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Graph Canonization
In graph theory, a branch of mathematics, graph canonization is the problem of finding a canonical form of a given graph ''G''. A canonical form is a labeled graph Canon(''G'') that is isomorphic to ''G'', such that every graph that is isomorphic to ''G'' has the same canonical form as ''G''. Thus, from a solution to the graph canonization problem, one could also solve the problem of graph isomorphism: to test whether two graphs ''G'' and ''H'' are isomorphic, compute their canonical forms Canon(''G'') and Canon(''H''), and test whether these two canonical forms are identical. The canonical form of a graph is an example of a complete graph invariant: every two isomorphic graphs have the same canonical form, and every two non-isomorphic graphs have different canonical forms... Conversely, every complete invariant of graphs may be used to construct a canonical form. The vertex set of an ''n''-vertex graph may be identified with the integers from 1 to ''n'', and using such an identifica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph theory), vertices'' (also called ''nodes'' or ''points'') which are connected by ''Glossary of graph theory terms#edge, edges'' (also called ''arcs'', ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a Set (mathematics), set of vertices (also called nodes or points); * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-polynomial Time
In computational complexity theory and the analysis of algorithms, an algorithm is said to take quasi-polynomial time if its time complexity is quasi-polynomially bounded. That is, there should exist a constant c such that the worst-case running time of the algorithm, on inputs of has an upper bound of the form 2^. The decision problems with quasi-polynomial time algorithms are natural candidates for being NP-intermediate, neither having polynomial time nor likely to be NP-hard. Complexity class The complexity class QP consists of all problems that have quasi-polynomial time algorithms. It can be defined in terms of DTIME as follows. :\mathsf = \bigcup_ \mathsf \left(2^\right) Examples An early example of a quasi-polynomial time algorithm was the Adleman–Pomerance–Rumely primality test. However, the problem of testing whether a number is a prime number has subsequently been shown to have a polynomial time algorithm, the AKS primality test. In some cases, quasi-polynomi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SMILES
A smile is a facial expression formed primarily by flexing the muscles at the sides of the mouth. Some smiles include a contraction of the muscles at the corner of the eyes, an action known as a Duchenne smile. Among humans, a smile expresses delight, sociability, happiness, joy, or amusement. It is distinct from a similar but usually involuntary expression of anxiety known as a grimace. Although cross-cultural studies have shown that smiling is a means of communication throughout the world, there are large differences among different cultures, religions, and societies, with some using smiles to convey confusion, embarrassment, or awkwardness. Evolutionary background Primatologist Signe Preuschoft traces the smile back over 30 million years of evolution to a "fear grin" stemming from monkeys and apes, who often used barely clenched teeth to portray to predators that they were harmless or to signal submission to more dominant group members. The smile may have evolved differen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chemical Substance
A chemical substance is a unique form of matter with constant chemical composition and characteristic properties. Chemical substances may take the form of a single element or chemical compounds. If two or more chemical substances can be combined without reacting, they may form a chemical mixture. If a mixture is separated to isolate one chemical substance to a desired degree, the resulting substance is said to be chemically pure. Chemical substances can exist in several different physical states or phases (e.g. solids, liquids, gases, or plasma) without changing their chemical composition. Substances transition between these phases of matter in response to changes in temperature or pressure. Some chemical substances can be combined or converted into new substances by means of chemical reactions. Chemicals that do not possess this ability are said to be inert. Pure water is an example of a chemical substance, with a constant composition of two hydrogen atoms bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identifier
An identifier is a name that identifies (that is, labels the identity of) either a unique object or a unique ''class'' of objects, where the "object" or class may be an idea, person, physical countable object (or class thereof), or physical mass noun, noncountable substance (or class thereof). The abbreviation ID often refers to identity, identification (the process of identifying), or an identifier (that is, an instance of identification). An identifier may be a word, number, letter, symbol, or any combination of those. The words, numbers, letters, or symbols may follow an code, encoding system (wherein letters, digits, words, or symbols ''stand for'' [represent] ideas or longer names) or they may simply be arbitrary. When an identifier follows an encoding system, it is often referred to as a code or id code. For instance the ISO/IEC 11179 metadata registry standard defines a code as ''system of valid symbols that substitute for longer values'' in contrast to identifiers wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chemical Database
