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Parallel Algorithms

Parallel Algorithm is the integral part of the computer science which is useful in processing huge volumes of data in quick time by dividing a problem into sub-problems and executed simultaneously on many different processing devices and combined together again to get the final desired result.  It focuses in the various terms like Parallelism, Concurrent Processing and Model of Computation etc.

Few Topics are:

•  Parallel Computation
• PRAM techniques
• Doubling Technique
• Summation trees and prefix summation
• Interconnection networks
• Sorting and Searching
• Pointer-based algorithms
• List ranking
• Tree contraction
• Connected components
• Minimum spanning tree
• Geometric Algorithms
• Distributed method

A parallel algorithm is an algorithm used for the execution of two or more processes in a computer. The operations in a parallel algorithm can be performed on multiple processors as well as on a single-processor by the use of multiple functional units, pipelined functions , pipelined memory systems.

The following parameters are considered while analyzing a parallel algorithm :-

• Total no. of processors
• Time complexity
• Total Cost

Parallel models are divided into two classes:

• Multiprocessor models: There are three types of multiprocessor models distinguished on the basis of memory access. They are: Local memory machine models ,modular memory machine models, Parallel random-access machine model.
• Work depth models: This model helps to calculate the parallelism of the algorithm. The work of an algorithm ‘W’ is the number of operations performed and ‘D’ is the depth of dependencies among various operations . This gives the parallelism P=W/D of the algorithm.
• Parallel programming in parallel algorithm makes use of different data structures like linked lists, arrays and hypercube network. Searching is the most important operation that is used to find the element in a list
• The number of searching techniques that are used to look for the element are divide and conquer, DFS,BFS, best- first search. And in order to arrange elements in ascending, descending or alphabetical order, sorting process is used. The various sorting techniques are : merge sort, enumeration sort, hyper quick sort.

It is defined as use of multiple compute resources simultaneously to solve a problem. It can be done by using multiple processors to run a program.

PROCESS: A problem is broken down into discrete parts so that they can be solved concurrently and each part is broken down further to a series of instructions and each Instruction  execute simultaneously on different processors & an overall coordination mechanism is also employed for proper coordination.

Practical use of parallel computing:

a)      Science & engineering

b)      Industry

Why parallel computing is used?

a)      It saves both time & money

b)      It solves large and complex problems

c)      It provides concurrency

Parallel computing can be achieved using different ways:

• SISD ( Single instruction Stream Single Data Stream ):  It is a serial computer where only one data stream is sent as an input on the CPU in one clock cycle and only one instruction stream is output in one clock cycle. It is the oldest system.
• SIMD (single instruction Stream multiple Data Stream):It is a type of parallel computer where all processing units execute the same instruction in one clock cycle and each processing unit can operate on different data items.
• MISD( multiple instruction Stream Single Data Stream): It is type of parallel computer where each processing unit operates on the data independently & a single data stream is fed as an input into multiple processing units. Example: Carnegie-Mellon C.mmp computer (1971).
• MIMD ( multiple instruction Stream multiple Data Stream): It is a type of parallel computer. In this every processor execute a different instruction stream & each processor work with different data stream.

PARALLEL ALGORITHMS

A parallel algorithm can be defined as an algorithm which makes use of concurrency for solving a problem in lesser time. Concurrency refers to a concept which means occurrence as well as processing of multiple activities at the same time.

An algorithm is designed in such a way that it runs on a parallel computer efficiently. A parallel algorithm can consists of a large number of arithmetic operations. However, its designing is done in such a way that many arithmetic operations are not dependent on each other & have the ability to be performed in simultaneously. In this algorithm, a piece can be executed on various processing devices & after that again put back finally for obtaining accurate result. It consists of many tasks that are executed simultaneously on various processors & is capable of communicating with one another.

For the analysis of parallel algorithms, time complexities, processor complexity and work complexity

The parallel computing is important for the application domains which are of large size and also because of the physical limitations of VLSI circuits. The main reason to design & use a parallel algorithm is decreasing entire process running time. Some applications include simulation of protein folding, weather forecasting & computational physics.

