1. Sorting Algorithms: A
Comprehensive Guide
A Sorting Algorithm is used to rearrange a given array or list of
elements in an order. Sorting is provided in library implementation
of most of the programming languages.
2. Types of Sorting Techniques
There are various sorting algorithms are used in
data structures. The following two types of sorting
algorithms can be broadly classified:
1. Comparison-based: We compare the elements
in a comparison-based sorting algorithm)
2. Non-comparison-based: We do not compare
the elements in a non-comparison-based sorting
algorithm)
3. Most Common Sorting Techniques
Some of the most common sorting algorithms are:
Selection sort, Bubble sort, Insertion Sort, Cycle Sort, Merge Sort, 3-way Merge Sort,
Quick sort, Heap sort and Counting sort.
Some other Sorting algorithms:
Radix sort, Bucket sort, Shell sort, Tim Sort, Comb Sort, Pigeonhole sorting, Cocktail
Sort, Strand sort, Stooge Sort, Tag Sort, Tree sort, Cartesian Sort, Odd-Even Sort / Brick
Sort, Gnome sort, Cocktail shaker sort
4. Comparison of Complexity Analysis of
Sorting Algorithms:
Name Best Case Average Case Worst Case Memory Method Used
Quick Sort nlogn nlogn n2
logn Partitioning
Merge Sort nlogn nlogn nlogn n Merging
Heap Sort nlogn nlogn nlogn 1 Selection
Insertion Sort n n2
n2 1 Insertion
Bubble Sort n n2
n2 1 Exchanging
Tree Sort nlogn nlogn nlogn n Insertion
5. Understanding Basic Sorting
Algorithms
1 Bubble Sort
Repeatedly compares and
swaps adjacent elements in
every pass. In the i-th pass,
the i-th largest element is
placed at the (N-i)-th
position. Simple to
implement but inefficient for
large datasets.
2 Selection Sort
Divides the array into sorted
(left) and unsorted (right)
subarrays. It selects the
smallest element from the
unsorted subarray and
places it at the beginning of
that subarray.
3 Insertion Sort
Builds a sorted array one element at a time. In the i-th iteration, the
i-th element is inserted into its proper place in the previously sorted
subarray. Efficient for small datasets and nearly sorted arrays.
Bubble Sort, Selection Sort, and Insertion Sort are simple sorting
algorithms that are commonly used to sort small datasets or as
building blocks for more complex sorting algorithms.
6. Bubble Sort
Bubble Sort is the simplest sorting algorithm that works by repeatedly
swapping the adjacent elements if they are in the wrong order. This
algorithm is not suitable for large data sets as its average and worst-case
time complexity are quite high.
• We sort the array using multiple passes. After the first pass, the maximum
element goes to end (its correct position). Same way, after second pass,
the second largest element goes to second last position and so on.
• In every pass, we process only those elements that have already not
moved to correct position. After k passes, the largest k elements must
have been moved to the last k positions.
• In a pass, we consider remaining elements and compare all adjacent and
swap if larger element is before a smaller element. If we keep doing this,
we get the largest (among the remaining elements) at its correct position.
8. Time and Space Complexity Analysis of Bubble Sort
• The time complexity of Bubble Sort is O(n^2) in the worst-case scenario and the space complexity of Bubble sort is O(1).
• Bubble Sort only needs a constant amount of additional space during the sorting process.
• The space complexity of Bubble Sort is O(1) This means that the amount of extra space (memory) required by the
algorithm remains constant regardless of the size of the input array being sorted.
• Bubble Sort only needs a constant amount of additional space to store temporary variables or indices during the sorting
process.
• Therefore, the space complexity of Bubble Sort is considered to be very efficient as it does not depend on the input size
and does not require additional space proportional to the input size.
Complexity Type Complexity
Time Complexity
Best: O(n)
Average: O(n^2)
Worst: O(n^2)
Space Complexity Worst: O(1)
9. Bubble Sort Deep Dive
Advantages
It does not require any additional memory space.
Disadvantages
Bubble sort is easy to understand and implement.
It is a stable sorting algorithm, meaning that elements with the same key value maintain their relative order
in the sorted output.
Bubble sort has a time complexity of O(n2) which makes it very slow for large data sets.
Bubble sort has almost no or limited real world applications. It is mostly used in academics to teach different
ways of sorting.
10. Selection Sort
Selection Sort is a comparison-based sorting algorithm. It sorts an array by
repeatedly selecting the smallest (or largest) element from the unsorted portion
and swapping it with the first unsorted element. This process continues until the
entire array is sorted.
1. First we find the smallest element and swap it with the first element. This way we get
the smallest element at its correct position.
2. Then we find the smallest among remaining elements (or second smallest) and swap
it with the second element.
3. We keep doing this until we get all elements moved to correct position.
13. Selection Sort Deep Dive
Find Minimum
Find the minimum element in the
unsorted subarray.
1
Swap Elements
Swap it with the first element of the
unsorted subarray.
2
Expand Sorted Portion
Move the boundary between sorted and
unsorted subarrays one element to the
right.
3
Repeat Process
Continue until the entire array is sorted.
4
Selection Sort always performs O(N²) comparisons, making it inefficient for large datasets. However, it makes only O(N) swaps,
which can be advantageous when swap operations are expensive.
Unlike Bubble Sort and Insertion Sort, Selection Sort doesn't have a best-case scenario where it performs better - it always
requires quadratic time.
