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 and memory storage requirements. The complexity of a problem is the complexity of the best algorithms that allow solving the problem.
In theoretical computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor.
In 3D computer graphics, solid objects are usually modeled by polyhedra. A face of a polyhedron is a planar polygon bounded by straight line segments, called edges. Curved surfaces are usually approximated by a polygon mesh. Computer programs for line drawings of opaque objects must be able to decide which edges or which parts of the edges are hidden by an object itself or by other objects, so that those edges can be clipped during rendering. This problem is known as hidden-line removal.
In mathematics, given a non-empty set of objects of finite extension in -dimensional space, for example a set of points, a bounding sphere, enclosing sphere or enclosing ball for that set is a -dimensional solid sphere containing all of these objects.
A Euclidean minimum spanning tree of a finite set of points in the Euclidean plane or higher-dimensional Euclidean space connects the points by a system of line segments with the points as endpoints, minimizing the total length of the segments. In it, any two points can reach each other along a path through the line segments. It can be found as the minimum spanning tree of a complete graph with the points as vertices and the Euclidean distances between points as edge weights.
In computer science, a k-d tree is a space-partitioning data structure for organizing points in a k-dimensional space. K-dimensional is that which concerns exactly k orthogonal axes or a space of any number of dimensions. k-d trees are a useful data structure for several applications, such as:
In computational geometry, Chan's algorithm, named after Timothy M. Chan, is an optimal output-sensitive algorithm to compute the convex hull of a set of points, in 2- or 3-dimensional space. The algorithm takes time, where is the number of vertices of the output. In the planar case, the algorithm combines an algorithm with Jarvis march, in order to obtain an optimal time. Chan's algorithm is notable because it is much simpler than the Kirkpatrick–Seidel algorithm, and it naturally extends to 3-dimensional space. This paradigm has been independently developed by Frank Nielsen in his Ph.D. thesis.
The closest pair of points problem or closest pair problem is a problem of computational geometry: given points in metric space, find a pair of points with the smallest distance between them. The closest pair problem for points in the Euclidean plane was among the first geometric problems that were treated at the origins of the systematic study of the computational complexity of geometric algorithms.
In computer science, the range searching problem consists of processing a set S of objects, in order to determine which objects from S intersect with a query object, called the range. For example, if S is a set of points corresponding to the coordinates of several cities, find the subset of cities within a given range of latitudes and longitudes.
In computer science, the maximum sum subarray problem, also known as the maximum segment sum problem, is the task of finding a contiguous subarray with the largest sum, within a given one-dimensional array A[1...n] of numbers. It can be solved in time and space.
Algorithms that construct convex hulls of various objects have a broad range of applications in mathematics and computer science.
In computer science, the segment tree is a data structure used for storing information about intervals or segments. It allows querying which of the stored segments contain a given point. A similar data structure is the interval tree.
In computer science, a range tree is an ordered tree data structure to hold a list of points. It allows all points within a given range to be reported efficiently, and is typically used in two or higher dimensions. Range trees were introduced by Jon Louis Bentley in 1979. Similar data structures were discovered independently by Lueker, Lee and Wong, and Willard. The range tree is an alternative to the k-d tree. Compared to k-d trees, range trees offer faster query times of but worse storage of , where n is the number of points stored in the tree, d is the dimension of each point and k is the number of points reported by a given query.
In computational geometry, the Bentley–Ottmann algorithm is a sweep line algorithm for listing all crossings in a set of line segments, i.e. it finds the intersection points of line segments. It extends the Shamos–Hoey algorithm, a similar previous algorithm for testing whether or not a set of line segments has any crossings. For an input consisting of line segments with crossings, the Bentley–Ottmann algorithm takes time . In cases where , this is an improvement on a naïve algorithm that tests every pair of segments, which takes .
Kenneth Lee Clarkson is an American computer scientist known for his research in computational geometry. He is a researcher at the IBM Almaden Research Center, and co-editor-in-chief of the Journal of Computational Geometry.
In the analysis of algorithms, several authors have studied the computation of the volume of high-dimensional convex bodies, a problem that can also be used to model many other problems in combinatorial enumeration. Often these works use a black box model of computation in which the input is given by a subroutine for testing whether a point is inside or outside of the convex body, rather than by an explicit listing of the vertices or faces of a convex polytope. It is known that, in this model, no deterministic algorithm can achieve an accurate approximation, and even for an explicit listing of faces or vertices the problem is #P-hard. However, a joint work by Martin Dyer, Alan M. Frieze and Ravindran Kannan provided a randomized polynomial time approximation scheme for the problem, providing a sharp contrast between the capabilities of randomized and deterministic algorithms.
In computational geometry, the largest empty rectangle problem,maximal empty rectangle problem or maximum empty rectangle problem, is the problem of finding a rectangle of maximal size to be placed among obstacles in the plane. There are a number of variants of the problem, depending on the particularities of this generic formulation, in particular, depending on the measure of the "size", domain, and the orientation of the rectangle.
David Mount is a professor at the University of Maryland, College Park department of computer science whose research is in computational geometry.
In geometry, a partition of a polygon is a set of primitive units, which do not overlap and whose union equals the polygon. A polygon partition problem is a problem of finding a partition which is minimal in some sense, for example a partition with a smallest number of units or with units of smallest total side-length.
In the design and analysis of algorithms for combinatorial optimization, parametric search is a technique invented by Nimrod Megiddo (1983) for transforming a decision algorithm into an optimization algorithm. It is frequently used for solving optimization problems in computational geometry.
