===========
Bubble Sort
===========
:doc:`/WorkProcessClassifiers/GlobalAlgorithm/index` - :doc:`/WorkProcessClassifiers/OneDimensionalAlgorithm/index`

*Bubble sort* algorithm is a simple sorting algorithm that sorts the target data elements by comparing and swapping the adjacent items. Its average and worst case performance is :math:`O(n^2)`\ . 

.. rubric:: Input Parameters

+--------------------+--------------------------------------------+----------------------------------------+-------------+---------+
| Parameter          | Type                                       | Constraint                             | Description | Remarks |
+====================+============================================+========================================+=============+=========+
| :math:`Y`          | :math:`Y \in \mathbb R^N`                  | :math:`N \in \mathbb{N}`               |             |         |
+--------------------+--------------------------------------------+----------------------------------------+-------------+---------+

.. rubric:: Output Parameters

+----------------------------+----------------------------------------------------+----------------------------------------+---------------------------------------------------+--------------------------------------------------------------------------------------+
| Parameter                  | Type                                               | Constraint                             | Description                                       | Remarks                                                                              |
+============================+====================================================+========================================+===================================================+======================================================================================+
| :math:`\hat{Y}`            | :math:`\hat{Y} \in \mathbb R^N`                    | :math:`N \in \mathbb{N}`               | Sorted data sequence of length :math:`N`          | The output result is sensitive to improper values such as 'nan', 'inf', 'null', etc. |
+----------------------------+----------------------------------------------------+----------------------------------------+---------------------------------------------------+--------------------------------------------------------------------------------------+

.. rubric:: Single Steps using the Algorithm

* :doc:`/DataPreprocessing/DataTransformation/DataSorting/DataSortingWithBubbleSort/index`

.. rubric:: References

- D.\  Knuth, The Art of Computer Programming, vol. 3: Sorting and Searching, 3rd Edition, Addison-Wesley, 1997.