Analysis of different sorting techniques
Last Updated :
23 Jan, 2024
In this article, we will discuss important properties of different sorting techniques including their complexity, stability and memory constraints. Before understanding this article, you should understand basics of different sorting techniques (See : Sorting Techniques).
Time complexity Analysis -
We have discussed the best, average and worst case complexity of different sorting techniques with possible scenarios.
Comparison based sorting -
In comparison based sorting, elements of an array are compared with each other to find the sorted array.
- Bubble sort and Insertion sort -
Average and worst case time complexity: n^2
Best case time complexity: n when array is already sorted.
Worst case: when the array is reverse sorted.
- Selection sort -
Best, average and worst case time complexity: n^2 which is independent of distribution of data.
- Merge sort -
Best, average and worst case time complexity: nlogn which is independent of distribution of data.
- Heap sort -
Best, average and worst case time complexity: nlogn which is independent of distribution of data.
- Quick sort -
It is a divide and conquer approach with recurrence relation:
T(n) = T(k) + T(n-k-1) + cn
- Worst case: when the array is sorted or reverse sorted, the partition algorithm divides the array in two subarrays with 0 and n-1 elements. Therefore,
T(n) = T(0) + T(n-1) + cn
Solving this we get, T(n) = O(n^2)
- Best case and Average case: On an average, the partition algorithm divides the array in two subarrays with equal size. Therefore,
T(n) = 2T(n/2) + cn
Solving this we get, T(n) = O(nlogn)
Non-comparison based sorting -
In non-comparison based sorting, elements of array are not compared with each other to find the sorted array.
- Radix sort -
Best, average and worst case time complexity: nk where k is the maximum number of digits in elements of array.
- Count sort -
Best, average and worst case time complexity: n+k where k is the size of count array.
- Bucket sort -
Best and average time complexity: n+k where k is the number of buckets.
Worst case time complexity: n^2 if all elements belong to same bucket.
In-place/Outplace technique -
A sorting technique is inplace if it does not use any extra memory to sort the array.
Among the comparison based techniques discussed, only merge sort is outplaced technique as it requires an extra array to merge the sorted subarrays.
Among the non-comparison based techniques discussed, all are outplaced techniques. Counting sort uses a counting array and bucket sort uses a hash table for sorting the array.
Online/Offline technique -
A sorting technique is considered Online if it can accept new data while the procedure is ongoing i.e. complete data is not required to start the sorting operation.
Among the comparison based techniques discussed, only Insertion Sort qualifies for this because of the underlying algorithm it uses i.e. it processes the array (not just elements) from left to right and if new elements are added to the right, it doesn't impact the ongoing operation.
Stable/Unstable technique -
A sorting technique is stable if it does not change the order of elements with the same value.
Out of comparison based techniques, bubble sort, insertion sort and merge sort are stable techniques. Selection sort is unstable as it may change the order of elements with the same value. For example, consider the array 4, 4, 1, 3.
In the first iteration, the minimum element found is 1 and it is swapped with 4 at 0th position. Therefore, the order of 4 with respect to 4 at the 1st position will change. Similarly, quick sort and heap sort are also unstable.
Out of non-comparison based techniques, Counting sort and Bucket sort are stable sorting techniques whereas radix sort stability depends on the underlying algorithm used for sorting.
Analysis of sorting techniques :
- When the array is almost sorted, insertion sort can be preferred.
- When order of input is not known, merge sort is preferred as it has worst case time complexity of nlogn and it is stable as well.
- When the array is sorted, insertion and bubble sort gives complexity of n but quick sort gives complexity of n^2.
Que - 1. Which sorting algorithm will take the least time when all elements of input array are identical? Consider typical implementations of sorting algorithms.
(A) Insertion Sort
(B) Heap Sort
(C) Merge Sort
(D) Selection Sort
Solution: As discussed, insertion sort will have the complexity of n when the input array is already sorted.
Que - 2. Consider the Quicksort algorithm. Suppose there is a procedure for finding a pivot element which splits the list into two sub-lists each of which contains at least one-fifth of the elements. Let T(n) be the number of comparisons required to sort n elements. Then, (GATE-CS-2012)
(A) T(n) <= 2T(n/5) + n
(B) T(n) <= T(n/5) + T(4n/5) + n
(C) T(n) <= 2T(4n/5) + n
(D) T(n) <= 2T(n/2) + n
Solution: The complexity of quick sort can be written as:
T(n) = T(k) + T(n-k-1) + cn
As given in question, one list contains 1/5th of total elements. Therefore, another list will have 4/5 of total elements. Putting values, we get:
T(n) = T(n/5) + T(4n/5) + cn, which matches option (B).
