1. August 20, 2002 (joint work with Umut Acar, and Guy Blelloch)
Convex Hull for Dynamic Data
Jorge L. Vittes
August 20, 2002
(joint work with Umut Acar, and Guy Blelloch)

2. Convex Hull for Dynamic Data
Motivation
Application data is dynamic
word processors: slowly changing text
graphics: render similar images
mobile phone networks: continuously moving hosts
Important to handle dynamic data efficiently
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Convex Hull for Dynamic Data

3. Dynamic Algorithms: Changing Datab b c c e d d a b c e d
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Convex Hull for Dynamic Data

4. Kinetic Algorithms: Moving Datab b c c d
Time = 1 d
Time = 0 b a c
Time = 1+e ... 1
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Convex Hull for Dynamic Data

5. How to invent Dynamic/Kinetic Algorithms
Just like any other algorithm. Think, ponder, divide, conquer...
Or, use adaptivity...
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Convex Hull for Dynamic Data

6. Convex Hull for Dynamic Data
Adaptivity
Makes a standard algorithm dynamic or kinetic
Requires little change to the standard algorithm
Can be done semi-automatically
Not all algorithms yield efficient adaptive algorithms
Will talk about this more
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Convex Hull for Dynamic Data

7. How does Adaptivity Work?
Represent a computation with a dynamic dependency graph
nodes $ data, edges $ dependencies
Sources $ input, sinks $ output,
The user can
change the input,
update the output
Update:
Take a changed node,
Update all its children (the children are now changed)
Repeat until no more changed nodes
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Convex Hull for Dynamic Data

8. Convex Hull for Dynamic Data
Adaptivity Example 1 2 3
fun f (a,b,c) =
let
u = a+b in
if (u > 0) then
g(u)
else
g(b+c)
end a b c
a+b
1+b u 3
if u>0 then
g(u)
else
g(b+c)
g(u) r
0.33
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Convex Hull for Dynamic Data

9. Adaptivity Example: Change-2 2 3
fun f (a,b,c) =
let
u = a+b in
if (u > 0) then
g(u)
else
g(b+c)
end a b c
a+b
1+b u 3
if u>0 then
g(u)
else
g(b+c)
g(u) r
0.33
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Convex Hull for Dynamic Data

10. Convex Hull for Dynamic Data
Update -2 2 3
fun f (a,b,c) =
let
u = a+b in
if (u > 0) then
g(u)
else
g(b+c)
end a b c
a+b
-2+b u v 5
if u>0 then
g(u)
else
g(b+c)
g(u)
g(v) r
0.5
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Convex Hull for Dynamic Data

11. Convex Hull for Dynamic Data
Update -2 2 3
fun f (a,b,c) =
let
u = a+b in
if (u > 0) then
g(u)
else
g(b+c)
end a b c
a+b
-2+b
b+c
2+c u v 5
g'(v)
if u>0 then
g(u)
else
g(b+c)
g(v) r
0.5
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Convex Hull for Dynamic Data

12. Adaptivity and Stability
Adaptivity updates the result by rerunning the parts of the computation affected by the input change
It is efficient when the computation is “stable”,I.e., computations on “similar” inputs are “similar”
We apply the adaptivity technique to convex hulls
Result: Efficient Dynamic and Kinetic convex hulls
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Convex Hull for Dynamic Data

13. 1-D Convex Hull: Max and Min
Just consider upper hull: Finding the maximum
Consider two algorithms:
The March: March through the list
The Tournament: pair up the elements and take the max of each pair
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Convex Hull for Dynamic Data

14. Convex Hull for Dynamic Data
Kinetic Maximum
Numbers increase/decrease continuously in time ni(t) = ni + ci t ni
n1(t) = 2+t
n2(t) = 5
n3(t) = 6-t
time
tred = 0
tgreen = 1
tblue = 3
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Convex Hull for Dynamic Data

15. Convex Hull for Dynamic Data
Sampling
n1(t) = 2+t ni
n2(t) = 5
n3(t) = 6-t
time
tred = 0
tgreen = 1
tblue = 3
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Convex Hull for Dynamic Data

16. Internal and External Events
External event: Final result changes
Internal event: Final result does not change
but a test fails
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Convex Hull for Dynamic Data

17. Proof Simulation via Certificates
A set of comparisons that prove the current maximum
Associate a certificate with each comparison
certificate = comparison result + failure time
Consider the times that a certificate fails
We need an algorithm that updates the result as well as the certificates
Use adaptivity to obtain to do the update efficiently
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Convex Hull for Dynamic Data

18. Convex Hull for Dynamic Data
Kinetic March
0.66
0.5 4
Time=0
4+t
5-t 1
7+1.5t 6 3 8 4
0.66
Time=0.5+e
4+t
5-t 1
7+1.5t 6 3 8 4
Time=0.66+e
4+t
5-t 1
7+1.5t 6 3 8 4
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Convex Hull for Dynamic Data

19. Kinetic March Performance
O(n): Because an item in the beginning of the list could become the maximum and it will be compared to the rest of the list
Not acceptable because computing from scratch takes O(n)
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Convex Hull for Dynamic Data

20. The Kinetic Tournament
Time = 0 8 7 8 7
5-t
f = 2 6 8 7 4
1+t
5-t 6 3 8 4
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Convex Hull for Dynamic Data

21. The Kinetic Tournament8
Time=2+e 7
f=6 8 7
1+t 6 8 7 4
1+t
5-t 6 3 8 4
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Convex Hull for Dynamic Data