A chemical database is a database specifically designed to store chemical information. This information is about chemical and crystal structures, spectra, reactions and syntheses, and thermophysical data. Types of chemical databases Bioactivity database Bioactivity databases correlate structures or other chemical information to bioactivity results taken from bioassays in literature, patents, and screening programs. Chemical structures Chemical structures are traditionally represented using lines indicating chemical bonds between atoms and drawn on paper (2D structural formulae). While these are ideal visual representations for the chemist, they are unsuitable for computational use and especially for search and storage. Small molecules (also called ligands in drug design applications), are usually represented using lists of atoms and their connections. Large molecules such as proteins are however more compactly represented using the sequences of their amino acid bui ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Data Mining
Data mining is the process of extracting and finding patterns in massive data sets involving methods at the intersection of machine learning, statistics, and database systems. Data mining is an interdisciplinary subfield of computer science and statistics with an overall goal of extracting information (with intelligent methods) from a data set and transforming the information into a comprehensible structure for further use. Data mining is the analysis step of the " knowledge discovery in databases" process, or KDD. Aside from the raw analysis step, it also involves database and data management aspects, data pre-processing, model and inference considerations, interestingness metrics, complexity considerations, post-processing of discovered structures, visualization, and online updating. The term "data mining" is a misnomer because the goal is the extraction of patterns and knowledge from large amounts of data, not the extraction (''mining'') of data itself. It also is a buzzwo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP-hard
In computational complexity theory, a computational problem ''H'' is called NP-hard if, for every problem ''L'' which can be solved in non-deterministic polynomial-time, there is a polynomial-time reduction from ''L'' to ''H''. That is, assuming a solution for ''H'' takes 1 unit time, ''H''s solution can be used to solve ''L'' in polynomial time. As a consequence, finding a polynomial time algorithm to solve a single NP-hard problem would give polynomial time algorithms for all the problems in the complexity class NP. As it is suspected, but unproven, that P≠NP, it is unlikely that any polynomial-time algorithms for NP-hard problems exist. A simple example of an NP-hard problem is the subset sum problem. Informally, if ''H'' is NP-hard, then it is at least as difficult to solve as the problems in NP. However, the opposite direction is not true: some problems are undecidable, and therefore even more difficult to solve than all problems in NP, but they are probably not NP- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjacency Matrix
In graph theory and computer science, an adjacency matrix is a square matrix used to represent a finite graph (discrete mathematics), graph. The elements of the matrix (mathematics), matrix indicate whether pairs of Vertex (graph theory), vertices are Neighbourhood (graph theory), adjacent or not in the graph. In the special case of a finite simple graph, the adjacency matrix is a (0,1)-matrix with zeros on its diagonal. If the graph is Glossary of graph theory terms#undirected, undirected (i.e. all of its Glossary of graph theory terms#edge, edges are bidirectional), the adjacency matrix is symmetric matrix, symmetric. The relationship between a graph and the eigenvalues and eigenvectors of its adjacency matrix is studied in spectral graph theory. The adjacency matrix of a graph should be distinguished from its incidence matrix, a different matrix representation whose elements indicate whether vertex–edge pairs are Incidence (graph), incident or not, and its degree matrix, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism Class
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is derived . The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may often be identified. In mathematical jargon, one says that two objects are the same up to an isomorphism. A common example where isomorphic structures cannot be identified is when the structures are substructures of a larger one. For example, all subspaces of dimension one of a vector space are isomorphic and cannot be identified. An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a cano ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lexicographical Order
In mathematics, the lexicographic or lexicographical order (also known as lexical order, or dictionary order) is a generalization of the alphabetical order of the dictionaries to sequences of ordered symbols or, more generally, of elements of a totally ordered set. There are several variants and generalizations of the lexicographical ordering. One variant applies to sequences of different lengths by comparing the lengths of the sequences before considering their elements. Another variant, widely used in combinatorics, orders subsets of a given finite set by assigning a total order to the finite set, and converting subsets into increasing sequences, to which the lexicographical order is applied. A generalization defines an order on an ''n''-ary Cartesian product of partially ordered sets; this order is a total order if and only if all factors of the Cartesian product are totally ordered. Definition The words in a lexicon (the set of words used in some language) have a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probabilistic Complexity Theory
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random determined by the random bits; thus either the running time, or the output (or both) are random variables. There is a distinction between algorithms that use the random input so that they always terminate with the correct answer, but where the expected running time is finite (Las Vegas algorithms, for example Quicksort), and algorithms which have a chance of producing an incorrect result (Monte Carlo algorithms, for example the Monte Carlo algorithm for the MFAS problem) or fail to produce a result either by signaling a failure or failing to terminate. In some cases, probabilistic algorithms are the only practical means of solving a problem. In common practice, randomized algor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
