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Unsolved Problems In Computer Science
This article is a list of notable unsolved problems in computer science. A problem in computer science is considered unsolved when no solution is known, or when experts in the field disagree about proposed solutions. Computational complexity * P versus NP problem * What is the relationship between BQP and NP? * NC = P problem * NP = co-NP problem * P = BPP problem * P = PSPACE problem * L = NL problem * PH = PSPACE problem * L = P problem * L = RL problem * Unique games conjecture * Is the exponential time hypothesis true? ** Is the strong exponential time hypothesis (SETH) true? * Do one-way functions exist? ** Is public-key cryptography possible? * Log-rank conjecture Polynomial versus nondeterministic-polynomial time for specific algorithmic problems * Can integer factorization be done in polynomial time on a classical (non-quantum) computer? * Can the discrete logarithm be computed in polynomial time on a classical (non-quantum) computer? * Can the shortest vector ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Unsolved Problems
List of unsolved problems may refer to several notable conjectures or open problems in various academic fields: Natural sciences, engineering and medicine * Unsolved problems in astronomy * Unsolved problems in biology * Unsolved problems in chemistry * Unsolved problems in engineering * Unsolved problems in geoscience * Unsolved problems in medicine * Unsolved problems in neuroscience * Unsolved problems in physics Mathematics, philosophy and information sciences * Unsolved problems in mathematics * Unsolved problems in statistics * Unsolved problems in computer science * Unsolved problems in philosophy * Unsolved problems in information theory Social sciences and languages * Unsolved problems in economics * Unsolved problems in fair division * Unsolved problems in linguistics See also * Cold case (unsolved crimes) * List of ciphertexts * List of NP-complete problems * List of PSPACE-complete problems * List of paradoxes * List of undecidable problems * List of unsolv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is generally expresse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simultaneous Embedding
Simultaneous embedding is a technique in graph drawing and information visualization for visualizing two or more different graphs on the same or overlapping sets of labeled vertices, while avoiding crossings within both graphs. Crossings between an edge of one graph and an edge of the other graph are allowed. If edges are allowed to be drawn as polylines or curves, then any planar graph may be drawn without crossing with its vertices in arbitrary positions in the plane, where the same vertex placement provides a simultaneous embedding. There are two restricted models: simultaneous geometric embedding, where each graph must be drawn planarly with line segments representing its edges rather than more complex curves, restricting the two given graphs to subclasses of the planar graphs, and simultaneous embedding with fixed edges, where curves or bends are allowed in the edges, but any edge in both graphs must be represented by the same curve in both drawings. In the unrestricted mode ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem Of The Three Geodesics
In differential geometry the theorem of the three geodesics, also known as Lyusternik–Schnirelmann theorem, states that every Riemannian manifold with the topology of a sphere has at least three simple closed geodesics (i.e. three embedded geodesic circles). The result can also be extended to quasigeodesics on a convex polyhedron, and to closed geodesics of reversible Finsler 2-spheres. The theorem is sharp: although every Riemannian 2-sphere contains infinitely many distinct closed geodesics, only three of them are guaranteed to have no self-intersections. For example, by a result of Morse if the lengths of three principal axes of an ellipsoid are distinct, but sufficiently close to each other, then the ellipsoid has only three simple closed geodesics. History and proof A geodesic, on a Riemannian surface, is a curve that is locally straight at each of its points. For instance, on the Euclidean plane the geodesics are lines, and on the surface of a sphere the geodesics are gre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
SIAM Journal On Discrete Mathematics
'' SIAM Journal on Discrete Mathematics'' is a peer-reviewed mathematics journal published quarterly by the Society for Industrial and Applied Mathematics (SIAM). The journal includes articles on pure and applied discrete mathematics. It was established in 1988, along with the ''SIAM Journal on Matrix Analysis and Applications'', to replace the ''SIAM Journal on Algebraic and Discrete Methods''. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.57. According to the ''Journal Citation Reports'', the journal has a 2016 impact factor of 0.755. Although its official ISO abbreviation is ''SIAM J. Discrete Math.'', its publisher and contributors frequently use the shorter abbreviation ''SIDMA''. References External links * Combinatorics journals Publications established in 1988 English-language journals Discrete Mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way ana ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique-width
