Parallel and Distributed Computation: Numerical Methods


Parallel and distributed computing systems offer the promise of a quantum leap in the computing power that can be brought to bear on many important problems whose requirements exceed the capabilities of the most powerful existing or planned serial computers. Whether and to what extent this promise can be fullfilled is still a matter of speculation, but several years of practical experience with both parallel computers and distributed data communication networks have brought about an understanding of the potential and limitations of parallel and distributed computation. The purpose of this book is to promote this understanding by focusing on algorithms that are naturally suited for large scale parallelization and that represent the best hope for solving problems which are much larger than those that can be solved at present.

Work on parallel and distributed computation spans several broad areas, such as the design of parallel machines, parallel programming languages, parallel algorithm development and analysis, and applications related issues. The focus of this book is on algorithms, and, even within this area, we restrict our attention primarily to numerical algorithms for solving systems of equations or optimization problems. Our choice of material is motivated by large problems for which it is essential to harness the power of massive parallelism, while keeping the communication overhead and delays within tolerable limits. Accordingly, we emphasize algorithms that admit a high degree of parallelization such as relaxation methods of the Jacobi and Gauss-Seidel type, and we address extensively issues of communication and synchronization. In particular, we discuss algorithms for interprocessor communication and we provide a comprehensive convergence analysis of asynchronous iterative methods.

The design of parallel algorithms can be pursued at several levels, and this explains to some extent the diversity of the scientific literature on the subject. For example:

We have mostly followed the first approach, concentrating on algorithmic analysis at a rather high level of abstraction. Our choice of algorithms, however, is such that in most cases, the methods of parallel implementation are either evident and straightforward, or else are covered by our broad discussion of parallel computation given in Chapter 1. We have not dealt with implementations in specific machines because types of machines are rapidly changing. Nonetheless, at several points, we have made reference to computations in regular architectures, such as mesh and hypercube, which are widely used. We carry out the analysis of various algorithms in considerable depth, guided by the belief that a thorough understanding of an algorithm is typically essential for its application to a challenging problem.

The book was developed through a course that we taught to graduate students at MIT. It is intended for use in graduate courses in engineering, computer science, operations research, and applied mathematics. We have assumed the equivalent of a first course in linear algebra and a grasp of advanced calculus and real analysis that most students are exposed to by the end of their undegraduate studies. Probabilistic notions beyond the theory of finite-state Markov chains are not needed with the exception of Section 7.8, which deals with stochastic gradient methods. We have not required any background in numerical analysis, graph algorithms, optimization, convexity, and duality, and we have consequently developed this material as needed in the main body of the text or the appendixes. We note, however, that the mathematically mature reader who has some background in some of these fields is likely to benefit more from the book, and to gain a deeper appreciation of the material.

The book can be used for several types of courses. One possibility is a course targeted on parallel algorithms, and intended for students who already have some knowledge of a subset of the fields of numerical analysis, graph theory, and optimization algorithms. Furthermore, such a course could have either a computer science flavor, by focusing on Chapters 1 and 8, and parts of Chapters 2, and 4 through 6 or alternatively, a numerical computation flavor by focusing on Chapters 2, 3, and parts of Chapters 1 and 4 through 7. Another possibility is a general course on numerical methods with a strong bias towards parallelizable algorithms. The book lends itself for such a course because it develops economically a broad variety of self-contained basic material in considerable depth.

Chapter 1 contains an exposition of some generic issues that arise in a broad variety of parallel algorithms and are best addressed without specific reference to any particular algorithm. In particular, we discuss the scheduling of a set of processors for the parallel execution of a prescribed algorithm, some basic issues relating to interprocessor communication and the effects of the communication penalty on the amount by which an algorithm can be speeded up. Special attention is paid on a few interesting architectures such as mesh and hypercube. We then consider issues of synchronization, and we contrast synchronous and asynchronous algorithms. In this chapter, we also introduce briefly relaxation methods of the Gauss-Seidel and Jacobi type and some of their associated issues of parallelization, communication, and synchronization that are recurring themes throughout the book.

Chapter 2 deals with parallel algorithms for systems of linear equations and matrix inversion. It covers direct methods, for general systems as well as systems with special structure, iterative methods, including their convergence analysis, and the conjugate gradient method.

Chapter 3 is devoted to iterative methods for nonlinear problems, such as finding fixed points of contraction mappings, unconstrained and constrained optimization, and variational inequalities. The convergence theory for such methods is developed in an economical way and emphasizes the case of Cartesian product constraint sets (in a primal and a dual setting), which lends itself naturally to parallelization and decomposition.

