Parallel Implementation of Sorting Algorithms
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Sort Algorithms 15-110 - Friday 2/28 Learning Objectives
Sort Algorithms 15-110 - Friday 2/28 Learning Objectives • Recognize how different sorting algorithms implement the same process with different algorithms • Recognize the general algorithm and trace code for three algorithms: selection sort, insertion sort, and merge sort • Compute the Big-O runtimes of selection sort, insertion sort, and merge sort 2 Search Algorithms Benefit from Sorting We use search algorithms a lot in computer science. Just think of how many times a day you use Google, or search for a file on your computer. We've determined that search algorithms work better when the items they search over are sorted. Can we write an algorithm to sort items efficiently? Note: Python already has built-in sorting functions (sorted(lst) is non-destructive, lst.sort() is destructive). This lecture is about a few different algorithmic approaches for sorting. 3 Many Ways of Sorting There are a ton of algorithms that we can use to sort a list. We'll use https://visualgo.net/bn/sorting to visualize some of these algorithms. Today, we'll specifically discuss three different sorting algorithms: selection sort, insertion sort, and merge sort. All three do the same action (sorting), but use different algorithms to accomplish it. 4 Selection Sort 5 Selection Sort Sorts From Smallest to Largest The core idea of selection sort is that you sort from smallest to largest. 1. Start with none of the list sorted 2. Repeat the following steps until the whole list is sorted: a) Search the unsorted part of the list to find the smallest element b) Swap the found element with the first unsorted element c) Increment the size of the 'sorted' part of the list by one Note: for selection sort, swapping the element currently in the front position with the smallest element is faster than sliding all of the numbers down in the list. -
Quick Sort Algorithm Song Qin Dept
Quick Sort Algorithm Song Qin Dept. of Computer Sciences Florida Institute of Technology Melbourne, FL 32901 ABSTRACT each iteration. Repeat this on the rest of the unsorted region Given an array with n elements, we want to rearrange them in without the first element. ascending order. In this paper, we introduce Quick Sort, a Bubble sort works as follows: keep passing through the list, divide-and-conquer algorithm to sort an N element array. We exchanging adjacent element, if the list is out of order; when no evaluate the O(NlogN) time complexity in best case and O(N2) exchanges are required on some pass, the list is sorted. in worst case theoretically. We also introduce a way to approach the best case. Merge sort [4]has a O(NlogN) time complexity. It divides the 1. INTRODUCTION array into two subarrays each with N/2 items. Conquer each Search engine relies on sorting algorithm very much. When you subarray by sorting it. Unless the array is sufficiently small(one search some key word online, the feedback information is element left), use recursion to do this. Combine the solutions to brought to you sorted by the importance of the web page. the subarrays by merging them into single sorted array. 2 Bubble, Selection and Insertion Sort, they all have an O(N2) In Bubble sort, Selection sort and Insertion sort, the O(N ) time time complexity that limits its usefulness to small number of complexity limits the performance when N gets very big. element no more than a few thousand data points. -
The Analysis and Synthesis of a Parallel Sorting Engine Redacted for Privacy Abstract Approv, John M
