The dynamic convex hull problem is a class of dynamic problems in

computational geometry Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems ar ...

. The problem consists in the maintenance, i.e., keeping track, of the

convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

for input data undergoing a sequence of discrete changes, i.e., when input data elements may be inserted, deleted, or modified. It should be distinguished from the

kinetic convex hull A kinetic convex hull data structure is a kinetic data structure that maintains the convex hull of a set of continuously moving points. It should be distinguished from dynamic convex hull data structures, which handle points undergoing discrete cha ...

, which studies similar problems for continuously moving points. Dynamic convex hull problems may be distinguished by the types of the input data and the allowed types of modification of the input data.

Planar point set

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 In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an element ...

for the worst-case

computational complexity In computer science, the computational complexity or simply complexity of an algorithm is the amount of resources required to run it. Particular focus is given to computation time (generally measured by the number of needed elementary operations) ...

of the recomputation of the convex hull is

\Omega(N)

, 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 ordered in some way and logarithmic-time methods for dynamic maintenance of ordered data are well-known. 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(log² ''n'') per update, but it has since been improved by

Timothy M. Chan Timothy Moon-Yew Chan is a Founder ProfessorTwo ...

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 In computational geometry, the point-in-polygon (PIP) problem asks whether a given point in the plane lies inside, outside, or on the boundary of a polygon. It is a special case of point location problems and finds applications in areas that deal ...

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 Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...

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

* * * * * * * * *{{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 The ''Journal of Computer and System Sciences'' (JCSS) is a peer-reviewed scientific journal in the field of computer science. ''JCSS'' is published by Elsevier, and it was started in 1967. Many influential scientific articles have been publishe ...

, pages = 166–204 , title = Maintenance of configurations in the plane , volume = 23 , year = 1981. Convex hull algorithms