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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
- compare-exchange
- O(n log n)

- Noncomparison based
- Uses known properties of the elements
- O(n) - bucket sort etc.

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.)

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}

Parallel Sorting Algorithms

- Merge Sort
- Quick Sort
- Bitonic Sort
- Others …

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

- 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.

- sequence of elements <a0, a1, … , an-1>

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

- 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 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.

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

- 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 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

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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