Parallel tree contraction and its application
21 Oct 1985-pp 478-489
TL;DR: A bottom-up algorithm to handle trees which has two major advantages over the top-down approach: the control structure is straight forward and easier to implement facilitating new algorithms using fewer processors and less time; and problems for which it was too difficult or too complicated to find polylog parallel algorithms are now easy.
Abstract: : Trees play a fundamental role in many computations, both for sequential as well as parallel problems. The classic paradigm applied to generate parallel algorithms in the presence of trees has been divide-conquer; finding a 1/3 - 2/3 separator and recursively solving the two subproblems. A now classic example is Brent's work on parallel evaluation of arithmetic expressions. This top-down approach has several complications, one of which is finding the separators. We define dynamic expression evaluation as the task of evaluating the expression with no free preprocessing. If we apply Brent's method, finding the separators seems to add a factor of log n to the running time. We give a bottom-up algorithm to handle trees. That is, all modifications to the tree are done locally. This bottom-up approach which we call CONTRACT has two major advantages over the top-down approach: (1) the control structure is straight forward and easier to implement facilitating new algorithms using fewer processors and less time; and (2) problems for which it was too difficult or too complicated to find polylog parallel algorithms are now easy.
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19 Oct 2009TL;DR: In this paper, the authors present a coherent and unified treatment of probabilistic techniques for obtaining high probability estimates on the performance of randomized algorithms, from the basic tool kit from the Chernoff-Hoeffding (CH) bounds to more sophisticated techniques like Martingales and isoperimetric inequalities, as well as some recent developments like Talagrand's inequality, transportation cost inequalities, and log-Sobolev inequalities.
Abstract: Randomized algorithms have become a central part of the algorithms curriculum based on their increasingly widespread use in modern applications. This book presents a coherent and unified treatment of probabilistic techniques for obtaining high- probability estimates on the performance of randomized algorithms. It covers the basic tool kit from the Chernoff-Hoeffding (CH) bounds to more sophisticated techniques like Martingales and isoperimetric inequalities, as well as some recent developments like Talagrand's inequality, transportation cost inequalities, and log-Sobolev inequalities. Along the way, variations on the basic theme are examined, such as CH bounds in dependent settings. The authors emphasize comparative study of the different methods, highlighting respective strengths and weaknesses in concrete example applications. The exposition is tailored to discrete settings sufficient for the analysis of algorithms, avoiding unnecessary measure-theoretic details, thus making the book accessible to computer scientists as well as probabilists and discrete mathematicians.
1,028 citations
TL;DR: A bibliographic survey on algorithms whose goal is to produce aesthetically pleasing drawings of graphs is presented, a first attempt to encompass both theoretical and application-oriented papers from disparate areas.
Abstract: Several data presentation problems involve drawing graphs so that they are easy to read and understand. Examples include circuit schematics and diagrams for information systems analysis and design. In this paper we present a bibliographic survey on algorithms whose goal is to produce aesthetically pleasing drawings of graphs. Research on this topic is spread over the broad spectrum of computer science. This bibliography constitutes a first attempt to encompass both theoretical and application-oriented papers from disparate areas.
959 citations
23 Feb 2013
TL;DR: This paper presents a lightweight graph processing framework that is specific for shared-memory parallel/multicore machines, which makes graph traversal algorithms easy to write and significantly more efficient than previously reported results using graph frameworks on machines with many more cores.
Abstract: There has been significant recent interest in parallel frameworks for processing graphs due to their applicability in studying social networks, the Web graph, networks in biology, and unstructured meshes in scientific simulation. Due to the desire to process large graphs, these systems have emphasized the ability to run on distributed memory machines. Today, however, a single multicore server can support more than a terabyte of memory, which can fit graphs with tens or even hundreds of billions of edges. Furthermore, for graph algorithms, shared-memory multicores are generally significantly more efficient on a per core, per dollar, and per joule basis than distributed memory systems, and shared-memory algorithms tend to be simpler than their distributed counterparts.In this paper, we present a lightweight graph processing framework that is specific for shared-memory parallel/multicore machines, which makes graph traversal algorithms easy to write. The framework has two very simple routines, one for mapping over edges and one for mapping over vertices. Our routines can be applied to any subset of the vertices, which makes the framework useful for many graph traversal algorithms that operate on subsets of the vertices. Based on recent ideas used in a very fast algorithm for breadth-first search (BFS), our routines automatically adapt to the density of vertex sets. We implement several algorithms in this framework, including BFS, graph radii estimation, graph connectivity, betweenness centrality, PageRank and single-source shortest paths. Our algorithms expressed using this framework are very simple and concise, and perform almost as well as highly optimized code. Furthermore, they get good speedups on a 40-core machine and are significantly more efficient than previously reported results using graph frameworks on machines with many more cores.
