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Download Bubble Sort, Selection Sort, Insertion Sort, Merge Sort, Quick Sort and Heap Sorts for free here. Computer Science and Engineering, Karnataka - Free Downloads.
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You may find this report with graphs at:
http://knol.google.com/k/sorting-algorithms#
The Architecture:
April 14, 2009
The basic architecture of sortings done here consists of a header file sor-April 14, 2009
tutil.h, which includes a class named SORT - containing the basic utilities on
array functions such as to generate random numbers - genRand(), printing the
array printArray() and to check if the array elements are in non-descending
order checkSort(). These functions are basic to every array containing integer
numbers. For sake of simplicity, only integer values are used.
The SORT class also inherits another class named sortdefinitions from file
sortdefinitions.h. The sortdefinitions.h file acts like a source of all the codes of
the sorting techniques done here. It includes the function definitions of all the
sorting techniques, heap sort, quick sort, bubble sort, selection sort, and thus
the two classes SORT and sortdefinitions forms the header file sortutil.h.
A main function is just written for every sorting technique by including the
header file sortutil.h. Thus there are four main’s for every sort. The basic idea
here was to use the resources provided by the sortutil.h file and perform the nec-
essary tasks. A word on main: A pointer to array has been used to dynamically
allocate the size. A file pointer (fp) has been used to write the runtimes to a file,
say, heap.dat, which eases to plot the graph. Each time the size of the array
is incremented by an OFFSET value, the value being between RANGELOW
and RANGEHIGH. A simple call to genRand with size ’i’ will fetch ’i’ random
numbers into the array ’a’. All printArray calls can be unmasked to view the
elements of the array, obviously when the size of array is very low. Then their
is call to get the present time in seconds and microseconds, the start time of
heap sorting. On calling heapSort, the ’i’ elements of the array are heap sorted.
Finally the end time is obtained from a call to gettimeofday. RunTime is cal-
culated by subtracting start time from end time. The runtime is displayed on
console as well as written to the file only if checkSort is successful. The array
size is incremented and the process is iterated.
Running script:
1. Change to the sorts directory
2. Run the shell script by typing ./run at command prompt
3. Type a filename in *.cpp format [say, heap.cpp]
4. Script terminates by plotting the graph
THE SORTING ALGORITHMS
By
VIKRAM
SHARAD
KIRANKUMAR
PRABHAKAR
4/2009
ContentsBy
VIKRAM
SHARAD
KIRANKUMAR
PRABHAKAR
4/2009
1 BUBBLE SORT: 2
1.1 The Algorithm: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Worst Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Best Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 SELECTION SORT 4
2.1 The Algorithm: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Worst Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Best Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 THE HEAP SORT 6
3.1 The Algorithm: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Analysis of Run Times: . . . . . . . . . . . . . . . . . . . . . . . 7
4 QUICK SORT 8
4.1 Worst Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2 Best Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1
1 BUBBLE SORT:
Bubble sort is a simple sorting algorithm use to sort the elements. 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.
1.1 The Algorithm:
BUBBLESORT(A ,n)
for i = 1 to n
for j = 0 to ( n – i )
if A[ j ] > A[j + 1]
A[ j ] ⇔ A[j + 1]
Bubble sort has worst-case and average complexity both (n2 ), where n is the
number of items being sorted. Even other (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.
Figure 1: Gnuplot for bubble sort
2
Consider test case with numbers – 4 2 5 1 2 and on each iteration of inner loop,
(2 4 5 1 2)→ (2 4 5 1 2)→ (2 4 1 5 2)→ (2 4 1 2 5)→ (2 4 1 2 5)→ (2 1 4 2 5)→
(2 1 2 4 5)→ (1 2 2 4 5)→ (1 2 2 4 5)→ (1 2 2 4 5)
1.2 Worst Case
The worst case occurs when the elements are sorted in non-ascending order,
because the largest element need to bubbled till last and hence the worst case
is O(n2 ) time. The graph for worst case is shown in the figure.
1.3 Best Case
The best case occurs when the elements are already sorted, because largest
element is already in right place and needs no swapping. This case takes O(n)
time. The graph for best case is shown in the figure.
3
2 SELECTION SORT
Selection sort is one of simplest sorting technique in which we select smallest
item in the list and exchange it with the first item.Obtain the second smallest
element in the list and exchange it with the second element and so on.Finally
all the items will be arranged in ascending order.In otherwords selection sort
performs sorting by repeatly putting the largest element in the unprocessed
portion of the array to the end of the this unprocessed portion until the whole
array is sorted.
2.1 The Algorithm:
for i←1 to n – 1
do pos ← i
for j ← i + 1 to n
if A[j] < A[pos]
then pos←j
if pos = i
then A[i] ⇔ A[pos]
Selection sort has two loops each of which execute n times. Hence the selection
sort is O(n2 ). The current element is assumed to be the smallest and is compared
with remaining elements. The position of the smallest element is found. If the
smallest element is not the current position, then the two elements are swapped.
Consider a test case with elements – 45 20 40 5 15
(45 20 40 5 15 )→ (5 20 40 45 15) → (5 15 40 45 20)→
(5 15 20 45 40) → (5 15 20 40 45)
2.2 Worst Case
Worst case occurs if the elements are already sorted in the non-ascending order,
because for each iteration the smallest element needs to be swapped and thus
the Worst Case is O(n2 ).
2.3 Best Case
Best case occurs if the elemsnts are already sorted in non-descending order, be-
cause current element is smallest and no element needs to be swapped and thus
the Best Case is O(n).
