http://parasol.tamu.edu

Open Problem #12Dynamic Planar Convex Hull

Juan BurgosParasol Lab, Texas A&M University

http://maven.smith.edu/~orourke/TOPP/P12.html#Problem.12

Static Planar Convex Hulls

• Lower bound of planar convex hull algorithms is Ω(n log n).

• Intuitively, this lower bound poses the following questions:

• Is possible to perform updates on a convex hull with at most logarithmic cost with respect to the current number of points?

• Following this trail of thought, is it also possible to perform useful queries of the convex hull with at most logarithmic cost with respect to the current number of points?

Subset of Useful Queries for Convex Hulls

• An extreme-point query asks to find the vertex of the convex hull that is extreme in a given direction.

• A tangent query asks to determine whether a given point is interior to the convex hull, and if not, to find the two tangent lines of the convex hull that passes through the given point.

• A gift-wrapping query asks to find the two vertices of the convex hull adjacent to a given vertex of the convex hull.

• A line-stabbing query asks to find the two edges of the convex hull (if any) that intersect a given line

Question to be Answered

• Can a planar convex hull be maintained to support both dynamic insertions and deletions of points and queries in O(log n) worst-case time time per operation?

• Note: Dynamic insertion and deletion means that new points can be added to our point set P and points can be deleted from our point set P. The convex hull must be able to adapt to the changes in its point set P as these insertion and deletions occur. Dynamic does NOT mean that points are moving in our plane.

Overmars’s and Leeuwen’s work

• Describes a data structure supporting insertions and deletions in O(log2 n) worst-case time and all types of queries described above in O(log n) worst-case time.

• First a favorite static planar convex hull algorithm was chosen to build the initial convex hull with no initial points at a O(no log no) cost.

• After construction, they progressively decomposed the convex hull into many left-convex hulls by dividing each successive hull in half as seen below in figure 5.

• Sadly, no major improvements came till 1999 that lowered the expected running times…

Improvements over the next 2 decades

• During the 1980’s – 1990’s many specialized results were found.

• Logarithmic insertion and deletion times were found for algorithms that only allowed one or the other.

• F. P. Preparata and M. I. Shamos. Computational Geometry: An Introduction. SpringerVerlag, Berlin, 1985.

• J. Hershberger and S. Suri. Applications of a semidynamic convex hull algorithm. BIT,32:249–267, 1992.

• Experiments with randomized insertions and deletions managed that they could achieve expected logarithmic update times.

• K. Mulmuley. Randomized multidimensional search trees: lazy balancing and dynamicshuffling. In Proc. 32nd Ann. Symp. on Foundations of Computer Science (FOCS), pages180–196, 1991.

• O. Schwarzkopf. Dynamic maintenance of geometric structures made easy. In Proc. 32 nd

Ann. Symp. on Foundations of Computer Science (FOCS), pages 197–206, 1991.

Major improvements 2000 – Today

• Discuss approximation methods that were used by other related works. Not in detail of course but in subtle detail… include their running times and assumptions and simplifications.

• Major improvements came in 1999-2000 from Chan and 2000-2002 from Gerth Stølting Brodal and Riko Jacob, the later pair basing their method off of both [Ch99] and [OvL81].

Upper Envelope

What is left in the Open Problem?• What still needs to be done in order to meet the O( log n ) runtime goal.

Year Insertion Time Deletion Time Query TimeM. H. Overmars and J. van

Leeuwen1981

O(log2 n)Worst-Case Time

O(log2 n)Worst-Case Time

O(log n)Worst-Case Time

Timothy M. Chan1999

O(log1+ɛ n)Amortized Time

O(log1+ɛ n)Amortized Time

O(log n)Worst-Case Time

Gerth Stølting Brodal and Riko Jacob

2000

O(log n ・ log log n)Amortized Time

O(log n ・ log log n)Amortized Time

O(log n)Worst-Case Time

Gerth Stølting Brodal and Riko Jacob

2002

O(log n)Amortized Time

O(log n)Amortized Time

O(log n)Wost-Case Time

Timothy M. Chan2011*

O(log1+ɛ n)O(poly log n)

O(nɛ)All Amortized Time

O(log1+ɛ n)O(poly log n)

O(nɛ)All Amortized Time

O(log1+ɛ n)O(log n+k)O(log n)

All Amortized Time

• * There are 3 rows for the 2011 paper since it had results for solving 3 problems related to dynamic convex hulls:

• Finding edges in a convex hull that intersect a query line.

• Supporting half-plane range reporting queries.

• Providing semi-dynamic data structure for maintaining line segments in the plane for determining whether or not a query line lies completely above the lower envelope.

Data Structure from:Gerth Stølting Brodal and Riko Jacob, 2002Dynamic planar convex hull.

Related Problem

• Related to open problem# 63: Dynamic Planar Nearest Neighbors.

• Will be presented later, Nov 8th, by Aditya Mahadevan.

References• [Cha11] Timothy M. Chan: Three problems about dynamic convex hulls. Symposium on Computational

Geometry 2011: 27-36

• [BJ02] Gerth Stølting Brodal and Riko Jacob.  Dynamic planar convex hull. In Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, November 2002.

• [BJ00] Gerth Stølting Brodal and Riko Jacob.  Dynamic planar convex hull with optimal query time and o(log n . loglog n) update time.  In Proc. 7th Scand. Workshop Algorithm

Theory, volume 1851 of Lecture Notes Comput. Sci., pages 57-70. Springer-Verlag, 2000.

• [Cha99] Timothy M. Chan.  Dynamic planar convex hull operations in near-logarithmic amortized time. 

In Proc. 40th Annu. IEEE Sympos. Found. Comput. Sci., pages 92-99, 1999.

• [OvL81] M. H. Overmars and J. van Leeuwen.  Maintenance of configurations in the plane.  J. Comput. Syst. Sci., 23:166-204, 1981.