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Bitonic Sort Algorithm

What is the Bitonic Sort Algorithm?

Bitonic Sort is a parallel sorting algorithm that is efficient for hardware implementation. It works by recursively sorting two halves of the array into a bitonic sequence (one half in ascending order and the other in descending order) and then merging them into a single sorted sequence.

Algorithm Steps

  1. Bitonic Sequence:

    • Divide the array into two halves.
    • Recursively sort one half in ascending order and the other half in descending order to form a bitonic sequence.
  2. Bitonic Merge:

    • Compare and swap elements to merge the bitonic sequence into a single sorted sequence.
  3. Recursive Sorting:

    • Apply the bitonic sequence and bitonic merge steps recursively until the entire array is sorted.

Complexity Analysis

  • Time Complexity: O(nlog2n)O(n \log^2 n) in the worst case.
  • Space Complexity: O(n)O(n) auxiliary space due to the recursive calls.

Example

Given an unsorted array:

array = [4, 3, 2, 1]

Using the Bitonic Sort Algorithm:

  • Create bitonic sequences: [4, 3] (ascending), [2, 1] (descending)
  • Merge bitonic sequences: [3, 4] and [1, 2] to [1, 2, 3, 4]

Implementation

def bitonic_sort(arr, low, cnt, direction):
if cnt > 1:
k = cnt // 2
bitonic_sort(arr, low, k, 1)
bitonic_sort(arr, low + k, k, 0)
bitonic_merge(arr, low, cnt, direction)

def bitonic_merge(arr, low, cnt, direction):
if cnt > 1:
k = cnt // 2
for i in range(low, low + k):
if (direction == 1 and arr[i] > arr[i + k]) or (direction == 0 and arr[i] < arr[i + k]):
arr[i], arr[i + k] = arr[i + k], arr[i]
bitonic_merge(arr, low, k, direction)
bitonic_merge(arr, low + k, k, direction)

def sort(arr, N, up):
bitonic_sort(arr, 0, N, up)

# Example usage:
array = [4, 3, 2, 1]
sort(array, len(array), 1)
print(array) # Output: [1, 2, 3, 4]

Conclusion

Bitonic Sort is an efficient parallel sorting algorithm with a consistent ( O(n \log^2 n) ) time complexity. It is well-suited for hardware implementations and can be useful in parallel computing environments.