AN ABSTRACT OF THE THESIS OF Byoungchul Ahn for the degree of Doctor of Philosophy in Electrical and Computer Engineering, presented on May 3. 1989. Title: The Analysis and Synthesis of a Parallel Sorting Engine Redacted for Privacy Abstract approv, John M. Murray / Thisthesisisconcerned withthe development of a unique parallelsort-mergesystemsuitablefor implementationinVLSI. Two new sorting subsystems, a high performance VLSI sorter and a four-waymerger,werealsorealizedduringthedevelopment process. In addition, the analysis of several existing parallel sorting architectures and algorithms was carried out. Algorithmic time complexity, VLSI processor performance, and chiparearequirementsfortheexistingsortingsystemswere evaluated.The rebound sorting algorithm was determined to be the mostefficientamongthoseconsidered. The reboundsorter algorithm was implementedinhardware asasystolicarraywith external expansion capability. The second phase of the research involved analyzing several parallel merge algorithms andtheirbuffer management schemes. The dominant considerations for this phase of the research were the achievement of minimum VLSI chiparea,design complexity, and logicdelay. Itwasdeterminedthattheproposedmerger architecture could be implemented inseveral ways. Selecting the appropriate microarchitecture for the merger, given the constraints of chip area and performance, was the major problem.The tradeoffs associated with this process are outlined. Finally,apipelinedsort-merge system was implementedin VLSI by combining a rebound sorter -
Coursenotes 4 Non-Adaptive Sorting Batcher's Algorithm
4. Non Adaptive Sorting Batcher’s Algorithm 4.1 Introduction to Batcher’s Algorithm Sorting has many important applications in daily life and in particular, computer science. Within computer science several sorting algorithms exist such as the “bubble sort,” “shell sort,” “comb sort,” “heap sort,” “bucket sort,” “merge sort,” etc. We have actually encountered these before in section 2 of the course notes. Many different sorting algorithms exist and are implemented differently in each programming language. These sorting algorithms are relevant because they are used in every single computer program you run ranging from the unimportant computer solitaire you play to the online checking account you own. Indeed without sorting algorithms, many of the services we enjoy today would simply not be available. Batcher’s algorithm is a way to sort individual disorganized elements called “keys” into some desired order using a set number of comparisons. The keys that will be sorted for our purposes will be numbers and the order that we will want them to be in will be from greatest to least or vice a versa (whichever way you want it). The Batcher algorithm is non-adaptive in that it takes a fixed set of comparisons in order to sort the unsorted keys. In other words, there is no change made in the process during the sorting. Unlike other methods of sorting where you need to remember comparisons (tournament sort), Batcher’s algorithm is useful because it requires less thinking. Non-adaptive sorting is easy because outcomes of comparisons made at one point of the process do not affect which comparisons will be made in the future. -
1 Lower Bound on Runtime of Comparison Sorts
CS 161 Lecture 7 { Sorting in Linear Time Jessica Su (some parts copied from CLRS) 1 Lower bound on runtime of comparison sorts So far we've looked at several sorting algorithms { insertion sort, merge sort, heapsort, and quicksort. Merge sort and heapsort run in worst-case O(n log n) time, and quicksort runs in expected O(n log n) time. One wonders if there's something special about O(n log n) that causes no sorting algorithm to surpass it. As a matter of fact, there is! We can prove that any comparison-based sorting algorithm must run in at least Ω(n log n) time. By comparison-based, I mean that the algorithm makes all its decisions based on how the elements of the input array compare to each other, and not on their actual values. One example of a comparison-based sorting algorithm is insertion sort. Recall that insertion sort builds a sorted array by looking at the next element and comparing it with each of the elements in the sorted array in order to determine its proper position. Another example is merge sort. When we merge two arrays, we compare the first elements of each array and place the smaller of the two in the sorted array, and then we make other comparisons to populate the rest of the array. In fact, all of the sorting algorithms we've seen so far are comparison sorts. But today we will cover counting sort, which relies on the values of elements in the array, and does not base all its decisions on comparisons. -
