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CS 584

CS 584. Sorting. One of the most common operations Definition: Arrange an unordered collection of elements into a monotonically increasing or decreasing order. Two categories of sorting internal (fits in memory) external (uses auxiliary storage). Sorting Algorithms. Comparison based

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CS 584

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  1. CS 584

  2. Sorting • One of the most common operations • Definition: • Arrange an unordered collection of elements into a monotonically increasing or decreasing order. • Two categories of sorting • internal (fits in memory) • external (uses auxiliary storage)

  3. Sorting Algorithms • Comparison based • compare-exchange • O(n log n) • Noncomparison based • Uses known properties of the elements • O(n) - bucket sort etc.

  4. Parallel Sorting Issues • Input and Output sequence storage • Where? • Local to one processor or distributed • Comparisons • How compare elements on different nodes • # of elements per processor • One (compare-exchange --> comm.) • Multiple (compare-split --> comm.)

  5. Compare-Exchange

  6. Compare-Split

  7. Sorting Networks • Specialized hardware for sorting • based on comparator x y x y max{x,y} min{x,y} min{x,y} max{x,y}

  8. Sorting Network

  9. Parallel Sorting Algorithms • Merge Sort • Quick Sort • Bitonic Sort • Others …

  10. Merge Sort • Simplest parallel sorting algorithm? • Steps • Distribute the elements • Everybody sort their own sequence • Merge the lists • Problem • How to merge the lists

  11. Bitonic Sort • Key operation: • rearrange a bitonic sequence to ordered • Bitonic Sequence • sequence of elements <a0, a1, … , an-1> • There exists i such that <a0, … ,ai> is monotonically increasing and <ai+1,… , an-1> is monotonically decreasing or • There exists a cyclic shift of indicies such that the above is satisfied.

  12. Bitonic Sequences • <1, 2, 4, 7, 6, 0> • First it increases then decreases • i = 3 • <8, 9, 2, 1, 0, 4> • Consider a cyclic shift • i will equal 3

  13. Rearranging a Bitonic Sequence • Let s = <a0, a1, … , an-1> • an/2 is the beginning of the decreasing seq. • Let s1= <min{a0, an/2}, min{a1, an/2 +1}…min{an/2-1,an-1}> • Let s2=<max{a0, an/2}, max{a1,an/2+1}… max{an/2-1,an-1} > • In sequence s1 there is an element bi = min{ai, an/2+i} • all elements before bi are from increasing • all elements after bi are from decreasing • Sequence s2 has a similar point • Sequences s1 and s2 are bitonic

  14. Rearranging a Bitonic Sequence • Every element of s1 is smaller than every element of s2 • Thus, we have reduced the problem of rearranging a bitonic sequence of size n to rearranging two bitonic sequences of size n/2 then concatenating the sequences.

  15. Rearranging a Bitonic Sequence

  16. Bitonic Merging Network

  17. What about unordered lists? • To use the bitonic merge for n items, we must first have a bitonic sequence of n items. • Two elements form a bitonic sequence • Any unsorted sequence is a concatenation of bitonic sequences of size 2 • Merge those into larger bitonic sequences until we end up with a bitonic sequence of size n

  18. Mapping onto a hypercube • One element per processor • Start with the sorting network maps • Each wire represents a processor • Map processors to wires to minimize the distance traveled during exchange

  19. Bitonic Merging Network

  20. Bitonic Merge on Hypercube

  21. Bitonic Sort Procedure BitonicSort for i = 0 to d -1 for j = i downto 0 if (i + 1)st bit of iproc <> jth bit of iproc comp_exchange_max(j, item) else comp_exchange_min(j, item) endif endfor endfor comp_exchange_max and comp_exchange_min compare and exchange the item with the neighbor on the jth dimension

  22. Bitonic Sort Stages

  23. Assignment • Pick 16 random integers • Draw the Bitonic Sort network • Step through the Bitonic sort network to produce a sorted list of integers. • Explain how the if statement in the Bitonic sort algorithm works.

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