SIMD- and Cache-Friendly Algorithm for Sorting an Array of Structures
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
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. -
PROC SORT (Then And) NOW Derek Morgan, PAREXEL International
Paper 143-2019 PROC SORT (then and) NOW Derek Morgan, PAREXEL International ABSTRACT The SORT procedure has been an integral part of SAS® since its creation. The sort-in-place paradigm made the most of the limited resources at the time, and almost every SAS program had at least one PROC SORT in it. The biggest options at the time were to use something other than the IBM procedure SYNCSORT as the sorting algorithm, or whether you were sorting ASCII data versus EBCDIC data. These days, PROC SORT has fallen out of favor; after all, PROC SQL enables merging without using PROC SORT first, while the performance advantages of HASH sorting cannot be overstated. This leads to the question: Is the SORT procedure still relevant to any other than the SAS novice or the terminally stubborn who refuse to HASH? The answer is a surprisingly clear “yes". PROC SORT has been enhanced to accommodate twenty-first century needs, and this paper discusses those enhancements. INTRODUCTION The largest enhancement to the SORT procedure is the addition of collating sequence options. This is first and foremost recognition that SAS is an international software package, and SAS users no longer work exclusively with English-language data. This capability is part of National Language Support (NLS) and doesn’t require any additional modules. You may use standard collations, SAS-provided translation tables, custom translation tables, standard encodings, or rules to produce your sorted dataset. However, you may only use one collation method at a time. USING STANDARD COLLATIONS, TRANSLATION TABLES AND ENCODINGS A long time ago, SAS would allow you to sort data using ASCII rules on an EBCDIC system, and vice versa. -
Radix Sort Comparison Sort Runtime of O(N*Log(N)) Is Optimal
Radix Sort Comparison sort runtime of O(n*log(n)) is optimal • The problem of sorting cannot be solved using comparisons with less than n*log(n) time complexity • See Proposition I in Chapter 2.2 of the text How can we sort without comparison? • Consider the following approach: • Look at the least-significant digit • Group numbers with the same digit • Maintain relative order • Place groups back in array together • I.e., all the 0’s, all the 1’s, all the 2’s, etc. • Repeat for increasingly significant digits The characteristics of Radix sort • Least significant digit (LSD) Radix sort • a fast stable sorting algorithm • begins at the least significant digit (e.g. the rightmost digit) • proceeds to the most significant digit (e.g. the leftmost digit) • lexicographic orderings Generally Speaking • 1. Take the least significant digit (or group of bits) of each key • 2. Group the keys based on that digit, but otherwise keep the original order of keys. • This is what makes the LSD radix sort a stable sort. • 3. Repeat the grouping process with each more significant digit Generally Speaking public static void sort(String [] a, int W) { int N = a.length; int R = 256; String [] aux = new String[N]; for (int d = W - 1; d >= 0; --d) { aux = sorted array a by the dth character a = aux } } A Radix sort example A Radix sort example A Radix sort example • Problem: How to ? • Group the keys based on that digit, • but otherwise keep the original order of keys. Key-indexed counting • 1. Take the least significant digit (or group of bits) of each key • 2. -
Overview of Sorting Algorithms
Unit 7 Sorting Algorithms Simple Sorting algorithms Quicksort Improving Quicksort Overview of Sorting Algorithms Given a collection of items we want to arrange them in an increasing or decreasing order. You probably have seen a number of sorting algorithms including ¾ selection sort ¾ insertion sort ¾ bubble sort ¾ quicksort ¾ tree sort using BST's In terms of efficiency: ¾ average complexity of the first three is O(n2) ¾ average complexity of quicksort and tree sort is O(n lg n) ¾ but its worst case is still O(n2) which is not acceptable In this section, we ¾ review insertion, selection and bubble sort ¾ discuss quicksort and its average/worst case analysis ¾ show how to eliminate tail recursion ¾ present another sorting algorithm called heapsort Unit 7- Sorting Algorithms 2 Selection Sort Assume that data ¾ are integers ¾ are stored in an array, from 0 to size-1 ¾ sorting is in ascending order Algorithm for i=0 to size-1 do x = location with smallest value in locations i to size-1 swap data[i] and data[x] end Complexity If array has n items, i-th step will perform n-i operations First step performs n operations second step does n-1 operations ... last step performs 1 operatio. Total cost : n + (n-1) +(n-2) + ... + 2 + 1 = n*(n+1)/2 . Algorithm is O(n2). Unit 7- Sorting Algorithms 3 Insertion Sort Algorithm for i = 0 to size-1 do temp = data[i] x = first location from 0 to i with a value greater or equal to temp shift all values from x to i-1 one location forwards data[x] = temp end Complexity Interesting operations: comparison and shift i-th step performs i comparison and shift operations Total cost : 1 + 2 + .. -
