CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2009 HIGH PERFORMANCE COMPUTING: MODELS, METHODS, & MEANS APPLIED PARALLEL ALGORITHMS 4 Dr. Hartmut

CSC 7600 Lecture 18: Applied Parallel Algorithms 4Spring 2009

HIGH PERFORMANCE COMPUTING: MODELS, METHODS, & MEANS

APPLIED PARALLEL ALGORITHMS 4

Dr. Hartmut Kaiser Department of Computer ScienceLouisiana State UniversityMarch 19th , 2009

1

http://www.csc.lsu.edu/


Topics

Fourier Transforms • Fourier analysis• Discrete Fourier transform• Fast Fourier transform• Parallel Implementation

Parallel Sorting • Bubble Sort • Merge Sort • Heap Sort • Quick Sort

2



Puzzle of the Day

void copy(char *to, char const *from, int count){ int n = (count + 3) / 4; switch (count % 4) { case 0: do { *to++ = *from++; case 3: *to++ = *from++; case 2: *to++ = *from++; case 1: *to++ = *from++; } while (--n > 0); } }

3

Duff‘s device: what is going on here?

'case' defines jump labels only!

Missing 'break' makes code 'fall through'



Topics



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Time and Frequency Domain Representation of Signals

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http://robots.freehostia.com/Radio/Image137.gif

•Two ways of looking at the same signal

Example 1: Time and frequency domain representations of a sine wave

http://www.theparticle.com/cs/bc/mcs/signalnotes.pdf



Example 2

Time and frequency domain representations of a 4Hz + 12Hz Sine Wave

6

http://www.theparticle.com/cs/bc/mcs/signalnotes.pdf



Fourier Analysis

• Fourier analysis: Represent continuous functions by potentially infinite series of sine and cosine functions

7http://zone.ni.com/cms/images/devzone/tut/a/8c34be30580.gif

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



Fourier Analysis

8

Nice demo: http://www.e-mri.org/image-formation/fourier-transform.html



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

9

http://www.engr.colostate.edu/~dga/mechatronics/figures/4-5.gif




• Synthesis of a square wave(of zero DC component) from its frequency domain components

• Ideal square wave is represented by the thick black line

10

http://mathworld.wolfram.com/FourierSeriesSquareWave.html




11

Nice demo: http://www.e-mri.org/image-formation/fourier-transform.html



Topics



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• 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

Digital Signals

13

http://www.solarisnetwork.com/term_Digital%20signalhttp://www.cdt.luth.se/~johnny/courses/smd074_1999_2/CodingCompression/kap27/slide2.html



Digital Signal Processing

• Processing of digital signals with the help of a computer

14

A/D Converter D/A Converter Digital Signal Processing

http://www.ece.rochester.edu/courses/ECE446/Introduction%20to%20Digital%20Signal%20Processing.pdf

Continuous Input

Continuous Output



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)

15

Adapted from http://www-sigproc.eng.cam.ac.uk/~op205/3F3_1_Introduction_to_DSP.pdf



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

16

Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn



Example 1

• Spectrogram of Speech Signal

17

NOTE: Spectrogram is a 3D representation of signal amplitude vs time and frequency

http://www.visualizationsoftware.com/gram.html

http://ccrma.stanford.edu/~jos/st/Spectrogram_Speech.html



Example 2

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DFT is used for converting image data in the spatial (2D) domain to the frequency domain before filtering andfor conversion back to spatialdomain afterwards

To filter an image in the frequency domain:1. Compute F(u,v) the DFT of the image2. Multiply F(u,v) by a filter function H(u,v)3. Compute the inverse DFT of the result

Adapted from www.comp.dit.ie/bmacnamee/materials/dip/lectures/ImageProcessing7-FrequencyFiltering.ppt

Output of different Gaussian low pass filters for removing blemishes

•Removing blemishes of a photograph



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

19




DFT Example (1/4)

16 data points representing signal strength over time

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DFT Example (2/4)

DFT yields amplitudes and frequencies of sine/cosine functions

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DFT Example (3/4)

Plot of four constituent sine/cosine functions and their sum

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DFT Example (4/4)

Continuous function and original 16 samples

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Formal Definition of DFT

24

• DFT of a discrete signal x[n] of N sample points is defined as

N

iN

n

nk enxkX

21

0

,*][][

• Direct implementation of this equation requires complex additions and multiplications

NOTE: DFT of an N point sequence gives N points in the transform domain

2N

for Nk 0

http://cas.ensmp.fr/~chaplais/wavetour_presentation/transformees/Fourier/FFTUS.html



Formal Definition of DFT

• Complex plane, relation of different powers of ω

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re

im

0,0

8

20

08

i

e

8

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Computing DFT

• Writing the previous definition of DFT in matrix form

• Matrix-vector product Fn x– x is input vector (signal samples)

– Each element of Fn

fi,j = nij for 0 i, j < n and n is primitive nth

root of unity

NOTE: n is a complex number defined as

n

i

e2

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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

1

5

3

2

11

11

1

0

112

012

102

002

x

x

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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

i

i

ii

ii

x

x

x

x

3

0

3

10

3

4

2

1

11

1111

11

1111

3

2

1

0

94

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04

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04

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04

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Topics



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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

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Fast Fourier Transform

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0100

0001

1000

0010

1100

1100

0011

0011

000

0101

010

0101

1

1

1

1111

1

1

1

1111

963

642

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94

64

34

64

44

24

34

24

14

i

i

iii

iii

iii

matrixnpermutatioP

matrixIdentityI

PF

F

DI

DIF

N

N

N

NN

:

:

0

0

2/

2/

2/

2/

Fourier matrix FN can be decomposed into half size Fourier matrices FN/2 :

Example (N = 4):

1

2

0000

0

000

000

0001

N

ND

oddthenrowsevenfirst

reorderingRowPN,

,:



Fast Fourier Transform

• Based on divide-and-conquer strategy

• Suppose we want to compute f(x)• We define two new functions, f[0] and f[1]

11

2210 ...)(

nn xaxaxaaxf

12/1

2531

]1[

12/2

2420

]0[

...

