This document discusses algorithms and parallel processing. It begins by defining algorithms and different types of algorithms like sequential and parallel algorithms. It then discusses analyzing parallel algorithms based on time complexity, number of processors required, and overall cost. Specific examples of parallel algorithms discussed include merge sort and parallel image processing. Fault tolerance in parallel systems is also covered, including load distribution, parallel region growing for image segmentation, and the process of system recovery from faults.
This document discusses parallel algorithms for sorting. It begins by defining parallel algorithms and explaining that the lower bound for comparison-based sorting of n elements is ฮ(n log n). It then discusses several parallel sorting algorithms: odd-even transposition sort on a linear array, quicksort, and sorting networks. It also covers sorting on different parallel models like CRCW, CREW, and EREW. An example is provided of applying an EREW sorting algorithm to a sample data set by recursively dividing it into subsequences until single elements remain to be sorted locally.
This document describes sequential and parallel quicksort algorithms. Sequential quicksort has average complexity of O(n log n) but worst case of O(n^2). Parallel quicksort partitions the array, sorts subarrays in parallel threads, and waits for threads to complete to improve performance over sequential quicksort. It uses a queue to store threads if processors are busy and chooses split keys through various methods like first element or median. New threads are only created if the subarray size is above a minimum to reduce overhead.
This document discusses four parallel searching algorithms: Alpha-Beta search, Jamboree search, Depth-First search, and PVS search. Alpha-Beta search prunes unpromising branches without missing better moves. Jamboree search parallelizes the testing of child nodes. Depth-First search recursively explores branches until reaching a dead end, then backtracks. PVS search splits the search tree across processors, backing up values in parallel at each level. However, load imbalance can occur if some branches are much larger than others.
The document summarizes a seminar presentation on parallel random access machine (PRAM) algorithms. It discusses the computational model of PRAM, algorithms like merging and sorting using odd-even merge. It also covers applications such as computing convex hulls and mentions that PRAM is a source of inspiration for parallel algorithms.
The document proposes a new parallel sorting algorithm called AA-sort that is suitable for multi-core SIMD processors. AA-sort avoids unaligned memory accesses to fully utilize SIMD instructions. It sorts data in-core using an optimized combsort and then merges sorted blocks out-of-core in parallel. Experimental results on PowerPC and Cell processors show AA-sort has better scalability and performance than other algorithms both sequentially and in parallel.
This document discusses parallel algorithms and models of parallel computation. It begins with an overview of parallelism and the PRAM model of computation. It then discusses different models of concurrent versus exclusive access to shared memory. Several parallel algorithms are presented, including list ranking in O(log n) time using an EREW PRAM algorithm and finding the maximum of n elements in O(1) time using a CRCW PRAM algorithm. It analyzes the performance of EREW versus CRCW models and shows how to simulate a CRCW algorithm using EREW in O(log p) time using p processors.
This document discusses parallel algorithms for linear algebra operations. It begins by defining parallel algorithms and linear algebra. It then describes dense matrix algorithms like matrix-vector multiplication and solving systems of linear equations using Gaussian elimination. It presents the serial algorithms for these operations and discusses parallel implementations using 1D row-wise partitioning among processes. It analyzes the computation and communication costs of the parallel Gaussian elimination algorithm.
Parallel algorithms can be specifically written to execute on computers with multiple processors. They are often modeled using parallel random-access machines (PRAM) which allow for an unlimited number of processors that can access shared memory uniformly. Common parallel algorithms include matrix multiplication, merge sort, and shortest path algorithms like Floyd's algorithm.
This document discusses parallel algorithms for sorting. It begins by defining parallel algorithms and explaining that the lower bound for comparison-based sorting of n elements is ฮ(n log n). It then discusses several parallel sorting algorithms: odd-even transposition sort on a linear array, quicksort, and sorting networks. It also covers sorting on different parallel models like CRCW, CREW, and EREW. An example is provided of applying an EREW sorting algorithm to a sample data set by recursively dividing it into subsequences until single elements remain to be sorted locally.
This document describes sequential and parallel quicksort algorithms. Sequential quicksort has average complexity of O(n log n) but worst case of O(n^2). Parallel quicksort partitions the array, sorts subarrays in parallel threads, and waits for threads to complete to improve performance over sequential quicksort. It uses a queue to store threads if processors are busy and chooses split keys through various methods like first element or median. New threads are only created if the subarray size is above a minimum to reduce overhead.
This document discusses four parallel searching algorithms: Alpha-Beta search, Jamboree search, Depth-First search, and PVS search. Alpha-Beta search prunes unpromising branches without missing better moves. Jamboree search parallelizes the testing of child nodes. Depth-First search recursively explores branches until reaching a dead end, then backtracks. PVS search splits the search tree across processors, backing up values in parallel at each level. However, load imbalance can occur if some branches are much larger than others.
The document summarizes a seminar presentation on parallel random access machine (PRAM) algorithms. It discusses the computational model of PRAM, algorithms like merging and sorting using odd-even merge. It also covers applications such as computing convex hulls and mentions that PRAM is a source of inspiration for parallel algorithms.
The document proposes a new parallel sorting algorithm called AA-sort that is suitable for multi-core SIMD processors. AA-sort avoids unaligned memory accesses to fully utilize SIMD instructions. It sorts data in-core using an optimized combsort and then merges sorted blocks out-of-core in parallel. Experimental results on PowerPC and Cell processors show AA-sort has better scalability and performance than other algorithms both sequentially and in parallel.
This document discusses parallel algorithms and models of parallel computation. It begins with an overview of parallelism and the PRAM model of computation. It then discusses different models of concurrent versus exclusive access to shared memory. Several parallel algorithms are presented, including list ranking in O(log n) time using an EREW PRAM algorithm and finding the maximum of n elements in O(1) time using a CRCW PRAM algorithm. It analyzes the performance of EREW versus CRCW models and shows how to simulate a CRCW algorithm using EREW in O(log p) time using p processors.