2.5 Classification of Parallel Computers
52 // Architectures 2.5 Classification of Parallel Computers 2.5 Classification of Parallel Computers 2.5.1 Granularity In parallel computing, granularity means the amount of computation in relation to communication or synchronisation Periods of computation are typically separated from periods of communication by synchronization events. • fine level (same operations with different data) ◦ vector processors ◦ instruction level parallelism ◦ fine-grain parallelism: – Relatively small amounts of computational work are done between communication events – Low computation to communication ratio – Facilitates load balancing 53 // Architectures 2.5 Classification of Parallel Computers – Implies high communication overhead and less opportunity for per- formance enhancement – If granularity is too fine it is possible that the overhead required for communications and synchronization between tasks takes longer than the computation. • operation level (different operations simultaneously) • problem level (independent subtasks) ◦ coarse-grain parallelism: – Relatively large amounts of computational work are done between communication/synchronization events – High computation to communication ratio – Implies more opportunity for performance increase – Harder to load balance efficiently 54 // Architectures 2.5 Classification of Parallel Computers 2.5.2 Hardware: Pipelining (was used in supercomputers, e.g. Cray-1) In N elements in pipeline and for 8 element L clock cycles =) for calculation it would take L + N cycles; without pipeline L ∗ N cycles Example of good code for pipelineing: §doi =1 ,k ¤ z ( i ) =x ( i ) +y ( i ) end do ¦ 55 // Architectures 2.5 Classification of Parallel Computers Vector processors, fast vector operations (operations on arrays). Previous example good also for vector processor (vector addition) , but, e.g. recursion – hard to optimise for vector processors Example: IntelMMX – simple vector processor. -
Distributed Algorithms with Theoretic Scalability Analysis of Radial and Looped Load flows for Power Distribution Systems
Electric Power Systems Research 65 (2003) 169Á/177 www.elsevier.com/locate/epsr Distributed algorithms with theoretic scalability analysis of radial and looped load flows for power distribution systems Fangxing Li *, Robert P. Broadwater ECE Department Virginia Tech, Blacksburg, VA 24060, USA Received 15 April 2002; received in revised form 14 December 2002; accepted 16 December 2002 Abstract This paper presents distributed algorithms for both radial and looped load flows for unbalanced, multi-phase power distribution systems. The distributed algorithms are developed from tree-based sequential algorithms. Formulas of scalability for the distributed algorithms are presented. It is shown that computation time dominates communication time in the distributed computing model. This provides benefits to real-time load flow calculations, network reconfigurations, and optimization studies that rely on load flow calculations. Also, test results match the predictions of derived formulas. This shows the formulas can be used to predict the computation time when additional processors are involved. # 2003 Elsevier Science B.V. All rights reserved. Keywords: Distributed computing; Scalability analysis; Radial load flow; Looped load flow; Power distribution systems 1. Introduction Also, the method presented in Ref. [10] was tested in radial distribution systems with no more than 528 buses. Parallel and distributed computing has been applied More recent works [11Á/14] presented distributed to many scientific and engineering computations such as implementations for power flows or power flow based weather forecasting and nuclear simulations [1,2]. It also algorithms like optimizations and contingency analysis. has been applied to power system analysis calculations These works also targeted power transmission systems. [3Á/14]. -
Scalability and Performance Management of Internet Applications in the Cloud