CS 758/858: Algorithms
CS 758/858: Algorithms ■ COVID Prof. Wheeler Ruml Algorithms TA Sumanta Kashyapi This Class Complexity http://www.cs.unh.edu/~ruml/cs758 4 handouts: course info, schedule, slides, asst 1 2 online handouts: programming tips, formulas 1 physical sign-up sheet/laptop (for grades, piazza) Wheeler Ruml (UNH) Class 1, CS 758 – 1 / 25 COVID ■ COVID Algorithms This Class Complexity ■ check your Wildcat Pass before coming to campus ■ if you have concerns, let me know Wheeler Ruml (UNH) Class 1, CS 758 – 2 / 25 ■ COVID Algorithms ■ Algorithms Today ■ Definition ■ Why? ■ The Word ■ The Founder This Class Complexity Algorithms Wheeler Ruml (UNH) Class 1, CS 758 – 3 / 25 Algorithms Today ■ ■ COVID web: search, caching, crypto Algorithms ■ networking: routing, synchronization, failover ■ Algorithms Today ■ machine learning: data mining, recommendation, prediction ■ Definition ■ Why? ■ bioinformatics: alignment, matching, clustering ■ The Word ■ ■ The Founder hardware: design, simulation, verification ■ This Class business: allocation, planning, scheduling Complexity ■ AI: robotics, games Wheeler Ruml (UNH) Class 1, CS 758 – 4 / 25 Definition ■ COVID Algorithm Algorithms ■ precisely defined ■ Algorithms Today ■ Definition ■ mechanical steps ■ Why? ■ ■ The Word terminates ■ The Founder ■ input and related output This Class Complexity What might we want to know about it? Wheeler Ruml (UNH) Class 1, CS 758 – 5 / 25 Why? ■ ■ COVID Computer scientist 6= programmer Algorithms ◆ ■ Algorithms Today understand program behavior ■ Definition ◆ have confidence in results, performance ■ Why? ■ The Word ◆ know when optimality is abandoned ■ The Founder ◆ solve ‘impossible’ problems This Class ◆ sets you apart (eg, Amazon.com) Complexity ■ CPUs aren’t getting faster ■ Devices are getting smaller ■ Software is the differentiator ■ ‘Software is eating the world’ — Marc Andreessen, 2011 ■ Everything is computation Wheeler Ruml (UNH) Class 1, CS 758 – 6 / 25 The Word: Ab¯u‘Abdall¯ah Muh.ammad ibn M¯us¯aal-Khw¯arizm¯ı ■ COVID 780-850 AD Algorithms Born in Uzbekistan, ■ Algorithms Today worked in Baghdad. -
Processamento Paralelo Em Cuda Aplicado Ao Modelo De Geração De Cenários Sintéticos De Vazões E Energias - Gevazp
PROCESSAMENTO PARALELO EM CUDA APLICADO AO MODELO DE GERAÇÃO DE CENÁRIOS SINTÉTICOS DE VAZÕES E ENERGIAS - GEVAZP André Emanoel Rabello Quadros Dissertação de Mestrado apresentada ao Programa de Pós-Graduação em Engenharia Elétrica, COPPE, da Universidade Federal do Rio de Janeiro, como parte dos requisitos necessários à obtenção do título de Mestre em Engenharia Elétrica. Orientadora: Carmen Lucia Tancredo Borges Rio de Janeiro Março de 2016 PROCESSAMENTO PARALELO EM CUDA APLICADO AO MODELO DE GERAÇÃO DE CENÁRIOS SINTÉTICOS DE VAZÕES E ENERGIAS - GEVAZP André Emanoel Rabello Quadros DISSERTAÇÃO SUBMETIDA AO CORPO DOCENTE DO INSTITUTO ALBERTO LUIZ COIMBRA DE PÓS-GRADUAÇÃO E PESQUISA DE ENGENHARIA (COPPE) DA UNIVERSIDADE FEDERAL DO RIO DE JANEIRO COMO PARTE DOS REQUISITOS NECESSÁRIOS PARA A OBTENÇÃO DO GRAU DE MESTRE EM CIÊNCIAS EM ENGENHARIA ELÉTRICA. Examinada por: ________________________________________________ Profa. Carmen Lucia Tancredo Borges, D.Sc. ________________________________________________ Prof. Antônio Carlos Siqueira de Lima, D.Sc. ________________________________________________ Dra. Maria Elvira Piñeiro Maceira, D.Sc. ________________________________________________ Prof. Sérgio Barbosa Villas-Boas, Ph.D. RIO DE JANEIRO, RJ – BRASIL MARÇO DE 2016 iii Quadros, André Emanoel Rabello Processamento Paralelo em CUDA Aplicada ao Modelo de Geração de Cenários Sintéticos de Vazões e Energias – GEVAZP / André Emanoel Rabello Quadros. – Rio de Janeiro: UFRJ/COPPE, 2016. XII, 101 p.: il.; 29,7 cm. Orientador(a): Carmen Lucia Tancredo Borges Dissertação (mestrado) – UFRJ/ COPPE/ Programa de Engenharia Elétrica, 2016. Referências Bibliográficas: p. 9 6-99. 1. Programação Paralela. 2. GPU. 3. CUDA. 4. Planejamento da Operação Energética. 5. Geração de cenários sintéticos. I. Borges, Carmen Lucia Tancredo. II. Universidade Federal do Rio de Janeiro, COPPE, Programa de Engenharia Elétrica. -