This document discusses parallel algorithms for linear algebra operations. It begins by defining parallel algorithms and linear algebra. It then describes dense matrix algorithms like matrix-vector multiplication and solving systems of linear equations using Gaussian elimination. It presents the serial algorithms for these operations and discusses parallel implementations using 1D row-wise partitioning among processes. It analyzes the computation and communication costs of the parallel Gaussian elimination algorithm.
Parallel algorithms can be specifically written to execute on computers with multiple processors. They are often modeled using parallel random-access machines (PRAM) which allow for an unlimited number of processors that can access shared memory uniformly. Common parallel algorithms include matrix multiplication, merge sort, and shortest path algorithms like Floyd's algorithm.
From the perspective of Design and Analysis of Algorithm. I made these slide by collecting data from many sites.
I am Danish Javed. Student of BSCS Hons. at ITU Information Technology University Lahore, Punjab, Pakistan.
This document discusses parallel algorithms for merging sorted arrays and ranking elements in a linked list. It presents:
1. A parallel merging algorithm that partitions the arrays into blocks, ranks the elements within matching blocks in parallel, and sequentially merges the smaller blocks. This takes O(log n) time using O(n) processors.
2. Two list ranking algorithms - one in O(log n) time and O(n log n) work, and an optimal one in O(log n log log n) time and O(n) work using independent sets and pointer jumping.
This document discusses parallel matrix multiplication algorithms on the Parallel Random Access Machine (PRAM) model. It describes algorithms that multiply matrices using different numbers of processors, from n3 processors down to n2 processors. The time complexity is O(log n) in all cases, while the processor and work complexities vary based on the number of processors. Block matrix multiplication is also introduced as a more efficient approach for shared memory machines by improving data locality.
These slides are about parallel sorting algorithms. In which four types of sorting algorithms are discussed with the comparison between their sequential and parallel ways. The four algorithms which are included are: Bubble sort, merge sort, Bitonic sort and Shear sort.
Elementary Parallel Algorithm - Sum of n numbers on Hypercube, Shuffle Exchange and Mesh SIMD computers, UMA multiprocessors, Broadcasting and pre-fix sum on multicomputer.
The document discusses the Random Access Machine (RAM) model of computation. The RAM model allows algorithms to be measured in a machine-independent way based on performance. It assumes a single processor where instructions are executed sequentially with no concurrency. The running time of an algorithm is the number of time steps, and space used is the number of memory cells. The RAM makes assumptions like each operation taking one time step and unlimited memory access.
The document discusses parallel algorithms and their analysis. It introduces a simple parallel algorithm for adding n numbers using log n steps. Parallel algorithms are analyzed based on their time complexity, processor complexity, and work complexity. For adding n numbers in parallel, the time complexity is O(log n), processor complexity is O(n), and work complexity is O(n log n). The document also discusses models of parallel computation like PRAM and designs of parallel architectures like meshes and hypercubes.
This document describes a parallel algorithm for computing prefix sums and merging sorted arrays. It presents:
1) A parallel prefix algorithm that computes prefix sums in O(log n) time using O(n) processors by performing prefix computations in parallel across a binary tree representation.
2) A parallel merging algorithm that partitions input arrays into blocks of size O(log n), ranks block elements in parallel, and sequentially merges matching blocks, achieving O(log n) time and O(n) processors.
3) An analysis showing the parallel prefix algorithm runs in O(log n) time using O(n) processors, and two sorted arrays can be merged in O(log n) time using O(n
The document describes algorithms for merging two sorted sequences in parallel. It discusses merging algorithms for the CREW and EREW models of parallel computation. For the CREW model, it presents a parallel merging algorithm that uses binary search. For the EREW model, it presents an adaptive and cost-optimal parallel merging algorithm that recursively finds the median element to partition the sequences.
The document discusses cost optimal parallel algorithms. It defines the cost of an algorithm as the product of its parallel time complexity and number of processors used. A cost optimal algorithm has the same complexity class as an optimal sequential algorithm. Several common problems like prefix sum and list ranking are shown to not have cost optimal parallel solutions. The document then discusses parallel reduction algorithms, proving using Brent's Theorem that a cost optimal parallel reduction algorithm exists with time complexity of log(n) and n/log(n) processors. An example of summing n numbers optimally in parallel is provided.
This document discusses parallel programming languages, including Fortran 90 and Sequent C. It provides an overview of parallel programming concepts like multicomputers, multiprocessors, and uniform memory access. It also gives examples of calculating variance and integration in parallel and describes language features for parallel programming like process creation, communication, and synchronization using barriers and mutual exclusion.
Parallel algorithms can be specifically written to execute on computers with multiple processing units. When designing parallel algorithms, the cost of communication and number of processors must be considered. Parallel algorithms are often modeled using the parallel random-access machine (PRAM) model, which simplifies issues like synchronization and communication. Common parallel algorithms include matrix multiplication, merge sort, and shortest path algorithms like Floyd's algorithm.
Performance analysis(Time & Space Complexity)swapnac12
ย
The document discusses algorithms analysis and design. It covers time complexity and space complexity analysis using approaches like counting the number of basic operations like assignments, comparisons etc. and analyzing how they vary with the size of the input. Common complexities like constant, linear, quadratic and cubic are explained with examples. Frequency count method is presented to determine tight bounds of time and space complexity of algorithms.
Lecture 3 insertion sort and complexity analysisjayavignesh86
ย
This document discusses algorithms and insertion sort. It begins by defining time complexity as the amount of computer time required by an algorithm to complete. Time complexity is measured by the number of basic operations like comparisons, not in physical time units. The document then discusses how to calculate time complexity by counting the number of times loops and statements are executed. It provides examples of calculating time complexities of O(n) for a simple for loop and O(n^2) for a nested for loop. Finally, it introduces insertion sort and divide-and-conquer algorithms.
