Bubble sort

Definition:

It is a simple sorting algorithm. It works by repeatedly stepping through the list to be sorted, comparing two items at a time and swapping them if they are in the wrong order. The pass through the list is repeated until no swaps are needed, which indicates that the list is sorted.
The algorithm gets its name from the way smaller elements "bubble" to the top of the list. Because it only uses comparisons to operate on elements, it is a comparison sort.


Run-time complexity Analysis:
♥ This is observing through the first two elements then swap the lesser to greater.
♥ Bubble sort has worst-case and average complexity both О(n²), where n is the number of items being sorted. There exist many sorting algorithms with the substantially better worst-case or average complexity of O(n log n). Even other О(n²) sorting algorithms, such as insertion sort, tend to have better performance than bubble sort. Therefore bubble sort is not a practical sorting algorithm when n is large.

Codes:
procedure bubbleSort( A : list of sortable items ) defined as:
do
swapped := false
for each i in 0 to length(A) - 2 inclusive do:
if A[ i ] > A[ i + 1 ] then
swap( A[ i ], A[ i + 1 ] )
swapped := true
end if
end for
while swapped
end procedure

Application:
♥ For example, swapping the height of the participants of the running event.

Reference:
http://en.wikipedia.org/wiki/Bubble_sort

Heapsort


Definition:

It is a comparison-based sorting algorithm, and is part of the selection sort family. Although somewhat slower in practice on most machines than a good implementation of quicksort.
It 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 aheap, 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.


Run-time Complexity Analysis:
♥ It has the advantage of a worst-case Θ(n log n) runtime. It is an in-place algorithm, but is not a stable sort.

Codes:
function heapSort(a, count) is
input: an unordered array a of length count

(first place a in max-heap order)
heapify(a, count)

end := count - 1
while end > 0 do
(swap the root(maximum value) of the heap with the last element of the heap)
swap(a[end], a[0])
(decrease the size of the heap by one so that the previous max value will
stay in its proper placement)
end := end - 1
(put the heap back in max-heap order)
siftDown(a, 0, end)

function heapify(a,count) is
(start is assigned the index in a of the last parent node)
start := (count - 2) / 2

while start ≥ 0 do
(sift down the node at index start to the proper place such that all nodes below
the start index are in heap order)
siftDown(a, start, count-1)
start := start - 1
(after sifting down the root all nodes/elements are in heap order)

function siftDown(a, start, end) is
input: end represents the limit of how far down the heap
to sift.
root := start

while root * 2 + 1 ≤ end do (While the root has at least one child)
child := root * 2 + 1 (root*2+1 points to the left child)
(If the child has a sibling and the child's value is less than its sibling's...)
if child + 1 ≤ end and a[child] < a[child + 1] then
child := child + 1 (... then point to the right child instead)
if a[root] < a[child] then (out of max-heap order)
swap(a[root], a[child])
root := child (repeat to continue sifting down the child now)
else
return

Application:
♥ Comparing the array of numbers in a sorted list.

Reference:
http://en.wikipedia.org/wiki/Sorting_algorithm#Heapsort

Insertion sort


Definition:

It is a simple sorting algorithm, a comparison sort in which the sorted array (or list) is built one entry at a time.
It 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.


Run-time Complexity Analysis:
♥ This is efficient and sequential.

Codes:
insertionSort(array A)
begin
for i := 1 to length[A]-1 do
begin
value := A[i];
j := i-1;
while j ≥ 0 and A[j] > value do
begin
A[j + 1] := A[j];
j := j-1;
end;
A[j+1] := value;
end;
end;

Application:
Most humans when sorting—ordering a deck of cards, for example—use a method that is similar to insertion sort.

Reference:
http://en.wikipedia.org/wiki/Sorting_algorithm#Insertion_sort

Shell sort


Definition:
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.
It is a sorting algorithm that is a generalization of insertion sort, with two observations:
insertion sort is efficient if the input is "almost sorted", and
insertion sort is typically inefficient because it moves values just one position at a time.
Run-time Complexity Analysis:
♥ This is an effective in terms of the efficiency of the sorted list.

Codes:
input: an array a of length n

inc ← round(n/2)
while inc > 0 do:
for i = inc .. n − 1 do:
temp ← a[i]
j ← i
while j ≥ inc and a[j − inc] > temp do:
a[j] ← a[j − inc]
j ← j − inc
a[j] ← temp
inc ← round(inc / 2.2)

Application:
♥ Sorting the numbers in a certain row.

Reference:
http://en.wikipedia.org/wiki/Sorting_algorithm#Shell_sort

Merge Sort
Definition:
An 0(n log n) comparison-based sorting algorithms.
In most implementations it is stable, meaning that it preserves the input order of equal elements in the sorted output.
It is an example of the divide and conquer algorithmic paradigm.
It was invented by John von Neumann in 1945.
Run-time Complexity Analysis:
♥ Efficient and effective

Code:
function merge_sort(m)
var list left, right, result
if length(m) ≤ 1
return m

// This calculation is for 1-based arrays.
For 0-based, use length(m)/2 - 1.
var middle = length(m) / 2
for each x in m up to middle
add x to left
for each x in m after middle
add x to right
left = merge_sort(left)
right = merge_sort(right)
result = merge(left, right)
return result

Application:
♥ Merging a bundle of something like sticks and other.

Reference:
en.wikipedia.org/wiki/Merge_sort
http://en.wikipedia.org/wiki/Sorting_algorithm#Merge_sort

Quick sort
Definition:
It is a well-known sorting algorithm developed by C.A.H Hoare.
It is a divide and conquer algorithm. It 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 andin-place We then recursively sort the lesser and greater sublists.
Run-time Complexity Analysis:
♥ this is performed through finding its pivot and sort it.

♥ typically unstable and somewhat complex but among the fastest sorting algorithms.
Codes:
function quicksort(array)
var list less, greater
if length(array) ≤ 1
return array
select and remove a pivot value pivot from array
for each x in array
if x ≤ pivot then append x to less
else append x to greater
return concatenate(quicksort(less), pivot, quicksort(greater))
Application:
finding the pivot of a given example and then sort it.
Reference:
http://en.wikipedia.org/wiki/Quicksort

Bucket Sort

Definition:
It is also called bin sort.

It is a sorting algorithm that works by partitioning it into a number of buckets. Each bucket is then sorted individually using the different sorting algorithm, or by recursively applying the bucket sorting algorithm.
Bucket sort works as follows:
Set up an array of initially empty "buckets."
Scatter: Go over the original array, putting each object in its bucket.
Sort each non-empty bucket.
Gather: Visit the buckets in order and put all elements back into the original array.
Run-time Complexity Analysis:
♥ efficient and effective in sorting the list.

Codes:
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 do
next-sort(buckets[i])
return the concatenation of buckets[0], ..., buckets[n-1]

Application:
♥ Given an array, put the array of numbers in a bucket where they must be placed then sort the list.

Reference:

commons.wikimedia.org/wiki/File:Bucket_sort_2.png
http://en.wikipedia.org/wiki/Bucket_sort