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Planar SAT
In computer science, the planar 3-satisfiability problem (abbreviated PLANAR 3SAT or PL3SAT) is an extension of the classical Boolean 3-satisfiability problem to a planar incidence graph. In other words, it asks whether the variables of a given Boolean formula—whose incidence graph consisting of variables and clauses can be embedded on a plane—can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called ''satisfiable''. On the other hand, if no such assignment exists, the function expressed by the formula is FALSE for all possible variable assignments and the formula is ''unsatisfiable''. For example, the formula "''a'' AND NOT ''b''" is satisfiable because one can find the values ''a'' = TRUE and ''b'' = FALSE, which make (''a'' AND NOT ''b'') = TRUE. In contrast, "''a'' AND NOT ''a''" is unsatisfiable. Like 3SAT, PLANAR-SAT is NP-complete, and is commo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar SAT Example
Planar is an adjective meaning "relating to a plane (geometry)". Planar may also refer to: Science and technology * Planar (computer graphics), computer graphics pixel information from several bitplanes * Planar (transmission line technologies), transmission lines with flat conductors * Planar, the structure resulting from the planar process used in the manufacture of semiconductor devices, such as planar transistors * Planar graph, graph that can be drawn in the plane so that no edges cross * Planar mechanism, a system of parts whose motion is constrained to a two-dimensional plane * Planar Systems, an Oregon-headquartered manufacturer of digital displays * Zeiss Planar, photographic lens designed by Paul Rudolph at Carl Zeiss in 1896 See also * List of planar symmetry groups * Planarity, a computer puzzle game * Plane (other) * Planer (other) The term planer may refer to several types of carpentry, carpentry tools, woodworking machines or metalworking machine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar SAT Crossover Gadget
Planar is an adjective meaning "relating to a plane (geometry)". Planar may also refer to: Science and technology * Planar (computer graphics), computer graphics pixel information from several bitplanes * Planar (transmission line technologies), transmission lines with flat conductors * Planar, the structure resulting from the planar process used in the manufacture of semiconductor devices, such as planar transistors * Planar graph, graph that can be drawn in the plane so that no edges cross * Planar mechanism, a system of parts whose motion is constrained to a two-dimensional plane * Planar Systems, an Oregon-headquartered manufacturer of digital displays * Zeiss Planar, photographic lens designed by Paul Rudolph at Carl Zeiss in 1896 See also * List of planar symmetry groups * Planarity, a computer puzzle game * Plane (other) Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tatamibari
Tatamibari () is a type of logic puzzle designed and published by Nikoli. The puzzle is based on Japanese tatami mats. Rules A Tatamibari puzzle is played on a rectangular grid with three different kinds of symbols in it: +, -. and , . The solver must partition the grid into rectangular or square regions according to the following rules: * Every partition must contain exactly one symbol in it. * A + symbol must be contained in a square. * A , symbol must be contained in a rectangle with a greater height than width. * A - symbol must be contained in a rectangle with a greater width than height. * Four pieces may never share the same corner. Computational Complexity The problem of finding a solution to a particular Tatamibari configuration is NP-complete In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can fin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shakashaka
is a logic puzzle developed by publisher Nikoli. Rules Shakashaka is played on a rectangular grid of white and black squares. Some black cells may contain a number. The objective of the puzzle is to place triangles in some of the white cells. There are four kinds of triangles which can be put in squares: In the resulting grid, * The white parts of the grid (uncovered by black triangles) must form a rectangle or a square. * Black cells with a number must be orthogonally adjacent to the specified number of black triangles. Computational complexity It is NP-complete In computational complexity theory, a problem is NP-complete when: # it is a problem for which the correctness of each solution can be verified quickly (namely, in polynomial time) and a brute-force search algorithm can find a solution by trying ... to decide whether a given Shakashaka puzzle has a solution. Furthermore, counting the number of solutions to a given Shakashaka puzzle is #P-complete. Reference ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nurikabe (puzzle)
Nurikabe (hiragana: ぬりかべ) is a binary determination puzzle named for Nurikabe, an invisible wall in Japanese folklore that blocks roads and delays foot travel. Nurikabe was apparently invented and named by Nikoli; other names (and attempts at localization) for the puzzle include ''Cell Structure'' and ''Islands in the Stream''. Rules The puzzle is played on a typically rectangular grid of cells, some of which contain numbers. Cells are initially of unknown color, but can only be black or white. Two same-color cells are considered "connected" if they are adjacent vertically or horizontally, but not diagonally. Connected white cells form "islands", while connected black cells form the "sea". The challenge is to paint each cell black or white, subject to the following rules: # Each numbered cell is an island cell, the number in it is the number of cells in that island. # Each island must contain exactly one numbered cell. # There must be only one sea, which is not all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fillomino