The document discusses parallel algorithms and parallel computing. It begins by defining parallelism in computers as performing more than one task at the same time. Examples of parallelism include I/O chips and pipelining of instructions. Common terms for parallelism are defined, including concurrent processing, distributed processing, and parallel processing. Issues in parallel programming such as task decomposition and synchronization are outlined. Performance issues like scalability and load balancing are also discussed. Different types of parallel machines and their classification are described.
The document provides an overview of algorithms, including definitions, types, characteristics, and analysis. It begins with step-by-step algorithms to add two numbers and describes the difference between algorithms and pseudocode. It then covers algorithm design approaches, characteristics, classification based on implementation and logic, and analysis methods like a priori and posteriori. The document emphasizes that algorithm analysis estimates resource needs like time and space complexity based on input size.
Algorithm And analysis Lecture 03& 04-time complexity.Tariq Khan
ย
This document discusses algorithm efficiency and complexity analysis. It defines key terms like algorithms, asymptotic complexity, Big O notation, and different complexity classes. It provides examples of analyzing time complexity for different algorithms like loops, nested loops, and recursive functions. The document explains that Big O notation allows analyzing algorithms independent of machine or input by focusing on the highest order term as the problem size increases. Overall, the document introduces methods for measuring an algorithm's efficiency and analyzing its time and space complexity asymptotically.
1) The document presents various algorithms for efficiently transposing matrices while minimizing memory accesses and cache misses.
2) It analyzes the algorithms under different memory models: RAM, I/O, cache, and cache-oblivious. The block transpose, half/full copying, and Morton layout algorithms improve performance by reusing data blocks.
3) Experimental results on a 300MHz system show the Morton layout and half copying algorithms have the fastest runtimes due to minimizing data references, L1 misses, and TLB misses. The relative performance of algorithms depends on cache miss latency.
All Pair Shortest Path Algorithm โ Parallel Implementation and AnalysisInderjeet Singh
ย
This project report discusses the parallel implementation of the all pair shortest path algorithm using MPI and OpenMP. The algorithm was implemented by decomposing the adjacency matrix row-wise across processes. Results show that the parallel algorithm achieves speedup over the sequential version, especially for large graph sizes. The MPI implementation performed better than the OpenMP version. Graphs in the report compare execution time, speedup, and efficiency for different problem sizes and numbers of processes/threads.
This document describes a parallel k-means clustering algorithm implemented with MPI. The algorithm partitions data points into k clusters based on their distances from centroid points. It was tested with various configurations of centroids, data points, processors, and tasks. Testing showed that processing time increases dramatically with more centroids and data, but only slowly increases or decreases with more processors and tasks, up to a point where adding more processors results in too little data per processor. The algorithm iterates until the difference between new and old centroid positions is minimal, converging on the cluster solutions.
El documento describe la importancia de las tecnologรญas de la informaciรณn y la comunicaciรณn (TIC) para las pequeรฑas y medianas empresas mexicanas. El uso de las TIC puede aumentar el crecimiento y la competitividad de una empresa al permitir la automatizaciรณn de tareas, el acceso a bases de datos, la mejora de procesos y la promociรณn a travรฉs de un sitio web. No emplear las TIC puede resultar en rezago para una empresa debido a limitaciones como conocimiento, financiamiento y habilidades limitadas.
From the perspective of Design and Analysis of Algorithm. I made these slide by collecting data from many sites.
I am Danish Javed. Student of BSCS Hons. at ITU Information Technology University Lahore, Punjab, Pakistan.
This document discusses parallel algorithms for merging sorted arrays and ranking elements in a linked list. It presents:
1. A parallel merging algorithm that partitions the arrays into blocks, ranks the elements within matching blocks in parallel, and sequentially merges the smaller blocks. This takes O(log n) time using O(n) processors.
2. Two list ranking algorithms - one in O(log n) time and O(n log n) work, and an optimal one in O(log n log log n) time and O(n) work using independent sets and pointer jumping.
This document discusses parallel matrix multiplication algorithms on the Parallel Random Access Machine (PRAM) model. It describes algorithms that multiply matrices using different numbers of processors, from n3 processors down to n2 processors. The time complexity is O(log n) in all cases, while the processor and work complexities vary based on the number of processors. Block matrix multiplication is also introduced as a more efficient approach for shared memory machines by improving data locality.
These slides are about parallel sorting algorithms. In which four types of sorting algorithms are discussed with the comparison between their sequential and parallel ways. The four algorithms which are included are: Bubble sort, merge sort, Bitonic sort and Shear sort.
Elementary Parallel Algorithm - Sum of n numbers on Hypercube, Shuffle Exchange and Mesh SIMD computers, UMA multiprocessors, Broadcasting and pre-fix sum on multicomputer.
The document discusses the Random Access Machine (RAM) model of computation. The RAM model allows algorithms to be measured in a machine-independent way based on performance. It assumes a single processor where instructions are executed sequentially with no concurrency. The running time of an algorithm is the number of time steps, and space used is the number of memory cells. The RAM makes assumptions like each operation taking one time step and unlimited memory access.
The document discusses parallel algorithms and their analysis. It introduces a simple parallel algorithm for adding n numbers using log n steps. Parallel algorithms are analyzed based on their time complexity, processor complexity, and work complexity. For adding n numbers in parallel, the time complexity is O(log n), processor complexity is O(n), and work complexity is O(n log n). The document also discusses models of parallel computation like PRAM and designs of parallel architectures like meshes and hypercubes.
