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Pancake Sorting
Pancake sorting is 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. In this form, the problem was first discussed by American geometer Jacob E. Goodman. A variant of the problem is concerned with ''burnt'' pancakes, where each pancake has a burnt side and all pancakes must, in addition, end up with the burnt side on bottom. All sorting methods require pairs of elements to be compared. For the traditional sorting problem, the usual problem studied is to minimize the number of comparisons required to sort a list. The number of actual operations, such as swapping two elements, is then irrelevant. For pancake sorting problems, in contrast, the aim is to minimize the number of operations, where the only allowed operations are reversals of the elements of some ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pancake Sort Operation
A pancake (or hotcake, griddlecake, or flapjack) is a flat cake, often thin and round, prepared from a Starch, starch-based batter (cooking), 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 leavening agent, 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 Brittany, 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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set , namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory. Permutations are used in almost every branch of mathematics, and in many other fields of scie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cayley Graph
In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley), and uses a specified set of generators for the group. It is a central tool in combinatorial and geometric group theory. The structure and symmetry of Cayley graphs makes them particularly good candidates for constructing families of expander graphs. Definition Let G be a group and S be a generating set of G. The Cayley graph \Gamma = \Gamma(G,S) is an edge-colored directed graph constructed as follows: In his Collected Mathematical Papers 10: 403–405. * Each element g of G is assigned a vertex: the vertex set of \Gamma is identified with G. * Each element s of S is assigned a color c_s. * For every g \in G and s \in S, there is a directed edge of color c_s from the vertex corresponding to g to the one corresponding to gs. Not ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Embedding
In topological graph theory, an embedding (also spelled imbedding) of a Graph (discrete mathematics), graph G on a surface (mathematics), surface \Sigma is a representation of G on \Sigma in which points of \Sigma are associated with graph theory, vertices and simple arcs (Homeomorphism, homeomorphic images of [0,1]) are associated with graph theory, edges in such a way that: * the endpoints of the arc associated with an edge e are the points associated with the end vertices of e, * no arcs include points associated with other vertices, * two arcs never intersect at a point which is interior to either of the arcs. Here a surface is a compact space, compact, connected space, connected 2-manifold. Informally, an embedding of a graph into a surface is a drawing of the graph on the surface in such a way that its edges may intersect only at their endpoints. It is well known that any finite graph can be embedded in 3-dimensional Euclidean space \mathbb^3.. A planar graph is one that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Girth (graph Theory)
In graph theory, the girth of an undirected graph is the length of a shortest cycle contained in the graph. If the graph does not contain any cycles (that is, it is a forest), its girth is defined to be infinity. For example, a 4-cycle (square) has girth 4. A grid has girth 4 as well, and a triangular mesh has girth 3. A graph with girth four or more is triangle-free. Cages A cubic graph (all vertices have degree three) of girth that is as small as possible is known as a -cage (or as a -cage). The Petersen graph is the unique 5-cage (it is the smallest cubic graph of girth 5), the Heawood graph is the unique 6-cage, the McGee graph is the unique 7-cage and the Tutte eight cage is the unique 8-cage. There may exist multiple cages for a given girth. For instance there are three nonisomorphic 10-cages, each with 70 vertices: the Balaban 10-cage, the Harries graph and the Harries–Wong graph. Image:Petersen1 tiny.svg, The Petersen graph has a girth of 5 Image:Heawood_Graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distance (graph Theory)
In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance or shortest-path distance. Notice that there may be more than one shortest path between two vertices. If there is no path connecting the two vertices, i.e., if they belong to different connected components, then conventionally the distance is defined as infinite. In the case of a directed graph the distance between two vertices and is defined as the length of a shortest directed path from to consisting of arcs, provided at least one such path exists. Notice that, in contrast with the case of undirected graphs, does not necessarily coincide with —so it is just a quasi-metric, and it might be the case that one is defined while the other is not. Related concepts A metric space defined over a set of points in terms of distances in a graph defined over th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Graph
In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each vertex are equal to each other. A regular graph with vertices of degree is called a graph or regular graph of degree . Also, from the handshaking lemma, a regular graph contains an even number of vertices with odd degree. Regular graphs of degree at most 2 are easy to classify: a graph consists of disconnected vertices, a graph consists of disconnected edges, and a graph consists of a disjoint union of cycles and infinite chains. A graph is known as a cubic graph. A strongly regular graph is a regular graph where every adjacent pair of vertices has the same number of neighbors in common, and every non-adjacent pair of vertices has the same number of neighbors in common. The smallest graphs that are regular but not strong ... [...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 o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pancake Graph G3
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 o ... [...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 abstr ... [...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 Persian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Futurama
''Futurama'' is an American animated science fiction sitcom created by Matt Groening for the Fox Broadcasting Company. The series follows the adventures of the professional slacker Philip J. Fry, who is cryogenically preserved for 1000 years and revived on December 31, 2999. Fry finds work at an interplanetary delivery company, working alongside the one-eyed Leela and robot Bender. The series was envisioned by Groening in the mid-1990s while working on ''The Simpsons''; he brought David X. Cohen aboard to develop storylines and characters to pitch the show to Fox. Following its initial cancelation by Fox, ''Futurama'' began airing reruns on Cartoon Network's Adult Swim programming block, which lasted from 2003 to 2007. It was revived in 2007 as four direct-to-video films, the last of which was released in early 2009. Comedy Central entered into an agreement with 20th Century Fox Television to syndicate the existing episodes and air the films as 16 new, half-hour episodes, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
