In graph theory, the shuffle-exchange network is an undirected cubic multigraph, whose vertices represent binary sequences of a given length and whose edges represent two operations on these sequence, circular shifts and flipping the lowest-order bit. [1]
In the version of this network introduced by Tomas Lang and Harold S. Stone in 1976, [2] simplifying earlier work of Stone in 1971, [3] the shuffle-exchange network of order consisted of an array of cells, numbered by the different binary numbers that can be represented with bits. These cells were connected by communications links in two different patterns: "exchange" links in which each cell is connected to the cell numbered with the opposite value in its lowest-order bit, and "shuffle" links in which each cell is connected to the cell whose number is obtained by a circular shift that shifts every bit to the next more significant position, except for the highest-order bit which shifts into the lowest-order position. The "exchange" links are bidirectional, while the "shuffle" links can only transfer information in one direction, from a cell to its circular shift. [2]
Subsequent work on networks with this topology removed the distinction between unidirectional and bidirectional communication links, allowing information to flow in either direction across each link. [1] [4]
The advantage of this communications pattern, over earlier methods, is that it allows information to be rapidly transferred through a small number of steps from any vertex in the network to any other vertex, while only requiring a single bit of control information (which of the two communications links to use) for each communications step. [2] Fast parallel algorithms for basic problems including sorting, matrix multiplication, polynomial evaluation, and Fourier transforms are known for parallel systems using this network. [4]
If this network is given a straightforward layout in the integer lattice, with the vertices placed on a line in numerical order, with each lattice edge carrying part of at most one communication link, and with each vertex or crossing of the network placed at a lattice point, the layout uses area , quadratic in its number of vertices. However, more compact and asymptotically optimal layouts with area were described by F. Thomson Leighton in his 1981 doctoral dissertation. [4]
A related communications network, the "omega network" or multi-stage shuffle-exchange network, consists of a given number of stages, each consisting of vertices, with the shuffle links connecting pairs of vertices in consecutive stages and the exchange links connecting pairs of vertices in the same stage as each other. [5]
The same operations on binary words, of rotation and flipping the first bit, can also be used to generate the cube-connected cycles, a different cubic parallel communications network with a greater number of vertices. Instead of having the binary words themselves as its vertices, the vertices of the cube-connected cycles represent operations on words that can be generated by rotation and flipping, and the edges represent the composition of one of these operations with an additional rotation or flip. [6]
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices which are connected by edges. A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics.
In mathematics, particularly graph theory, and computer science, a directed acyclic graph is a directed graph with no directed cycles. That is, it consists of vertices and edges, 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 to information science to computation (scheduling).
In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices.
In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called vertices and each of the related pairs of vertices is called an edge. Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics.
Graph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional depictions of graphs arising from applications such as social network analysis, cartography, linguistics, and bioinformatics.
In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix.
In the mathematical theory of matroids, a graphic matroid is a matroid whose independent sets are the forests in a given finite undirected graph. The dual matroids of graphic matroids are called co-graphic matroids or bond matroids. A matroid that is both graphic and co-graphic is sometimes called a planar matroid ; these are exactly the graphic matroids formed from planar graphs.
In computer science, a topological sort or topological ordering of a directed graph is a linear ordering of its vertices such that for every directed edge uv from vertex u to vertex v, u comes before v in the ordering. For instance, the vertices of the graph may represent tasks to be performed, and the edges may represent constraints that one task must be performed before another; in this application, a topological ordering is just a valid sequence for the tasks. Precisely, a topological sort is a graph traversal in which each node v is visited only after all its dependencies are visited. A topological ordering is possible if and only if the graph has no directed cycles, that is, if it is a directed acyclic graph (DAG). Any DAG has at least one topological ordering, and algorithms are known for constructing a topological ordering of any DAG in linear time. Topological sorting has many applications especially in ranking problems such as feedback arc set. Topological sorting is possible even when the DAG has disconnected components.
In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. In other words, there is no vertex outside the independent set that may join it because it is maximal with respect to the independent set property.
