From Wikipedia, the free encyclopedia
In computer science and mathematics, a sorting algorithm is an algorithm that puts elements of a list in a certain order. The most-used orders are numerical order and lexicographical order. Efficient sorting is important to optimizing the use of other algorithms (such as search and merge algorithms) that require sorted lists to work correctly; it is also often useful for canonicalizing data and for producing human-readable output. More formally, the output must satisfy two conditions:
- The output is in nondecreasing order (each element is no smaller than the previous element according to the desired total order);
- The output is a permutation, or reordering, of the input.
Summaries of popular sorting algorithms
Bubble sort
Main article: Bubble sort
Bubble sort is a straightforward and simplistic method of sorting data that is used in computer science education. The algorithm starts at the beginning of the data set. It compares the first two elements, and if the first is greater than the second, it swaps them. It continues doing this for each pair of adjacent elements to the end of the data set. It then starts again with the first two elements, repeating until no swaps have occurred on the last pass. While simple, this algorithm is highly inefficient and is rarely used except in education. For example, if we have 100 elements then the total number of comparisons will be 10000. A slightly better variant, cocktail sort, works by inverting the ordering criteria and the pass direction on alternating passes. Its average case and worst case are both O(n²).
Source Code
Below is the basic bubble sort algorithm.
void bubbleSort(int numbers[], int array_size) |
(http://linux.wku.edu/~lamonml/algor/sort/bubble.html)[1].
Insertion sort
Main article: Insertion sort
Insertion sort is a simple sorting algorithm that is relatively efficient for small lists and mostly-sorted lists, and often is used as part of more sophisticated algorithms. It works by taking elements from the list one by one and inserting them in their correct position into a new sorted list. In arrays, the new list and the remaining elements can share the array's space, but insertion is expensive, requiring shifting all following elements over by one. Shell sort (see below) is a variant of insertion sort that is more efficient for larger lists.
Source Code
Below is the basic insertion sort algorithm.
void insertionSort(int numbers[], int array_size) |
(http://linux.wku.edu/~lamonml/algor/sort/insertion.html)[2].
Shell sort
Main article: Shell sort
Shell sort was invented by Donald Shell in 1959. It improves upon bubble sort and insertion sort by moving out of order elements more than one position at a time. One implementation can be described as arranging the data sequence in a two-dimensional array and then sorting the columns of the array using insertion sort. Although this method is inefficient for large data sets, it is one of the fastest algorithms for sorting small numbers of elements (sets with fewer than 1000 or so elements). Another advantage of this algorithm is that it requires relatively small amounts of memory.
Source CodeBelow is the basic shell sort algorithm.
void shellSort(int numbers[], int array_size){
int i, j, increment, temp;
increment = 3;
while (increment > 0)
{ for (i=0; i < j =" i;" temp =" numbers[i];">= increment) && (numbers[j-increment] > temp)){ numbers[j] = numbers[j - increment];
j = j - increment; } numbers[j] = temp;
} if (increment/2 != 0) increment = increment/2;
else if (increment == 1)
increment = 0;
else increment = 1; }}
(http://linux.wku.edu/~lamonml/algor/sort/merge.html)[3].
Merge sort
Main article: Merge sort
Merge sort takes advantage of the ease of merging already sorted lists into a new sorted list. It starts by comparing every two elements (i.e., 1 with 2, then 3 with 4...) and swapping them if the first should come after the second. It then merges each of the resulting lists of two into lists of four, then merges those lists of four, and so on; until at last two lists are merged into the final sorted list. Of the algorithms described here, this is the first that scales well to very large lists, because its worst-case running time is O(n log n).
Source CodeBelow is the basic shell sort algorithm.
void shellSort(int numbers[], int array_size){
int i, j, increment, temp;
increment = 3;
while (increment > 0) {
for (i=0; i <>
j = i; temp = numbers[i];
while ((j >= increment) && (numbers[j-increment] > temp)) {
numbers[j] = numbers[j - increment];
j = j - increment; }
numbers[j] = temp; }
if (increment/2 != 0)
increment = increment/2;
else if (increment == 1)
increment = 0;
else increment = 1; }}
(http://linux.wku.edu/~lamonml/algor/sort/merge.html)[4].