22. The Kinetic Tournament
Time=6+e 8
f=7
1+t 8 7
1+t 6 8 7 4
1+t
5-t 6 3 8 4
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Convex Hull for Dynamic Data

23. The Kinetic Tournament
Time=7+e
1+t
f=7
1+t 8 7
1+t 6 8 7 4
1+t
5-t 6 3 8 4
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Convex Hull for Dynamic Data

24. Performance of Kinetic Tournament
Worst case log n time per event
This kinetic algorithm is an adaptive version of the standard tournament algorithm for finding maximum
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Convex Hull for Dynamic Data

25. Convex Hull for Dynamic Data
2-D Convex Hull
Many algorithms: Quick Hull, Graham Scan, Incremental, Merge Hull, Ultimate, Improved Ultimate...
We will focus on the Quick Hull algorithm
Input: A list of points P
Output: The boundary points on the hull of P
Example: Input = [a,b,c,d] Output = [a,b,d] a b c d
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Convex Hull for Dynamic Data

26. Convex Hull for Dynamic Data
Quick Hull Example J O B F M H K P D G A L I E C N
[A B C D E F G H I J K L M N O P]
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Convex Hull for Dynamic Data

27. Quick Hull Example - FilterJ O B F M H K P D G A L I E C N
[ A B D F G H J K M O P]
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Convex Hull for Dynamic Data

28. Quick Hull Example - MaximumJ O B F M H K P D G A L I E C N
[A B D F G H J K M O P]
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Convex Hull for Dynamic Data

29. Quick Hull Example - FilterJ O B F M H K P D G A L I E C N
[ [A B F J ] [J O P] ]
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Convex Hull for Dynamic Data

30. Quick Hull Example - MaximumJ O B F M H K P D G A L I E C N
[[A B F J] [J O P] ]
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Convex Hull for Dynamic Data

31. Quick Hull Example - Base CaseJ O B F M H K P D G A L I E C N
[ [A B] [B J] [J O P] ]
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Convex Hull for Dynamic Data

32. Quick Hull Example - DoneJ O B F M H K P D G A L I E C N
[ [A B] [B J] [J O] [O P] ]
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Convex Hull for Dynamic Data

33. Convex Hull for Dynamic Data
Kinetic Quick Hull
Two kinds of tests: Line-side and distance comparisons
Filtering => Line Side
Finding the furthest point => Distance comparisons
Have certificates for these two events that is all
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Convex Hull for Dynamic Data

34. Convex Hull for Dynamic Data
Line Side Test Fails J O B F M H K P D G A L E I C N
[ A B D F G H J K M O P]
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Convex Hull for Dynamic Data

35. “I” is inserted in the middle of the listJ O B F M H K P D G I A L E C N
[A B D F G I H J K M O P]
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Convex Hull for Dynamic Data

36. Convex Hull for Dynamic Data
Recompute Maximum J O B F M H K P D G I A L E C N
[A B D F G I H J K M O P]
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Convex Hull for Dynamic Data

37. Dynamic Tournament - Random Trees8 7 8 7 5 6 8 7 1 6 3 8 4 7 4
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Convex Hull for Dynamic Data

38. Dynamic Tournament - Random Trees8 7 8 7 5 6 8 7 1 6 3 8 4 7 4 6 9
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Convex Hull for Dynamic Data

39. Dynamic Tournament - Random Trees9 7 9 7 5 9 8 7 1 3 8 4 9 7 4 6 9
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Convex Hull for Dynamic Data

40. Distance Comparison Fails - Case 1J O B F M H K P D G A L I E C N
[A B D F G H J K M O P]
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Convex Hull for Dynamic Data

41. Distance Comparison Fails - Case 2J O B F M H K P D G A L I E C N
[A B D F G H J K M O P]
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Convex Hull for Dynamic Data

42. Convex Hull for Dynamic Data
“B” is the new maximum B J O F M H K P D G A L I E C N
[A B D F G H J K M O P]
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Convex Hull for Dynamic Data

43. Convex Hull for Dynamic Data
New recursive calls B J O F M H K P D G A L I E C N
[[A B] [J M O]]
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Convex Hull for Dynamic Data

44. Convex Hull for Dynamic Data
Experiments
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Convex Hull for Dynamic Data

45. Convex Hull for Dynamic Data
Experiments
Events per External Event 25 20
events/external event
5*logn
5*logn^2 15
event/external 10 5 20 40 60 80
100
120
Points
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Convex Hull for Dynamic Data

46. Convex Hull for Dynamic Data
Experiments
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Convex Hull for Dynamic Data

47. Convex Hull for Dynamic Data
Summary of my Work
Kinetic Algorithms for convex hulls using adaptivity
Bounce events: Can maintain convex hull of points in a box - the points bounce off of the walls of the box
Streamlined library for kinetic convex hulls in the SML language
A standard algorithm can be made kinetic in a few hours of work
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Convex Hull for Dynamic Data

48. Convex Hull for Dynamic Data
Work In Progress
Timothy Chan’s O(h log n) algorithm: Improved “Ultimate Convex Hull”: Have a working version
Graham Scan Algorithm: Have a working version
Merge Hull
Incremental Convex Hull
Graham Scan, Merge and Incremental Hull algorithms require an extension to adaptivity
Performance analysis of various convex hull algorithms
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Convex Hull for Dynamic Data

49. Conclusion and Future Work
Used adaptivity to solve the kinetic convex hull problem.
Encouraging results.
Adaptivity makes writing dynamic/kinetic algorithms a simple edition on the standard algorithm
The quickhull algorithm updates based on events efficiently in the expected case.
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Convex Hull for Dynamic Data