Fillomino (フィルオミノ) is a type of logic puzzle published by many publishers. Other published titles for the puzzle include ''Allied Occupation''. Rules ''Fillomino'' is played on a rectangular grid with no standard size; the internal grid lines are often dotted. (When published as ''Allied Occupation'' in the World Puzzle Championship, the cells of the grid are circular, but this is purely an aesthetic concern.) Some cells of the grid start containing numbers, referred to as "givens". The goal is to divide the grid into regions called polyominoes (by filling in their boundaries) such that each given number ''n'' in the grid satisfies the following constraints: * Each clue ''n'' is part of a polyomino of size n; * no two polyominoes of matching size (number of cells) are orthogonally adjacent (share a side). It is possible for two givens with matching number to belong to the same polyomino in the solution, and for a polyomino to have no given at all. Solution metho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Directed Acyclic Graph
In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one vertex to another, such that following those directions will never form a closed loop. A directed graph is a DAG if and only if it can be topologically ordered, by arranging the vertices as a linear ordering that is consistent with all edge directions. DAGs have numerous scientific and computational applications, ranging from biology (evolution, family trees, epidemiology) to information science (citation networks) to computation (scheduling). Directed acyclic graphs are sometimes instead called acyclic directed graphs or acyclic digraphs. Definitions A graph is formed by vertices and by edges connecting pairs of vertices, where the vertices can be any kind of object that is connected in pairs by edges. In the case of a directed graph, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circuit Satisfiability Problem
In theoretical computer science, the circuit satisfiability problem (also known as CIRCUIT-SAT, CircuitSAT, CSAT, etc.) is the decision problem of determining whether a given Boolean circuit has an assignment of its inputs that makes the output true. In other words, it asks whether the inputs to a given Boolean circuit can be consistently set to 1 or 0 such that the circuit outputs 1. If that is the case, the circuit is called ''satisfiable''. Otherwise, the circuit is called ''unsatisfiable.'' In the figure to the right, the left circuit can be satisfied by setting both inputs to be 1, but the right circuit is unsatisfiable. CircuitSAT is closely related to Boolean satisfiability problem (SAT), and likewise, has been proven to be NP-complete. It is a prototypical NP-complete problem; the Cook–Levin theorem is sometimes proved on CircuitSAT instead of on the SAT, and then CircuitSAT can be reduced to the other satisfiability problems to prove their NP-completeness. The satisfiabi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Cut
For a graph, a maximum cut is a cut whose size is at least the size of any other cut. That is, it is a partition of the graph's vertices into two complementary sets and , such that the number of edges between and is as large as possible. Finding such a cut is known as the max-cut problem. The problem can be stated simply as follows. One wants a subset of the vertex set such that the number of edges between and the complementary subset is as large as possible. Equivalently, one wants a bipartite subgraph of the graph with as many edges as possible. There is a more general version of the problem called weighted max-cut, where each edge is associated with a real number, its weight, and the objective is to maximize the total weight of the edges between and its complement rather than the number of the edges. The weighted max-cut problem allowing both positive and negative weights can be trivially transformed into a weighted minimum cut problem by flipping the sign in all weig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1-in-3-SAT
In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called ''satisfiable''. On the other hand, if no such assignment exists, the function expressed by the formula is FALSE for all possible variable assignments and the formula is ''unsatisfiable''. For example, the formula "''a'' AND NOT ''b''" is satisfiable because one can find the values ''a'' = TRUE and ''b'' = FALSE, which make (''a'' AND NOT ''b'') = TRUE. In contrast, "''a'' AND NOT ''a''" is unsatisfiable. SAT is the first problem that was proved to be NP-complet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamic Programming
Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. In both contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a recursive manner. While some decision problems cannot be taken apart this way, decisions that span several points in time do often break apart recursively. Likewise, in computer science, if a problem can be solved optimally by breaking it into sub-problems and then recursively finding the optimal solutions to the sub-problems, then it is said to have ''optimal substructure''. If sub-problems can be nested recursively inside larger problems, so that dynamic programming methods are applicable, then there is a relation between the value of the larger problem and the values of the sub-problems.Cormen, T. H.; Leiserson, C. E.; Rives ... [...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 sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