In graph theory, the clique-width of a graph is a parameter that describes the structural complexity of the graph; it is closely related to treewidth, but unlike treewidth it can be bounded even for dense graphs. It is defined as the minimum number of labels needed to construct by means of the following 4 operations : #Creation of a new vertex with label (denoted by ) #Disjoint union of two labeled graphs and (denoted by G \oplus H) #Joining by an edge every vertex labeled to every vertex labeled (denoted by ), where #Renaming label to label (denoted by ) Graphs of bounded clique-width include the cographs and distance-hereditary graphs. Although it is NP-hard to compute the clique-width when it is unbounded, and unknown whether it can be computed in polynomial time when it is bounded, efficient approximation algorithms for clique-width are known. Based on these algorithms and on Courcelle's theorem, many graph optimization problems that are NP-hard for arbitrary graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Tree
In computer science, a binary tree is a k-ary k = 2 tree data structure in which each node has at most two children, which are referred to as the ' and the '. A recursive definition using just set theory notions is that a (non-empty) binary tree is a tuple (''L'', ''S'', ''R''), where ''L'' and ''R'' are binary trees or the empty set and ''S'' is a singleton set containing the root. Some authors allow the binary tree to be the empty set as well. From a graph theory perspective, binary (and K-ary) trees as defined here are arborescences. A binary tree may thus be also called a bifurcating arborescence—a term which appears in some very old programming books, before the modern computer science terminology prevailed. It is also possible to interpret a binary tree as an undirected, rather than a directed graph, in which case a binary tree is an ordered, rooted tree. Some authors use rooted binary tree instead of ''binary tree'' to emphasize the fact that the tree is rooted, bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 rotation distance at most , 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 leads to the root ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity Game
A parity game is played on a colored directed graph, where each node has been colored by a priority – one of (usually) finitely many natural numbers. Two players, 0 and 1, move a (single, shared) token along the edges of the graph. The owner of the node that the token falls on selects the successor node, resulting in a (possibly infinite) path, called the play. The winner of a finite play is the player whose opponent is unable to move. The winner of an infinite play is determined by the priorities appearing in the play. Typically, player 0 wins an infinite play if the largest priority that occurs infinitely often in the play is even. Player 1 wins otherwise. This explains the word "parity" in the title. Parity games lie in the third level of the Borel hierarchy, and are consequently determined. Games related to parity games were implicitly used in Rabin's proof of decidability of the monadic second-order theory of ''n'' successors ( S2S for ''n'' = 2), where determinacy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leaf Power
In the mathematical area of graph theory, a -leaf power of a tree is a graph whose vertices are the leaves of and whose edges connect pairs of leaves whose distance in is at most . That is, is an induced subgraph of the graph power , induced by the leaves of . For a graph constructed in this way, is called a -leaf root of . A graph is a leaf power if it is a -leaf power for some . These graphs have applications in phylogeny, the problem of reconstructing evolutionary trees. Related classes of graphs Since powers of strongly chordal graphs are strongly chordal and trees are strongly chordal, it follows that leaf powers are strongly chordal graphs. Actually, leaf powers form a proper subclass of strongly chordal graphs; a graph is a leaf power if and only if it is a fixed tolerance NeST graph and such graphs are a proper subclass of strongly chordal graphs. In it is shown that interval graphs and the larger class of rooted directed path graphs are leaf powers. The indif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Isomorphism Problem
The graph isomorphism problem is the computational problem of determining whether two finite graphs are isomorphic. The problem is not known to be solvable in polynomial time nor to be NP-complete, and therefore may be in the computational complexity class NP-intermediate. It is known that the graph isomorphism problem is in the low hierarchy of class NP, which implies that it is not NP-complete unless the polynomial time hierarchy collapses to its second level. At the same time, isomorphism for many special classes of graphs can be solved in polynomial time, and in practice graph isomorphism can often be solved efficiently. This problem is a special case of the subgraph isomorphism problem, which asks whether a given graph ''G'' contains a subgraph that is isomorphic to another given graph ''H''; this problem is known to be NP-complete. It is also known to be a special case of the non-abelian hidden subgroup problem over the symmetric group. In the area of image recognition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
