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Pebble Game
In mathematics and computer science, a pebble game is a type of mathematical game played by placing "pebbles" or "markers" on a directed graph, directed acyclic graph according to certain rules: * A given step of the game consists of either placing a pebble on an empty vertex or removing a pebble from a previously pebbled vertex. * A vertex may be pebbled only if all its predecessors have pebbles. * The objective of the game is to successively pebble each vertex of ''G'' (in any order) while minimizing the number of pebbles that are ever on the graph simultaneously. Running time The trivial solution is to pebble an ''n''-vertex graph in ''n'' steps using ''n'' pebbles. Hopcroft, Paul and Valiant showed that any vertex of an ''n''-vertex graph can be pebbled with O(''n''/log ''n'') pebbles where the constant depends on the maximum in-degree. This enabled them to prove that DTIME(''f''(''n'')) is contained in DSPACE(''f''(''n'')/log ''f''(''n'')) for all time-constructible fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jakob Nordström
{{disambiguation ...
Jakob may refer to: People * Jakob (given name), including a list of people with the name * Jakob (surname), including a list of people with the name Other * Jakob (band), a New Zealand band, and the title of their 1999 EP * Max Jakob Memorial Award, annual award to scholars in the field of heat transfer * Ohel Jakob synagogue (Munich) See also * Jacob (other) * St. Jacob (other) St. Jacob is James, son of Zebedee, or Saint James the Great. James is used as a translation of the Hebrew name Jacob (Ya'akov). St. Jacob, St. Jacobs or St. Jakob may also refer to: People *Saint James (other) * Saint Jacob of Alaska, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
IBM Watson Research Center
The Thomas J. Watson Research Center is the headquarters for IBM Research. The center comprises three sites, with its main laboratory in Yorktown Heights, New York, Yorktown Heights, New York (state), New York, U.S., 38 miles (61 km) north of New York City, Albany, New York, Albany, New York (state), New York and with offices in Cambridge, Massachusetts, Cambridge, Massachusetts. Overview The center, headquarters of IBM's IBM Research, Research division, is named for both Thomas J. Watson, Thomas J. Watson, Sr. and Thomas Watson, Jr., who led IBM as president and CEO, respectively, from 1915 (when it was known as the Computing-Tabulating-Recording Company) to 1971. The research is intended to improve hardware (Physical Sciences, physical sciences and semiconductors research), services (business modelling, consulting, and operations research), software (programming languages, security, speech recognition, data management, and collaboration tools), and systems (operating ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicholas Pippenger
Nicholas John Pippenger is a researcher in computer science. He has produced a number of fundamental results many of which are being widely used in the field of theoretical computer science, database processing and compiler optimization. He has also achieved the rank of IBM Fellow at Almaden IBM Research Center in San Jose, California. He has taught at the University of British Columbia in Vancouver, British Columbia, Canada and at Princeton University in the US. In the Fall of 2006 Pippenger joined the faculty of Harvey Mudd College. Pippenger holds a B.S. in Natural Sciences from Shimer College and a PhD from the Massachusetts Institute of Technology. He is married to Maria Klawe, President of Harvey Mudd College. In 1997 he was inducted as a Fellow of the Association for Computing Machinery. In 2013 he became a fellow of the American Mathematical Society. The complexity class, Nick's Class (NC), of problems quickly solvable on a parallel computer, was named by Stephen Cook ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Separator Theorem
In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split into smaller pieces by removing a small number of vertices. Specifically, the removal of vertices from an -vertex graph (where the invokes big O notation) can partition the graph into disjoint subgraphs each of which has at most vertices. A weaker form of the separator theorem with vertices in the separator instead of was originally proven by , and the form with the tight asymptotic bound on the separator size was first proven by . Since their work, the separator theorem has been reproven in several different ways, the constant in the term of the theorem has been improved, and it has been extended to certain classes of nonplanar graphs. Repeated application of the separator theorem produces a separator hierarchy which may take the form of either a tree decomposition or a branch-decomposition of the graph. Separator hierarc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chip-firing Game
The chip-firing game is a one-player game on a graph which was invented around 1983 and since has become an important part of the study of structural combinatorics. Each vertex has the number of "chips" indicated by its state variable. On each firing, a vertex is selected and one of its chips is transferred to each neighbour (vertex it shares an edge with). The number of chips on each vertex cannot be negative. The game ends when no firing is possible. Definition Let the finite graph ''G'' be connected and loopless, with vertices ''V'' = . Let deg(''v'') be the degree of a vertex, and e(''v,w'') the number of edges between vertices ''v'' and ''w''. A configuration or state of the game is defined by assigning each vertex a nonnegative integer ''s''(''v''), representing the number of chips on this vertex. A move starts with selecting a vertex ''w'' which has at least as many chips as its degree: ''s''(''w'') ≥ deg(''w''). The vertex ''w'' is fired, moving one chip from w al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Pebbling
Graph pebbling is a mathematical game played on a graph with zero or more pebbles on each of its vertices. 'Game play' is composed of a series of pebbling moves. A pebbling move on a graph consists of choosing a vertex with at least two pebbles, removing two pebbles from it, and adding one to an adjacent vertex (the second removed pebble is discarded from play). π(''G''), the pebbling number of a graph ''G'', is the lowest natural number ''n'' that satisfies the following condition: Given any target or 'root' vertex in the graph and any initial configuration of ''n'' pebbles on the graph, it is possible, after a series of pebbling moves, to reach a new configuration in which the designated root vertex has one or more pebbles. For example, on a graph with 2 vertices and 1 edge connecting them the pebbling number is 2. No matter how the two pebbles are placed on the vertices of the graph it is always possible to move a pebble to any vertex in the graph. One of the central quest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ehrenfeucht–Fraïssé Game
In the mathematical discipline of model theory, the Ehrenfeucht–Fraïssé game (also called back-and-forth games) is a technique based on game semantics for determining whether two structures are elementarily equivalent. The main application of Ehrenfeucht–Fraïssé games is in proving the inexpressibility of certain properties in first-order logic. Indeed, Ehrenfeucht–Fraïssé games provide a complete methodology for proving inexpressibility results for first-order logic. In this role, these games are of particular importance in finite model theory and its applications in computer science (specifically computer aided verification and database theory), since Ehrenfeucht–Fraïssé games are one of the few techniques from model theory that remain valid in the context of finite models. Other widely used techniques for proving inexpressibility results, such as the compactness theorem, do not work in finite models. Ehrenfeucht–Fraïssé-like games can also be defined for othe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIAM Journal On Computing
The ''SIAM Journal on Computing'' is a scientific journal focusing on the mathematical and formal aspects of computer science. It is published by the Society for Industrial and Applied Mathematics (SIAM). Although its official ISO abbreviation is ''SIAM J. Comput.'', its publisher and contributors frequently use the shorter abbreviation ''SICOMP''. SICOMP typically hosts the special issues of the IEEE Annual Symposium on Foundations of Computer Science (FOCS) and the Annual ACM Symposium on Theory of Computing (STOC), where about 15% of papers published in FOCS and STOC each year are invited to these special issues. For example, Volume 48 contains 11 out of 85 papers published in FOCS 2016. References * External linksSIAM Journal on Computing on |
Computational Complexity Theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying their computational complexity, i.e., the amount of resources needed to solve them, such as time and storage. Other measures of complexity are also used, such as the amount of communication (used in communication complexity), the number of gates in a circuit (used in circuit complexity) and the number of processors (used in parallel computing). One of the roles of computationa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
