Partition Algorithms

Partition algorithms are an essential concept in computer science, particularly in sorting and searching algorithms. These algorithms divide a set of data into smaller, more manageable subsets (or partitions), which can then be processed or manipulated more efficiently. The most commonly known use of partition algorithms is in the QuickSort algorithm, but they have broader applications in various algorithms.

Partitioning refers to the process of dividing an array or a list into two or more parts (sub-arrays or sub-lists), based on a certain condition or pivot element. Each part should satisfy some specific properties relative to the pivot.

Ex: [3, 2, 5, 9, 8, 10, 66, 8] after partitions considering pivot = 8(last element) the resultant array will be [3, 2, 5, 8, 8, 10, 28, 9]

Simple Partition Algorithm:

Step-by-step Process:

1. Choose a pivot: Select an element from the array as the pivot. Common choices are the first element, the last element, or a random element.

2. Rearrange elements: Rearrange the array such that:

   • All elements smaller than the pivot are moved to the left of the pivot.

   • All elements greater than the pivot are moved to the right.

3. Return the index: Once the array is partitioned, the pivot is placed in its correct position. The index of the pivot is returned, which is used to further divide the array into two sub-arrays for recursive sorting.

Note: Choosing pivot also makes the difference in complexity in the algorithm

Algorithms Used and their complexities

1. Naive Partition Algorithm

   • Simple, but inefficient. Less commonly used in practice.

   • Time Complexity: O(n)

   • Space Complexity: O(1)

2. Lomuto Partition Algorithm (Most Used)

   • Easy to implement but can degrade to quadratic time in the worst case, making it inefficient for large data sets.

   • Time Complexity: O(n) - for QuickSort it goes to O(n²)

   • Space Complexity: O(1)

3. Hoare Partition Scheme (Efficient Approach)

   • More efficient than Lomuto due to fewer swaps and better partitioning behavior.

   • Time Complexity: O(n) - for QuickSort it goes to O(n²)

   • Space Complexity: O(1)

Application

• Sorting Algorithms (QuickSort)

• Binary Search

• Median Finding

• Merge Sort and External Sorting

Conclusion

• Partitioning algorithms are crucial for efficient data manipulation, particularly in sorting.

• The key goal is to divide a collection into smaller parts based on some pivot element.

• Common partitioning schemes are Lomuto and Hoare.

• Partitioning helps reduce the complexity of problems (like sorting) by dividing them into smaller, easier sub-problems.