816 citations
02 Jan 1991
TL;DR: In this paper, the authors discuss parallel algorithms for shared-memory machines and discuss the theoretical foundations of parallel algorithms and parallel architectures, and present a theoretical analysis of the appropriate logical organization of a massively parallel computer.
Abstract: Publisher Summary
This chapter discusses parallel algorithms for shared-memory machines. Parallel computation is rapidly becoming a dominant theme in all areas of computer science and its applications. It is estimated that, within a decade, virtually all developments in computer architecture, systems programming, computer applications and the design of algorithms will be taking place within the context of parallel computation. In preparation for this revolution, theoretical computer scientists have begun to develop a body of theory centered on parallel algorithms and parallel architectures. As there is no consensus yet on the appropriate logical organization of a massively parallel computer, and as the speed of parallel algorithms is constrained as much by limits on interprocessor communication as it is by purely computational issues, it is not surprising that a variety of abstract models of parallel computation have been pursued. Closest to the hardware level are the VLSI models, which focus on the technological limits of today's chips, in which gates and wires are packed into a small number of planar layers.
812 citations
Book•
01 Jan 1990TL;DR: A model of parallelism that extends and formalizes the Data-Parallel model on which the Connection Machine and other supercomputers are based is described, and it is argued that data-parallel models are not only practical and can be applied to a surprisingly wide variety of problems, they are also well suited for very-high-level languages and lead to a concise and clear description of algorithms and their complexity.
Abstract: "Vector Models for Data-Parallel Computing "describes a model of parallelism that extends and formalizes the Data-Parallel model on which the Connection Machine and other supercomputers are based. It presents many algorithms based on the model, ranging from graph algorithms to numerical algorithms, and argues that data-parallel models are not only practical and can be applied to a surprisingly wide variety of problems, they are also well suited for very-high-level languages and lead to a concise and clear description of algorithms and their complexity. Many of the author's ideas have been incorporated into the instruction set and into algorithms currently running on the Connection Machine.The book includes the definition of a parallel vector machine; an extensive description of the uses of the scan (also called parallel-prefix) operations; the introduction of segmented vector operations; parallel data structures for trees, graphs, and grids; many parallel computational-geometry, graph, numerical and sorting algorithms; techniques for compiling nested parallelism; a compiler for Paralation Lisp; and details on the implementation of the scan operations.Guy E. Blelloch is an Assistant Professor of Computer Science and a Principal Investigator with the Super Compiler and Advanced Language project at Carnegie Mellon University.Contents: Introduction. Parallel Vector Models. The Scan Primitives. Computational-Geometry Algorithms. Graph Algorithms. Numerical Algorithms. Languages and Compilers. Correction-Oriented Languages. Flattening Nested Parallelism. A Compiler for Paralation Lisp. Paralation-Lisp Code. The Scan Vector Model. Data Structures. Implementing Parallel Vector Models. Implementing the Scan Operations. Conclusions. Glossary.
571 citations
Cites methods from "Parallel tree contraction and its a..."
...This random mate technique is similar to the method discussed by Miller and Reif [75]....
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References
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01 Jan 1963
TL;DR: These notes cover the basic definitions of discrete probability theory, and then present some results including Bayes' rule, inclusion-exclusion formula, Chebyshev's inequality, and the weak law of large numbers.
