A Discussion of Sorting Algorithms and Their Implementation
by Michael Dipperstein
The more basic stuff I publish source for, the more requests for source I seem to get. This time my Huffman code page landed me a request for C++ code which implements Huffman coding and merge sort. The request obviously came from a student trying to get me to do his/her homework. I didn't do the homework assignment, but it started me thinking about sorting algorithms, and off I went on a trip down sorting memory lane.
I thought that I could put everything to rest after reading about some of the algorithms I remembered the names of. After all, there are people that have made a career out of studying sorting algorithms, what would I want to do that hadn't already been done? I thought wrong, there's a lot of discussion about the algorithms, and even some code. Most of the code I saw had errors. Other code snippets were just obfuscated by object oriented programming techniques. So I decided to write my own library of sorting algorithms.
As time goes on, I continue to discover new algorithms and sometimes even add them to my library. I recently discovered the Radix Sort while reading about optimizations to the Burrows-Wheeler Transformation. I'm sure I'll find another sort to add in the future.
The rest of this page discusses some of the issues concerning the sorting algorithms that I chose to implement.
Click here to skip the discussion and go to instructions on downloading my sorting library.
There are several popular and well researched sorting algorithms. My sort library implements the traditional version of the following algorithms:
Each of the algorithms has the following order of operations:
|Best Case||Average Case||Worst Case|
|Shell Sort1||O(n × log(n))||O(n3/2)||O(n3/2)|
|Quick Sort||O(n)||O(n × log(n))||O(n2)|
|Merge Sort||O(n × log(n))||O(n × log(n))||O(n × log(n))|
|Heap Sort||O(n × log(n))||O(n × log(n))||O(n × log(n))|
|O(k × n)||O(k × n)||O(k × n)|
|1 When using an increment of the form (3k - 1) / 2|
|2 Here k is the number of keys|
Insertion sort is probably the first sorting algorithm that is taught in programming classes. It is far from efficient in terms of the number of comparisons, but is very compact in terms of code space required. As such, the insertion sort algorithm may be useful for sorting small lists.
Insertion sort is an iterative algorithm that makes n - 1 passes on a list of n items. On the first pass the second item in the unsorted list is inserted in the proper position for a sorted list containing the first two items. The next pass, the third item is inserted in a the proper position for a list sorted list containing the first three items. After the kth iteration the first k + 1 items in the list are in sorted order. The algorithm continues until the nth item is inserted into its position.
The insertion sort algorithm is implemented by my
sort library function
When I first learned the bubble sort algorithm, it was described as "cute". I believe that cuteness is the only thing this algorithm has going for it. It typically requires the same order of operations as insertion sort, and is not any more code efficient.
Bubble sort is an iterative algorithm that makes n - 1 passes on a list of n items. On the first pass, bubble sort starts at the head of the list and compares the first two items, if the items are out of order, they are swapped, then the second and third items are compared and swapped if necessary. The comparison and swapping continue to the end of the list. After the first iteration, the largest item is in the nth position.
The algorithm repeats itself, stopping one item earlier each pass. The algorithm completes when only the first two items are compared. This sort library includes an additional optimization which will halt the algorithm on any pass which finds the list to be already sorted (no swaps are performed).
The bubble sort algorithm is implemented by my sort
Shell sort is an optimization of the insertion sort proposed by D. L. Shell. Shell observed that performing an insertion sort on sublist of items that are separated by some distance (an increment) may allow items to be moved further across the list.
For a list of n items [x1 … xn], select an
increment, i, and start by sorting the resulting n/i sub lists:
[x1, xi + 1, x2i + 1, …],
[x2, xi + 2, x2i + 2, …], …
[xi, xi + i, x2i + i, …].
Next decrease i and repeat the process until i equals 1. At the time of my writing this, no optimal increment value is known. However, D.E. Knuth has demonstrated that using increments of the form (3k - 1) / 2 results in an O(n3/2) Shell sort.
The Shell sort algorithm is implemented by my sort
ShellSort. The Shell sort function
provided in this library uses increments of the form
(3k - 1) / 2.
ANSI C provides a quick sort function,
stdlib.h, yet I felt compelled to include a quick sort function
in my library.
Quick sort is a divide and conquer algorithm with two phases, a partition phase and a sort phase.
In the partition phase list of two or more elements are partitioned into two smaller lists. All list items less than or equal to a pivot item are placed in one partition, while all items greater than a pivot are placed in another partition. For my implementation, I always designate the first item in the list as the pivot. To partition, I maintain left and right search pointers and perform the following steps.
Step 1. Start the left search pointer at the second item in the list and the right pointer at the last item in the list.
Step 2. Move the left pointer to the right until it finds an item greater than the pivot or points to the same item as the right pointer.
- If the left pointer crosses the right pointer, partition the list at the crossing point and quit.
Step 3. Move the right pointer left until it crosses the left pointer or finds an item less than or equal to the pivot.
- If the right pointer crosses the left pointer, partition the list at the crossing point and quit.
Step 4. Swap the items pointed to by the two pointers and go to step 2.
The sort phase is really simple. Just call the quick sort function twice, once passing the left partition as the list to be sorted, then passing the right partition as the list to be sorted.
Quick sort has an average order of operation of O(n × log(n)), but when the partitions are not anywhere near even, the order of operation approaches O(n2). For sorted lists and reverse sorted lists quick sort is an O(n2) algorithm.
My sort library function
implements the quick sort algorithm. It is a recursive implementation of
quick sort which makes it pretty inefficient with stack space. The stack
space consumed may be equal to the size of the list. A web search should
reveal several memory optimizations for quick sort.
