The dynamic convex hull problem is a class of dynamic problems in computational geometry. Dynamic problems in Computational complexity theory are problems stated in terms of the changing input data Computational geometry is a branch of Computer science devoted to the study of algorithms which can be stated in terms of Geometry. The problem consists in the maintenance, i. e. , keeping track, of the convex hull for the dynamically changing input data, i. In Mathematics, the convex hull or convex envelope for a set of points X in a Real Vector space V is the minimal Convex 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.
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 for the worst-case computational complexity of the recomputation of the convex hull is Ω(N), since this time is required for a mere reporting of the output. In Mathematics, especially in Order theory, an upper bound of a Subset S of some Partially ordered set ( P, &le Computational complexity theory, as a branch of the Theory of computation in Computer science, investigates the problems related to the amounts of resources 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. Sorting is any process of arranging items in some sequence and/or in different sets and accordingly it has two common yet distinct meanings ordering: arranging
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 and others. Timothy M Chan (born 1976, in Michigan is Professor in Computer Science at the University of Waterloo, where he currently holds a University
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. 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 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
- Overmars, M. H.; van Leeuwen, J. Markus Hendrik Overmars (born 29 September 1958 in Zeist, Netherlands) is a Dutch computer scientist and teacher of games programming (1981). "Maintenance of configurations in the plane". J. Comput. Sys. Sci. 23 (2): 166–204. doi:. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
- Jacob Rico, Dynamic Planar Convex Hull (2002), a BRICS dissertation
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