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The '''dynamic convex hull problem''' is a class of [[dynamic problem (algorithms)|dynamic problem]]s in [[computational geometry]]. The problem consists in the maintenance, i.e., keeping track, of the [[convex hull]] for the dynamically changing input data, i.e., when input data elements may be inserted, deleted, or modified. Problems of this class may be distinguished by the types of the input data and the allowed types of modification of the input data.
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==Planar point set==
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It is easy to construct an example for which the convex hull contains all input points, but after the insertion of a single point the convex hull becomes a triangle. And conversely, the deletion of a single point may produce the opposite drastic change of the size of the output. Therefore if the convex hull is required to be reported in traditional way as a polygon, the [[lower bound]] for the worst-case [[computational complexity]] of the recomputation of the convex hull is <math>\Omega(N)</math>, since this time is required for a mere reporting of the output. This lower bound is attainable, because several general-purpose convex hull algorithms run in linear time when input points are [[sorting|ordered]] in some way and logarithmic-time methods for dynamic maintenance of ordered data are well-known.
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This problem may be overcome by eliminating the restriction on the output representation. There are data structures that can maintain representations of the convex hull in an amount of time per update that is much smaller than linear. For many years the best algorithm of this type was that of Overmars and van Leeuwen (1981), which took time O(log<sup>2</sup> ''n'') per update, but it has since been improved by [[Timothy M. Chan]] and others.
 
In a number of applications finding the convex hull is a step in an algorithm for the solution of the overall problem. The selected representation of the convex hull may influence on the computational complexity of further operations of the overall algorithm. For example, the [[point in polygon]] query for a convex polygon represented by the ordered set of its vertices may be answered in logarithmic time, which would be impossible for convex hulls reported by the set of it vertices without any additional information. Therefore some research of dynamic convex hull algorithms involves the  computational complexity of various [[geometric search problems]] with convex hulls stored in specific kinds of data structures. The mentioned approach of Overmars and van Leeuwen allows for logarithmic complexity of various common queries.
 
==References==
{{reflist}}
*{{citation
| last1 = Overmars | first1 = M. H. | author1-link = Mark Overmars
| last2 = van Leeuwen | first2 = J. | author2-link = Jan van Leeuwen
| doi = 10.1016/0022-0000(81)90012-X
| issue = 2
| journal = [[Journal of Computer and System Sciences]]
| pages = 166–204
| title = Maintenance of configurations in the plane
| volume = 23
| year = 1981}}.
*Jacob Riko, ''[http://www.brics.dk/DS/02/3/BRICS-DS-02-3.pdf Dynamic Planar Convex Hull]'' (2002), a [http://www.brics.dk/ BRICS] dissertation
 
[[Category:Convex hull algorithms]]

Latest revision as of 04:10, 4 March 2014

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