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Reconfiguration
In discrete mathematics and theoretical computer science, reconfiguration problems are computational problems involving reachability or Connectivity (graph theory), connectivity of state spaces. Types of problems Here, a state space is a discrete set of configurations of a system or solutions of a combinatorial problem, called states, together with a set of allowed moves linking one state to another. Reconfiguration problems may ask: *For a given class of problems, is the state space always connected? That is, can one transform every pair of states into each other with a sequence of moves? If not, what is the computational complexity of determining whether the state space for a particular problem is connected? *What is the diameter of the state space, the smallest number such that every two states can be transformed into each other with at most moves? *Given two states, what is the complexity of determining whether they can be transformed into each other, or of finding the shortes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Token Reconfiguration
In computational complexity theory and combinatorics, the token reconfiguration problem is a reconfiguration problem on a graph with both an initial and desired state for tokens. Given a graph G, an initial state of tokens is defined by a subset V \subset V(G) of the vertices of the graph; let n = , V, . Moving a token from vertex v_1 to vertex v_2 is valid if v_1 and v_2 are joined by a path in G that does not contain any other tokens; note that the distance traveled within the graph is inconsequential, and moving a token across multiple edges sequentially is considered a single move. A desired end state is defined as another subset V' \subset V(G). The goal is to minimize the number of valid moves to reach the end state from the initial state. Motivation The problem is motivated by so-called sliding puzzles, which are in fact a variant of this problem, often restricted to rectangular grid graphs with no holes. The most famous such puzzle, the 15 puzzle, is a variant of thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Nondeterministic Constraint Logic
In theoretical computer science, nondeterministic constraint logic is a combinatorial system in which an orientation is given to the edges of a weighted undirected graph, subject to certain constraints. One can change this orientation by steps in which a single edge is reversed, subject to the same constraints. This is a form of reversible logic in that each sequence of edge orientation changes can be undone. Reconfiguration problems for constraint logic, asking for a sequence of moves to connect certain states, connect all states, or reverse a specified edge have been proven to be PSPACE-complete. These hardness results form the basis for proofs that various games and puzzles are PSPACE-hard or PSPACE-complete. Constraint graphs In the simplest version of nondeterministic constraint logic, each edge of an undirected graph has weight either one or two. (The weights may also be represented graphically by drawing edges of weight one as red and edges of weight two as blue.) The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Cereceda's Conjecture
In the mathematics of graph coloring, Cereceda’s conjecture is an unsolved problem on the distance between pairs of colorings of sparse graphs. It states that, for two different colorings of a graph of Degeneracy (graph theory), degeneracy , both using at most colors, it should be possible to reconfiguration, reconfigure one coloring into the other by changing the color of one vertex at a time, using a number of steps that is quadratic in the size of the graph. The conjecture is named after Luis Cereceda, who formulated it in his 2007 doctoral dissertation. Background The Degeneracy (graph theory), degeneracy of an undirected graph is the smallest number such that every non-empty subgraph of has at least one vertex of degree at most . If one repeatedly removes a minimum-degree vertex from until no vertices are left, then the largest of the degrees of the vertices at the time of their removal will be exactly , and this method of repeated removal can be used to compute the de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Rotation Distance
In discrete mathematics and theoretical computer science, the rotation distance between two binary trees with the same number of nodes is the minimum number of tree rotations needed to reconfigure one tree into another. Because of a combinatorial equivalence between binary trees and triangulations of convex polygons, rotation distance is equivalent to the flip distance for triangulations of convex polygons. Rotation distance was first defined by Karel Čulík II and Derick Wood in 1982. Every two -node binary trees have a rotation distance of at most for all , and some pairs of trees have exactly this distance. The computational complexity of computing the rotation distance is unknown. Definition A binary tree is a structure consisting of a set of nodes, one of which is designated as the root node, in which each remaining node is either the ''left child'' or ''right child'' of some other node, its ''parent'', and in which following the parent links from any node eventually lea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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PSPACE-complete
In computational complexity theory, a decision problem is PSPACE-complete if it can be solved using an amount of memory that is polynomial in the input length (PSPACE, polynomial space) and if every other problem that can be solved in polynomial space can be Polynomial-time reduction, transformed to it in polynomial time. The problems that are PSPACE-complete can be thought of as the hardest problems in PSPACE, the class of decision problems solvable in polynomial space, because a solution to any one such problem could easily be used to solve any other problem in PSPACE. Problems known to be PSPACE-complete include determining properties of regular expressions and context-sensitive grammars, determining the truth of quantified Boolean formula problem, quantified Boolean formulas, step-by-step changes between solutions of combinatorial optimization problems, and many puzzles and games. Theory A problem is defined to be PSPACE-complete if it can be solved using a polynomial amount o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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State Space