Hasso-Plattner-Institut University of Potsdam Internet Technology and Systems Group Scalability and Performance Management of Internet Applications in the Cloud A thesis submitted for the degree of "Doktors der Ingenieurwissenschaften" (Dr.-Ing.) in IT Systems Engineering Faculty of Mathematics and Natural Sciences University of Potsdam By: Wesam Dawoud Supervisor: Prof. Dr. Christoph Meinel Potsdam, Germany, March 2013 This work is licensed under a Creative Commons License: Attribution – Noncommercial – No Derivative Works 3.0 Germany To view a copy of this license visit http://creativecommons.org/licenses/by-nc-nd/3.0/de/ Published online at the Institutional Repository of the University of Potsdam: URL http://opus.kobv.de/ubp/volltexte/2013/6818/ URN urn:nbn:de:kobv:517-opus-68187 http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-68187 To my lovely parents To my lovely wife Safaa To my lovely kids Shatha and Yazan Acknowledgements At Hasso Plattner Institute (HPI), I had the opportunity to meet many wonderful people. It is my pleasure to thank those who sup- ported me to make this thesis possible. First and foremost, I would like to thank my Ph.D. supervisor, Prof. Dr. Christoph Meinel, for his continues support. In spite of his tight schedule, he always found the time to discuss, guide, and motivate my research ideas. The thanks are extended to Dr. Karin-Irene Eiermann for assisting me even before moving to Germany. I am also grateful for Michaela Schmitz. She managed everything well to make everyones life easier. I owe a thanks to Dr. Nemeth Sharon for helping me to improve my English writing skills. -
An Efficient Evolutionary Algorithm for Solving Incrementally Structured
An Efficient Evolutionary Algorithm for Solving Incrementally Structured Problems Jason Ansel Maciej Pacula Saman Amarasinghe Una-May O'Reilly MIT - CSAIL July 14, 2011 Jason Ansel (MIT) PetaBricks July 14, 2011 1 / 30 Our goal is to make programs run faster We use evolutionary algorithms to search for faster programs The PetaBricks language defines search spaces of algorithmic choices Who are we? I do research in programming languages (PL) and compilers The PetaBricks language is a collaboration between: A PL / compiler research group A evolutionary algorithms research group A applied mathematics research group Jason Ansel (MIT) PetaBricks July 14, 2011 2 / 30 The PetaBricks language defines search spaces of algorithmic choices Who are we? I do research in programming languages (PL) and compilers The PetaBricks language is a collaboration between: A PL / compiler research group A evolutionary algorithms research group A applied mathematics research group Our goal is to make programs run faster We use evolutionary algorithms to search for faster programs Jason Ansel (MIT) PetaBricks July 14, 2011 2 / 30 Who are we? I do research in programming languages (PL) and compilers The PetaBricks language is a collaboration between: A PL / compiler research group A evolutionary algorithms research group A applied mathematics research group Our goal is to make programs run faster We use evolutionary algorithms to search for faster programs The PetaBricks language defines search spaces of algorithmic choices Jason Ansel (MIT) PetaBricks July 14, 2011 -
SIMD Extensions
SIMD Extensions PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Sat, 12 May 2012 17:14:46 UTC Contents Articles SIMD 1 MMX (instruction set) 6 3DNow! 8 Streaming SIMD Extensions 12 SSE2 16 SSE3 18 SSSE3 20 SSE4 22 SSE5 26 Advanced Vector Extensions 28 CVT16 instruction set 31 XOP instruction set 31 References Article Sources and Contributors 33 Image Sources, Licenses and Contributors 34 Article Licenses License 35 SIMD 1 SIMD Single instruction Multiple instruction Single data SISD MISD Multiple data SIMD MIMD Single instruction, multiple data (SIMD), is a class of parallel computers in Flynn's taxonomy. It describes computers with multiple processing elements that perform the same operation on multiple data simultaneously. Thus, such machines exploit data level parallelism. History The first use of SIMD instructions was in vector supercomputers of the early 1970s such as the CDC Star-100 and the Texas Instruments ASC, which could operate on a vector of data with a single instruction. Vector processing was especially popularized by Cray in the 1970s and 1980s. Vector-processing architectures are now considered separate from SIMD machines, based on the fact that vector machines processed the vectors one word at a time through pipelined processors (though still based on a single instruction), whereas modern SIMD machines process all elements of the vector simultaneously.[1] The first era of modern SIMD machines was characterized by massively parallel processing-style supercomputers such as the Thinking Machines CM-1 and CM-2. These machines had many limited-functionality processors that would work in parallel. -
Scalability and Optimization Strategies for GPU Enhanced Neural Networks (Genn)
