These slides are about parallel sorting algorithms. In which four types of sorting algorithms are discussed with the comparison between their sequential and parallel ways. The four algorithms which are included are: Bubble sort, merge sort, Bitonic sort and Shear sort.
2. Sorting in Parallel
● Why?
it is a frequent operation in many applications.
● Goal?
sorting a sequence of values in increasing order using n processors
● Potential speedup?
best sequential algorithm has complexity= O(n log n)
● the best we can aim with a parallel algorithm, using n processors is:
optimal complexity of a parallel sorting algorithm: O(n log n)/n = O(log n)
5. Bubble Sort
● One of the straight-forward sorting methods.
– Cycles through the list.
– Compares consecutive elements and swaps them if necessary.
– Stops when no more out of order pair.
● Slow & inefficient.
● Average performance is O(n^2 ).
7. Parallel Bubble Sort:
● Also known as Odd-Even Bubble sort.
● operates in two alternate phases:
Phase-even:
even processes exchange values with right neighbors.
Phase-odd:
odd processes exchange values with right neighbors.
9. Parallel Bubble Sort:
Algorithm:
1. For k = 0 to n-1
2. If k is even then
3. for i = 0 to (n/2) do in parallel
4. If A[2i] > A[2i+1] then
5. Exchange A[2i] ↔ A[2i+1]
6. Else
7. for i = 0 to (n/2)-1 do in parallel
8. If A[2i+1] > A[2i+2] then
9. Exchange A[2i+1] ↔ A[2i+2]
10. Next k
10. Parallel bubble sort : Analysis
● Steps 1-10 is a one big loop that is represented n times.
● Therefore, the parallel time complexity is O(n).
● The algorithm, odd-numbered steps need (n/2) - 1 processors and even-numbered
steps require (n/2) processors.
● Therefore, this needs O(n) processors.
12. Mergesort:
● Example of a divide-and-conquer algorithm.
● Sorting method to sort a vector; first subdivides it in two parts,
● applies again the same method to each part and when they are both sorted (2 sorted
vectors/lists) with m and n elements,
● They are merged to produce a sorted vector that contains m + n elements of the initial
vector.
● The average complexity is O(n log n).
● T (n) = b n=1
=2T(n/2)+b(n) n>1
● Solve the recurrence relation, T(n) = O(nlogn)
13. Parallel Merge Sort:
● Collects sorted list onto one processor.
● Merges elements as they come together.
● Simple tree structure.
● Parallelism is limited when near the root.
Divide:
14. Parallel Merge Sort:
● Collects sorted list onto one processor.
● Merges elements as they come together.
● Simple tree structure.
● Parallelism is limited when near the root.
Conquer
:
15. Parallel Merge sort: Using a strategy to assign work to
processors organized in a tree.
16. Merge sort - Time complexity
● Sequential Merge sort, O(nlogn)
● In Parallel,We have n processors
● logn time is required to divide sequence
● logn is for merging it.
● logn+logn= 2 logn
● O(logn)
18. Bitonic Sort :
● A Bitonic List is defined as a list with
no more than one LOCAL MAXIMUM and no
more than one LOCAL MINIMUM.
● (Endpoints must be considered - wraparound )
A sequence is bitonic if it contains two
sequences, one increasing and one decreasing,
i.e.
● a1
< a2
< . . . < ai−1
< ai
> ai+1
> ai+2
> . . . > an
for
some i such that (0 ≤ i ≤ n)
● a sequence is bitonic if the property described
is attained by a circular rotation to the right of
its elements.
19. Bitonic sort:
How many parallel steps does it take to sort ?
--> logN
How would you sort a bitonic list WHEN N is NOT A POWER OF 2 ?
-->Use as many processes as N' where N' is a power of 2 and the
smallest number larger than N. Assign a special value such as
"infinity" to all the extra processes (some kind of a padding).
20. Bitonic Sort:
How to divide a bitonic sequence into two
bitonic subsequences ??
● The compare-and-exchange
operation moves smaller values to
the left and greater values to the
right.
● Given a bitonic sequence, if we apply
recursively these operations we get a
sorted sequence.
21. Bitonic Sort:
How to divide a bitonic sequence into two
bitonic subsequences ??
● The compare-and-exchange
operation moves smaller values to
the left and greater values to the
right.
● Given a bitonic sequence, if we apply
recursively these operations we get a
sorted sequence.
22. Number of steps (P=n)
● In order to form a sorted sequence of length n from two sorted sequences of length n/2,
there are log(n) comparator stages required.
● T(n) = log(n) + T(n/2)
● The solution of this recurrence equation is,
● T(n) = log(n) + log(n)-1 + log(n)-2 + ... + 1 = log(n) · (log(n)+1) / 2
● Each stage of the sorting network consists of n/2 comparators. On the whole, these are
Θ(n·log2
(n)) comparators.
● For P=n, T(n)= Θ(n·log2
(n)) / n = Θ(log2
(n))
25. Two-Dimensional Sorting on a Mesh
● Two network structures have received
special attention:
mesh and hypercube
● The layout of a sorted sequence on a
mesh could be row by row or snake-like:
26. Shear Sort:
Input: unsorted nn-array.
Output: the array sorted in snake-like order
Method:
repeat log(n) times
i. sort the rows (in alternating direction);
ii. sort the columns;
sort the rows;
Sorting the rows in alternating direction means that even rows are sorted from left to right
and odd rows from right to left.
27. Shear sort:
Alternate row and column sorting until list is fully sorted.
Alternate row directions to get snake-like sorting:
28. Shear sort – Time complexity
ODD-PHASE:
Even Rows: sort ascending order (use O-E Transposition sort) O(n)
Odd Rows: sort descending order (use O-E Transposition sort) O(n)
EVEN-PHASE: Sort each column in ascending order.
It takes O(logn) phases to sort (n x n) numbers: T = O(nlogn)
Tpar
= O(logn)
