Convex Hull Algorithms for Dynamic Data

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Convex Hull Algorithmsfor Dynamic Data

• Kanat Tangwongsan
• Joint work with
• Guy Blelloch and Umut Acar (TTI-C)

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The Convex Hull Problem
Output CH(S)
Input S

• Extensively studied
• Graham Scan, Gift-Wrapping, Incremental Hull,
  Overmars, Quick-Hull, Ultimate Hull.
• Matched Lower-bound already!

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Convex Hull Dynamic Case

• Maintain the Hull under insertion and deletion.

insert
delete
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Convex Hull Kinetic Case

• each point has velocity (v)
• maintain the hull (efficiently)

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Known Solutions and Issues

• Dynamic Convex Hull
• Overmars, Brodal, Mulmuley, Schwarzkopf,
• Preparata, Hershberger and Suri, Chan
• complicated!!
• hard to compose algorithms together
• Kinetic Convex Hull
• Guibas, de Berg,
• None existed for 3 or higher dimensions

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Self-Adjusting Computation
Idea Record who reads what and when
main
main
g
f
g
f
h
h
k
m
k
m
Memory
Some parts of the input change
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SAC vs. Dynamic Convex Hull

• Automatic dynamization
• Allow composition of programs
• Down-side not always efficient!
• Stability (input-sensitivity) Acar et al.
  SODA04
• Our Job
• Design a stable static convex hull algorithm
• Focus 3-D case.

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Our Approach
Self-Adjusting Tree
Suitable Convex Hull in 3D
Dynamic Convex Hull in 3D

j
Face
Give orientation to edges
i
Store (i,j) as key and k as data
k
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Self-Adjusting Binary Tree
Initial Run
Time
Re-execution
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Kinetic Convex Hull

• 1-D Case
• S(t) x1(t), x2(t), , xn(t), where xi(t)
  xi(0) vit
• Maintain min and max
• Observation

x1
Value
x2
x3
Time
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Idea Certificates

• Each comparison is associated with a certificate,
  where
• Self-Adjusting Computation
• account for changes
• in order of expiration times

certificate comparsion result expiration time
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Summary

• Focus 3D dynamic, kinetic convex hull
• Dynamic Convex Hull
• Amortized expected O(log n) in certain models.
• Worst-case bound O(n log n)
• More performance analysis to come
• Real-time bound can it be better than O(n)?
• Practical performance implemented, being
  evaluated.
• Kinetic Convex Hull
• in progress!

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Questions?
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Thank you!!