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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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Cs 584

CS 584



  • 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

Sorting Algorithms

  • Comparison based

    • compare-exchange

    • O(n log n)

  • Noncomparison based

    • Uses known properties of the elements

    • O(n) - bucket sort etc.

Parallel sorting issues

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.)

Compare exchange


Compare split


Sorting networks

Sorting Networks

  • Specialized hardware for sorting

    • based on comparator









Sorting network

Sorting Network

Parallel sorting algorithms

Parallel Sorting Algorithms

  • Merge Sort

  • Quick Sort

  • Bitonic Sort

  • Others …

Merge sort

Merge Sort

  • Simplest parallel sorting algorithm?

  • Steps

    • Distribute the elements

    • Everybody sort their own sequence

    • Merge the lists

  • Problem

    • How to merge the lists

Bitonic sort

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.

Bitonic sequences

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

Rearranging a bitonic sequence

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

Rearranging a bitonic sequence1

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.

Rearranging a bitonic sequence2

Rearranging a Bitonic Sequence

Bitonic merging network

Bitonic Merging Network

What about unordered lists

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

Mapping onto a hypercube

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

Bitonic merging network1

Bitonic Merging Network

Bitonic merge on hypercube

Bitonic Merge on Hypercube

Bitonic sort1

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)


comp_exchange_min(j, item)




comp_exchange_max and comp_exchange_min compare and

exchange the item with the neighbor on the jth dimension

Bitonic sort stages

Bitonic Sort Stages



  • 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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