1. June 12, 2003 (joint work with Umut Acar, and Guy Blelloch)
Convex Hull for Dynamic Data
Convex Hull and Parallel Tree Contraction
Jorge L. Vittes
June 12, 2003
(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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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44. Convex Hull for Dynamic Data
Experiments
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45. Summary of ConvexHull Work
Kinetic Algorithms for convex hulls using adaptivity
Timothy Chan’s O(h log n) algorithm: Improved “Ultimate Convex Hull”: Have a working version
QuickHull
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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46. Convex Hull for Dynamic Data
Parallel Tree Contraction
Fundamental technique [Miller & Reif ‘85]
Contraction proceeeds in rounds
Each round shrinks the tree by a constant factor
Expected O(logn) rounds
Innovative Idea: Shrink the tree by local operations
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47. Parallel Tree Contraction
Start with a tree
In each round:
Each node flips a coin
If leaf node then rake
If degree=2 and flip = H, and neighbors = T then contract
Expected O(logn) rounds.
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48. Convex Hull for Dynamic Data
Example H H H T T H T H T
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49. Contracting and RakingH H H T T H T H T
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Convex Hull for Dynamic Data

50. Contracting and RakingH T T
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51. Contracting and Raking (cont.)H T T
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52. Convex Hull for Dynamic Data
Done H
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53. Convex Hull for Dynamic Data
Dynamic Trees Problem
Given a forest of weighted trees
Operations
Link: edge insertion
cut: edge deletion
Queries
Heaviest edge in a subtree?
Heaviest edge on a path?
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54. Convex Hull for Dynamic Data
Data Structures for Dynamic Trees
Sleator Tarjan ‘85
Amortized O(logn) and worst-case O(logn)
Topology Trees [Frederickson ‘93]
Ternary (degree-tree) trees
Worst case O(logn)
Top Trees [AlstrupHoLiTh ‘97]
Generalize Topology Trees for arbitrary degree
Idea: Trees as paths
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Convex Hull for Dynamic Data

55. Dynamic Parallel Tree Contraction
Keep a copy of each round of the initial run.
Each round affects next round.
The nodes that “live” to the next round copy their neighbors scars, and pointers to them.
Dependencies are based on what the node reads to do its work.
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56. Dynamic Parallel Tree ContractionH H T T H H H H T T T T H H T T H H T T x x T H T H T T x x x H T T x x
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57. Convex Hull for Dynamic Data
Propagation
If any data changes nodes whose action depend on that data are woken up.
Wake-up only those nodes that get affected by a change.
Run same code as in original run.
Expected constant amount of nodes woken up per round.
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58. Convex Hull for Dynamic Data
Change an edge H H T T H H H H T T T T H H T T H H T T x x T H T H T T x x x H T T x x
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59. Convex Hull for Dynamic Data
Three nodes woken up H H T T H H H H T T T T H H T T H H T T x x T H T H T T x x x H T T x x
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60. Nodes rerun code, more nodes woken upH H T T H H H H T T T T H H T T H H T T x x T H T H T T x x x H T T x x
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61. Propagation continuedH H T T H H H H T T T T H H T T H H T T x x T H T H T T x x x H T T x x
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62. Convex Hull for Dynamic Data
Experimental Results
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63. Convex Hull for Dynamic Data
Work In Progress
Analyzing Power of the data structure (what it can and cannot do)
Different Applications
Analyzing Running times for:
Different changes.
Unbalanced Trees.
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64. Conclusion and Future Work
ConvexHull
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.
Parrallel Tree Contraction
Efficient times O(log(n)) expected time for an update.
Future Work:
More Applications
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