This document describes a parallel algorithm for computing prefix sums and merging sorted arrays. It presents:
1) A parallel prefix algorithm that computes prefix sums in O(log n) time using O(n) processors by performing prefix computations in parallel across a binary tree representation.
2) A parallel merging algorithm that partitions input arrays into blocks of size O(log n), ranks block elements in parallel, and sequentially merges matching blocks, achieving O(log n) time and O(n) processors.
3) An analysis showing the parallel prefix algorithm runs in O(log n) time using O(n) processors, and two sorted arrays can be merged in O(log n) time using O(n
The document describes algorithms for merging two sorted sequences in parallel. It discusses merging algorithms for the CREW and EREW models of parallel computation. For the CREW model, it presents a parallel merging algorithm that uses binary search. For the EREW model, it presents an adaptive and cost-optimal parallel merging algorithm that recursively finds the median element to partition the sequences.
The document discusses cost optimal parallel algorithms. It defines the cost of an algorithm as the product of its parallel time complexity and number of processors used. A cost optimal algorithm has the same complexity class as an optimal sequential algorithm. Several common problems like prefix sum and list ranking are shown to not have cost optimal parallel solutions. The document then discusses parallel reduction algorithms, proving using Brent's Theorem that a cost optimal parallel reduction algorithm exists with time complexity of log(n) and n/log(n) processors. An example of summing n numbers optimally in parallel is provided.
This document discusses parallel programming languages, including Fortran 90 and Sequent C. It provides an overview of parallel programming concepts like multicomputers, multiprocessors, and uniform memory access. It also gives examples of calculating variance and integration in parallel and describes language features for parallel programming like process creation, communication, and synchronization using barriers and mutual exclusion.
Parallel algorithms can be specifically written to execute on computers with multiple processing units. When designing parallel algorithms, the cost of communication and number of processors must be considered. Parallel algorithms are often modeled using the parallel random-access machine (PRAM) model, which simplifies issues like synchronization and communication. Common parallel algorithms include matrix multiplication, merge sort, and shortest path algorithms like Floyd's algorithm.
Performance analysis(Time & Space Complexity)swapnac12
ย
The document discusses algorithms analysis and design. It covers time complexity and space complexity analysis using approaches like counting the number of basic operations like assignments, comparisons etc. and analyzing how they vary with the size of the input. Common complexities like constant, linear, quadratic and cubic are explained with examples. Frequency count method is presented to determine tight bounds of time and space complexity of algorithms.
Lecture 3 insertion sort and complexity analysisjayavignesh86
ย
This document discusses algorithms and insertion sort. It begins by defining time complexity as the amount of computer time required by an algorithm to complete. Time complexity is measured by the number of basic operations like comparisons, not in physical time units. The document then discusses how to calculate time complexity by counting the number of times loops and statements are executed. It provides examples of calculating time complexities of O(n) for a simple for loop and O(n^2) for a nested for loop. Finally, it introduces insertion sort and divide-and-conquer algorithms.
The document discusses parallel algorithms and parallel computing. It begins by defining parallelism in computers as performing more than one task at the same time. Examples of parallelism include I/O chips and pipelining of instructions. Common terms for parallelism are defined, including concurrent processing, distributed processing, and parallel processing. Issues in parallel programming such as task decomposition and synchronization are outlined. Performance issues like scalability and load balancing are also discussed. Different types of parallel machines and their classification are described.
The document provides an overview of algorithms, including definitions, types, characteristics, and analysis. It begins with step-by-step algorithms to add two numbers and describes the difference between algorithms and pseudocode. It then covers algorithm design approaches, characteristics, classification based on implementation and logic, and analysis methods like a priori and posteriori. The document emphasizes that algorithm analysis estimates resource needs like time and space complexity based on input size.
Algorithm And analysis Lecture 03& 04-time complexity.Tariq Khan
ย
This document discusses algorithm efficiency and complexity analysis. It defines key terms like algorithms, asymptotic complexity, Big O notation, and different complexity classes. It provides examples of analyzing time complexity for different algorithms like loops, nested loops, and recursive functions. The document explains that Big O notation allows analyzing algorithms independent of machine or input by focusing on the highest order term as the problem size increases. Overall, the document introduces methods for measuring an algorithm's efficiency and analyzing its time and space complexity asymptotically.
1) The document presents various algorithms for efficiently transposing matrices while minimizing memory accesses and cache misses.
2) It analyzes the algorithms under different memory models: RAM, I/O, cache, and cache-oblivious. The block transpose, half/full copying, and Morton layout algorithms improve performance by reusing data blocks.
3) Experimental results on a 300MHz system show the Morton layout and half copying algorithms have the fastest runtimes due to minimizing data references, L1 misses, and TLB misses. The relative performance of algorithms depends on cache miss latency.
All Pair Shortest Path Algorithm โ Parallel Implementation and AnalysisInderjeet Singh
ย
This project report discusses the parallel implementation of the all pair shortest path algorithm using MPI and OpenMP. The algorithm was implemented by decomposing the adjacency matrix row-wise across processes. Results show that the parallel algorithm achieves speedup over the sequential version, especially for large graph sizes. The MPI implementation performed better than the OpenMP version. Graphs in the report compare execution time, speedup, and efficiency for different problem sizes and numbers of processes/threads.
This document describes a parallel k-means clustering algorithm implemented with MPI. The algorithm partitions data points into k clusters based on their distances from centroid points. It was tested with various configurations of centroids, data points, processors, and tasks. Testing showed that processing time increases dramatically with more centroids and data, but only slowly increases or decreases with more processors and tasks, up to a point where adding more processors results in too little data per processor. The algorithm iterates until the difference between new and old centroid positions is minimal, converging on the cluster solutions.