Main article: Heapsort
Heapsort is a much more efficient version of selection sort. It also works by determining the largest (or smallest) element of the list, placing that at the end (or beginning) of the list, then continuing with the rest of the list, but accomplishes this task efficiently by using a data structure called a heap, a special type of binary tree. Once the data list has been made into a heap, the root node is guaranteed to be the largest element. When it is removed and placed at the end of the list, the heap is rearranged so the largest element remaining moves to the root. Using the heap, finding the next largest element takes O(log n) time, instead of O(n) for a linear scan as in simple selection sort. This allows Heapsort to run in O(n log n) time.
Source CodeBelow is the basic heap sort algorithm. The siftDown() function builds and reconstructs the heap.
void heapSort(int numbers[], int array_size){
int i, temp;
for (i = (array_size / 2)-1; i >= 0; i--)
siftDown(numbers, i, array_size);
for (i = array_size-1; i >= 1; i--) {
temp = numbers[0];
numbers[0] = numbers[i];
numbers[i] = temp;
siftDown(numbers, 0, i-1); }}
void siftDown(int numbers[], int root, int bottom){
int done, maxChild, temp;
done = 0; while ((root*2 <= bottom) && (!done)) {
if (root*2 == bottom) maxChild = root * 2;
else if (numbers[root * 2] > numbers[root * 2 + 1])
maxChild = root * 2; else maxChild = root * 2 + 1;
if (numbers[root] <>
temp = numbers[root];
numbers[root] = numbers[maxChild];
numbers[maxChild] = temp; root = maxChild; }
else done = 1; }}
(http://linux.wku.edu/~lamonml/algor/sort/heap.html)[5].
Quicksort
Main article: Quicksort
Quicksort is a divide and conquer algorithm which relies on a partition operation: to partition an array, we choose an element, called a pivot, move all smaller elements before the pivot, and move all greater elements after it. This can be done efficiently in linear time and in-place. We then recursively sort the lesser and greater sublists. Efficient implementations of quicksort (with in-place partitioning) are typically unstable sorts and somewhat complex, but are among the fastest sorting algorithms in practice. Together with its modest O(log n) space usage, this makes quicksort one of the most popular sorting algorithms, available in many standard libraries. The most complex issue in quicksort is choosing a good pivot element; consistently poor choices of pivots can result in drastically slower O(n²) performance, but if at each step we choose the median as the pivot then it works in O(n log n).
function quicksort(array)
var list less, greater
if length(array) ≤ 1 return array
select and remove a pivot value pivot from arrayfor each x in array
if x ≤ pivot then append x to lesselse append x to greater
return concatenate(quicksort(less), pivot, quicksort(greater))
Bucket sort
Main article: Bucket sort
Bucket sort is a sorting algorithm that works by partitioning an array into a finite number of buckets. Each bucket is then sorted individually, either using a different sorting algorithm, or by recursively applying the bucket sorting algorithm. A variation of this method called the single buffered count sort is faster than the quicksort and takes about the same time to run on any set of data.
function bucket-sort(array, n) is
buckets ← new array of n empty lists
for i = 0 to (length(array)-1) do
insert array[i] into
buckets[msbits(array[i], k)]
for i = 0 to n - 1 donext-sort(buckets[i])
return the concatenation of buckets[0], ..., buckets[n-1]
Radix sort
Main article: Radix sort
Radix sort is an algorithm that sorts a list of fixed-size numbers of length k in O(n · k) time by treating them as bit strings. We first sort the list by the least significant bit while preserving their relative order using a stable sort. Then we sort them by the next bit, and so on from right to left, and the list will end up sorted. Most often, the counting sort algorithm is used to accomplish the bitwise sorting, since the number of values a bit can have is small.
Distribution sort
Distribution sort refers to any sorting algorithm where data is distributed from its input to multiple intermediate structures which are then gathered and placed on the output. It is typically not considered to be very efficient because the intermediate structures need to be created, but sorting in smaller groups is more efficient than sorting one larger group.
Shuffle sort
Shuffle sort is a type of distribution sort algorithm (see above) that begins by removing the first 1/8 of the n items to be sorted, sorts them recursively, and puts them in an array. This creates n/8 "buckets" to which the remaining 7/8 of the items are distributed. Each "bucket" is then sorted, and the "buckets" are concatenated into a sorted array.
(http://en.wikipedia.org/wiki/Sorting_algorithm)[2].