Sorting Algorithms Correcness, Complexity and Other Properties
Sorting Algorithms Correcness, Complexity and other Properties Joshua Knowles School of Computer Science The University of Manchester COMP26912 - Week 9 LF17, April 1 2011 The Importance of Sorting Important because • Fundamental to organizing data • Principles of good algorithm design (correctness and efficiency) can be appreciated in the methods developed for this simple (to state) task. Sorting Algorithms 2 LF17, April 1 2011 Every algorithms book has a large section on Sorting... Sorting Algorithms 3 LF17, April 1 2011 ...On the Other Hand • Progress in computer speed and memory has reduced the practical importance of (further developments in) sorting • quicksort() is often an adequate answer in many applications However, you still need to know your way (a little) around the the key sorting algorithms Sorting Algorithms 4 LF17, April 1 2011 Overview What you should learn about sorting (what is examinable) • Definition of sorting. Correctness of sorting algorithms • How the following work: Bubble sort, Insertion sort, Selection sort, Quicksort, Merge sort, Heap sort, Bucket sort, Radix sort • Main properties of those algorithms • How to reason about complexity — worst case and special cases Covered in: the course book; labs; this lecture; wikipedia; wider reading Sorting Algorithms 5 LF17, April 1 2011 Relevant Pages of the Course Book Selection sort: 97 (very short description only) Insertion sort: 98 (very short) Merge sort: 219–224 (pages on multi-way merge not needed) Heap sort: 100–106 and 107–111 Quicksort: 234–238 Bucket sort: 241–242 Radix sort: 242–243 Lower bound on sorting 239–240 Practical issues, 244 Some of the exercise on pp. -
Quicksort Z Merge Sort Z T(N) = Θ(N Lg(N)) Z Not In-Place CSE 680 Z Selection Sort (From Homework) Prof
Sorting Review z Insertion Sort Introduction to Algorithms z T(n) = Θ(n2) z In-place Quicksort z Merge Sort z T(n) = Θ(n lg(n)) z Not in-place CSE 680 z Selection Sort (from homework) Prof. Roger Crawfis z T(n) = Θ(n2) z In-place Seems pretty good. z Heap Sort Can we do better? z T(n) = Θ(n lg(n)) z In-place Sorting Comppgarison Sorting z Assumptions z Given a set of n values, there can be n! 1. No knowledge of the keys or numbers we permutations of these values. are sorting on. z So if we look at the behavior of the 2. Each key supports a comparison interface sorting algorithm over all possible n! or operator. inputs we can determine the worst-case 3. Sorting entire records, as opposed to complexity of the algorithm. numbers,,p is an implementation detail. 4. Each key is unique (just for convenience). Comparison Sorting Decision Tree Decision Tree Model z Decision tree model ≤ 1:2 > z Full binary tree 2:3 1:3 z A full binary tree (sometimes proper binary tree or 2- ≤≤> > tree) is a tree in which every node other than the leaves has two c hildren <1,2,3> ≤ 1:3 >><2,1,3> ≤ 2:3 z Internal node represents a comparison. z Ignore control, movement, and all other operations, just <1,3,2> <3,1,2> <2,3,1> <3,2,1> see comparison z Each leaf represents one possible result (a permutation of the elements in sorted order). -
An Evolutionary Approach for Sorting Algorithms
ORIENTAL JOURNAL OF ISSN: 0974-6471 COMPUTER SCIENCE & TECHNOLOGY December 2014, An International Open Free Access, Peer Reviewed Research Journal Vol. 7, No. (3): Published By: Oriental Scientific Publishing Co., India. Pgs. 369-376 www.computerscijournal.org Root to Fruit (2): An Evolutionary Approach for Sorting Algorithms PRAMOD KADAM AND Sachin KADAM BVDU, IMED, Pune, India. (Received: November 10, 2014; Accepted: December 20, 2014) ABstract This paper continues the earlier thought of evolutionary study of sorting problem and sorting algorithms (Root to Fruit (1): An Evolutionary Study of Sorting Problem) [1]and concluded with the chronological list of early pioneers of sorting problem or algorithms. Latter in the study graphical method has been used to present an evolution of sorting problem and sorting algorithm on the time line. Key words: Evolutionary study of sorting, History of sorting Early Sorting