In graph theory, a book embedding is a generalization of planar embedding of a graph to embeddings into a book, a collection of half-planes all having the same line as their boundary. Usually, the vertices of the graph are required to lie on this boundary line, called the spine, and the edges are required to stay within a single half-plane. The book thickness of a graph is the smallest possible number of half-planes for any book embedding of the graph. Book thickness is also called pagenumber, stacknumber or fixed outerthickness. Book embeddings have also been used to define several other graph invariants including the pagewidth and book crossing number.
In graph theory, the hypercube graphQn is the graph formed from the vertices and edges of an n-dimensional hypercube. For instance, the cube graph Q3 is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube. Qn has 2n vertices, 2n – 1n edges, and is a regular graph with n edges touching each vertex.
In graph theory, a cactus is a connected graph in which any two simple cycles have at most one vertex in common. Equivalently, it is a connected graph in which every edge belongs to at most one simple cycle, or in which every block is an edge or a cycle.
In graph theory, a random geometric graph (RGG) is the mathematically simplest spatial network, namely an undirected graph constructed by randomly placing N nodes in some metric space and connecting two nodes by a link if and only if their distance is in a given range, e.g. smaller than a certain neighborhood radius, r.
In graph theory, the cube-connected cycles is an undirected cubic graph, formed by replacing each vertex of a hypercube graph by a cycle. It was introduced by Preparata & Vuillemin (1981) for use as a network topology in parallel computing.
In the mathematical field of graph theory, the Fibonacci cubes or Fibonacci networks are a family of undirected graphs with rich recursive properties derived from its origin in number theory. Mathematically they are similar to the hypercube graphs, but with a Fibonacci number of vertices. Fibonacci cubes were first explicitly defined in Hsu (1993) in the context of interconnection topologies for connecting parallel or distributed systems. They have also been applied in chemical graph theory.
In graph theory, a k-degenerate graph is an undirected graph in which every subgraph has a vertex of 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 sparse it is, and is within a constant factor of other sparsity measures such as the arboricity of a graph.
In graph theory, a partial cube is a graph that is isometric to a subgraph of a hypercube. In other words, a partial cube can be identified with a subgraph of a hypercube in such a way that the distance between any two vertices in the partial cube is the same as the distance between those vertices in the hypercube. Equivalently, a partial cube is a graph whose vertices can be labeled with bit strings of equal length in such a way that the distance between two vertices in the graph is equal to the Hamming distance between their labels. Such a labeling is called a Hamming labeling; it represents an isometric embedding of the partial cube into a hypercube.
An arc diagram is a style of graph drawing, in which the vertices of a graph are placed along a line in the Euclidean plane, with edges being drawn as semicircles in one or both of the two halfplanes bounded by the line, or as smooth curves formed by sequences of semicircles. In some cases, line segments of the line itself are also allowed as edges, as long as they connect only vertices that are consecutive along the line. Variations of this drawing style in which the semicircles are replaced by convex curves of some other type are also commonly called arc diagrams.
Rocha–Thatte algorithm is a distributed algorithm in graph theory for detecting cycles on large-scale directed graphs based on the bulk synchronous message passing abstraction. This algorithm for detecting cycles by message passing is suitable to be implemented in distributed graph processing systems, and it is also suitable for implementations in systems for disk-based computations, such as the GraphChi, where the computation is mainly based on secondary memory. Disk-based computations are necessary when we have a single computer for processing large-scale graphs, and the computation exceeds the primary memory capacity.
The Mixed Chinese Postman problem is the search for the shortest traversal of a graph with a set of vertices V, a set of undirected edges E with positive rational weights, and a set of directed arcs A with positive rational weights that covers each edge or arc at least once at minimal cost. The problem has been proven to be NP-complete by Papadimitriou. The mixed Chinese postman problem often arises in arc routing problems such as snow ploughing, where some streets are too narrow to traverse in both directions while other streets are bidirectional and can be plowed in both directions. This article assumes , which relates the computational complexity class of polynomial time with deterministic time to the set of NP complete problems. It is easy to check if a mixed graph has a postman tour of any size is easy to check by verifying if the graph is strongly connected. The problem is NP hard if we restrict the postman tour to traverse each arc exactly once or if we restrict it to traverse each edge exactly once, as proved by Zaragoza Martinez.
