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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 satisfiability problem#3-satisfiability, Boolean 3-satisfiability problem to a planar graph, 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 graph embedding, embedded on a plane (geometry), plane—can be consistently replaced by the values TRUE or FALSE in such a way that the formula Validity (logic), evaluates to TRUE. If this is the case, the formula is called ''Satisfiability, satisfiable''. On the other hand, if no such assignment exists, the function expressed by the formula is Contradiction#Contradiction in formal logic, 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''&n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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) * {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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. See also * List of Nikoli puzzle types is a Japanese publisher that specializes in games and, especially, logic puzzles. ''Nikoli'' is also the nickname of a quarterly magazine (whose full name is ''Puzzle Communication ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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, not sharing an edge with other white squares/rectangles. * 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, NP-complete problems are the hardest of the problems to which ''solutions'' can be verified ''quickly''. Somewhat more precisely, a problem is NP-complete when: # It is a decision problem, meaning that for any ... to decide whether a given Shakashaka puzzle has a solution. Furthermore, counting the number of solutions ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Nurikabe (puzzle)
Nurikabe (hiragana: ) is a binary determination puzzle named for Nurikabe, Nurikabe, an invisible wall in Japanese folklore that blocks roads and delays foot travel. Nurikabe was apparently invented and named by the publisher Nikoli (publisher), 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 (touching diagonally doesn't count). 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. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Fillomino
Fillomino (フィルオミノ) is a type of logic puzzle initially published in the 1980's in Japan in the magazine ''Puzzle Communication Nikoli'', and since replicated by many publishers in different countries. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 also 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, each edg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 satisfiab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Maximum Cut
In 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
1-in-3-SAT
In computational complexity, one-in-three 3-SAT (also known variously as 1-in-3-SAT and exactly-1 3-SAT) is an NP-complete variant of the Boolean satisfiability problem. Given a conjunctive normal form with three literals per clause, the problem is to determine whether there exists a truth assignment to the variables so that each clause has ''exactly'' one TRUE literal (and thus exactly two FALSE literals). In contrast, ordinary 3-SAT requires that every clause has ''at least'' one TRUE literal. Formally, a one-in-three 3-SAT problem is given as a generalized conjunctive normal form with all generalized clauses using a ternary operator ''R'' that is TRUE just if exactly one of its arguments is. When all the variables of a one-in-three 3-SAT formula have the same literal, the satisfiability problem is called one-in-three monotone 3-SAT. One-in-three 3-SAT, together with its monotone case, is listed as NP-complete problem "LO4" in the standard reference ''Computers and Intractability: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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. * ''n'' is called the length of the circuit resp. length of the cycle. 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 grap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