El documento describe la importancia de las tecnologรญas de la informaciรณn y la comunicaciรณn (TIC) para las pequeรฑas y medianas empresas mexicanas. El uso de las TIC puede aumentar el crecimiento y la competitividad de una empresa al permitir la automatizaciรณn de tareas, el acceso a bases de datos, la mejora de procesos y la promociรณn a travรฉs de un sitio web. No emplear las TIC puede resultar en rezago para una empresa debido a limitaciones como conocimiento, financiamiento y habilidades limitadas.
This document provides a description and target audience for various events at the Central Fest festival. It lists events such as a bouncy castle, live performances, Olympic ambassador talks, workshops on applique, community land trusts, citizenship, archery, cookery, multicultural food, beauty, Bollywood dance, rock band, and film clips. For each event it identifies the target audience, such as families for the bouncy castle, everyone for performances, those interested in the Olympics, those wanting to learn new skills, and people open to new experiences like food from different cultures.
Este documento resume la evoluciรณn del concepto de administraciรณn estratรฉgica desde las definiciones iniciales de Chandler hasta los enfoques mรกs modernos de Hofer y Schendel. Explica que la administraciรณn estratรฉgica se centra en el establecimiento de objetivos, la formulaciรณn de estrategias, la implementaciรณn y el logro de objetivos. Tambiรฉn describe los orรญgenes y premisas bรกsicas de la planeaciรณn empresarial asรญ como diferentes tipos de estrategias que una empresa puede adoptar.
Cannabis Juice and its amazing benefits should be shared with the public. In its raw form the plant has no psychoactive effects and it can be the ultimate pain killer.
Knitting boar - Toronto and Boston HUGs - Nov 2012Josh Patterson
ย
1) The document discusses machine learning and parallel iterative algorithms like stochastic gradient descent. It introduces the Mahout machine learning library and describes an implementation of parallel SGD called Knitting Boar that runs on YARN.
2) Knitting Boar parallelizes Mahout's SGD algorithm by having worker nodes process partitions of the training data in parallel while a master node merges their results.
3) The author argues that approaches like Knitting Boar and IterativeReduce provide better ways to implement machine learning algorithms for big data compared to traditional MapReduce.
El documento habla sobre la importancia de la privacidad y la seguridad en lรญnea. Explica que los usuarios deben tomar medidas para proteger su informaciรณn personal en Internet, como usar contraseรฑas seguras y actualizadas, y estar atentos al phishing.
This document discusses parallel computation models and algorithms. It describes the circuit model, which is a basic parallel model represented as a directed acyclic graph. Circuits can perform computations in parallel through fixed sequences of elementary operations. The document also introduces comparison networks, which are circuits composed only of comparator nodes used to sort or merge input sequences. Comparison networks correspond to oblivious parallel sorting and merging algorithms, where the network size and depth determine the algorithm's parallel and sequential complexity.
This document discusses parallel hashing algorithms for key-value stores. It describes several approaches: separate chaining, open addressing, linear probing, cuckoo hashing, and lock-free cuckoo hashing. Separate chaining uses linked lists in each bucket but has poor cache locality. Open addressing uses probing to find vacant slots but has poor memory efficiency above 70% capacity. Linear probing resolves collisions by probing subsequent slots but performance degrades with high load. Cuckoo hashing uses two hash tables and relocates keys to resolve collisions, allowing high load factors over 90%. Lock-free cuckoo hashing allows concurrent mutations using compare-and-swap without locking entire paths.
Balanced Scorecard Templates - Version 3Clive Keyte
ย
The Balanced Scorecard is a strategic planning and management method used to align business activities to a vision and strategy of an organisation and improve internal and external communications monitor organisational performance against strategic goals. This presentation contains a set of useful templates
The document discusses implementing Dijkstra's algorithm for finding the shortest path between nodes in a graph in parallel using MPI (Message Passing Interface). It provides the sequential pseudo code for Dijkstra's algorithm and evaluates the performance of the parallel MPI implementation versus the serial and Hadoop implementations by measuring execution time on graphs of increasing size with varying numbers of nodes and tasks. The MPI implementation showed improved performance over the serial and Hadoop implementations due to its ability to parallelize the algorithm across multiple nodes.
A Parallel Algorithm for Approximate Frequent Itemset Mining using MapReduce Fabio Fumarola
ย
The document presents MrAdam, a parallel algorithm for approximate frequent itemset mining using MapReduce. MrAdam avoids expensive communication and synchronization costs by mining approximate frequent itemsets from big data with statistical error guarantees. It combines a statistical approach based on the Chernoff bound with MapReduce-based local model discovery and global combination through an SE-tree and structural interpolation. Experiments show MrAdam is 2 to 100 times faster than previous frequent itemset mining algorithms using MapReduce.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
An Algorithm for Optimized Cost in a Distributed Computing SystemIRJET Journal
ย
This document summarizes an algorithm for optimized cost allocation in a distributed computing system. The algorithm considers a set of tasks that need to be assigned to processors across multiple phases. It calculates execution costs, residing costs, communication costs, and reallocation costs to determine the optimal allocation that minimizes overall system costs. The algorithm is demonstrated on a sample problem involving 4 tasks to be allocated across 2 processors over 5 phases. Cost matrices are provided and the algorithm partitions the problem into subproblems to determine the lowest cost allocation for each phase and overall.
Performance analysis and randamized agorithamlilyMalar1
ย
The document discusses performance analysis of algorithms in terms of space and time complexity. It provides examples to show how to calculate the space and time complexity of algorithms. Specifically, it analyzes the space and time complexity of a sum algorithm. For space complexity, it identifies the fixed and variable components, showing the space complexity is O(n). For time complexity, it analyzes the number of steps and their frequency to determine the time complexity is O(2n+3). The document also discusses other algorithm analysis topics like asymptotic notations, amortized analysis, and randomized algorithms.
