|
Pursuit–evasion
Pursuit–evasion (variants of which are referred to as cops and robbers and graph searching) is a family of problems in mathematics and computer science in which one group attempts to track down members of another group in an environment. Early work on problems of this type modeled the environment geometrically. In 1976, Torrence Parsons introduced a formulation whereby movement is constrained by a graph. The geometric formulation is sometimes called continuous pursuit–evasion, and the graph formulation discrete pursuit–evasion (also called graph searching). Current research is typically limited to one of these two formulations. Discrete formulation In the discrete formulation of the pursuit–evasion problem, the environment is modeled as a graph. Problem definition There are innumerable possible variants of pursuit–evasion, though they tend to share many elements. A typical, basic example is as follows (cops and robber games): Pursuers and evaders occupy nodes of a gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angel Problem
The angel problem is a question in combinatorial game theory proposed by John Horton Conway. The game is commonly referred to as the angels and devils game.John H. Conway, The angel problem', in: Richard Nowakowski (editor) ''Games of No Chance'', volume 29 of MSRI Publications, pages 3–12, 1996. The game is played by two players called the angel and the devil. It is played on an infinite chessboard (or equivalently the points of a 2D lattice). The angel has a power ''k'' (a natural number 1 or higher), specified before the game starts. The board starts empty with the angel in one square. On each turn, the angel jumps to a different empty square which could be reached by at most ''k'' moves of a chess king, i.e. the distance from the starting square is at most ''k'' in the infinity norm. The devil, on its turn, may add a block on any single square not containing the angel. The angel may leap over blocked squares, but cannot land on them. The devil wins if the angel is u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pursuit Curve
In geometry, a curve of pursuit is a curve constructed by analogy to having a point (geometry), point or points representing pursuers and pursuees; the curve of pursuit is the curve traced by the pursuers. Definition With the paths of the pursuer and pursuee Parametrization (geometry), parameterized in time, the pursuee is always on the pursuer's tangent. That is, given , the pursuer (follower), and , the pursued (leader), for every with there is an such that :L(t) = F(t) + xF'\!(t). History The pursuit curve was first studied by Pierre Bouguer in 1732. In an article on navigation, Bouguer defined a curve of pursuit to explore the way in which one ship might maneuver while pursuing another. Leonardo da Vinci has occasionally been credited with first exploring curves of pursuit. However Paul J. Nahin, having traced such accounts as far back as the late 19th century, indicates that these anecdotes are unfounded. Single pursuer The path followed by a single pursuer, followi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haven (graph Theory)
In graph theory, a haven is a certain type of function on sets of vertices in an undirected graph. If a haven exists, it can be used by an evader to win a pursuit–evasion game on the graph, by consulting the function at each step of the game to determine a safe set of vertices to move into. Havens were first introduced by as a tool for characterizing the treewidth of graphs. Their other applications include proving the existence of small separators on minor-closed families of graphs, and characterizing the ends and clique minors of infinite graphs... Definition If is an undirected graph, and is a set of vertices, then an -flap is a nonempty connected component of the subgraph of formed by deleting . A haven of order in is a function that assigns an -flap to every set of fewer than vertices. This function must also satisfy additional constraints which are given differently by different authors. The number is called the ''order'' of the haven.. In the original ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cop-win Graph
In graph theory, a cop-win graph is an undirected graph on which the pursuer (cop) can always win a pursuit–evasion game against a robber, with the players taking alternating turns in which they can choose to move along an edge of a graph or stay put, until the cop lands on the robber's vertex. Finite cop-win graphs are also called dismantlable graphs or constructible graphs, because they can be dismantled by repeatedly removing a dominated vertex (one whose Neighbourhood (graph theory), closed neighborhood is a subset of another vertex's neighborhood) or constructed by repeatedly adding such a vertex. The cop-win graphs can be recognized in polynomial time by a greedy algorithm that constructs a dismantling order. They include the chordal graphs, and the graphs that contain a universal vertex. Definitions Pursuit–evasion Cop-win graphs can be defined by a pursuit–evasion game in which two players, a cop and a robber, are positioned at different initial vertices of a given u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scotland Yard (board Game)
''Scotland Yard'' is a board game in which a team of players controlling different detectives cooperate to track down a player controlling a criminal as they move around a board representing the streets of London. It was first published in 1983 by Ravensburger and is named after Scotland Yard which is the headquarters of London's Metropolitan Police Service in real-life. ''Scotland Yard'' is an symmetric game, asymmetric board game, during which the detective players cooperatively solve a variant of the pursuit–evasion problem. The game is published by Ravensburger in most of Europe and Canada and by Milton_Bradley_Company, Milton Bradley in the United States. It received the ''Spiel des Jahres'' (Game of the Year) award in 1983, the same year that it was published. Gameplay One player controls 'Mr. X': a criminal whose location is only revealed periodically throughout gameplay. The other players each control at least one detective, all of which are always present on the board ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rufus Isaacs (game Theorist)