Chapter 4 deals with the shortest path problem and other, more general, dynamic programming problems. The dynamic programming algorithm can be viewed as a relaxation method and lends itself well for parallelization. We establish (and strengthen somewhat) the classical results for discounted and undiscounted Markovian decision problems, and we also discuss the associated parallel computation issues.

Chapter 5 is devoted to network flow problems. In the first four sections we deal with the important class of linear problems, and we present some easily parallelizable algorithms, that are conceptually related to the Gauss-Seidel and Jacobi relaxation methods. We then discuss related algorithms for network problems with nonlinear convex cost. The methods of the first five sections can be viewed as relaxation methods applied in a space of dual (price) variables. In the last section we consider relaxation-like methods applied to nonlinear multicommodity flow problems in the primal space of flow variables.

The last three chapters deal with asynchronous algorithms in which each processor computes at its own pace, using intermediate results of other processors that are possibly outdated due to unpredictable communication delays. Among other topics, we develop asynchronous versions of all the major types of synchronous parallel algorithms that were discussed in previous chapters.

In Chapter 6, we introduce a general class of asynchronous iterative methods (called "totally asynchronous"), and we develop a general theorem for proving their convergence. This theorem is used repeatedly to establish the validity of a broad variety of asynchronous algorithms involving iteration mappings that are either monotone or contractions with respect to a weighted maximum norm. In particular, we show convergence of linear and nonlinear iterations involving weighted maximum norm contractions arising in the solution of systems of equations, discounted dynamic programming, unconstrained and constrained optimization, and variational inequalities. We also discuss iterations involving monotone mappings arising in shortest path problems, undiscounted dynamic programming, and linear and nonlinear network flow problems.

In Chapter 7, we consider "partially asynchronous" algorithms in which some mild restrictions are placed on the amount of asynchronism present. We prove convergence of a variety of algorithms for fixed points of nonexpansive mappings, detrministic and stochastic optimization, Markov chains, load balancing in a computer network, and optimal routing in data networks.

Chapter 8 is similar in philosophy with Chapter 1 in that it deals with generic issues of parallel and distributed computation. It discusses the organization of an inherently asynchronous network of processors for the purpose of executing a general type of parallel algorithm. It addresses issues like termination detection, processor scheduling, methods for taking a "snapshot" of the global state of a computing system, synchronization via "rollback", and methods for maintaining communication with a center in the face of topological changes.

Several subjects of the book can be covered independently of each other, thereby allowing the reader or an instructor to use material selectively to suit his/her needs. For example the following groups of sections can each be omitted without loss of continuity:

Each major section contains several exercises that, for the most part, illustrate and supplement the theoretical developments of the text. They include algorithmic variations, convergence and complexity analysis, examples, and counterexamples. Some of the exercises are quite challenging, occasionally representing original research. The serious reader will benefit a great deal from these exercises, which constitute one of the principal components of the text. Solutions of all the exercises are provided in a manual that will be available to instructors.

A substantial portion of our material is recent, and presents subjects that have not been addressed in other textbooks. This includes most of the last two sections of Chapter 1, much of the last two sections of Chapter 3, Sections 5.2 through 5.5, and the entire Chapters 6 through 8. Some of the material presented is original research that was conducted as the textbook was being written and has not yet been published elsewhere.

The literature on our subject is trully enormous, and our references are not comprehensive. We thus apologize in advance to the many authors whose work has not been cited. We have restricted ourselves to listing the sources that we have used, together with a selection of sources that contain material supplementing the text.

We are thankful to a number of individuals and institutions for their help. The inquisitive spirit of our students motivated us to think harder about many issues. We learned a great deal about distributed computation through our long association and collaboration with Bob Gallager and Pierre Humblet. We have appreciated our research collaboration with Michael Athans, David Castanon, Jon Eckstein, Eli Gafni, and Christos Papadimitriou, that produced some of the material included in the book. Tom Luo and Cuneyt Ozveren contributed research material that was incorporated in exercises. We are thankful for the helpful comments of a number of people, including Chee-Seng Chow, George Cybenko, Stratis Gallopoulos, George Hart, and Tom Richardson. Our greatest debt of gratitude to a single individual goes to Paul Tseng who worked closely with us on several of the topics presented, particularly the communication algorithms of Section 1.3, the network flow algorithms of Chapter 5, and the partially asynchronous algorithms of Section 7.2. In addition, Paul reviewed the entire manuscript, sharpened several proofs and results, and contributed much original research in the form of exercises. We were fortunate to work at the Laboratory for Information and Decision Systems of M.I.T., which provided us with a stimulating research environment. Funding for our research was provided by the Army Research Office through the Center for Intelligent Control Systems, Bellcore Inc., the National Science Foundation, and the Office of Naval Research.

Dimitri P. Bertsekas

John N. Tsitsiklis

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