Scalability and Optimization Strategies for GPU Enhanced Neural Networks (GeNN) Naresh Balaji1, Esin Yavuz2, Thomas Nowotny2 {T.Nowotny, E.Yavuz}@sussex.ac.uk, [email protected] 1National Institute of Technology, Tiruchirappalli, India 2School of Engineering and Informatics, University of Sussex, Brighton, UK Abstract: Simulation of spiking neural networks has been traditionally done on high-performance supercomputers or large-scale clusters. Utilizing the parallel nature of neural network computation algorithms, GeNN (GPU Enhanced Neural Network) provides a simulation environment that performs on General Purpose NVIDIA GPUs with a code generation based approach. GeNN allows the users to design and simulate neural networks by specifying the populations of neurons at different stages, their synapse connection densities and the model of individual neurons. In this report we describe work on how to scale synaptic weights based on the configuration of the user-defined network to ensure sufficient spiking and subsequent effective learning. We also discuss optimization strategies particular to GPU computing: sparse representation of synapse connections and occupancy based block-size determination. Keywords: Spiking Neural Network, GPGPU, CUDA, code-generation, occupancy, optimization 1. Introduction The computational performance of traditional single-core CPUs has steadily increased since its arrival, primarily due to the combination of increased clock frequency, process technology advancements and compiler optimizations. The clock frequency was pushed higher and higher, from 5 MHz to 3 GHz in the years from 1983 to 2002, until it reached a stall a few years ago [1] when the power consumption and dissipation of the transistors reached peaks that could not compensated by its cost [2]. -
An Introduction to Gpus, CUDA and Opencl
An Introduction to GPUs, CUDA and OpenCL Bryan Catanzaro, NVIDIA Research Overview ¡ Heterogeneous parallel computing ¡ The CUDA and OpenCL programming models ¡ Writing efficient CUDA code ¡ Thrust: making CUDA C++ productive 2/54 Heterogeneous Parallel Computing Latency-Optimized Throughput- CPU Optimized GPU Fast Serial Scalable Parallel Processing Processing 3/54 Why do we need heterogeneity? ¡ Why not just use latency optimized processors? § Once you decide to go parallel, why not go all the way § And reap more benefits ¡ For many applications, throughput optimized processors are more efficient: faster and use less power § Advantages can be fairly significant 4/54 Why Heterogeneity? ¡ Different goals produce different designs § Throughput optimized: assume work load is highly parallel § Latency optimized: assume work load is mostly sequential ¡ To minimize latency eXperienced by 1 thread: § lots of big on-chip caches § sophisticated control ¡ To maXimize throughput of all threads: § multithreading can hide latency … so skip the big caches § simpler control, cost amortized over ALUs via SIMD 5/54 Latency vs. Throughput Specificaons Westmere-EP Fermi (Tesla C2050) 6 cores, 2 issue, 14 SMs, 2 issue, 16 Processing Elements 4 way SIMD way SIMD @3.46 GHz @1.15 GHz 6 cores, 2 threads, 4 14 SMs, 48 SIMD Resident Strands/ way SIMD: vectors, 32 way Westmere-EP (32nm) Threads (max) SIMD: 48 strands 21504 threads SP GFLOP/s 166 1030 Memory Bandwidth 32 GB/s 144 GB/s Register File ~6 kB 1.75 MB Local Store/L1 Cache 192 kB 896 kB L2 Cache 1.5 MB 0.75 MB -
Lecture 8.Key
CSC 391/691: GPU Programming Fall 2015 Parallel Sorting Algorithms Copyright © 2015 Samuel S. Cho Sorting Algorithms Review 2 • Bubble Sort: O(n ) 2 • Insertion Sort: O(n ) • Quick Sort: O(n log n) • Heap Sort: O(n log n) • Merge Sort: O(n log n) • The best we can expect from a sequential sorting algorithm using p processors (if distributed evenly among the n elements to be sorted) is O(n log n) / p ~ O(log n). Compare and Exchange Sorting Algorithms • Form the basis of several, if not most, classical sequential sorting algorithms. • Two numbers, say A and B, are compared between P0 and P1. P0 P1 A B MIN MAX Bubble Sort • Generic example of a “bad” sorting 0 1 2 3 4 5 algorithm. start: 1 3 8 0 6 5 0 1 2 3 4 5 Algorithm: • after pass 1: 1 3 0 6 5 8 • Compare neighboring elements. • Swap if neighbor is out of order. 0 1 2 3 4 5 • Two nested loops. after pass 2: 1 0 3 5 6 8 • Stop when a whole pass 0 1 2 3 4 5 completes without any swaps. after pass 3: 0 1 3 5 6 8 0 1 2 3 4 5 • Performance: 2 after pass 4: 0 1 3 5 6 8 Worst: O(n ) • 2 • Average: O(n ) fin. • Best: O(n) "The bubble sort seems to have nothing to recommend it, except a catchy name and the fact that it leads to some interesting theoretical problems." - Donald Knuth, The Art of Computer Programming Odd-Even Transposition Sort (also Brick Sort) • Simple sorting algorithm that was introduced in 1972 by Nico Habermann who originally developed it for parallel architectures (“Parallel Neighbor-Sort”).