The document discusses algorithms, data abstraction, asymptotic analysis, arrays, polynomials, and sparse matrices. It defines algorithms and discusses their advantages and disadvantages. It explains how to design an algorithm and describes iterative and recursive algorithms. It defines data abstraction and gives an example using smartphones. It discusses time and space complexity analysis and different asymptotic notations like Big O, Omega, and Theta. It describes what arrays are, different types of arrays, and applications of arrays. It explains how to represent and add polynomials using linked lists. Finally, it defines sparse matrices and two methods to represent them using arrays and linked lists.
This document provides an overview of the "High Performance Computing" module, including its syllabus, aim, objectives, outcomes, contents, and descriptions of parallel algorithms and parallel programming languages. The module covers parallel algorithms and their complexity analyses, models of computation, and shared memory parallel programming. It also discusses selection sort, merge sort, searching algorithms, and parallel programming languages like OpenMP. The overall document provides foundational information on key concepts in high performance and parallel computing.
The document discusses analyzing algorithms by predicting their resource requirements, such as time complexity. It introduces the random-access machine (RAM) model for analyzing algorithms, which assumes basic instructions like arithmetic, data movement, and control take constant time. The document then analyzes the time complexity of insertion sort, finding its best-case running time is linear but its worst-case running time is quadratic, making worst-case analysis most important.
Design & Analysis of Algorithm course .pptxJeevaMCSEKIOT
ย
This document discusses algorithms and their analysis. It defines an algorithm as a well-defined computational procedure that takes inputs and produces outputs. Key characteristics of algorithms include definiteness, finiteness, effectiveness, correctness, simplicity, and unambiguousness. The document discusses two common algorithm design techniques: divide and conquer, which divides a problem into subproblems, and greedy techniques, which make locally optimal choices to find a near-optimal solution. It also covers analyzing algorithms, including asymptotic time and space complexity analysis to determine how resource usage grows with problem size.
This document discusses complexity analysis of algorithms. It defines an algorithm and lists properties like being correct, unambiguous, terminating, and simple. It describes common algorithm design techniques like divide and conquer, dynamic programming, greedy method, and backtracking. It compares divide and conquer with dynamic programming. It discusses algorithm analysis in terms of time and space complexity to predict resource usage and compare algorithms. It introduces asymptotic notations like Big-O notation to describe upper bounds of algorithms as input size increases.
The document discusses asymptotic analysis and algorithm complexity. It defines asymptotic notations like Big-O, Omega, and Theta notations which are used to describe the time complexity of algorithms. Big-O gives the upper bound/worst case, Omega gives the lower bound/best case, and Theta gives both upper and lower bounds for average case. It also discusses space and time complexity analysis of algorithms and different types of time complexities including constant, logarithmic, linear, quadratic, and nlogn time. Finally, it covers arrays, types of arrays, dynamic memory allocation functions in C like malloc, calloc, free, and realloc.
The document discusses the limits of parallelism and different memory organizations of parallel computers. It introduces Amdahl's law and Gustafson-Barsi's law, which describe the theoretical speedup limits based on how much of a program can be parallelized. Shared memory multiprocessors can provide shared access to memory but do not scale well. Distributed memory machines partition memory across nodes but require message passing between nodes.
This document provides an overview of a lecture on designing and analyzing computer algorithms. It discusses key concepts like what an algorithm and program are, common algorithm design techniques like divide-and-conquer and greedy methods, and how to analyze algorithms' time and space complexity. The goals of analyzing algorithms are to understand their behavior, improve efficiency, and determine whether problems can be solved within a reasonable time frame.
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1. Page 01
Algorithms
1. Introduction
An algorithm is defined as a sequence of computational steps required to
accomplish a specific task. The algorithm works for a given input and will
terminate in a well-defined state. The basic conditions of an algorithm are: input,
output, definiteness, effectiveness and finiteness. The purpose of the developing an
algorithm is to solve a general, well specified problem.
A concern while designing an algorithm also pertains to the kind of computer
on which the algorithm would be executed. The two forms of architectures of
computers are: sequential computer and parallel computer. Therefore, depending
upon the architecture of the computers, we have sequential as well as parallel
algorithms.
The algorithms which are executed on the sequential computers simply
perform according to sequence of steps for solving a given problem. Such
algorithms are known as sequential algorithms.
However, a problem can be solved after dividing it into sub-problems and those
in turn are executed in parallel. Later on, the results of the solutions of these sub
problems can be combined together and the final solution can be achieved. In such
situations, the number of processors required would be more than one and they
would be communicating with each other for producing the final output. This
environment operates on the parallel computer and the special kind of algorithms
called parallel algorithms are designed for these computers. The parallel algorithms
depend on the kind of parallel computer they are designed for. Hence, for a given
problem, there would be a need to design the different kinds of parallel algorithms
depending upon the kind of parallel architecture.
A parallel computer is a set of processors that are able to work cooperatively to
solve a computational problem. This definition is broad enough to include parallel
supercomputers that have hundreds or thousands of processors, networks of
workstations, multi-processor workstations, and embedded systems. The parallel
computers can be represented with the help of various kinds of models such as
random access machine (RAM), parallel random access machine (PRAM),
Interconnection Networks etc. While designing a parallel algorithm, the
2. Page 02
computational power of various models can be analyzed and compared, parallelism
can be involved for a given problem on a specific model after understanding the
characteristics of a model. The analysis of parallel algorithm on different models
assist in determining the best model for a problem after receiving the results in
terms of the time and space complexity.
2. Analysis of Parallel Algorithms
A generic algorithm is mainly analyzed on the basis of the following
parameters: the time complexity (execution time) and the space complexity
(amount of space required). Usually we give much more importance to time
complexity in comparison with space complexity. The subsequent section
highlights the criteria of analyzing the complexity of parallel algorithms. The
fundamental parameters required for the analysis of parallel algorithms are as
follow:
โข Time Complexity
โข The Total Number of Processors Required
โข The Cost Involved.