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• Basic Techniques, Lists,Trees, Searching, Sorting,Graphs, PRAM Limitations, Performance Analysis Methodology, Eos, Parallel Programming and Performance Models, Amdahl's Law, PRAM, CREW, EREW, Network Models, Hypercube, Mesh, Fat Trees
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The Design and Analysis of Parallel Algorithms

• Models of parallel computation,Generalities,Sorting on an EREW-SIMD PRAM computer,Bitonic Sorting Algorithm,Relations between PRAM models
• \Complexity Classes and the Parallel Processing Thesis,P-Completeness and Inherently Sequential Problems,Brent’s Theorem
• SIMD Algorithms,Doubling Algorithms,The Brent Scheduling Principle.,Pipelining,Divide and Conquer,MIMD Algorithms,Generalities,Race-conditions
• Optimization of loops,Deadlocks,Comparison of the SIMD and MIMD models of computation
• Distributed-Memory Models,Generic Parallel Algorithms,The Butterfly Network,Description,Cube-Connected Cycles,Dataflow Computers,
• Existing Parallel Computers,Asynchronous Parallel Programming,Portable Programming Packages, Linda, Automatically Parallelizing Compilers
• SIMD Programming: the Connection Machine, Generalities, Algorithm-Design Considerations, The CProgramming Language, Running a C program,Semantics of C
• Shapes and parallel allocation of data, Parallel Types and Procedures,Parallel Operators,Sample programs,Pointers
• Subtleties of Communication,Collisions, A Critique of C,Programming a MIMD-SIMD Hybrid Computer, Data declaration, Elementary data types,Parallel data types.
• The FORALL Statement.,The asynchronous case, The synchronous case,A Critique of Modula,Numerical Algorithms

Design and Analysis of Parallel Algorithms

• Linear algebra,Matrix-multiplication, Systems of linear equations, Generalities on vectors and matrices, Jacobi Method, JOR method
• SOR and Consistently Ordered Methods, Power-series methods: the Pan-Reif Algorithm,The main algorithm, Proof of the main result
• Nonlinear Problems,A Parallel Algorithm for Computing Determinants,Discrete Fourier Transform,Fast Fourier Transform Algorithm,Eigenvalues of cyclic matrices
• JPEG Algorithm,Wavelets,Discrete Wavelet Transforms,Numerical Evaluation of Definite Integrals
• One-dimensional case,Higher-dimensional integrals,Partial Differential Equations,Elliptic Differential Equations,Self-adjoint equations,Rate of convergence
• Parabolic Differential Equations,Error Analysis,Implicit Methods,Error Analysis,Hyperbolic Differential Equations
• Finite differences,A Survey of Symbolic Algorithms,Doubling Algorithms,General Principles,Recurrence Relations,Deterministic Finite Automata
• Parallel Interpolation,Euler Tour Algorithm,Parallel Tree Contractions,Shortest Paths,Connected Components,Algorithm for a CREW Computer
• Algorithm for a CRCW computer,Spanning Trees and Forests,An algorithm for an inverted spanning forest,Minimal Spanning Trees and Forests
• Cycles in a Graph,A simple algorithm for a cycle basis,Lowest Common Ancestors,Parsing and the Evaluation of arithmetic expressions
• algorithm for building syntax-trees,An algorithm for evaluating a syntax tree,Parallel searching,Sorting Algorithms for a PRAM computer,Cole Sorting Algorithm — CREW version
• Cole Sorting Algorithm — EREW version,Detailed proof of the correctness of the algorithm,Expander Graphs,Number-Theoretic Considerations
• Probabilistic Algorithms,Numerical algorithms,Monte Carlo Integration,Monte Carlo algorithms,Las Vegas algorithms, The class textbfRNC,Work-efficient parallel prefix computation, The Valiant and Brebner Sorting Algorithm
• Maximal Matchings in Graphs,A Partitioning Lemma, Perfect Matchings,The General Case,The Maximal Independent-Set Problem
Parallel algorithms

• Shared-memory
• Distributed-memory
• Gpgpu-based high-end computing systems
• Computational geometry
• Image analysis
• Graph theory