Linear Array

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LINEAR ARRAY

JYOTIKA JAIN

MTECH 2009 IS 13ABV-IIITM Gwalior

Gwalior-474 010, MP, India

September 12, 2010

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OUTLINE

Introduction

Odd-Even Transposition Sort

Merge-Splitting Sort

Merge Sort on a Pipeline

Enumeration Sort

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INTRODUCTION

Parallel sorting algo for SIMD machines in which processorsare interconnected to form a linear array

Pi linked by a communication path to processors Pi1 andPi+1

No other link available

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ODD-EVEN TRANSPOSITION SORT

no of processors = no of elements in the input sequence Algorithm

for k=1 to [n/2] do

1. for i=1,3,...,2[n/2]-1 do in parallelif yi>yi+1 then yiyi+1 end ifend for

2. for i=2,4,...,2[(n-1)/2] do in parallelif yi>yi+1 then yiyi+1 end ifend for

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EXAMPLE

Figure: S=7,6,5,4,3,2,1

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ANALYSIS

Running time of algo t(n) = O(n)

Cost c(n) = t(n) p(n) = o(n) n = O(n2)

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MERGE SPLITTING SORT

Algorithm:

Preprocessing stepfor i=1,2,..,p do in parallelprocessor Pi sorts Si using a sequential algoend forend of preprocessing

for k=1 to [p/2] do1. for i=1,3,...,2[p/2]-1 do in parallel

1.1 merge Si and Si+1 into a sorted subsequence Ai1.2 Si first(n/p) elements ofAi1.3 Si+1second(n/p) elements ofAi

end for2. for i=2,4,...,2[?(p-1)/2] do in parallel

2.1 merge Si and Si+1 into a sorted subsequence Ai2.2 Si first(n/p) elements ofAi2.3 Si+1second(n/p) elements ofAi

end for

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EXAMPLE

Figure: Sort (12,9,10,11,7,4,3,6,2,1,8,5)

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ANALYSIS

By using heap sort in preprocessing step

The total running time is

t(n) = o[(n/p)log(n/p)] + O(n) = O((nlogn)/p+ O(n)) Cost of the algorithm

c(n) = t(n) p= O(nlogn) + o(np)optimal for p logn

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MERGING SORT ON A PIPELINE

DO steps 1,2,3 in parallel

1. P1 performs the following steps

1.1 read x1 from q1

1.2 j01.3 for i=2 to n do place xi1 on q2+j read xi from q1 jj+1 mod 2

end for1.4 place xn on q3

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STEP 2

for i=2 to r do in parallel

1. j02. k13. while k n do

if q2i2 is 2i2 elements long and q2(i1)+1 contains one

element

then3.1 for m=1 to 2i1 do

Pi compares the first element in q2(i1) to the first element inq2(i1)+1removes the larger of the two and places it on q2i+j

end for3.2 jj+1 mod 23.3 kk + 2i1

end ifend while

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STEP 3

if q2r is 2r1 elements long and q2r+1 contains one element then

for m=1 to 2r do

Pr+1 compares the first element in q2r to the first element inq2r+1,removes the larger of the two and places it on q2(r+1)

end for

end if

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EXAMPLE

Figure: Sort (1,5,3,2,8,7,4,6)

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Figure: Sort (1,5,3,2,8,7,4,6)

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ANALYSIS

Running time O(n)

Cost is given by:c(n) = t(n) p(n) = O(n) (logn + 1) = O(nlogn)

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O SO


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ENUMERATION SORT

ALGORITHM

1. for i=1 to n do in parallelPi sets its register C to 1end for

2. for k=1 to 2n do2.1 if kn then h1 else hkn end if2.2 for i=hto n do in parallel

if its registers X and Y are non empty and X


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CONTD..

2.3 for i=h to n-1 do in parallel

if its register Y is nonempty then processor Pi shifts the integer init to Pi+1 which stores it in its own register Y end if

2.4 if kn then processors P1 and Pk read the next integer xkfrom the input queue and store it in their registers Y andX,respectively end if

2.5 if k>n then processor Pkn stores in register Z ofPj thecontent of its register X, where j is the value stored in its register

C end if

end for

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CONTD


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CONTD..

Step 3 for k=1 to n do

1. processor Pn places the contents of its register Z on theoutput queue

2. for i=k to n-1 do in parallelprocessor Pi shifts the contents of its register Z to the registerZ ofPi+1end for

end for

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EXAMPLE


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EXAMPLE

Figure: Sort (8,9,7)

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Figure: Sort (8,9,7)

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ANALYSIS


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ANALYSIS

Running time O(n)

Cost is given by:c(n) = t(n) p(n) = O(n) n = O(n2)

cannot handle sequences with repeated numbers

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THANK YOU

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Linear Array

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