2.1 Time Complexity
As it happens, most people who implement algorithms want to know how much
of a particular resource (such as time or storage) is required for a given algorithm.
The parallel architectures have been designed for improving the computation
power of the various algorithms. Thus, the major concern of evaluating an
algorithm is the determination of the amount of time required to execute. Usually,
the time complexity is calculated on the basis of the total number of steps executed
to accomplish the desired output.
The resource consumption in parallel algorithms is both processor cycles on
each processor and also the communication overhead between the processors.
Thus, first in the computation step, the local processor performs an arithmetic
and logic operation. Thereafter, the various processors communicate with each
other for exchanging messages and/or data. Hence, the time complexity can be
calculated on the basis of computational cost and communication cost involved.
The time complexity of an algorithm varies depending upon the instance of the
input for a given problem. For example, the already sorted list (10, 17, 19, 21, 22,
3. Page 03
and 33) will consume less amount of time than the reverse order of list (33, 22, 21,
19, 17, and 10). The time complexity of an algorithm has been categorized into
three forms:
i) Best Case Complexity;
ii) Average Case Complexity; and
iii) Worst Case Complexity.
The best case complexity is the least amount of time required by the algorithm
for a given input. The average case complexity is the average running time
required by the algorithm for a given input. Similarly, the worst case complexity
can be defined as the maximum amount of time required by the algorithm for a
given input.
The main factors involved for analyzing the time complexity depends upon the:
๏ผ Algorithm
๏ผ Parallel computer model
๏ผ Specific set of inputs.
Mostly the size of the input is a function of time complexity of the algorithm.
2.2 Number of Processors
One of the other factors that assist in analysis of parallel algorithms is the total
number of processors required to deliver a solution to a given problem. Thus, for a
given input of size say n, the number of processors required by the parallel
algorithm is a function of n, usually denoted by TP (n).
2.3 Overall Cost
Finally, the total cost of the algorithm is a product of time complexity of the
parallel algorithm and the total number of processors required for computation.
Cost = Time Complexity * Total Number of Processors
The other form of defining the cost is that it specifies the total number of steps
executed collectively by the n number of processors, i.e., summation of steps.
Another term related with the analysis of the parallel algorithms is efficiency of the
algorithm. It is defined as the ratio of the worst case running time of the best
4. Page 04
sequential algorithm and the cost of the parallel algorithm. The efficiency would be
mostly less than or equal to 1. In a situation, if efficiency is greater than 1 then it
means that the sequential algorithm is faster than the parallel algorithm.
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3. Merge Sort Algorithm
First, divide the given sequence of n numbers into two parts, each consisting of
n/2 numbers. Thereafter, recursively split the sequence into two parts until each
number acts as an independent sequence. Consequently, the independent numbers
are first sorted and recursively merged until a sorted sequence of n numbers is not
achieved.
In order to perform the above-mentioned task, there will be two kinds of circuits
which would be used in the following manner: the first one for sorting and another
one for merging the sorted list of numbers.
Odd-Even Merging Circuit
Let us firstly illustrate the concept of merging two sorted sequences using an
odd-even merging circuit. The working of a merging circuit is as follows:
1) Let there be two sorted sequences A=(a1, a2, a3, a4โฆโฆโฆ am) and B=(b1, b2, b3,
b4โฆโฆโฆ bm) which are required to be merged.
2) With the help of a merging circuit (m/2,m/2), merge the odd indexed numbers of
the two sub sequences i.e. (a1, a3, a5โฆโฆ am-1) and (b1, b3, b5โฆโฆโฆ bm-1) and thus
resulting in sorted sequence (c1, c2, c3โฆโฆโฆ cm).
3) Thereafter, with the help of a merging circuit (m/2,m/2), merge the even
indexed numbers of the two sub sequences i.e. (a2, a4, a6โฆโฆโฆ am) and (b2, b4, b6โฆโฆโฆ
bm) and thus resulting in sorted sequence (d1, d2, d3โฆโฆโฆ dm).
4) The final output sequence O=(o1, o2, o3โฆโฆโฆ o2m ) is achieved in the following
manner:
5. Page 05
o1 = a1 and o2m = bm .In general the formula is as given below: o2i = min(ai+1,bi ) and
o2I+1 = max(ai+1,bi ) for i=1,2,3,4โฆโฆโฆ.m-1.
Now, let us take an example for merging the two sorted sequences of length 4,
i.e., A=(a1, a2, a3, a4) and B=(b1, b2, b3, b4). Suppose the numbers of the sequence are
A=(4,6,9,10) and B=(2,7,8,12). The circuit of merging the two given sequences is
illustrated in Figure 7.
Sorting Circuit along with Odd-Even Merging Circuit
As we already know, the merge sort algorithm requires two circuits, i.e. one for
merging and another for sorting the sequences. Therefore, the sorting circuit has
been derived from the above-discussed merging circuit. The basic steps followed
by the circuit are highlighted below:
i) The given input sequence of length n is divided into two sub-sequences of length
n/2 each.
ii) The two sub sequences are recursively sorted.
iii) The two sorted sub sequences are merged (n/2,n/2) using a merging circuit in order
to finally get the sorted sequence of length n.
6. Page 06
Now, let us take an example for sorting the n numbers say 4,2,10,12,8,7,6,9. The
circuit of sorting + merging given sequence is illustrated in Figure 8.
Analysis of Merge Sort
i) The width of the sorting + merging circuit is equal to the maximum number of
devices required in a stage is O(n/2). As in the above figure the maximum number
of devices for a given stage is 4 which is 8/2 or n/2.
ii) The circuit contains two sorting circuits for sorting sequences of length n/2 and
thereafter one merging circuit for merging of the two sorted sub sequences (see
7. Page 07
stage 4th
in the above figure). Let the functions Ts and Tm denote the time
complexity of sorting and merging in terms of its depth.
