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Distributed computing is a field of computer science that studies distributed systems. A distributed system is a system whose components are located on different networked computers, which communicate and coordinate their actions by passing messages to one another. The components interact with one another in order to achieve a common goal. Three significant characteristics of distributed systems are: concurrency of components, lack of a global clock, and independent failure of components. Examples of distributed systems vary from SOA-based systems to massively multiplayer online games to peer-to-peer applications.
A minimum spanning tree (MST) or minimum weight spanning tree is a subset of the edges of a connected, edge-weighted undirected graph that connects all the vertices together, without any cycles and with the minimum possible total edge weight. That is, it is a spanning tree whose sum of edge weights is as small as possible. More generally, any edge-weighted undirected graph has a minimum spanning forest, which is a union of the minimum spanning trees for its connected components.
In the mathematical field of graph theory, a spanning treeT of an undirected graph G is a subgraph that is a tree which includes all of the vertices of G, with minimum possible number of edges. In general, a graph may have several spanning trees, but a graph that is not connected will not contain a spanning tree. If all of the edges of G are also edges of a spanning tree T of G, then G is a tree and is identical to T.
In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly connected components of an arbitrary directed graph form a partition into subgraphs that are themselves strongly connected. It is possible to test the strong connectivity of a graph, or to find its strongly connected components, in linear time.
In computer science, a parallel random-access machine (PRAM) is a shared-memory abstract machine. As its name indicates, the PRAM was intended as the parallel-computing analogy to the random-access machine (RAM). In the same way that the RAM is used by sequential-algorithm designers to model algorithmic performance, the PRAM is used by parallel-algorithm designers to model parallel algorithmic performance. Similar to the way in which the RAM model neglects practical issues, such as access time to cache memory versus main memory, the PRAM model neglects such issues as synchronization and communication, but provides any (problem-size-dependent) number of processors. Algorithm cost, for instance, is estimated using two parameters O(time) and O(time × processor_number).
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 biconnected component is a maximal biconnected subgraph. Any connected graph decomposes into a tree of biconnected components called the block-cut tree of the graph. The blocks are attached to each other at shared vertices called cut vertices or articulation points. Specifically, a cut vertex is any vertex whose removal increases the number of connected components.
In computer science, the prefix sum, cumulative sum, inclusive scan, or simply scan of a sequence of numbers x0, x1, x2, ... is a second sequence of numbers y0, y1, y2, ..., the sums of prefixes of the input sequence:
In graph theory and computer science, the lowest common ancestor (LCA) of two nodes v and w in a tree or directed acyclic graph (DAG) T is the lowest node that has both v and w as descendants, where we define each node to be a descendant of itself.
In mathematics, a graph partition is the reduction of a graph to a smaller graph by partitioning its set of nodes into mutually exclusive groups. Edges of the original graph that cross between the groups will produce edges in the partitioned graph. If the number of resulting edges is small compared to the original graph, then the partitioned graph may be better suited for analysis and problem-solving than the original. Finding a partition that simplifies graph analysis is a hard problem, but one that has applications to scientific computing, VLSI circuit design, and task scheduling in multiprocessor computers, among others. Recently, the graph partition problem has gained importance due to its application for clustering and detection of cliques in social, pathological and biological networks. For a survey on recent trends in computational methods and applications see Buluc et al. (2013).
In computer science, a Cartesian tree is a binary tree derived from a sequence of numbers; it can be uniquely defined from the properties that it is heap-ordered and that a symmetric (in-order) traversal of the tree returns the original sequence. Introduced by Vuillemin (1980) in the context of geometric range searching data structures, Cartesian trees have also been used in the definition of the treap and randomized binary search tree data structures for binary search problems. The Cartesian tree for a sequence may be constructed in linear time using a stack-based algorithm for finding all nearest smaller values in a sequence.
In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split into smaller pieces by removing a small number of vertices. Specifically, the removal of O(√n) vertices from an n-vertex graph can partition the graph into disjoint subgraphs each of which has at most 2n/3 vertices.
Uzi Vishkin is a computer scientist at the University of Maryland, College Park, where he is Professor of Electrical and Computer Engineering at the University of Maryland Institute for Advanced Computer Studies (UMIACS). Uzi Vishkin is known for his work in the field of parallel computing. In 1996, he was inducted as a Fellow of the Association for Computing Machinery, with the following citation: "One of the pioneers of parallel algorithms research, Dr. Vishkin's seminal contributions played a leading role in forming and shaping what thinking in parallel has come to mean in the fundamental theory of Computer Science."
