (cont.)
A point with the max. x coordinate is
extreme?Why ? Homework
of extreme pointsIf , then
Jarvis outperforms Graham.