The Ts can be calculated as follows:
Ts(n) =Ts(n/2) + Tm(n/2)
Ts(n) =Ts(n/2) + log(n/2) ,
Therefore, Ts (n) is equal to O(log2
n).
4. Image Processing
Image is the 2 dimensional distributions of the small image points called
pixels. It can be considered as a function of two real variables, for example, f(x,y)
with f as the amplitude (e.g. brightness) of the image at position (x,y). Image
Processing is the process of enhancing the image and extraction of meaningful
information from an image.
Image Processing with parallel computing is an alternative way to solve image
processing problems that require large times of processing or handling large
amounts of information in "acceptable time" (according to each criterion).
The main idea of parallel image processing is to divide the problem into simple
tasks and solve them concurrently, in such a way the total time can be divided
between the total tasks (in the best case). Parallel image processing cannot be
applied to all problems, in other words we can say that not all the problems can be
coded in a parallel form. A parallel program must have some features for a correct
and efficient operation; otherwise, it is possible that runtime or operation does not
have the expected performance. These features include the following:
๏ Granularity: Itโs defined as the number of basic units and it is classified as:
๏ท Coarse-grained: Few tasks of more intense computing.
๏ท Fine-grained: A large number of small parts and less intense computing.
๏ Synchronization: This prevents the overlap of two or more processes.
๏ Latency: This is the time transition of information from request to receipt.
8. Page 08
๏ Scalability: Itโs defined as the ability of an algorithm to maintain its efficiency by
increasing the number of processors and the size of the problem in the same
proportion.
5. Proposed Algorithms for Parallel Image Processing Tasks
We will go through the following sequential image processing algorithms and
develop its parallel versions.
Load Distribution between Processors:
Suppose we are taking the following image (1) as an example. The main step of
these algorithms is to determine the number of tiles to be generated. The number of
tiles corresponds to the amount of threads. If only a thread exists, the computation
is just sequential computation. Otherwise if there are two or more threads then the
image is divided into distinct areas, as shown in Figure 2, 3, 4. Each thread is
responsible for processing the pixels included in its tile and to execute different
tasks but considering maintaining synchronization between all the processor
otherwise there will be the situation of deadlock between processors.
Parallel Segmentation by Region Growing Technique and
Calculation of different Features of Segmented Regions:
Region growing is one of the very famous techniques of segmentation. Main
drawback of the region growing is that it is time consuming. This algorithm is
designed to reduce timing of this algorithm. These are the steps involved in it.
1. Take Input image on client processor.
2. Select the Seed pixels (It is based on some user criterion like pixels in a certain
gray level range, pixels evenly spaced on a grid, etc.) for different regions on
client processor.
3. Count number of seed points and activate same number of kernels.
4. Send copy of image and respective seed pixels on each kernel processor.
5. Regions are then grown from these seed pixels to adjacent depending on pixel
intensity, gray level texture, or color, etc. on each kernel processor for this
9. Page 09
image information is also important because region grown is based on the
membership criteria.
6. Calculate different features of different ROI (region of interest) on individual
kernel.
7. Send segmented regions and information of ROIs to the client processor and
deactivation of worker kernels.
8. Reconstruct the image on client processor and Display of all the information.
6. Fault Tolerance
To achieve the needed reliability and availability, we need fault-tolerant
computers. They have the ability to tolerate faults by detecting failures, and
isolate defect modules so that the rest of the system can operate correctly.
10. Page 10
Reliability techniques have also become of increasing interest to general-
purpose computer systems.
Parallel computing can also be applied to the design of fault-tolerance
computer systems, particularly via lockstep systems performing the same
operation in parallel. This provides redundancy in case one component should
fail, and also allows automatic error detection and error correction if the results
differ. These methods can be used to help prevent single event upsets caused by
transient errors. Although additional measures may be required in embedded or
specialized systems, this method can provide a cost effective approach to
achieve n-modular redundancy in commercial off-the-shelf systems.
7. Process of System Recovery from Fault Tolerance
Faults in different parts of parallel system have different importance. Letโs
think about a fault processor, line or switch. The most important is fault on
processor. In this case the processes allocated on this processor have to be
moved to other processor, recovered and initialized one more time. Usually we
can think about that processor memory content is lost after fault appearing, or
lack of access. It is necessary to remove and to redirect all communications
lines going through this process.
Every process of parallel system from the moment when the fault appears till
the end of the recovery is getting a new attribute (fig.1). When processor
element PE failed, then:
โข Every process allocated on the processor element PE is called locked
process, main and copy too,
โข Every process except locked process, communicating with locked
process is called fault influenced process,
โข Every process except locked process, not communicating with locked
process is called free process.
11. Page 11
The process of system recovery is known. But there is a question how and
who controls recovery of kernel of processor. Control can be either centralized
or decentralized. In case of decentralized control it is necessary to build on the
fact that all kernels dispose the same data, according to which they determine
final processors. Every kernel determines final processors for those locked
processes which have on its processor allocated copies of processes. If the copy
of process is located on more than one processor then the corresponding
processor transmit message about system recovery to other processors where
the other copies are located. Content of the message is about final processor for
the exact copy of process and time mark of begin of recovery. The kernel of the
system after receiving all messages about system recovery compares this time
mark with its own time of recovery. Lately the kernel doesnโt realize any code
reallocation of the relevant processes. In case of equality of time marks can be
decisive by another criterion, like for example identification number of a
processor. There is a question how many copies of processes are enough for
sufficient resistance against faults. In case of active and passive processes it
depends on requested security. One passive copy of the process is sufficient if
we assume, that fault doesnโt appear on two processors occupied by the same
process at the same time or in time of recovery of the system.