Abstract: These notes cover the basic definitions of discrete probability theory, and then present some results including Bayes' rule, inclusion-exclusion formula, Chebyshev's inequality, and the weak law of large numbers. 1 Sample spaces and events To treat probability rigorously, we define a sample space S whose elements are the possible outcomes of some process or experiment. For example, the sample space might be the outcomes of the roll of a die, or flips of a coin. To each element x of the sample space, we assign a probability, which will be a non-negative number between 0 and 1, which we will denote by p(x). We require that x∈S p(x) = 1, so the total probability of the elements of our sample space is 1. What this means intuitively is that when we perform our process, exactly one of the things in our sample space will happen. Example. The sample space could be S = {a, b, c}, and the probabilities could be p(a) = 1/2, p(b) = 1/3, p(c) = 1/6. If all elements of our sample space have equal probabilities, we call this the uniform probability distribution on our sample space. For example, if our sample space was the outcomes of a die roll, the sample space could be denoted S = {x 1 , x 2 ,. .. , x 6 }, where the event x i correspond to rolling i. The uniform distribution, in which every outcome x i has probability 1/6 describes the situation for a fair die. Similarly, if we consider tossing a fair coin, the outcomes would be H (heads) and T (tails), each with probability 1/2. In this situation we have the uniform probability distribution on the sample space S = {H, T }. We define an event A to be a subset of the sample space. For example, in the roll of a die, if the event A was rolling an even number, then A = {x 2 , x 4 , x 6 }. The probability of an event A, denoted by P(A), is the sum of the probabilities of the corresponding elements in the sample space. For rolling an even number, we have P(A) = p(x 2) + p(x 4) + p(x 6) = 1 2 Given an event A of our sample space, there is a complementary event which consists of all points in our sample space that are not …
6,236 citations
TL;DR: In this paper, it was shown that the likelihood ratio test for fixed sample size can be reduced to this form, and that for large samples, a sample of size $n$ with the first test will give about the same probabilities of error as a sample with the second test.
Abstract: In many cases an optimum or computationally convenient test of a simple hypothesis $H_0$ against a simple alternative $H_1$ may be given in the following form. Reject $H_0$ if $S_n = \sum^n_{j=1} X_j \leqq k,$ where $X_1, X_2, \cdots, X_n$ are $n$ independent observations of a chance variable $X$ whose distribution depends on the true hypothesis and where $k$ is some appropriate number. In particular the likelihood ratio test for fixed sample size can be reduced to this form. It is shown that with each test of the above form there is associated an index $\rho$. If $\rho_1$ and $\rho_2$ are the indices corresponding to two alternative tests $e = \log \rho_1/\log \rho_2$ measures the relative efficiency of these tests in the following sense. For large samples, a sample of size $n$ with the first test will give about the same probabilities of error as a sample of size $en$ with the second test. To obtain the above result, use is made of the fact that $P(S_n \leqq na)$ behaves roughly like $m^n$ where $m$ is the minimum value assumed by the moment generating function of $X - a$. It is shown that if $H_0$ and $H_1$ specify probability distributions of $X$ which are very close to each other, one may approximate $\rho$ by assuming that $X$ is normally distributed.
3,760 citations
TL;DR: An algorithm for dividing a graph into triconnected components is presented and is both theoretically optimal to within a constant factor and efficient in practice.
Abstract: An algorithm for dividing a graph into triconnected components is presented. When implemented on a random access computer, the algorithm requires $O(V + E)$ time and space to analyze a graph with V vertices and E edges. The algorithm is both theoretically optimal to within a constant factor and efficient in practice.
903 citations
TL;DR: It is shown that arithmetic expressions with n ≥ 1 variables and constants; operations of addition, multiplication, and division; and any depth of parenthesis nesting can be evaluated in time 4 log 2 + 10(n - 1) using processors which can independently perform arithmetic operations in unit time.
Abstract: It is shown that arithmetic expressions with n ≥ 1 variables and constants; operations of addition, multiplication, and division; and any depth of parenthesis nesting can be evaluated in time 4 log2n + 10(n - 1)/p using p ≥ 1 processors which can independently perform arithmetic operations in unit time. This bound is within a constant factor of the best possible. A sharper result is given for expressions without the division operation, and the question of numerical stability is discussed.
864 citations
"Parallel tree contraction and its a..." refers background in this paper
...These bounds are commonly known as Chernoff bounds [ 6 ]....
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29 Oct 1979
TL;DR: Results are derived suggesting that the undirected reachability problem is structurally different from, and easier than, the directed version of NSPACE(logn), an affirmative answer to a question of S. Cook.
Abstract: It is well known that the reachability problem for directed graphs is logspace-complete for the complexity class NSPACE(log n) , and thus holds the key to the open question of whether DSPACE(logn)= NSPACE(logn) ([3,4,5,6]). Here as usual OSPACE(logn) is the class of languages that are accepted in logn space by deterministic Turing Ma chi nes, wh i 1eNSPACE( log n) i s the c1ass 0 f 1anguages that are accepted in log n space by nondeterministic ones. The reachability problem for undirected graphs has also been considered ([5]), but it has remained an open question whether undirected graph reachability is logspace-complete for NSPACE(logn). Here we derive results suggesting that the undirected reachability problem is structurally different from, and easier than, the directed version. These results are an affirmative answer to a question of S. Cook.
757 citations