Time and Space Complexity Comparison Table :
|
|
| Bubble Sort | Ω(N) | Θ(N2) | O(N2) | O(1) |
| Selection Sort | Ω(N2) | Θ(N2) | O(N2) | O(1) |
| Insertion Sort | Ω(N) | Θ(N2) | O(N2) | O(1) |
| Merge Sort | Ω(N log N) | Θ(N log N) | O(N log N) | O(N) |
| Heap Sort | Ω(N log N) | Θ(N log N) | O(N log N) | O(1) |
| Quick Sort | Ω(N log N) | Θ(N log N) | O(N2) | O(log N) |
| Radix Sort | Ω(N k) | Θ(N k) | O(N k) | O(N + k) |
| Count Sort | Ω(N + k) | Θ(N + k) | O(N + k) | O(k) |
| Bucket Sort | Ω(N + k) | Θ(N + k) | O(N2) | O(N) |
Sort stability, Efficiency, Passes Comparison Table :
|
0(n2)
| n-1
| stable
|
0(n2)
| n-1
| unstable
(can be stable using Linked List)
|
0(n)
0(n2)
|
n-1
n-1
|
stable
|
0(n log n)
0(n2)
|
log n
n-1
|
unstable
|
0(n log n)
| log n
| stable
|
0(n)
0(n2)
|
log n
log n
|
unstable
|
0(n)
| No. of digits in the largest number
| stable
|
Similar Reads
Program for sorting variables of any data type
Given an array of any data type, the task is to sort the given array without using in-built sorting functions.Examples:Input: arr[] = [64, 34, 25, 12, 22, 11, 90]Output: 11 12 22 25 34 64 90 Input: arr[] = ["banana", "apple", "orange", "grape", "kiwi"]Output: apple banana grape kiwi orange Input: ar
7 min read
Sort a String in Java (2 different ways)
The string class doesn't have any method that directly sorts a string, but we can sort a string by applying other methods one after another. The string is a sequence of characters. In java, objects of String are immutable which means a constant and cannot be changed once created. Creating a String T
6 min read
The Slowest Sorting Algorithms
A Sorting Algorithm is used to rearrange a given array or list elements according to a comparison operator on the elements. The comparison operator is used to decide the new order of the element in the respective data structure. But Below is some of the slowest sorting algorithms: Stooge Sort: A Sto
15+ min read
Time and Space complexity analysis of Selection Sort
The Selection sort algorithm has a time complexity of O(n^2) and a space complexity of O(1) since it does not require any additional memory space apart from a temporary variable used for swapping. Time Complexity Analysis of Selection Sort:Best-case: O(n2), best case occurs when the array is already
2 min read
Time Complexities of all Sorting Algorithms
The efficiency of an algorithm depends on two parameters:Time ComplexityAuxiliary SpaceBoth are calculated as the function of input size(n). One important thing here is that despite these parameters, the efficiency of an algorithm also depends upon the nature and size of the input. Time Complexity:T
2 min read
When to use each Sorting Algorithms | Set 2
Sorting is the process of arranging a set of data in a specific order, which may be numerical (ascending, descending) or lexicographical (alphabetical) order. Why Sorting Is Required? Sorting is very essential when there is a need to highly optimize the searching algorithm. For example, let's assume
5 min read
Commonly Asked Data Structure Interview Questions on Sorting
Sorting is a fundamental concept in computer science and data structures, often tested in technical interviews. Sorting algorithms are essential for organizing data in a specific order, whether it's ascending or descending. Understanding various sorting techniques—like Quick Sort, Merge Sort, Bubble
4 min read
Application and uses of Quicksort
Quicksort: Quick sort is a Divide Conquer algorithm and the fastest sorting algorithm. In quick sort, it creates two empty arrays to hold elements less than the pivot element and the element greater than the pivot element and then recursively sort the sub-arrays. There are many versions of Quicksort
2 min read
Sorting collection of String and StringBuffer in Java
Sorting a collection of objects can be done in two ways:Â By implementing Comparable (Natural Order) e.g. Strings are sorted ASCIIbetically. meaning B comes before A and 10 is before 2.By implementing Comparator (Custom order) and it can be in any order.Using Collections.sort() method of Collections
2 min read
Types of Sorting Algorithm in R Programming
There are multiple ways by which data can be sorted in the R language. It’s up to the data Analyst to consider the most suitable method based upon the structure of the data. There are multiple algorithms for performing sorting on the data in the R programming language. Below different types of sorti
6 min read