algorithms, list of inventors for sorting. IntroDUCTION name and their contribution may skipped from the study. Therefore readers have all the rights to In spite of plentiful literature and research extent this study with the valid proofs. Ultimately in sorting algorithmic domain there is mess our objective behind this research is very much found in documentation as far as credential clear, that to provide strength to the evolutionary concern2. Perhaps this problem found due to lack study of sorting algorithms and shift towards a good of coordination and unavailability of common knowledge base to preserve work of our forebear platform or knowledge base in the same domain. for upcoming generation. Otherwise coming Evolutionary study of sorting algorithm or sorting generation could receive hardly information about problem is foundation of futuristic knowledge sorting problems and syllabi may restrict with some base for sorting problem domain1. -
Overview Parallel Merge Sort
CME 323: Distributed Algorithms and Optimization, Spring 2015 http://stanford.edu/~rezab/dao. Instructor: Reza Zadeh, Matriod and Stanford. Lecture 4, 4/6/2016. Scribed by Henry Neeb, Christopher Kurrus, Andreas Santucci. Overview Today we will continue covering divide and conquer algorithms. We will generalize divide and conquer algorithms and write down a general recipe for it. What's nice about these algorithms is that they are timeless; regardless of whether Spark or any other distributed platform ends up winning out in the next decade, these algorithms always provide a theoretical foundation for which we can build on. It's well worth our understanding. • Parallel merge sort • General recipe for divide and conquer algorithms • Parallel selection • Parallel quick sort (introduction only) Parallel selection involves scanning an array for the kth largest element in linear time. We then take the core idea used in that algorithm and apply it to quick-sort. Parallel Merge Sort Recall the merge sort from the prior lecture. This algorithm sorts a list recursively by dividing the list into smaller pieces, sorting the smaller pieces during reassembly of the list. The algorithm is as follows: Algorithm 1: MergeSort(A) Input : Array A of length n Output: Sorted A 1 if n is 1 then 2 return A 3 end 4 else n 5 L mergeSort(A[0, ..., 2 )) n 6 R mergeSort(A[ 2 , ..., n]) 7 return Merge(L, R) 8 end 1 Last lecture, we described one way where we can take our traditional merge operation and translate it into a parallelMerge routine with work O(n log n) and depth O(log n). -
Bubble Sort: an Archaeological Algorithmic Analysis
Bubble Sort: An Archaeological Algorithmic Analysis Owen Astrachan 1 Computer Science Department Duke University [email protected] Abstract 1 Introduction Text books, including books for general audiences, in- What do students remember from their first program- variably mention bubble sort in discussions of elemen- ming courses after one, five, and ten years? Most stu- tary sorting algorithms. We trace the history of bub- dents will take only a few memories of what they have ble sort, its popularity, and its endurance in the face studied. As teachers of these students we should ensure of pedagogical assertions that code and algorithmic ex- that what they remember will serve them well. More amples used in early courses should be of high quality specifically, if students take only a few memories about and adhere to established best practices. This paper is sorting from a first course what do we want these mem- more an historical analysis than a philosophical trea- ories to be? Should the phrase Bubble Sort be the first tise for the exclusion of bubble sort from books and that springs to mind at the end of a course or several courses. However, sentiments for exclusion are sup- years later? There are compelling reasons for excluding 1 ported by Knuth [17], “In short, the bubble sort seems discussion of bubble sort , but many texts continue to to have nothing to recommend it, except a catchy name include discussion of the algorithm after years of warn- and the fact that it leads to some interesting theoreti- ings from scientists and educators.