In computer science, a state space is a discrete space representing the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory. For instance, the toy problem Vacuum World has a discrete finite state space in which there are a limited set of configurations that the vacuum and dirt can be in. A "counter" system, where states are the natural numbers starting at 1 and are incremented over time has an infinite discrete state space. The angular position of an undamped pendulum is a continuous (and therefore infinite) state space. Definition State spaces are useful in computer science as a simple model of machines. Formally, a state space can be defined as a tuple [''N'', ''A'', ''S'', ''G''] where: * ''N'' is a Set (mathematics), set of states * ''A'' is a set of arcs connecting the states * ''S'' is a nonempty subset of ''N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Flip Distance
In discrete mathematics and theoretical computer science, the flip distance between two triangulations of the same point set is the number of flips required to transform one triangulation into another. A flip removes an edge between two triangles in the triangulation and then adds the other diagonal in the edge's enclosing quadrilateral, forming a different triangulation of the same point set. This problem is known to be NP-hard. However, the computational complexity of determining the flip distance between convex polygons, a special case of this problem, is unknown. Computing the flip distance between convex polygon triangulations is also equivalent to rotation distance, the number of rotations required to transform one binary tree into another. Definition Given a family of triangulations of some geometric object, a ''flip'' is an operation that transforms one triangulation to another by removing an edge between two triangles and adding the opposite diagonal to the resulting q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Reconfigurability
{{inline, date=June 2024 Reconfigurability denotes the Reconfigurable Computing capability of a system, so that its behavior can be changed by reconfiguration, i. e. by loading different configware code. This static reconfigurability distinguishes between reconfiguration time and run time. Dynamic reconfigurability denotes the capability of a dynamically reconfigurable system that can dynamically change its behavior during run time, usually in response to dynamic changes in its environment. In the context of wireless communication dynamic reconfigurability tackles the changeable behavior of wireless networks and associated equipment, specifically in the fields of radio spectrum, radio access technologies, protocol stacks, and application services. Research regarding the (dynamic) reconfigurability of wireless communication systems is ongoing for example in working group 6 of the Wireless World Research Forum (WWRF), in the Wireless Innovation Forum (WINNF) (formerly Software Def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Co-NP-complete
In complexity theory, computational problems that are co-NP-complete are those that are the hardest problems in co-NP, in the sense that any problem in co-NP can be reformulated as a special case of any co-NP-complete problem with only polynomial overhead. If P is different from co-NP, then all of the co-NP-complete problems are not solvable in polynomial time. If there exists a way to solve a co-NP-complete problem quickly, then that algorithm can be used to solve all co-NP problems quickly. Each co-NP-complete problem is the complement of an NP-complete problem. There are some problems in both NP and co-NP, for example all problems in P or integer factorization. However, it is not known if the sets are equal, although inequality is thought more likely. See co-NP and NP-complete for more details. Fortune showed in 1979 that if any sparse language is co-NP-complete (or even just co-NP-hard), then , a critical foundation for Mahaney's theorem. Formal definition A decisio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Degeneracy (graph Theory)
In graph theory, a -degenerate graph is an undirected graph in which every subgraph has at least one vertex of degree (graph theory), degree at most k. That is, some vertex in the subgraph touches k or fewer of the subgraph's edges. The degeneracy of a graph is the smallest value of k for which it is k-degenerate. The degeneracy of a graph is a measure of how dense graph, sparse it is, and is within a constant factor of other sparsity measures such as the arboricity of a graph. Degeneracy is also known as the -core number, width, and linkage, and is essentially the same as the coloring number or Szekeres–Wilf number (named after ). The k-degenerate graphs have also been called -inductive graphs. The degeneracy of a graph may be computed in linear time by an algorithm that repeatedly removes minimum-degree vertices. The Connected component (graph theory), connected components that are left after all vertices of degree less than k have been (repeatedly) removed are called the - ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Kempe Chain
In mathematics, a Kempe chain is a device used mainly in the study of the four colour theorem. Intuitively, it is a connected chain of vertices on a graph with alternating colours. History Kempe chains were first used by Alfred Kempe in his attempted proof of the four colour theorem. Even though his proof turned out to be incomplete, the method of Kempe chains is crucial to the success of valid modern proofs, such as the first successful one by Kenneth Appel and Wolfgang Haken. Furthermore, the method is used in the proof of the five color theorem by Percy John Heawood, a weaker but more easily proven version of the four colour theorem. Formal definition The term "Kempe chain" is used in two different but related ways. Suppose ''G'' is a graph with vertex set ''V'', with a given colouring function : c : V \to S, where ''S'' is a finite set of colours, containing at least two distinct colours ''a'' and ''b''. If ''v'' is a vertex with colour ''a'', then the (''a'', ''b'')-Kem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Graph Coloring
In graph theory, graph coloring is a methodic assignment of labels traditionally called "colors" to elements of a Graph (discrete mathematics), graph. The assignment is subject to certain constraints, such as that no two adjacent elements have the same color. Graph coloring is a special case of graph labeling. In its simplest form, it is a way of coloring the Vertex (graph theory), 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 (graph theory), edges 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 (graph theory), 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |