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Prof. Thomas Sterling Dr. Hartmut Kaiser Department of Computer Science Louisiana State University March 24th , 2011 HIGH PERFORMANCE COMPUTING: MODELS, METHODS, & MEANS APPLIED PARALLEL ALGORITHMS 4 CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 1 Topics Fourier Transforms • Fourier analysis • Discrete Fourier transform • Fast Fourier transform • Parallel Implementation Parallel Sorting • Bubble Sort • Merge Sort • Heap Sort • Quick Sort CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 2 Puzzle of the Day Duff‘s device: what is going on here? void copy(char *to, char const *from, int count) { int n = (count + 3) / 4; switch (count % 4) { case 0: do { *to++ = *from++; 'case' defines jump labels only! case 3: *to++ = *from++; case 2: *to++ = *from++; Missing 'break' makes code 'fall through' case 1: *to++ = *from++; } while (--n > 0); } } CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 3 Topics Fourier Transforms • Fourier analysis • Discrete Fourier transform • Fast Fourier transform • Parallel Implementation Parallel Sorting • Bubble Sort • Merge Sort • Heap Sort • Quick Sort CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 4 Time and Frequency Domain Representation of Signals •Two ways of looking at the same signal Example 1: Time and frequency domain representations of a sine wave http://robots.freehostia.com/Radio/Image137.gif http://www.theparticle.com/cs/bc/mcs/signalnotes.pdf CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 5 Example 2 Time and frequency domain representations of a 4Hz + 12Hz Sine Wave http://www.theparticle.com/cs/bc/mcs/signalnotes.pdf CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 6 Fourier Analysis • Fourier analysis: Represent continuous functions by potentially infinite series of sine and cosine functions NOTE: The signal sum is composed from sine and cosine functions Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn http://zone.ni.com/cms/images/devzone/tut/a/8c34be30580.gif CSC 7600 Lecture 18: Applied Parallel Algorithms 4 7 Spring 2011 Fourier Analysis Nice demo: http://www.imaios.com/en/e-Courses/e-MRI/image-formation/Fourier-transform CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 8 Fourier Representation of Square Wave • Spectrum extends to infinity • As we move from left to right on the frequency axis amplitude(of components) decreases monotonically http://www.engr.colostate.edu/~dga/mechatronics/figures/4-5.gif CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 9 Fourier Representation of Square Wave • Synthesis of a square wave(of zero DC component) from its frequency domain components • Ideal square wave is represented by the thick black line http://mathworld.wolfram.com/FourierSeriesSquareWave.html CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 10 Fourier Representation of Square Wave Nice demo: http://www.imaios.com/en/e-Courses/e-MRI/image-formation/Fourier-transform CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 11 Topics Fourier Transforms • Fourier analysis • Discrete Fourier transform • Fast Fourier transform • Parallel Implementation Parallel Sorting • Bubble Sort • Merge Sort • Heap Sort • Quick Sort CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 12 Digital Signals • Digital signal: A digital signal is a signal that is both discrete and quantized • Digital signals can be obtained by sampling analog signals • The figure represents an analog to digital converter that does sampling and quantization CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 13 Digital Signal Processing • Processing of digital signals with the help of a computer Continuous Continuous Input Digital Signal Output A/D Converter D/A Converter Processing http://www.ece.rochester.edu/courses/ECE446/Introduction%20to%20Digital%20Signal%20Processing.pdf CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 14 Advantages of Digital Signal Processing • Digital system can be simply reprogrammed for other applications / ported to different hardware / duplicated (Reconfiguring analog system means hardware redesign, testing, verification) • DSP provides better control of accuracy requirements (Analog system depends on strict components tolerance, response may drift with temperature) • Digital signals can be easily stored without deterioration (Analog signals are not easily transportable and often can’t be processed off-line) • More sophisticated signal processing algorithms can be implemented (Difficult to perform precise mathematical operations in analog form) Adapted from http://www-sigproc.eng.cam.ac.uk/~op205/3F3_1_Introduction_to_DSP.pdf CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 15 Why use Discrete Fourier Transform? • Digital Signal Processing applications often require mapping of data in the time domain to its frequency domain counterparts • Many applications in science, engineering Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 16 Example 1 • Spectrogram of Speech Signal NOTE: Spectrogram is a 3D representation of signal amplitude vs time and frequency http://ccrma.stanford.edu/~jos/st/Spectrogram_Speech.html CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 17 Example 2 •Removing blemishes of a photograph DFT is used for converting image data in the spatial (2D) To filter an image in the frequency domain: domain to the frequency 1. Compute F(u,v) the DFT of the image domain before filtering and 2. Multiply F(u,v) by a filter function H(u,v) for conversion back to spatial 3. Compute the inverse DFT of the result domain afterwards Output of different Gaussian low pass filters for removing blemishes Adapted from www.comp.dit.ie/bmacnamee/materials/dip/lectures/ImageProcessing7-FrequencyFiltering.ppt CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 18 Discrete Fourier Transform(Qualitative) • Discrete Fourier transform: Map a sequence over time to another sequence over frequency – Signal strength as a function of time – Fourier coefficients as a function of frequency Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 19 DFT Example (1/4) 16 data points representing signal strength over time Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 20 DFT Example (2/4) DFT yields amplitudes and frequencies of sine/cosine functions Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 21 DFT Example (3/4) Plot of four constituent sine/cosine functions and their sum Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 22 DFT Example (4/4) Continuous function and original 16 samples Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 23 Formal Definition of DFT • DFT of a discrete signal x[n] of N sample points is defined as N 1 2i X [k ] x[n] nk , e N for 0k N n 0 2 • Direct implementation of this equation requires N complex additions and multiplications NOTE: DFT of an N point sequence gives N points in the transform domain http://cas.ensmp.fr/~chaplais/wavetour_presentation/transformees/Fourier/FFTUS.html CSC 7600 Lecture 18: Applied Parallel Algorithms 4 24 Spring 2011 Formal Definition of DFT • Complex plane, relation of different powers of ω for N = 8 N 1 2i im X [k ] x[n] nk , e N 2i n 0 0k N 2 3 2i e 2 8 8 2i 83 e 8 8 e 1 1 8 2i 2i 0 e 4 84 e 0 8 8 8 re 0,0 2i 7 5 2i e 7 8 8 85 e 8 2i N 1 2i 6 1 e 6 X [k ] nk , e 8 8 x[n] N N k 0 0n N CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 25 Computing DFT • Writing the previous definition of DFT in matrix form N 1 2i • Matrix-vector product X = Fn x X [k ] x[n] nk , e N n 0 – x is input vector (signal samples) 0k N – Each element of Fn fi,j = n for 0 i, j < n and n is primitive nth ij root of unity – X is output vector (discrete Fourier coefficients) 2i NOTE: n is a complex number defined as e N Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 26 Example 1 How to compute the DFT of a vector having two elements? • Example Vector: (2, 3) • 2, the primitive square root of unity, is -1 2 0 2 1 x0 1 1 2 5 0 0 10 11 1 1 3 1 2 x1 2 200 201 X 0 1 1 1 5 2 10 11 2 1 1 1 3 2 2 X1 Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 27 Example 2 How to compute the DFT of a vector having four elements? • Example Vector:(1, 2, 4, 3) • The primitive 4th root of unity is i 4 4 4 4 x0 1 1 1 1 1 10 0 0 0 0 0 4 4 4 4 x1 1 i 1 i 2 3 i 1 2 3 0 2 4 6 x 1 1 1 1 4 0 4 4 4 4 2 0 3 6 9 x 1 i 1 i 3 3 i 4 4 4 4 3 Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 28 Topics Fourier Transforms • Fourier analysis • Discrete Fourier transform • Fast Fourier transform • Parallel Implementation Parallel Sorting • Bubble Sort • Merge Sort • Heap Sort • Quick Sort CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 29 Why Fast Fourier Transform(FFT)? • Reduce the computational operations required • Straightforward implementation: (n2) • Fast Fourier transform: (n log n) - (n log n) << (n2) for large values of n Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 30 Fast Fourier Transform Fourier matrix FN can be decomposed into half size Fourier matrices FN/2 : 1 0 0 0 I DN / 2 FN / 2 0 FN I D 0 P 0 0 0 N / 2 FN / 2 DN 0 0 2 0 I : Identity matrix 0 P : permutation matrix 0 0 0 0 N 1 PN : Row reordering, first even rows, then odd Example (N = 4): 1 1 1 1 1 1 1 1 1 0 1 0 1 1 0 0 0 1 0 0 1 4 4 4 1 i 1 2 3 i2 i3 0 1 0 i 1 1 0 0 0 0 0 1 1 2 4 6 1 i 2 i4 i6 1 0 1 0 0 0 1 1 1 0 0 0 4 4 4 1 3 6 9 1 i 3 i6 9 i 0 1 0 i 0 0 1 1 0 0 1 0 4 4 4 CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 31 Fast Fourier Transform • Based on divide-and-conquer strategy • We want to compute f(x), a polynomial of degree n (n is power of 2) at the n complex nth roots of unity • We define two new functions, f [0] and f [1] f ( x) a0 a1 x a2 x 2 ... an 1 x n 1 f [ 0] ( x) a0 a2 x a4 x2 ... an2 xn / 21 x x 2 f [1] ( x) a1 a3 x a5 x2 ... an1 xn / 21 f ( x) f [ 0 ] ( x 2 ) x f [1] ( x 2 ) Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 32 FFT (Cont…) • Problem of evaluating f (x) at n values of reduces to a) Evaluating f [0](x) and f [1](x) at n/2 values of That is, computing f(x) at points n , n , n , ... , n 1 0 1 2 n becomes evaluating f [0] & f [1] at (n )2 , (n )2 , (n )2 , ... , (n / 21 )2 0 1 2 n b) Performing f [0](x2) + x f [1](x2) • Leads to recursive algorithm with time complexity (n log n) Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 33 Spring 2011 Recursive Sequential Implementation of FFT Recursive_FFT(a,n) Parameter n Number of elements in a a[0……(n-1)] Coefficients Local n Primitive nth root of unity Evaluate polynomial at this point a [0] Even numbered coefficients a[1] Odd numbered coefficients y Result of transform y [0] Result of FFT of a [0] y[1] Result of FFT of a [1] Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 34 Recursive Sequential Implementation of FFT (Cont…) if n=1 then return a else 2 i n e n 1 a [0] (a[0],a[2],….,a[n-2]) a [1] (a[1],a[3],….,a[n-1]) y [0] Recursive_FFT(a [0],n/2) y [1] Recursive_FFT(a [1],n/2) for k0 to n/2 -1 do y[k] y [0] [k]+* y [1] [k] y[k+n/2] y [0] [k]- * y [1] [k] * n end for return y Adapted from slides(and text) of Parallel Programming in C with MPI and endif OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 35 Iterative Implementation Preferable • Well-written iterative version performs fewer index computations than recursive version • Iterative version evaluates key common sub- expression only once • Easier to derive parallel FFT algorithm when sequential algorithm in iterative form Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 36 Recursive Iterative (1/4) We now discuss the derivation of an iterative algorithm starting with the recursive one • Each rounded rectangle indicates an fft function call • The function goes on dividing the vector Recursive implementation of FFT for the into half until a scalar is obtained input sequence (1,2,4,3) is shown below (NOTE: DFT of a scalar is the scalar itself) (10,-3-i,0,-3+i) • The values returned as result of each function call is indicated on the curved arrows fft(1,2,4,3) (5,-3) (5,-1) Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn fft(1,4) fft(2,3) (1) (4) (2) (3) fft(1) fft(4) fft(2) fft(3) CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 Recursive Iterative (2/4) • Determining which computations are performed for each function invocation • For each rounded rectangle, the computation is of the form x+y(z) x-y(z) which corresponds to the following statements of the recursive algorithm y[k] y [0] [k]+* y [1] [k] y[k+n/2] y [0] [k]- * y [1] [k] Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn 5+1(5) -3+i(-1) 5-1(5) -3-i(-1) (5, -3) (5, -1) 1+1(4) 1-1(4) 2+1(3) 2-1(3) 1 4 2 3 CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 38 Recursive Iterative (3/4) • This diagram tracks the propagation of data values (input vector at the bottom and FFT output at the top) • Permutation stage: Index i of the input vector is replaced by rev(i), where rev(i) is the binary value of i read in the reverse order (00=>00, 01=>10, 10=>01, 11=>11) 10 -3-i 0 -3+i 5+1*5 -3+i*(-1) 5-1*5 -3-i*(-1) 5 -3 5 -1 1+1*4 1-1*4 2+1*3 2-1*3 1 4 2 3 1 4 2 3 Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn 1 2 4 3 CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 39 Recursive Iterative (4/4) • Initially, the scalars are simply forwarded upwards as the DFT of a scalar is the scalar itself • For other stages, computation of the output is performed using two values forwarded from the previous stage • The arrows depicting data flow form butterfly patterns • An iterative algorithm can be deduced from the previous diagram • The computation represented in each row (excluding the bottommost row) corresponds to one iteration of the algorithm • Hence log(n) iterations should be performed (log(4)=2 in the previous example) • For each iteration the algorithm modifies the value of every index (here n indices) Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 40 Topics Fourier Transforms • Fourier analysis • Discrete Fourier transform • Fast Fourier transform • Parallel Implementation Parallel Sorting • Bubble Sort • Merge Sort • Heap Sort • Quick Sort CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 41 Stages of Parallel Program Design • Partition – Divide problem into tasks • Communicate – Determine amount and pattern of communication • Agglomerate – Combine tasks • Map – Assign agglomerated tasks to processors • Efficiency analysis Adapted from http://nereida.deioc.ull.es/html/openmp/minnesotatutorial/content_openMP.html CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 42 Parallel FFT Program Design • Domain decomposition – Associate primitive task with each element of input vector a and corresponding element of output vector y • Add channels to handle communications between tasks Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 43 FFT Task/Channel Graph (n=8) •Long rounded rectangles represent tasks and arrows indicate communication between processes Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 44 FFT Task/Channel Graph (n=8) Cont… Steps: •Permute vector as follows (000=>000, 001=>100, …, 110=>011, 111=>111) •Perform log(n) iterations (log(8)=3) - stage 1 completed after iteration 1 - stage 2 completed after iteration 2 - stage 3 completed after iteration 3 (Vector y after stage 3 gives the output) NOTE: Vector y will contain the intermediate results of stage 1 and stage 2 Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn stage 1 stage 2 stage 3 CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 45 Diagrammatic Representation of Profiling Results Conventions: C represents a function compute (args) that accepts the propagated values and performs the following computation (refer slide 33) x+y(z) x-y(z) S represents the MPI_Send(args) command R represents the MPI_Receive(args) command P represents the function permute(args) which is basically permute(args) { ……… MPI_Send(args) ……… } represents the time for which the process is idle http://www.cs.uoregon.edu/research/paracomp/tau/tauprofile/images/petsc/ 46 CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 Diagrammatic Representation of Profiling Results Permutation Stage 1 Stage 2 Stage 3 Phase y[0] P0 S R C S R C S R C y[1] P1 P R R S C S R C S R C NOTE: The diagram is oversimplified to P2 y[2] S R C R S C S R C enhance understanding of butterfly diagram P3 P R R S C R S C S R C y[3] P4 R P S R C S R C R S C y[4] P5 y[5] R S C S R C R S C P6 y[6] R P S R C R S C R S C P7 R S C R S C R S C y[7] CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 47 Agglomeration and Mapping • Agglomerate primitive tasks associated with contiguous elements of vector to reduce communication • Map one agglomerated task to each process Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 48 After Agglomeration, Mapping Input In general, an n point FFT can be implemented on a multicomputer supporting p processes In this case, n=16 and p=4. a[0], a[1], a[2], a[3] process 1 a[4], a[5], a[6], a[7] process 2 and so on Adapted from slides(and text) of Parallel Output Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 49 Phases of Parallel FFT Algorithm • Phase 1: Processes permute a’s (all-to-all communication) • Phase 2: – First log n – log p iterations of FFT – No message passing is required • Phase 3: – Final log p iterations – Processes organized as logical hypercube – In each iteration every process swaps values with partner across a hypercube dimension Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 50 Computation Complexity Analysis • Each process performs equal share of computation – Sequential complexity: Θ(n log n) • Hence the complexity of parallel implementation is Θ(n log n / p) Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 Communication Complexity Analysis • A maximum of ceil(n / p) elements of the vector associated with a process • In the all to all communication stage, every process swaps about n/p values with its counterpart – Time complexity: Θ(n/p log p) • A total of log p iterations that need communication with other processes (average n/p swaps) – Time complexity: Θ(n/p log p) • Hence the total communication complexity of parallel implementation is Θ(n/p log p) Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 52 Topics Fourier Transforms • Fourier analysis • Discrete Fourier transform • Fast Fourier transform • Parallel Implementation Parallel Sorting • Bubble Sort • Merge Sort • Heap Sort • Quick Sort CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 53 Parallel Sorting • Finding a permutation of a sequence [a1, a2, ...an-1], such that a1 <= a2 <= … an-1 • Often we sort records based on key • Parallel sort results in: – Partial sequences are sorted on all nodes – Largest value on node N-1 is smaller or equal to smallest value on node N • Several ways to parallelize – Chunk sequence, sort locally, merge back (bubblesort) – Project algorithm structure onto cmmunication and distribution scheme (quicksort) CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 54 Bubble Sort • The bubble sort is the oldest and simplest sort in use. Unfortunately, it's also the slowest. • The bubble sort works by comparing each item in the list with the item next to it, and swapping them if required. • The algorithm repeats this process until it makes a pass all the way through the list without swapping any items (in other words, all items are in the correct order). • This causes larger values to "bubble" to the end of the list while smaller values "sink" towards the beginning of the list. The bubble sort is generally considered to be the most inefficient sorting algorithm in common usage. Under best-case conditions (the list is already sorted), the bubble sort can approach a constant O(n) level of complexity. General-case is O(n2). Pros: Simplicity and ease of implementation. Cons: Extremely inefficient. Reference http://math.hws.edu/TMCM/java/xSortLab/ Source http://www.sci.hkbu.edu.hk/tdgc/tutorial/ExpClusterComp/sorting/bubblesort.c http://www.sci.hkbu.edu.hk CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 55 Bubblesort void sort(int *v, int n) { int i, j; for(i = n-2; i >= 0; i--) for(j = 0; j <= i; j++) if(v[j] > v[j+1]) swap(v[j], v[j+1]); } CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 56 Bubblesort CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 57 Discussion • Bubble sort takes time proportional to N*N/2 for N data items • This parallelization splits N data items into N/P so time on one of the P processors now proportional to (N/P*N/P)/2 – i.e. have reduced time by a factor of P*P! • Bubble sort is much slower than quick sort! – Better to run quick sort on single processor than bubble sort on many processors! http://www.sci.hkbu.edu.hk CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 58 Topics Fourier Transforms • Fourier analysis • Discrete Fourier transform • Fast Fourier transform • Parallel Implementation Parallel Sorting • Bubble Sort • Merge Sort • Heap Sort • Quick Sort CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 59 Merge Sort • The merge sort splits the list to be sorted into two equal halves, and places them in separate arrays. • Each array is recursively sorted, and then merged back together to form the final sorted list. • Like most recursive sorts, the merge sort has an algorithmic complexity of O(n log n). • Elementary implementations of the merge sort make use of three arrays - one for each half of the data set and one to store the sorted list in. The below algorithm merges the arrays in-place, so only two arrays are required. There are non-recursive versions of the merge sort, but they don't yield any significant performance enhancement over the recursive algorithm on most machines. Pros: Marginally faster than the heap sort for larger sets. Cons: At least twice the memory requirements of the other sorts; recursive. Reference http://math.hws.edu/TMCM/java/xSortLab/ Source http://www.sci.hkbu.edu.hk/tdgc/tutorial/ExpClusterComp/sorting/mergesort.c CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 60 Merge Sort [cdekate@celeritas sort]$ mpiexec -np 4 ./mergesort 1000000; 4 processors; 0.250000 secs [cdekate@celeritas sort]$ CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 61 Mergesort void msort(int *A, int min, int max) { int *C; /* dummy, just to fit the function */ int mid = (min+max)/2; int lowerCount = mid - min + 1; int upperCount = max - mid; /* If the range consists of a single element, it's already sorted */ if (max == min) { return; } else { /* Otherwise, sort the first half */ sort(A, min, mid); /* Now sort the second half */ sort(A, mid+1, max); /* Now merge the two halves */ C = merge(A + min, lowerCount, A + mid + 1, upperCount); } } CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 62 Mergesort CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 63 Topics Fourier Transforms • Fourier analysis • Discrete Fourier transform • Fast Fourier transform • Parallel Implementation Parallel Sorting • Bubble Sort • Merge Sort • Heap Sort • Quick Sort CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 64 Heap Sort • The heap sort is the slowest of the O(n log n) sorting algorithms, but unlike the merge and quick sorts it doesn't require massive recursion or multiple arrays to work. This makes it the most attractive option for very large data sets of millions of items. • The heap sort works as it name suggests 1. It begins by building a heap out of the data set, 2. Then removing the largest item and placing it at the end of the sorted array. 3. After removing the largest item, it reconstructs the heap and removes the largest remaining item and places it in the next open position from the end of the sorted array. 4. This is repeated until there are no items left in the heap and the sorted array is full. Elementary implementations require two arrays - one to hold the heap and the other to hold the sorted elements. To do an in-place sort and save the space the second array would require, the algorithm below "cheats" by using the same array to store both the heap and the sorted array. Whenever an item is removed from the heap, it frees up a space at the end of the array that the removed item can be placed in. Pros: In-place and non-recursive, making it a good choice for extremely large data sets. Cons: Slower than the merge and quick sorts. Reference http://ciips.ee.uwa.edu.au/~morris/Year2/PLDS210/heapsort.html Source http://www.sci.hkbu.edu.hk/tdgc/tutorial/ExpClusterComp/heapsort/heapsort.c CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 65 Heapsort CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 66 Topics Fourier Transforms • Fourier analysis • Discrete Fourier transform • Fast Fourier transform • Parallel Implementation Parallel Sorting • Bubble Sort • Merge Sort • Heap Sort • Quick Sort CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 67 Quick Sort • The quick sort is an in-place, divide-and-conquer, massively recursive sort. • Divide and Conquer Algorithms – Algorithms that solve (conquer) problems by dividing them into smaller sub- problems until the problem is so small that it is trivially solved. • In Place – In place sorting algorithms don't require additional temporary space to store elements as they sort; they use the space originally occupied by the elements. • Quicksort takes time proportional to (worst case) N*N for N data items, usually n log n, but most of the time much faster – for 1,000,000 items, Nlog2N ~ 1,000,000*20 • Constant communication cost – 2*N data items – for 1,000,000 must send/receive 2*1,000,000 from/to root • In general, processing/communication proportional to N*log2N/2*N = log2N/2 – so for 1,000,000 items, only 20/2 =10 times as much processing as communication • Suggests can only get speedup, with this parallelization, for very large N Reference http://ciips.ee.uwa.edu.au/~morris/Year2/PLDS210/qsort.html Source http://www.sci.hkbu.edu.hk/tdgc/tutorial/ExpClusterComp/qsort/qsort.c http://www.sci.hkbu.edu.hk CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 68 Quick Sort • The recursive algorithm consists of four steps (which closely resemble the merge sort): 1. If there are one or less elements in the array to be sorted, return immediately. 2. Pick an element in the array to serve as a "pivot" point. (Usually the left-most element in the array is used.) 3. Split the array into two parts - one with elements larger than the pivot and the other with elements smaller than the pivot. 4. Recursively repeat the algorithm for both halves of the original array. • The efficiency of the algorithm is majorly impacted by which element is chosen as the pivot point. • The worst-case efficiency of the quick sort, O(n2), occurs when the list is sorted and the left-most element is chosen. • If the data to be sorted isn't random, randomly choosing a pivot point is recommended. As long as the pivot point is chosen randomly, the quick sort has an algorithmic complexity of O(n log n). Pros: Extremely fast. Cons: Very complex algorithm, massively recursive http://www.sci.hkbu.edu.hk CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 69 Quicksort CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 70 Summary : Material for the Test • Discrete Fourier Transform: Slides 24-26 • Fast Fourier Transform (FFT): Slides 30-40 • Parallel FFT: Slides 41-52 CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 71 CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2011 72

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