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Pancake Graph
In the mathematical field of graph theory, the pancake graph ''P''''n'' or ''n''-pancake graph is a graph whose vertices are the permutations of ''n'' symbols from 1 to ''n'' and its edges are given between permutations transitive by prefix reversals. Pancake sorting is the colloquial term for the mathematical problem of sorting a disordered stack of pancakes in order of size when a spatula can be inserted at any point in the stack and used to flip all pancakes above it. A ''pancake number'' is the minimum number of flips required for a given number of pancakes. Obtaining the pancake number is equivalent to the problem of obtaining the diameter of the pancake graph. The pancake graph of dimension ''n'', ''P''''n'', is a regular graph with n! vertices. Its degree is ''n'' − 1, hence, according to the handshaking lemma, it has \dfrac ~ n! \left(n-1\right) edges. ''P''''n'' can be constructed recursively from n copies of ''P''''n''−1, by assigning a diffe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pancake Graph G4
A pancake (or hotcake, griddlecake, or flapjack) is a flat cake, often thin and round, prepared from a starch-based batter that may contain eggs, milk and butter and cooked on a hot surface such as a griddle or frying pan, often frying with oil or butter. It is a type of batter bread. Archaeological evidence suggests that pancakes were probably eaten in prehistoric societies. The pancake's shape and structure varies worldwide. In the United Kingdom, pancakes are often unleavened and resemble a crêpe. In North America, a leavening agent is used (typically baking powder) creating a thick fluffy pancake. A ''crêpe'' is a thin Breton pancake of French origin cooked on one or both sides in a special pan or crepe maker to achieve a lacelike network of fine bubbles. A well-known variation originating from southeast Europe is a ''palačinke'', a thin moist pancake fried on both sides and filled with jam, cream cheese, chocolate, or ground walnuts, but many other fillings—sweet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Coloring
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as-is. This is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dense Graph
In mathematics, a dense graph is a graph in which the number of edges is close to the maximal number of edges (where every pair of vertices is connected by one edge). The opposite, a graph with only a few edges, is a sparse graph. The distinction of what constitutes a dense or sparse graph is ill-defined, and depends on context. The graph density of simple graphs is defined to be the ratio of the number of edges with respect to the maximum possible edges. For undirected simple graphs, the graph density is: :D = \frac = \frac For directed, simple graphs, the maximum possible edges is twice that of undirected graphs (as there are two directions to an edge) so the density is: :D = \frac = \frac where is the number of edges and is the number of vertices in the graph. The maximum number of edges for an undirected graph is = \frac2, so the maximal density is 1 (for complete graphs) and the minimal density is 0 . Upper density ''Upper density'' is an extension of the concept of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manuel Blum
Manuel Blum (born 26 April 1938) is a Venezuelan-American computer scientist who received the Turing Award in 1995 "In recognition of his contributions to the foundations of computational complexity theory and its application to cryptography and program checking". Education Blum was born to a Jewish family in Venezuela. Blum was educated at MIT, where he received his bachelor's degree and his master's degree in electrical engineering in 1959 and 1961 respectively, and his Ph.D. in mathematics in 1964 supervised by Marvin Minsky.. Career Blum worked as a professor of computer science at the University of California, Berkeley until 2001. From 2001 to 2018, he was the Bruce Nelson Professor of Computer Science at Carnegie Mellon University, where his wife, Lenore Blum, was also a professor of Computer Science. In 2002, he was elected to the United States National Academy of Sciences. In 2006, he was elected a member of the National Academy of Engineering for contributions to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David X
David X ( ka, დავით X) (1482–1526) was a king of the Georgian kingdom of Kartli from 1505 to 1525. Life David was the eldest son of Constantine II, whom he succeeded as king of Kartli in 1505. Although Constantine had recognised the independence of the breakaway Georgian kingdoms of Imereti and Kakheti, the rivalry among these polities continued under David. He had to defend his kingdom against the attacks by Alexander II of Imereti in the west, and George II of Kakheti in the east. In August 1509, Alexander took fort-city Gori and the northwestern corner of Kartli, but had to abandon the occupied lands to David due to the Ottoman raid on Imereti in 1510. A year later, George of Kakheti surged into Kartli, but failed to capture the king in a besieged castle of Ateni. In 1513, George invaded again, only to be defeated and taken prisoner by David’s younger brother Bagrat I of Mukhrani. He died in captivity and his kingdom was annexed to Kartli. In 1518, the Pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hal Sudborough
