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Guillotine Partition
Guillotine partition is the process of partitioning a rectilinear polygon, possibly containing some holes, into rectangles, using only guillotine-cuts. A guillotine-cut (also called an edge-to-edge cut) is a straight bisecting line going from one edge of an existing polygon to the opposite edge, similarly to a paper guillotine. Guillotine partition is particularly common in designing floorplans in microelectronics. An alternative term for a guillotine-partition in this context is a slicing partition or a slicing floorplan. Guillotine partitions are also the underlying structure of binary space partitions. There are various optimization problems related to guillotine partition, such as: minimizing the number of rectangles or the total length of cuts. These are variants of polygon partitioning problems, where the cuts are constrained to be guillotine cuts. A related but different problem is '' guillotine cutting''. In that problem, the original sheet is a plain rectangle without ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial-time Approximation Scheme
In computer science (particularly algorithmics), a polynomial-time approximation scheme (PTAS) is a type of approximation algorithm for optimization problems (most often, NP-hard optimization problems). A PTAS is an algorithm which takes an instance of an optimization problem and a parameter and produces a solution that is within a factor of being optimal (or for maximization problems). For example, for the Euclidean traveling salesman problem, a PTAS would produce a tour with length at most , with being the length of the shortest tour. Sanjeev Arora, Polynomial-time Approximation Schemes for Euclidean TSP and other Geometric Problems, Journal of the ACM 45(5) 753–782, 1998. The running time of a PTAS is required to be polynomial in the problem size for every fixed ε, but can be different for different ε. Thus an algorithm running in time or even counts as a PTAS. Variants Deterministic A practical problem with PTAS algorithms is that the exponent of the polynomial co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optimization Algorithms And Methods
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing Argument of a function, input values from within an allowed set and computing the Value (mathematics), value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. Optimization problems Optimization problems can be divided into two categ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Face (graph Theory)
This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges. Symbols A B C D E F G H I J K L M N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Graph
In graph theory, a planar graph is a graph (discrete mathematics), graph that can be graph embedding, embedded in the plane (geometry), plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph, or a planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with addit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polychromatic Coloring
Polychrome is the "practice of decorating architectural elements, sculpture, etc., in a variety of colors." The term is used to refer to certain styles of architecture, pottery, or sculpture in multiple colors. When looking at artworks and architecture from antiquity and the European Middle Ages, people tend to believe that they were monochrome. In reality, the pre-Renaissance past was full of colour, and Greco-Roman sculptures and Gothic cathedrals, that are now white, beige, or grey, were initially painted in a variety of colours. As André Malraux stated: "Athens was never white but her statues, bereft of color, have conditioned the artistic sensibilities of Europe ..the whole past has reached us colorless." Polychrome was and is a practice not limited only to the Western world. Non-Western artworks, like Chinese temples, Oceanian Uli figures, or Maya ceramic vases, were also decorated with colours. Ancient Near East Similarly to the ancient art of other regions, Ancient ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Mathematics (journal)
''Discrete Mathematics'' is a biweekly peer-reviewed scientific journal in the broad area of discrete mathematics, combinatorics, graph theory, and their applications. It was established in 1971 and is published by North-Holland Publishing Company. It publishes both short notes, full length contributions, as well as survey articles. In addition, the journal publishes a number of special issues each year dedicated to a particular topic. Although originally it published articles in French and German, it now allows only English language articles. The editor-in-chief is Douglas West ( University of Illinois, Urbana). History The journal was established in 1971. The first article it published was written by Paul Erdős, who went on to publish a total of 84 papers in the journal. Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact facto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schröder Number
In mathematics, the Schröder number S_n, also called a ''large Schröder number'' or ''big Schröder number'', describes the number of lattice paths from the southwest corner (0,0) of an n \times n grid to the northeast corner (n,n), using only single steps north, (0,1); northeast, (1,1); or east, (1,0), that do not rise above the SW–NE diagonal. The first few Schröder numbers are :1, 2, 6, 22, 90, 394, 1806, 8558, ... . where S_0=1 and S_1=2. They were named after the German mathematician Ernst Schröder (mathematician), Ernst Schröder. Examples The following figure shows the 6 such paths through a 2 \times 2 grid: Related constructions A Schröder path of length n is a lattice path from (0,0) to (2n,0) with steps northeast, (1,1); east, (2,0); and southeast, (1,-1), that do not go below the x-axis. The nth Schröder number is the number of Schröder paths of length n. The following figure shows the 6 Schröder paths of length 2. Similarly, the Schröder numbers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Donald Knuth
Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist and mathematician. He is a professor emeritus at Stanford University. He is the 1974 recipient of the ACM Turing Award, informally considered the Nobel Prize of computer science. Knuth has been called the "father of the analysis of algorithms". Knuth is the author of the multi-volume work '' The Art of Computer Programming''. He contributed to the development of the rigorous analysis of the computational complexity of algorithms and systematized formal mathematical techniques for it. In the process, he also popularized the asymptotic notation. In addition to fundamental contributions in several branches of theoretical computer science, Knuth is the creator of the TeX computer typesetting system, the related METAFONT font definition language and rendering system, and the Computer Modern family of typefaces. As a writer and scholar, Knuth created the WEB and CWEB computer programming systems des ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Geometry (journal)
''Computational Geometry'', also known as ''Computational Geometry: Theory and Applications'', is a peer-reviewed mathematics journal for research in theoretical and applied computational geometry, its applications, techniques, and design and analysis of geometric algorithms. All aspects of computational geometry are covered, including the numerical, graph theoretical and combinatorial aspects, as well as fundamental problems in various areas of application of computational geometry: in computer graphics, pattern recognition, image processing, robotics, electronic design automation, CAD/CAM, and geographical information systems. The journal was founded in 1991 by Jörg-Rüdiger Sack and Jorge Urrutia.. It is indexed by ''Mathematical Reviews'', Zentralblatt MATH, Science Citation Index, and Current Contents ''Current Contents'' is a rapid alerting service database from Clarivate, formerly the Institute for Scientific Information and Thomson Reuters. It is published online ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