...

n

n

nn

xaxaxaaf

xaxaxaaf

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NOTE: Different FFT implementations exist Adapted from slides(and text) of Parallel Programming in C with MPI and OpenMP by Michael Quinn



FFT (Cont…)

• Note: f(x) = f [0](x2) + x f [1](x2)

• Problem of evaluating f (x) at n values of reduces toa) Evaluating f [0](x) and f [1](x) at n/2 values of That is, computing f(x) at points

becomes evaluating f [0] & f [1] at

b) Performing f [0](x2) + x f [1](x2)

• Leads to recursive algorithm with time complexity (n log

n)

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212/222120 )(....,,.........)(,)(,)( nnnnn

1210 ...,,.........,, nnnnn




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]

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Recursive Sequential Implementation of FFT (Cont…)

if n=1 then

return a

else

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

endif

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n

i

e2




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

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Recursive Iterative (1/4)

Recursive implementation of FFT for the

input sequence (1,2,4,3) is shown below

fft(1) fft(4) fft(3)fft(2)

fft(1,2,4,3)

fft(2,3)fft(1,4)

(5,-3)

(10,-3-i,0,-3+i)

(5,-1)

(2) (3)(4)(1)

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 into half until a scalar is obtained(NOTE: DFT of a scalar is the scalar itself)

• The values returned as result of each function call is indicated on the curved arrows





• 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]

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5+1*5) -3+i*(-1) 5-1*5 -3-i*(-1)

4

1+1*4 1-1*4 2+1*3 2-1*3

1 2 3




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)

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5+1*5 -3+i*(-1) 5-1*5 -3-i*(-1)

5 -3 -15

1+1*4 1-1*4 2+1*3 2-1*3

1 4 32

1 2

1 4 2 3

4 3

10 -3+i-3-i 0





• 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)

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Topics



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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

42

Adapted from http://nereida.deioc.ull.es/html/openmp/minnesotatutorial/content_openMP.html



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

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FFT Task/Channel Graph (n=8)

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•Long rounded rectangles representtasks and arrows indicate communication between processes




FFT Task/Channel Graph (n=8) Cont…

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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

stage 1 stage 2 stage 3




Diagrammatic Representation of Profiling Results

46

Conventions:

represents a function compute (args) that accepts the propagated values and performs the following computation (refer slide 33) x+y(z) x-y(z)

represents the MPI_Send(args) command

represents the MPI_Receive(args) command

represents the function permute(args) which is basically permute(args) { ……… MPI_Send(args) ……… }

represents the time for which the process is idle

C

S

R

http://www.cs.uoregon.edu/research/paracomp/tau/tauprofile/images/petsc/

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Diagrammatic Representation of Profiling Results

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Permutation Phase

Stage 1

NOTE: The diagram is oversimplified to enhance understandingof butterfly diagram

P

P

P0

P1

P2

P3

P4

P5

P6

P7

P

S

SR

S

R

S

R

S

R

Stage 2 Stage 3

y[0]

y[1]

y[2]

y[3]

y[4]

y[6]

y[5]

y[7]

R

R

R

RP

R

S

R

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R

C

C

C

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Agglomeration and Mapping

• Agglomerate primitive tasks associated with contiguous elements of vector to reduce communication

• Map one agglomerated task to each process

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After Agglomeration, MappingInput

Output

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In general, an n point FFT can beimplemented on a multicomputersupporting 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




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

50




Computation Complexity Analysis

• Each process performs equal share of computation– Sequential complexity: Θ(n log n / p)

• Hence the complexity of parallel implementation is

Θ(n log n / p)




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)

52




Topics



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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)

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.

Referencehttp://math.hws.edu/TMCM/java/xSortLab/

Sourcehttp://www.sci.hkbu.edu.hk/tdgc/tutorial/ExpClusterComp/sorting/bubblesort.c

http://www.sci.hkbu.edu.hk

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]);}

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Bubblesort

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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!


58



Topics



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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

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Merge Sort

[cdekate@celeritas sort]$ mpirun -np 4 mergesort1000000; 4 processors; 0.250000 secs[cdekate@celeritas sort]$

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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);

}}

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Mergesort

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Topics



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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 suggests1. 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.

Referencehttp://ciips.ee.uwa.edu.au/~morris/Year2/PLDS210/heapsort.html

Sourcehttp://www.sci.hkbu.edu.hk/tdgc/tutorial/ExpClusterComp/heapsort/heapsort.c

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Heapsort

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Topics



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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

Referencehttp://ciips.ee.uwa.edu.au/~morris/Year2/PLDS210/qsort.html

Sourcehttp://www.sci.hkbu.edu.hk/tdgc/tutorial/ExpClusterComp/qsort/qsort.c


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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


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Quicksort

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CSC 7600 Lecture 18: Applied Parallel Algorithms 4Spring 2009 71


Documents

CSC 7600 Lecture 18: Applied Parallel Algorithms 4 Spring 2009 HIGH PERFORMANCE COMPUTING: MODELS, METHODS, & MEANS APPLIED PARALLEL ALGORITHMS 4 Dr. Hartmut