HAL may refer to: Aviation * Halali Airport (IATA airport code: HAL) Halali, Oshikoto, Namibia * Hawaiian Airlines (ICAO airline code: HAL) * HAL Airport, Bangalore, India * Hindustan Aeronautics Limited an Indian aerospace manufacturer of fighter aircraft and helicopters Businesses * HAL Allergy, a Dutch pharmaceutical company * HAL Computer Systems, a defunct computer manufacturer * HAL Laboratory, a Japanese video game developer * Halliburton's New York Stock Exchange ticker symbol * Hamburg America Line, a shipping company * Hindustan Aeronautics Limited, an Indian aerospace manufacturer of fighter aircraft and helicopters * Hindustan Antibiotics Limited, an Indian public sector pharmaceutical manufacturer * Holland America Line, a cruise ship operator * HAL FM, or CHNS-FM, a classic rock station in Halifax, Nova Scotia Computing * Hardware abstraction layer, a layer of software that hides hardware differences from higher level programs * HAL (software), an imp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Texas At Dallas
The University of Texas at Dallas (UTD or UT Dallas) is a public research university in Richardson, Texas. It is one of the largest public universities in the Dallas area and the northernmost institution of the University of Texas system. It was initially founded in 1961 as a private research arm of Texas Instruments. The young university has been characterized by rapid growth in research output and its competitive undergraduate admissions policies since its inception. Less than 47 years after its founding, the Carnegie Foundation had classified the university as a doctoral research university with "Highest Research Activity"—faster than any other school in Texas. The university is associated with four Nobel Prizes and has members of the National Academy of Sciences and National Academy of Engineering on its faculty with the most notable research projects including the areas of Space Science, Bioengineering, Cybersecurity, Nanotechnology, and Behavioral and Brain Sciences. UT ... [...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 very 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 of 0.87. Notable publications * The 1972 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christos Papadimitriou
Christos Charilaos Papadimitriou ( el, Χρήστος Χαρίλαος "Χρίστος" Παπαδημητρίου; born August 16, 1949) is a Greek theoretical computer scientist and the Donovan Family Professor of Computer Science at Columbia University. Education Papadimitriou studied at the National Technical University of Athens, where in 1972 he received his Bachelor of Arts degree in electrical engineering. He then pursued graduate studies at Princeton University, where he received his Ph.D. in electrical engineering and computer science in 1976 after completing a doctoral dissertation titled "The complexity of combinatorial optimization problems." Career Papadimitriou has taught at Harvard, MIT, the National Technical University of Athens, Stanford, UCSD, University of California, Berkeley and is currently the Donovan Family Professor of Computer Science at Columbia University. Papadimitriou co-authored a paper on pancake sorting with Bill Gates, then a Harvard under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bill Gates
William Henry Gates III (born October 28, 1955) is an American business magnate and philanthropist. He is a co-founder of Microsoft, along with his late childhood friend Paul Allen. During his career at Microsoft, Gates held the positions of chairman, chief executive officer (CEO), president and chief software architect, while also being the largest individual shareholder until May 2014. He was a major entrepreneur of the microcomputer revolution of the 1970s and 1980s. Gates was born and raised in Seattle. In 1975, he and Allen founded Microsoft in Albuquerque, New Mexico. It became the world's largest personal computer software company. Gates led the company as chairman and CEO until stepping down as CEO in January 2000, succeeded by Steve Ballmer, but he remained chairman of the board of directors and became chief software architect. During the late 1990s, he was criticized for his business tactics, which have been considered anti-competitive. This opinion has b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Path
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding one more edge to form a Hamiltonian cycle, and removing any edge from a Hamiltonian cycle produces a Hamiltonian path. Determining whether such paths and cycles exist in graphs (the Hamiltonian path problem and Hamiltonian cycle problem) are NP-complete. Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as ''Hamilton's puzzle'', which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. Hamilton solved this problem using the icosian calculus, an algebraic structure based on roots of unity with many similarities to the quaternions (also invented by Ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle (graph Theory)
In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal. A graph without cycles is called an ''acyclic graph''. A directed graph without directed cycles is called a '' directed acyclic graph''. A connected graph without cycles is called a ''tree''. Definitions Circuit and cycle * A circuit is a non-empty trail in which the first and last vertices are equal (''closed trail''). : Let be a graph. A circuit is a non-empty trail with a vertex sequence . * A cycle or simple circuit is a circuit in which only the first and last vertices are equal. Directed circuit and directed cycle * A directed circuit is a non-empty directed trail in which the first and last vertices are equal (''closed directed trail''). : Let be a directed graph. A directed circuit is a non-empty directed trail with a vertex seq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
