Sorting network

About: Sorting network is a research topic. Over the lifetime, 802 publications have been published within this topic receiving 27724 citations.

Sorting networks and their applications

Abstract: To achieve high throughput rates today's computers perform several operations simultaneously. Not only are I/O operations performed concurrently with computing, but also, in multiprocessors, several computing operations are done concurrently. A major problem in the design of such a computing system is the connecting together of the various parts of the system (the I/O devices, memories, processing units, etc.) in such a way that all the required data transfers can be accommodated. One common scheme is a high-speed bus which is time-shared by the various parts; speed of available hardware limits this scheme. Another scheme is a cross-bar switch or matrix; limiting factors here are the amount of hardware (an m × n matrix requires m × n cross-points) and the fan-in and fan-out of the hardware.

Introduction to parallel computing: design and analysis of algorithms

Abstract: Introduction. What is Parallel Computing? The Scope of Parallel Computing. Issues in Parallel Computing. Organization and Contents of The Text. Bibliographic Remarks. Problems. References. Models of Parallel Computers. A Taxonomy of Parallel Architectures. An Idealized Parallel Computer. Dynamic Interconnection Networks. Static Interconnection Networks. Embedding Other Networks Into a Hypercube. Routing Mechanisms For Static Networks. Communication Costs in Static Interconnection Networks. Cost-Performance Tradeoffs. Architectural Models For Parallel Algorithm Design. Bibliographic Remarks. References. Basic Communication Operations. Simple Message Transfer Between Two Processors. One-To-All Broadcast. All-To-All Broadcast, Reduction, and Prefix Sums. One-To-All Personalized Communications. All-To-All Personalized Communications. Circular Shift. Faster Methods For Some Communication Operations. Summary. Bibliographic Remarks. Problems. References. Performance and Scalability of Parallel Systems. Performance Metrics For Parallel Systems. The Effect of Granularity and Data Mapping On Performance. The Scalability of Parallel Systems. The Isoefficiency Metric of Scalability. Sources of Parallel Overhead. Minimum Execution Time and Minimum Cost-Optimal Execution Time. Other Scalability Metrics and Bibliographic Remarks. Problems. References. Dense Matrix Algorithms. Mapping Matrices Onto Processors. Matrix Transpositon. Matrix-Vector Multiplication. Matrix Multiplication. Solving a System of Linear Equations. Bibliographic Remarks. Problems. References. Sorting. Issues in Sorting On Parallel Computers. Sorting Networks. Bubble Sort and Its Variants. Quicksort. Other Sorting Algorithms. Bibliographic Remarks. Problems. References. Graph Algorithms. Definitions and Representation. Minimum Spanning Tree: Prim's Algorithm. Single-Source Shortest Paths: Dijkstra's Algorithms. All-Pairs Shortest Paths. Transitive Closure. Connected Components. Algorithms For Sparse Graphs. Bibliographic Remarks. Problems. References. Search Algorithms For Discrete Optimization Problems. Definitions and Examples. Sequential Search Algorithms. Search Overhead Factor. Parallel Depth-First Search. Parallel Best-First Search. Speedup Anomalies in Parallel Search Algorithms. Bibliographic Remarks. Problems. References. Dynamic Programming. Serial Monadic Dp Formulations. Nonserial Monadic Dp Formulations. Serial Polyadic Dp Formulations. Nonserial Polyadic Dp Formulations. Summary and Discussion. Bibliographic Remarks. Problems. References. Fast Fourier Transform. The Serial Algorithm. The Binary-Exchange Algorithm. The Transpose Algorithm. Cost-Effectiveness of Meshes and Hypercubes For Fft. Bibliographic Remarks. Problems. References. Solving Sparse Systems of Linear Equations. Basic Operations. Iterative Methods. Finite Element Method. Direct Methods For Sparse Linear Systems. Multigrid Methods. Bibliographic Remarks. Problems. References. Systolic Algorithms and Their Mapping Onto Parallel Computers. Examples of Systolic Systems. General Issues in Mapping Systolic Systems Onto Parallel Computers. Mapping One-Dimensional Systolic Arrays. Bibliographic Remarks. Problems. References. Parallel Programming. Parallel Programming Paradigms. Primitive For The Message-Passing Programming Paradigm. Data-Parallel Languages. Primitives For The Shared-Address-Space Programming Paradigm. Fortran D. Bibliographic Remarks. References. Appendix A. Complexity of Functions and Order Analysis. Author Index. Subject Index. 0805331700T04062001

The input/output complexity of sorting and related problems

Abstract: We provide tight upper and lower bounds, up to a constant factor, for the number of inputs and outputs (I/OS) between internal memory and secondary storage required for five sorting-related problems: sorting, the fast Fourier transform (FFT), permutation networks, permuting, and matrix transposition. The bounds hold both in the worst case and in the average case, and in several situations the constant factors match. Secondary storage is modeled as a magnetic disk capable of transferring P blocks each containing B records in a single time unit; the records in each block must be input from or output to B contiguous locations on the disk. We give two optimal algorithms for the problems, which are variants of merge sorting and distribution sorting. In particular we show for P = 1 that the standard merge sorting algorithm is an optimal external sorting method, up to a constant factor in the number of I/Os. Our sorting algorithms use the same number of I/Os as does the permutation phase of key sorting, except when the internal memory size is extremely small, thus affirming the popular adage that key sorting is not faster. We also give a simpler and more direct derivation of Hong and Kung's lower bound for the FFT for the special case B = P = O(1).

Access and Alignment of Data in an Array Processor

Abstract: This paper discusses the design of a primary memory system for an array processor which allows parallel, conflict-free access to various slices of data (e.g., rows, columns, diagonals, etc.), and subsequent alignment of these data for processing. Memory access requirements for an array processor are discussed in general terms and a set of common requirements are defined. The ability to meet these requirements is shown to depend on the number of independent memory units and on the mapping of the data in these memories. Next, the need to align these data for processing is demonstrated and various alignment requirements are defined. Hardware which can perform this alignment function is discussed, e.g., permutation, indexing, switching or sorting networks, and a network (the omega network) based on Stone's shuffle-exchange operation [1] is presented. Construction of this network is described and many of its useful properties are proven. Finally, as an example of these ideas, an array processor is shown which allows conflict-free access and alignment of rows, columns, diagonals, backward diagonals, and square blocks in row or column major order, as well as certain other special operations.