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

I'm very curious, why stability is or is not important in sorting algorithms?

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)

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.

It depends on what you do.

Imagine you've got some people records with a first and a last name field. First you sort the list by first name. If you then sort the list with a stable algorithm by last name, you'll have a list sorted by first name AND last name.

There's a few reasons why stability can be important. One is that, if two records don't need to be swapped by swapping them you can cause a memory update, a page is marked dirty, and needs to be re-written to disk (or another slow medium).

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 unsorted array. 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.

However, any given sorting algo which is not stable can be modified to be stable. There can be sorting algo specific ways to make it stable, but in general, any comparison based sorting algorithm which is not stable by nature can be modified to be stable by changing the key comparison operation so that the comparison of two keys considers position as a factor for objects with equal keys.

References: http://www.math.uic.edu/~leon/cs-mcs401-s08/handouts/stability.pdf http://en.wikipedia.org/wiki/Sorting_algorithm#Stability

I know there are many answers for this, but to me, this answer, by Robert Harvey, summarized it much more clearly:

> A stable sort is one which preserves the original order of the input set, where the [unstable] algorithm does not distinguish between two or more items.

Source

If you assume what you are sorting are just numbers and only their values identify/distinguish them (e.g. elements with same value are identicle), then the stability-issue of sorting is meaningless.

However, objects with same priority in sorting may be distinct, and sometime their relative order is meaningful information. In this case, unstable sort generates problems.

For example, you have a list of data which contains the time cost [T] of all players to clean a maze with Level [L] in a game. Suppose we need to rank the players by how fast they clean the maze. However, an additional rule applies: players who clean the maze with higher-level always have a higher rank, no matter how long the time cost is.

Of course you might try to map the paired value [T,L] to a real number [R] with some algorithm which follows the rules and then rank all players with [R] value.

However, if stable sorting is feasible, then you may simply sort the entire list by [T] (Faster players first) and then by [L]. In this case, the relative order of players (by time cost) will not be changed after you grouped them by level of maze they cleaned.

PS: of course the approach to sort twice is not the best solution to the particular problem but to explain the question of poster it should be enough.

Stable sort will always return same solution (permutation) on same input.

For instance [2,1,2] will be sorted using stable sort as permutation [2,1,3] (first is index 2, then index 1 then index 3 in sorted output) That mean that output is always shuffled same way. Other non stable, but still correct permutation is [2,3,1].

Quick sort is not stable sort and permutation differences among same elements depends on algorithm for picking pivot. Some implementations pick up at random and that can make quick sort yielding different permutations on same input using same algorithm.

Stable sort algorithm is necessary deterministic.

Some more examples of the reason for wanting stable sorts. Databases are a common example. Take the case of a transaction data base than includes last|first name, date|time of purchase, item number, price. Say the data base is normally sorted by date|time. Then a query is made to make a sorted copy of the data base by last|first name, since a stable sort preserves the original order, even though the inquiry compare only involves last|first name, the transactions for each last|first name will be in data|time order.

A similar example is classic Excel, which limited sorts to 3 columns at a time. To sort 6 columns, a sort is done with the least significant 3 columns, followed by a sort with the most significant 3 columns.

A classic example of a stable radix sort is a card sorter, used to sort by a field of base 10 numeric columns. The cards are sorted from least significant digit to most significant digit. On each pass, a deck of cards is read and separated into 10 different bins according to the digit in that column. Then the 10 bins of cards are put back into the input hopper in order ("0" cards first, "9" cards last). Then another pass is done by the next column, until all columns are sorted. Actual card sorters have more than 10 bins since there are 12 zones on a card, a column can be blank, and there is a mis-read bin. To sort letters, 2 passes per column are needed, 1st pass for digit, 2nd pass for the 12 11 zone.

Later (1937) there were card collating (merging) machines that could merge two decks of cards by comparing fields. The input was two already sorted decks of cards, a master deck and an update deck. The collator merged the two decks into a a new mater bin and an archive bin, which was optionally used for master duplicates so that the new master bin would only have update cards in case of duplicates. This was probably the basis for the idea behind the original (bottom up) merge sort.