14. Time and Space Complexity Analysis of Selection Sort
• Time Complexity: O(n^2) ,as there are two nested loops:
One loop to select an element of Array one by one = O(n)
Another loop to compare that element with every other Array element = O(n)
Therefore overall complexity = O(n) * O(n) = O(n*n) = O(n^2)
• Space complexity: O(1) as the only extra memory used is for temporary variables.
Complexity Type Complexity
Time Complexity
Best: O(n^2)
Average: O(n^2)
Worst: O(n^2)
Space Complexity Worst: O(1)
15. Selection Sort Deep Dive
Advantages
Requires only a constant O(1) extra memory space.
Disadvantages
Easy to understand and implement, making it ideal for teaching basic sorting concepts.
It requires less number of swaps (or memory writes) compared to many other standard algorithms. Only cycle
sort beats it in terms of memory writes. Therefore it can be simple algorithm choice when memory writes are
costly.
Selection sort has a time complexity of O(n^2) makes it slower compared to algorithms like Quick Sort or
Merge Sort.
Does not maintain the relative order of equal elements which means it is not stable.
16. Insertion Sort
• Insertion sort is a simple sorting algorithm that works by iteratively inserting each element of an unsorted list into its
correct position in a sorted portion of the list.
• It is like sorting playing cards in your hands. You split the cards into two groups: the sorted cards and the unsorted cards.
Then, you pick a card from the unsorted group and put it in the right place in the sorted group.
17. Insertion Sort Deep Dive
Start with Second Element
We start with second element of the array as first element in the array is
assumed to be sorted.
Compare with Sorted Elements
Compare second element with the first element and check if the second element is
smaller then swap them.
Shift Elements
Move to the third element and compare it with the first two elements
and put at its correct position
Insert Key
Insert the key at its correct position in the sorted subarray and
expand the sorted portion. Repeat until the entire array is sorted.
Insertion Sort is efficient for small datasets and nearly sorted arrays. It's an
adaptive algorithm, meaning it performs better when the array is partially sorted,
with a best-case time complexity of O(N).
18. Insertion Sort Deep Dive
Advantages
Space-efficient as it is an in-place algorithm. Efficient for small lists and nearly sorted lists.
Disadvantages
Simple and easy to implement. Also, Stable sorting algorithm.
Adoptive. the number of inversions is directly proportional to number of swaps. For example, no swapping
happens for a sorted array and it takes O(n) time only.
Inefficient for large lists.
Not as efficient as other sorting algorithms (e.g., merge sort, quick sort) for most cases.
19. Time and Space Complexity
Comparison
Algorithm Best Case Average
Case
Worst
Case
Space
Complexit
y
Bubble
Sort
O(N) O(N²) O(N²) O(1)
Selection
Sort
O(N²) O(N²) O(N²) O(1)
Insertion
Sort
O(N) O(N²) O(N²) O(1)
All three algorithms have quadratic worst-case time complexity, making
them suitable only for small datasets.
20. Performance Comparison
Based on the implementation example provided, when sorting a list of random integers, Insertion Sort outperforms both Bubble Sort and Selection Sort. Bubble Sort is
significantly slower due to its excessive swapping operations, while Selection Sort performs better than Bubble Sort but worse than Insertion Sort.
However, it's important to note that performance can vary depending on the specific characteristics of the dataset being sorted.
21. Advanced Sorting Algorithms
Merge Sort
A divide-and-conquer algorithm that divides the
input array into two halves, recursively sorts them,
and then merges the sorted halves. It has a
consistent O(N log N) time complexity but requires
O(N) extra space.
Quick Sort
Another divide-and-conquer algorithm that selects a
'pivot' element and partitions the array around it. It
has an average case time complexity of O(N log N)
but can degrade to O(N²) in the worst case.
Heap Sort
Uses a binary heap data structure to sort elements. It
has a consistent O(N log N) time complexity and O(1)
space complexity, making it more space-efficient
than Merge Sort.
Radix Sort
A non-comparative sorting algorithm that sorts data
with integer keys by grouping keys by individual
digits. It has a time complexity of O(d * (n + k)),
where d is the number of digits, n is the number of
elements, and k is the range of input.
22. Choosing the Right Sorting Algorithm
Data Size
For small datasets (fewer
than 10-20 elements),
simple algorithms like
Insertion Sort may be
more efficient due to
lower overhead. For larger
datasets, advanced
algorithms like Quick Sort
or Merge Sort are
preferred.
Stability Requirements
If preserving the relative
order of equal elements is
important, choose stable
algorithms like Bubble
Sort, Insertion Sort, or
Merge Sort. Selection Sort
and Quick Sort are not
stable by default.
Memory Constraints
In memory-constrained
environments, in-place
sorting algorithms like
Insertion Sort, Selection
Sort, or Heap Sort are
preferred over algorithms
requiring additional space
like Merge Sort.
Data Characteristics
For nearly sorted data,
Insertion Sort performs
exceptionally well. For
random data, Quick Sort is
often the fastest. For data
with a small range of
values, counting-based
sorts like Radix Sort can be
very efficient.
23. Key Takeaways and Next Steps
1
Master Advanced Algorithms
Explore Quick Sort, Merge Sort, and other O(N log N) algorithms
2
Analyze Algorithm Tradeoffs
Consider time complexity, space requirements, and stability
3
Implement Basic Sorting Algorithms
Practice coding Bubble, Selection, and Insertion Sort
4
Understand Sorting Fundamentals
Learn time/space complexity and algorithm characteristics
Sorting algorithms are fundamental to computer science and form the basis for many more complex algorithms.
Understanding their strengths, weaknesses, and implementation details will help you choose the right tool for each specific
situation.
