Convex Hull Algorithms for Dynamic Data Kanat Tangwongsan Joint work with Guy Blelloch and Umut Acar (TTI-C)

The Convex Hull Problem Extensively studied: Graham Scan, Gift-Wrapping, Incremental Hull, Overmars, Quick-Hull, Ultimate Hull. Matched Lower-bound already! Input: S Output: CH(S)

Convex Hull ++ : Dynamic Case Maintain the Hull under insertion and deletion. insert delete

Convex Hull # : Kinetic Case each point has velocity ( v ) maintain the hull (efficiently)

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

Self-Adjusting Computation main f g h k m f g h k m Memory Some parts of the input change Idea: Record who reads what and when

SAC vs. Dynamic Convex Hull Automatic “dynamization” Allow composition of programs Down-side: not always efficient! Stability (=input-sensitivity) [Acar et al. SODA’04] Our Job: Design a stable static convex hull algorithm Focus: 3-D case.

Our Approach Self-Adjusting Tree Suitable Convex Hull in 3D + Dynamic Convex Hull in 3D k i j Face Store (i,j) as key and k as data Give orientation to edges

Self-Adjusting Binary Tree Time Initial Run: Re-execution:

Kinetic Convex Hull 1-D Case S(t) = {x 1 (t), x 2 (t), …, x n (t)}, where x i (t) = x i (0) + v i t Maintain min and max Observation: Value Time x1x1 x2x2 x3x3

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

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!

Questions?

Thank you!!