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Parallel Sorting: Strategies for Distributed Data Values and Sequential Algorithms

This article explores parallel sorting techniques for distributed data values, including sequential algorithms like Naive Solution, Distributed Sequential Sort, Odd-Even Sort, Shearsort, Bucketsort, Parallel Mergesort, and Parallel Quicksort. The motivation behind parallel sorting is to achieve speedup in sorting large data sets when memory must be distributed. The article discusses the complexities and communication costs involved in each approach and provides insights into optimizing memory space.

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Parallel Sorting: Strategies for Distributed Data Values and Sequential Algorithms

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  1. Parallel sorting… • n data values • sequential algorithms ~ O( n.log n) except special cases. • motivation: • as n huge, memory must be distributed • want some parallel speedup. • on p processes  nlocal  nglobal / nprocs. • O(nglobal(log nglobal+1) ) possible with parallel sort. Naive solution… • Distributed sequential sort method. • Still ~ O( nglobal.log nglobal )+ communication cost: • as n huge gets painful because many short messages passed. • no parallel speedup • optimal for memory space • Consider lightly optimized bubble sort. • O(n2)

  2. Distributed sequential sort… S S O O R R T T I I N N G G E E X X A A M M P P L L E E O O N N E E A A Partition Sort I N O R S T A E G M P X A E E L N O Compare/swap I A N O R S T A I E G M P X A E E L N O Compare/swap A E A I N O R S T I N E I G M P X E A E E L N O Compare/swap Sort X O P R S T I L M N N O A A E E E G Compare/swap I P R S T X O O I L M N N O A A E E E G Compare/swap Sort I L M N N O O X O T P P R R S S R S T T P X X O A A E E E G DONE

  3. Odd-even sort… S S O O R R T T I I N N G G E E X X A A M M P P L L E E O O N N E E A B Partition odd/even R I N S T T M A A E M P X E P A L N E N E O N A A E G T I N A E G A M P A M P E X E First iteration p1 R S T p3 p5 Evens merge Odds return high Odds merge Odds send left Evens send left Evens return high Sort data[0,…,nlocal-1] I S S O N R O G E A R E G S X X T L M E E L P X O O O O R S R R S T G E E G X L A E E E O A L N E N O p0 I N T p2 M A P M X P p4 Second iteration O L N P O X A A I N E G O A R S E T E O L R A S M E E T N L R M S M P T X N E E I N Evens send left Odds merge Evens return high Odds return high Evens merge Odds send left O I N R S T A R E S E T M A P E E X A E G I N O I N O R A E S G T L R M M S P X T N L L O N P N O O X Third iteration S T X O P X G N E E O L N M O N R S T A E E I N Evens return high Odds send left Evens merge O I N L R M S N T N O A E I L M N A A I E N E O A E G E I E G O P R R S T O P X S T X DONE ceil(nprocs/2) iterations

  4. Odd-even sort… (subtleties) • Allocate contiguous memory for merge • int * data, *buffer; • data = (int*)calloc(2*nlocal, sizeof(int)); // 2*nlocal ints allocated. data [0,…,nlocal-1] is significant • buffer = &data[nlocal]; // buffer points to second half of the data[ ] array • Need to figure out the left and right neighbors: • left_neighbor = mype-1; // mype will send nlocal ints from &data[0] to left_neighbor’s &buffer[0] • right_neighbor = mype+1; // mype will send nlocal ints from &buffer[0] to right_neighbor’s &data[0] • BUT… • …the left-most pe (mype == 0) has nowhere to send the low values in data[0,…,nlocal-1] since it has no left neighbor. • This pe’s left_neighbor should be set to MPI_PROC_NULL. Sending to MPI_PROC_NULL returns immediately without doing anything. • AND … • …the right-most pe (mype == nprocs-1) has nowhere to return the high data pointed to by buffer because it has no right neighbor. • This pe should send the high values in &buffer[0,nlocal-1] to MPI_PROC_NULL, and • EITHER • When initializing data[ ], this pe should also pad buffer[0,…,nlocal-1] with INT_MAX (assuming we’re sorting ints) so that the merge step operates correctly, • OR • This pe should not execute the merge step of the iteration.

  5. Shearsort… • Have seen treating processes as 1D array decomposed into odd/even. • How about odd-even sort on 2D array of processes? • Big speed-up requires special ordering Smallest number O 1 2 3 7 6 5 4 8 9 10 11 15 14 13 12 Largest number

  6. Shearsort… • For i = 1,2,3,…,log(n)+1 • If i is odd • sort even rows biggest at left, smallest at right • sort odd rows smallest at left, biggest at right • If i is even • sort all columnsso smallest number is at top, and biggest at bottom S A A A A O O E R I E E E R G G S G T G E I E T N N N N N E M I M M M G L N L G L L I E E I I E n=4 n=2 n=5 n=3 n=1 Done N N N N X A N A M N N N O P M P P P T T P X P O X N O O X X S T R N O S E S R S L R L X E R O T

  7. Other sorts… • Bucketsort: • Pe’s partition their data into small buckets. • Pe’s send appropriate chunks to “large buckets” on each pe. • Pe’s sort “large buckets”. • (Optional: preprocessing stage where master collects info on distribution, assigns buckets to slaves. This procedure deals with non-uniform distribution problems). • Parallel mergesort O(log2n) • Odd-even implementation of standard mergesort. • Parallel quicksort: O(n) • master-slave, difficult to balance sub-tasks. • tree implementation – even harder to balance tree • hypercube topology can be optimal! • still O(n2) worst case.

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