4
Figure 2: Gnuplot for selection sort
5
3 THE HEAP SORT
This sorting technique builds a max heap tree, in which none of the parent
nodes are smaller than their children, and then sorts the elements by extracting
the topmost element. A sequence of events takes place in this sorting. First the
main function calls ’heapSort’, which in turn calls buildMaxHeap. This is the
basic tool for building the max heap. It builds the max heap tree from the lower
nodes to the root by calling the maxHeap function. maxHeap function checks
always that parent is larger than their children, and if the property is violated
the largest child is exchanged with the parent.
3.1 The Algorithm:
LEFT(i)
return 2i + 1
RIGHT(i)
return 2i + 2
MAX-HEAPIFY(A, i)
l ← LEFT(i)
r ← RIGHT(i)
if l ≤ size[A] and A[l] > A[i]
then largest ← l
else largest ← i
if r ≤ size[A] and A[r] > A[largest]
thenlargest ← r
if largest = i
then exchange A[i] ⇔ A[largest]
MAX-HEAPIFY(A,largest)
The recurence relation for MAX-HEAPIFY is T(n) ≤ T(2n/3) + Θ(1), since
each childern’s subtrees have a size at most 2n/3 – the worst case being when
the last row of the tree is exactly half full. Thus MAX-HEAPIFY takes T(n) =
O(lg n).
BUILD-MAX-HEAP(A)
for i ← n/2 downto 1
do MAX-HEAPIFY(A,i)
BUILD-MAX-HEAP takes O(n) time.
6
HEAP-SORT(A)
BUILD-MAX-HEAP(A)
for i ← n downto 2
do exchange A[1] ⇔ A[i]
n←n–1
MAX-HEAPIFY(A,1)
HEAP-SORT takes O(n lg n) time since call to BUILD-MAX-HEAP takes O(n)
time and each of the n–1 calls to MAX-HEAPIFY takes O(lg n) time.
For the elements - 10, 5, 16, 8, 68, 21, 43; the heapSort procedure results in a
non-descendingly sorted elements. The state of the elements at each instance
within heapSort iterative with i from number of elements to 2 is:
(10 5 16 8 68 21 43)→ (16 10 43 8 5 21 68)→ (16 10 21 8 5 43 68)→
( 5 10 16 8 21 43 68)→ ( 8 10 5 16 21 43 68)→ ( 5 8 10 16 21 43 68)→
( 5 8 10 16 21 43 68)
Figure 3: GnuPlot for Heap Sort
3.2 Analysis of Run Times:
Any of General, Worst or Best Cases of inputs need O(n) time to BUILD-MAX-
HEAP. Further, any of ”add” or ”remove” operations takes O(lg n) time. Hence,
runtimes for heap sorting in all the three cases is always same with O(n lg n)
time.
7
4 QUICK SORT
Quick Sort is based on the technique ’Divide and Conquer’. It divides the ele-
ments into two parts and then sorts them. This sorting is continued recursively
till all the elements are sorted. Quick Sort has been implemented using two
functions - Partition and QuickSort.
Paritition routine takes the middle element as pivot, rearranging elements such
that elements smaller than the pivot is moved onto left of pivot and elements
greater than pivot are moved onto its right. This sorting technique swaps the
elements even if they are equal, hence is considered as an unstable algorithm.
PARTITION (A, p, r)
x ← A[(p + r) / 2]
i←p–1
j←r+1
while TRUE
do repeat j ← j – 1
until A[j] ≤ x
repeat i ← i + 1
until A[i] ≥ x
if i < j
then exchange A[i] ⇔ A[j]
else return j
Quick Sort calls the partition subroutine to obtain the pivot element and recur-
sively call itself to sort elements from pth element to pivot and then from pivot
to rth element. This recursive call is done until p < r.
QUICKSORT (A, p, r)
if p < r
then q ← PARTITION(A, p, r)
QUICKSORT(A, p, q)
QUICKSORT(A, q+1, r)
The steps for sorting of an input is given below :-
(4 2 3 5 1)→ (1 2 3 5 4)→ (1 2 3 5 4)→ (1 2 3 5 4)→ (1 2 3 5 4)→ (1 2 3 5 4)→
(1 2 3 5 4)→ (1 2 3 4 5)→ (1 2 3 4 5)→ (1 2 3 4 5)→ (1 2 3 4 5)
4.1 Worst Case
The Worst case happens when the elements are already sorted. The partition
routine returns the value as i+1. Thus the quicksort has to call partition n
times and there is a loop of n times inside the partition.Thus the order of the
quickSort becomes
T(N) = T(N-1) + O(N)
T(N) = O(N2 )
8
Figure 4: Plot for Quick Sort Worst Case
4.2 Best Case
The Best case is when the partiton function returns the middle element. when
this happens, the array is split into half and N reduces by half every time. Thus
in the best case, We get the time of QuickSort as
T(N) = ≤ 2T(n/2) + O(n) T(N) = O(n lg n)
9
Figure 5: Plot for Quick Sort Best Case
Figure 6: Comparing All Sorts
Comparing all the sorts—Bubble, Selection, Heap and Quick, there was huge
difference in order of time.
10
References
[1] Cormen, Thomas H et al, 2005, Introduction to Algorithms—Second Edi-
tion, Prentice-Hall of India.
[2] Levitin, Anany, 2003, Introduction to The Design & Analysis of Algo-
rithms, Pearson Education
LATEX Reference
[3] LESLIE LAMPORT, 1985, LATEX—A Document Preparation System—
User’s Guide And Reference Manual,Addison-Wesley, Reading.
[4] LATEX Tutorials, 2003, A Primer—Indian TEX Users Group, India.
[5] Greenberg, Harvey J, 1999, A Simplified Introduction to LATEX , Denver.
[6] Internet Sources
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