What is stability in sorting algorithms and why is it important?

Sorting stability means that records with the same key retain their relative order before and after the sort.

So stability matters if, and only if, the problem you're solving requires retention of that relative order.

If you don't need stability, you can use a fast, memory-sipping algorithm from a library, like heapsort or quicksort, and forget about it.

If you need stability, it's more complicated. Stable algorithms have higher big-O CPU and/or memory usage than unstable algorithms. So when you have a large data set, you have to pick between beating up the CPU or the memory. If you're constrained on both CPU and memory, you have a problem. A good compromise stable algorithm is a binary tree sort; the Wikipedia article has a pathetically easy C++ implementation based on the STL.

You can make an unstable algorithm into a stable one by adding the original record number as the last-place key for each record.

A sorting algorithm is said to be stable if two objects with equal keys appear in the same order in sorted output as they appear in the input array to be sorted. Some sorting algorithms are stable by nature like Insertion sort, Merge Sort, Bubble Sort, etc. And some sorting algorithms are not, like Heap Sort, Quick Sort, etc.

Background: a "stable" sorting algorithm keeps the items with the same sorting key in order. Suppose we have a list of 5-letter words:

peach
straw
apple
spork

If we sort the list by just the first letter of each word then a stable-sort would produce:

apple
peach
straw
spork

In an unstable sort algorithm, straw or spork may be interchanged, but in a stable one, they stay in the same relative positions (that is, since straw appears before spork in the input, it also appears before spork in the output).

We could sort the list of words using this algorithm: stable sorting by column 5, then 4, then 3, then 2, then 1. In the end, it will be correctly sorted. Convince yourself of that. (by the way, that algorithm is called radix sort)

Now to answer your question, suppose we have a list of first and last names. We are asked to sort "by last name, then by first". We could first sort (stable or unstable) by the first name, then stable sort by the last name. After these sorts, the list is primarily sorted by the last name. However, where last names are the same, the first names are sorted.

You can't stack unstable sorts in the same fashion.

A stable sorting algorithm is the one that sorts the identical elements in their same order as they appear in the input, whilst unstable sorting may not satisfy the case. - _{^{I thank my algorithm lecturer Didem Gozupek to have provided insight into algorithms}}.

I again needed to edit the question due to some feedback that some people don't get the logic of the presentation. It illustrates sorting w.r.t. first elements. On the other hand, you can either consider the illustration consisting of key-value pairs.

Stable Sorting Algorithms:

• Insertion Sort
• Merge Sort
• Bubble Sort
• Tim Sort
• Counting Sort
• Block Sort
• Quadsort
• Library Sort
• Cocktail shaker Sort
• Gnome Sort
• Odd–even Sort

Unstable Sorting Algorithms:

• Heap sort
• Selection sort
• Shell sort
• Quick sort
• Introsort (subject to Quicksort)
• Tree sort
• Cycle sort
• Smoothsort
• Tournament sort(subject to Hesapsort)