Like quick sort, merge sort is a divide and conquer algorithm with two phases. In the case of merge sort, the phases are a partition phase, and a merge phase.
In the partition phase, if the list of items to be sorted contains more than one item, the list is split in half and the merge sort algorithm is applied on each half of the list.
At the end of the partition phase, the algorithm has two sorted list halves. The merge phase merges the two sorted halves into one list. For my implementation of this algorithm, I perform the merge by allocating a block of memory equal to the size of the combined lists, and merge the lists into the new memory block. Allocation of a new memory block means that this implementation requires additional storage space equal to the size of the original list.
To merge two lists, compare the lowest value in one list with the lowest value in the other list. Remove the lowest of the two values from its list and place it at the end of the merged list. Repeat this process until there are no more items in either list.
Unlike quick sort, merge sort always partitions the list in half, so it is always an order O(n × log(n)) algorithm.
The merge sort algorithm is implemented by my sort
Heap sort is a two phase algorithm, it has a heap building phase and a promotion phase. A heap is a binary tree with the propriety that every node is greater than it's children. For a list of 1 .. N items this means that item i is greater than it's children, items (i / 2) and ((i / 2) + 1).
Once the heap is built, the largest item will be at the root of the heap. To sort, simply swap the first item in the list with Nth item in the list. Next rebuild the heap with items 1 .. (N - 1). Repeat the process of swapping and rebuilding the heap with one less item, until only two items remain.
The heap sort algorithm is implemented by my sort
Radix sort is an unusual sort algorithm. All the sort algorithms discussed so far compare the elements being sorted to eachother. The radix sort doesn't. A radix sort uses one or more keys to sort values. The key of a value indicates the sort grouping for a value on a particular pass. Successive passes with different keys may be used to sort a list of elements.
On a radix sort pass, a list of items to be sorted is processed from beginning to end. One at a time, the items in the list are placed at the end of a list of items with the same key value. After all items have been processed, the lists of key values are reassembled smallest key to largest key.
A somewhat intuitive example of how to use a radix sort might be alphabetizing strings of length N. Each pass of the radix sort should use a character in the string as it's key. The less intuitive part about alphabetizing is that radix sort passes should be from performed from the last character to the first character. The first pass will use the Nth character, the second pass will use the (N - 1)th character, …, the Nth pass will use the first character.
Example: Sorting 3 Character Strings (start with third character). Each cell represents a bin.
|First Pass||asd qwd||vbn||vgy||qaz|
|Third Pass||asd||qaz||qwd||vbn vgy|
Sorting unsigned integers is similar to sorting strings. If we use a radix of 10, we can sort integers one digits at a time. Like strings, integers should be sorted least significant digit to most significant digit.
Example: Sorting 4 Digit Integers (start with least significant digit). Each cell represents a bin.
|First Pass||2453||5624 1234||1356||7579|
|Second Pass||5624||1234||2453 1356||7579|
|Fourth Pass||1234 1356||2453||5624||7579|
Pierre Terdiman's article Radix Sort Revisited discusses radix sort techniques for sorting signed integers and floating point values.
The radix sort algorithm is implemented by my sort
RadixSort. Each time
RadixSort is called, it makes two passes on the data being
sorted. The first pass counts the number of values with each possible key,
the second pass places the data in its sorted order. Counting the number of
values for each key allows the data to be sorted into a fixed size array
instead of a linked list or similar data structure.
The library archive that I have provided includes
a program file, sample.c, which demonstrates the usage of each sort
function. I have also provided the function
may be used to determine if a list is actually sorted. All of my sort
functions share the same parameter list as the
- A pointer to the list of items to be sorted
- The number of items in the list
- The size of the list items
- A pointer to a comparison function
By utilizing the same format, code which calls
qsort may be
modified to use use my functions simply by changing the function name.
Conversely, code that uses my functions may be made to use
qsort simply by changing the function name.
It is possible to sacrifice flexibility for speed by rewriting the functions so that a fixed comparison is performed inline instead of calling the comparison function passed as a parameter. Greater speed may also be obtained by using arrays of a known type instead of void pointers with an item size determined at runtime.
I am releasing my implementation of the sorting algorithms under the LGPL. All of the functions have been combined into a single library. As I add enhancements or fix bugs, I will post them here as newer versions. The larger the version number, the newer the version. I'm retaining the older versions for historical reasons. I recommend that most people download the newest version unless there is a compelling reason to do otherwise.
Each version is contained in its own zipped archive which includes, C source, a sample program demonstrating the usage of each sort function, and a Makfile for building the library with GNU make and gcc. None of the archives contain executable programs. A copy of the archives may be obtained by clicking on the links below.
|Version 0.3||Added the "-d" option to print the unsorted and sorted results when the sample code is executed.|
|Added C99 sample64.c which sorts on
|Cleaner code in hopes of fixing bugs seen on 64-bit PowerPC.|
|Version 0.2||Eliminates warnings generated when more compiler warnings are enabled.|
|Corrects error in QuickSort which caused unnecessary comparisons.|
|Makes RadixSort compatible with 64-bit architectures.|
|Version 0.1||Initial release.|
The most of the source code that I have provided is written in ANSI C.
sample64.c are written in C99 inorder to force the use of
64-bit integers in the 64-bit sample code (
I have received a report that sorting large sets (>224 values) of 64-bit integers frequently fails when attempted on PowerPC system running 64-bit Linux. I have been unable to duplicate the problem on 32-bit x86 platforms and do not have a 64-bit platform to test on.
Another error may occur when the number of bytes being sorted exceeds the
If you have any further questions or comments, you may contact me by e-mail. My e-mail address is: email@example.com