In computer science, the all nearest smaller values problem is the following task: for each position in a sequence of numbers, search among the previous positions for the last position that contains a smaller value. This problem can be solved efficiently both by parallel and non-parallel algorithms: Berkman, Schieber & Vishkin (1993), who first identified the procedure as a useful subroutine for other parallel programs, developed efficient algorithms to solve it in the Parallel Random Access Machine model; it may also be solved in linear time on a non-parallel computer using a stack-based algorithm. Later researchers have studied algorithms to solve it in other models of parallel computation.
Explicit Multi-Threading (XMT) is a computer science paradigm for building and programming parallel computers designed around the parallel random-access machine (PRAM) parallel computational model. A more direct explanation of XMT starts with the rudimentary abstraction that made serial computing simple: that any single instruction available for execution in a serial program executes immediately. A consequence of this abstraction is a step-by-step (inductive) explication of the instruction available next for execution. The rudimentary parallel abstraction behind XMT, dubbed Immediate Concurrent Execution (ICE) in Vishkin (2011), is that indefinitely many instructions available for concurrent execution execute immediately. A consequence of ICE is a step-by-step (inductive) explication of the instructions available next for concurrent execution. Moving beyond the serial von Neumann computer, the aspiration of XMT is that computer science will again be able to augment mathematical induction with a simple one-line computing abstraction.
The Euler tour technique (ETT), named after Leonhard Euler, is a method in graph theory for representing trees. The tree is viewed as a directed graph that contains two directed edges for each edge in the tree. The tree can then be represented as a Eulerian circuit of the directed graph, known as the Euler tour representation (ETR) of the tree. The ETT allows for efficient, parallel computation of solutions to common problems in algorithmic graph theory. It was introduced by Tarjan and Vishkin in 1984.
In graph theory, Robbins' theorem, named after Herbert Robbins (1939), states that the graphs that have strong orientations are exactly the 2-edge-connected graphs. That is, it is possible to choose a direction for each edge of an undirected graph G, turning it into a directed graph that has a path from every vertex to every other vertex, if and only if G is connected and has no bridge.
In graph theory, a bipolar orientation or st-orientation of an undirected graph is an assignment of a direction to each edge that causes the graph to become a directed acyclic graph with a single source s and a single sink t, and an st-numbering of the graph is a topological ordering of the resulting directed acyclic graph.
Pointer jumping or path doubling is a design technique for parallel algorithms that operate on pointer structures, such as linked lists and directed graphs. Pointer jumping allows an algorithm to follow paths with a time complexity that is logarithmic with respect to the length of the longest path. It does this by "jumping" to the end of the path computed by neighbors.
This article discusses the analysis of parallel algorithms. Like in the analysis of "ordinary", sequential, algorithms, one is typically interested in asymptotic bounds on the resource consumption, but the analysis is performed in the presence of multiple processor units that cooperate to perform computations. Thus, one can determine not only how many "steps" a computation takes, but also how much faster it becomes as the number of processors goes up. Interestingly, the analysis approach works by first suppressing the number of processors. The next background paragraph explains how the abstraction of the number of processors first emerged.
References
- Anderson, Richard J.; Miller, Gary L. (1990), "A simple randomized parallel algorithm for list-ranking", Information Processing Letters, 33 (5): 269–273, doi:10.1016/0020-0190(90)90196-5 .
- Cole, Richard; Vishkin, Uzi (1989), "Faster optimal parallel prefix sums and list ranking", Information and Computation, 81 (3): 334–352, doi: 10.1016/0890-5401(89)90036-9 .
- Tarjan, Robert E.; Vishkin, Uzi (1985), "An efficient parallel biconnectivity algorithm", SIAM Journal on Computing, 14 (4): 862–874, CiteSeerX 10.1.1.465.8898 , doi:10.1137/0214061 .
- Vishkin, Uzi (1984), "Randomized speed-ups in parallel computation", Annual ACM Symposium on Theory of Computing: 230–239, doi:10.1145/800057.808686, ISBN 0-89791-133-4 .
- Wyllie, J. C. (1979), The Complexity of Parallel Computation, Ph.D. thesis, Department of Computer Science, Cornell University .