Rufus Philip Isaacs (June 11, 1914 – January 18, 1981) was an American game theorist especially prominent in the 1950s and 1960s with his work on differential games. Biography Isaacs was born on June 11, 1914, in New York City. He worked for the RAND Corporation from 1948 until winter 1954/1955. His investigation stemmed from classic pursuit–evasion type zero-sum dynamic two-player games such as the Princess and monster game. In 1942, he married Rose Bicov, and they had two daughters. His work in pure mathematics included working with monodiffric functions, fractional-order mappings, graph theory, analytic functions, and number theory. In graph theory he constructed the first two infinite families of snarks. In applied mathematics, he worked with aerodynamics, elasticity, optimization, and differential games, which he is most known for. He received his bachelor's degree from MIT in 1936, and received his MA and PhD from Columbia University in 1942 and 1943 respectiv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chases And Escapes
''Chases and Escapes: The Mathematics of Pursuit and Evasion'' is a mathematics book on continuous pursuit–evasion problems. It was written by Paul J. Nahin, and published by the Princeton University Press in 2007. It was reissued as a paperback reprint in 2012. The Basic Library List Committee of the Mathematical Association of America has rated this book as essential for inclusion in undergraduate mathematics libraries. Topics The book has four chapters, covering the solutions to 21 continuous pursuit–evasion problems, with an additional 10 "challenge problems" left for readers to solve, with solutions given in an appendix. The problems are presented as entertaining stories that "breathe life into the mathematics and invite wider engagement", and their solutions use varied methods, including the computer calculation of numerical solutions for differential equations whose solutions have no closed form. Most of the material was previously known, but is collected here for the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homicidal Chauffeur Problem
{{Short description, Mathematical pursuit problem In game theory, the homicidal chauffeur problem is a mathematical pursuit problem which pits a hypothetical runner, who can only move slowly, but is highly maneuverable, against the driver of a motor vehicle, which is much faster but far less maneuverable, who is attempting to run him down. Both runner and driver are assumed to never tire. The question to be solved is: under what circumstances, and with what strategy, can the driver of the car guarantee that he can always catch the pedestrian, or the pedestrian guarantee that he can indefinitely elude the car? The problem is often used as an unclassified proxy for missile defense and other military targeting, allowing scientists to publish on it without security implications. The problem was proposed by Rufus Isaacs in a 1951 report for the RAND Corporation, and in the book ''Differential Games''.R. Isaacs, ''Differential Games: A Mathematical Theory with Applications to Warfare a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Princess And Monster Game
A princess and monster game is a pursuit–evasion game played by two players in a region. Formal definition In his book ''Differential Games'' (1965), Rufus Isaacs defined the game as: This game remained a well-known open problem until it was solved by Shmuel Gal in the late 1970s. His optimal strategy for the princess is to move to a random location in the room and stay still for a time interval which is neither too short nor too long, before going to another (independent) random location and repeating the procedure. The proposed optimal search strategy for the monster is based on subdividing the room into many narrow rectangles, picking a rectangle at random and searching it in some specific way, after some time picking another rectangle randomly and independently, and so on. Princess and monster games can be played on a pre-selected graph. It can be demonstrated that for any finite graph an optimal mixed search strategy exists that results in a finite payoff. This game ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torrence Parsons
Torrence Douglas Parsons (7 March 1941 – 2 April 1987) was an American mathematician. He worked mainly in graph theory, and is known for introducing a graph-theoretic view of pursuit–evasion problems (Parsons 1976, 1978). He obtained his Ph.D. from Princeton University Princeton University is a private university, private Ivy League research university in Princeton, New Jersey, United States. Founded in 1746 in Elizabeth, New Jersey, Elizabeth as the College of New Jersey, Princeton is the List of Colonial ... in 1966 under the supervision of Albert W. Tucker. Selected publications * * Notes Further reading Memorial articles in *''Journal of Graph Theory'' vol. 12 *''Discrete Mathematics'' vol. 78 1941 births 1987 deaths 20th-century American mathematicians Graph theorists Princeton University alumni {{US-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Christos H
Christos may refer to: * Jesus of Nazareth * Christ (title), a title for the Jewish Messiah in Christianity * Christos (surname) * Christos (given name) *, a Greek owned, Liberian flagged cargo ship in service 1962-71 See also * Christ (other) * Christo (other) * Christa (other) Christa may refer to: * Christa (given name), a female given name * Janusz Christa (1934–2008), Polish comics author * '' Swedish Fly Girls'', a 1971 film also known as ''Christa'' * 1015 Christa, an asteroid See also * Christ (disambiguation ... * Christus (other) {{Disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
