Sorting algorithms

Selection Sort

The selection sort algorithm sorts an array by repeatedly finding the minimum element (considering ascending order) from unsorted part and putting it at the beginning. The algorithm maintains two subarrays in a given array. 1) The subarray which is already sorted. 2) Remaining subarray which is unsorted. In every iteration of selection sort, the minimum element (considering ascending order) from the unsorted subarray is picked and moved to the sorted subarray. Time Complexity: O(n2) as there are two nested loops. Auxiliary Space: O(1) The good thing about selection sort is it never makes more than O(n) swaps and can be useful when memory write is a costly operation. Exercise : Sort an array of strings using Selection Sort Stability : The default implementation is not stable. However it can be made stable. Please see stable selection sort for details. In Place : Yes, it does not require extra space.

Bubble Sort

Bubble Sort is the simplest sorting algorithm that works by repeatedly swapping the adjacent elements if they are in wrong order. Worst and Average Case Time Complexity: O(n*n). Worst case occurs when array is reverse sorted. Best Case Time Complexity: O(n). Best case occurs when array is already sorted. Auxiliary Space: O(1) Boundary Cases: Bubble sort takes minimum time (Order of n) when elements are already sorted. Sorting In Place: Yes Stable: Yes

HeapSort

Heap sort is a comparison based sorting technique based on Binary Heap data structure. It is similar to selection sort where we first find the maximum element and place the maximum element at the end. We repeat the same process for remaining element. What is Binary Heap? Let us first define a Complete Binary Tree. A complete binary tree is a binary tree in which every level, except possibly the last, is completely filled, and all nodes are as far left as possible (Source Wikipedia) A Binary Heap is a Complete Binary Tree where items are stored in a special order such that value in a parent node is greater(or smaller) than the values in its two children nodes. The former is called as max heap and the latter is called min heap. The heap can be represented by binary tree or array. Notes: Heap sort is an in-place algorithm. Its typical implementation is not stable, but can be made stable (See this) Time Complexity: Time complexity of heapify is O(Logn). Time complexity of createAndBuildHeap() is O(n) and overall time complexity of Heap Sort is O(nLogn). Applications of HeapSort 1. Sort a nearly sorted (or K sorted) array 2. k largest(or smallest) elements in an array Heap sort algorithm has limited uses because Quicksort and Mergesort are better in practice. Nevertheless, the Heap data structure itself is enormously used. See Applications of Heap Data Structure

Merge Sort

Like QuickSort, Merge Sort is a Divide and Conquer algorithm. It divides input array in two halves, calls itself for the two halves and then merges the two sorted halves. The merge() function is used for merging two halves. The merge(arr, l, m, r) is key process that assumes that arr[l..m] and arr[m+1..r] are sorted and merges the two sorted sub-arrays into one. Time Complexity: Sorting arrays on different machines. Merge Sort is a recursive algorithm and time complexity can be expressed as following recurrence relation. T(n) = 2T(n/2) + \Theta(n) The above recurrence can be solved either using Recurrence Tree method or Master method. It falls in case II of Master Method and solution of the recurrence is \Theta(nLogn). Time complexity of Merge Sort is \Theta(nLogn) in all 3 cases (worst, average and best) as merge sort always divides the array into two halves and take linear time to merge two halves. Auxiliary Space: O(n) Algorithmic Paradigm: Divide and Conquer Sorting In Place: No in a typical implementation Stable: Yes