Parallel Algorithm - Sorting

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Sorting is a process of arranging elements in a group in a particular order, i.e., ascending order, descending order, alphabetic order, etc. Here we will discuss the following −

  • Enumeration Sort
  • Odd-Even Transposition Sort
  • Parallel Merge Sort
  • Hyper Quick Sort

Sorting a list of elements is a very common operation. A sequential sorting algorithm may not be efficient enough when we have to sort a huge volume of data. Therefore, parallel algorithms are used in sorting.

Enumeration Sort

Enumeration sort is a method of arranging all the elements in a list by finding the final position of each element in a sorted list. It is done by comparing each element with all other elements and finding the number of elements having smaller value.

Therefore, for any two elements, ai and aj any one of the following cases must be true −

  • ai < aj
  • ai > aj
  • ai = aj

Algorithm

procedure ENUM_SORTING (n)

begin
   for each process P1,j do
      C[j] := 0;
		
   for each process Pi, j do
	
      if (A[i] < A[j]) or A[i] = A[j] and i < j) then
         C[j] := 1;
      else
         C[j] := 0;
			
   for each process P1, j do
      A[C[j]] := A[j];
		
end ENUM_SORTING

Odd-Even Transposition Sort

Odd-Even Transposition Sort is based on the Bubble Sort technique. It compares two adjacent numbers and switches them, if the first number is greater than the second number to get an ascending order list. The opposite case applies for a descending order series. Odd-Even transposition sort operates in two phases − odd phase and even phase. In both the phases, processes exchange numbers with their adjacent number in the right.

Odd-Even Transposition Sort

Algorithm

procedure ODD-EVEN_PAR (n) 

begin 
   id := process's label 
	
   for i := 1 to n do 
   begin 
	
      if i is odd and id is odd then 
         compare-exchange_min(id + 1); 
      else 
         compare-exchange_max(id - 1);
			
      if i is even and id is even then 
         compare-exchange_min(id + 1); 
      else 
         compare-exchange_max(id - 1);
			
   end for
	
end ODD-EVEN_PAR

Parallel Merge Sort

Merge sort first divides the unsorted list into smallest possible sub-lists, compares it with the adjacent list, and merges it in a sorted order. It implements parallelism very nicely by following the divide and conquer algorithm.

Parallel Merge Sort

Algorithm

procedureparallelmergesort(id, n, data, newdata)

begin
   data = sequentialmergesort(data)
	
      for dim = 1 to n
         data = parallelmerge(id, dim, data)
      endfor
		
   newdata = data
end

Hyper Quick Sort

Hyper quick sort is an implementation of quick sort on hypercube. Its steps are as follows −

  • Divide the unsorted list among each node.
  • Sort each node locally.
  • From node 0, broadcast the median value.
  • Split each list locally, then exchange the halves across the highest dimension.
  • Repeat steps 3 and 4 in parallel until the dimension reaches 0.

Algorithm

procedure HYPERQUICKSORT (B, n)
begin

   id := process’s label;
	
   for i := 1 to d do
      begin
      x := pivot;
      partition B into B1 and B2 such that B1 ≤ x < B2;
      if ith bit is 0 then
		
      begin
         send B2 to the process along the ith communication link;
         C := subsequence received along the ith communication link;
         B := B1 U C;
      endif
      
      else
         send B1 to the process along the ith communication link;
         C := subsequence received along the ith communication link;
         B := B2 U C;
         end else
      end for
		
   sort B using sequential quicksort;
	
end HYPERQUICKSORT
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