arrays1.ppt python programme arrays insertion

Arrays
 The simplest type of data structure is a one-dimensional array or list
or linear array.
 Linear array is a list of finite number (n) of similar data elements
referenced respectively by a set of n consecutive numbers , usually
1,2…..n.
 If we choose a name A for the array , then the elements of A are
denoted by subscript notation
a1,a2,a3……,an
or by the parenthesis notation
A(1),A(2)………. ,A(n)
or by the bracket notation
A[1],A[2]………. , A[n]
 The value K in A[K] is called a subscript and A[K] is called a
subscripted variable.
 The elements of an array are stored in consecutive memory locations.

Linear Array
 A linear array LA can be pictured as
(LB) 1
2 1 2 3 4 5
3
4
(UB) 5
Notations :-
 LB :- Lower Bound
 UB :- Upper Bound
 Length of an array :- UB – LB + 1
 LOC(LA[K]) :- Address of the element LA[K] of the array LA
 Base (LA) :- Address of the first element of LA.
 LOC (LA [K]) =Base (LA) + w (K-lower bound) ,where w is the number of words per memory
cell for the array LA.

Traversing Linear Arrays
Let LA be a collection of elements stored
in memory of the computer. Suppose we
want to print the contents of the array or
to count the number of elements of LA
with a given property. This can be
accomplished by traversing LA ,that is by
accessing and processing each element of
exactly once.

Algorithm
 Here LA is a linear array with lower bound LB
and upper bound UB. This algorithm traverses
LA applying an operation to each element of
LA.
1. [Initialize counter] Set K = LB.
2. Repeat steps 3 and 4 while K<=UB
3. [Visit Element] Apply process to LA[K].
4. [Increment counter.] K = K+1.
End of step 2 loop.
5. Exit.

Insertion
Suppose LA is an array containing 5
names in alphabetical order
Brown
Davis
Johnson
Smith
Wagner
 Suppose we want to add an element “Ford” into the
array.
 Then Johnson, Smith, and wagner must each be moved
downward one location.

Algorithm
 (Inserting into a Linear Array) INSERT(LA,N,K,ITEM)
Here LA is a linear array with N elements and K is a
positive integer such that K<=N. This algorithm inserts an
element ITEM into the Kth
position inLA.
1. [Initialize counter.] Set J=N.
2. Repeat steps 3 and 4 while J>=K.
3. [Move Jth
element downward.] Set LA[J+1] = LA[J].
4. [Decrease Counter.] Set J = J-1.
[End of step 2 loop.]
5. [Insert element.] set LA[K] = ITEM.
6. [Reset N] Set N= N + 1.
7. Exit

Deletion
Suppose we want to delete an element
“smith” from the array.
Brown
Davis
Johnson
Smith
Wagner
The “wagner” should be moved upwards.

Deletion
 (Deleting from a Linear array)DELETE(LA,N,K,ITEM)
Here LA is a linear array with N elements and K is
a positive integer such that K<=N. This algorithm
deletes the Kth
element from LA.
 Set ITEM = LA[K]
 Repeat for J=K to N-1:
[Move J+1 th
element upward] Set LA[J] =
LA[J+1].
[End of loop.]
3. [Reset the number N of elements in LA.] Set N=N-1
4. Exit .

Searching
Let A be a collection of data elements in
memory and suppose a specific ITEM of
information is given. Searching refers to the
operation of finding the location LOC of
ITEM in A or printing some message that
ITEM does not appear there.
The search is said to be successful if ITEM
does appear in A and unsuccessful
otherwise.

Searching Techniques
There are different searching techniques
Linear Search
Binary Search
Binary search is more efficient than linear
search. It take lesser time for execution.
The complexity of searching algorithms is
measured in terms of the number of
comparisons required to find ITEM in array
.

Linear Search
 Suppose A is an array with N elements.
 To search for a given item in A , compare ITEM
with each element of A one by one.
 This method , which traverses A sequentially to
locate ITEM , is called linear search or sequential
search.
4 5 6 7 8
4 5 6 7 8 6
1 2 3 4 5 N+1
Step 1: Assign ITEM to N+1
Step 1: Assign ITEM to N+1 th
th
location
location

Algorithm
 (Linear search) LINEAR(A,N,ITEM,LOC)
Here A is a linear array with N elements, and ITEM is a given
item of information. This algorithm finds the location LOC of
ITEM in DATA , or sets LOC=0 if search is unsuccessful.
1. [Insert ITEM at the end of A]. Set A[N+1]=ITEM.
2. [Initialize Counter] Set LOC=1.
3. [Search for ITEM]
Repeat while A[LOC] != ITEM
Set LOC=LOC+1
[End of loop]
4. [Successful?] Print LOC
5. [UnSuccessful?] If LOC=N+1 , then Set LOC=0.
6. Exit

Binary Search
 Suppose A is an array which is sorted in increasing numerical
order or alphabetically.
 Then there is an extremely efficient searching algorithm, called
binary search.
 The algorithm compares ITEM with the middle element
A[MID] where MID is obtained by
MID=((BEG+END)/2)
 If A[MID]=ITEM , then search is successful and set LOC=MID.
Otherwise a new segment of A is obtained as follows:
 If ITEM<A[MID], then ITEM can appear only in the left half of the segment :
A[BEG],A[BEG+1],…….A[MID-1]
• So reset END = MID-1 and search again.
 If ITEM>A[MID], then ITEM can appear only in the right half of the segment :
A[MID+1],A[MID+2]……..A[END].
• So reset BEG=MID+1 and search again.

Algorithm
 (Binary search) BINARY(A,LB,UB,ITEM,LOC)
Here A is a sorted array with a lower bound LB and upper bound UB, and ITEM
is a given ITEM of information. The variables BEG,MID and END denote,
respectively the beginning, end and middle locations of a segment of elements of A.
This algorithm finds the location Loc of ITEM in A or sets LOC=NULL.
1. [Initialize segment variables.]
Set BEG=LB,END=UB AND MID = INT((BEG+END)/2)
2. Repeat Steps 3 and 4 while BEG<= END and A[MID]!=ITEM
3. If ITEM<A[MID], then:
Set END= MID – 1
Else
Set BEG = MID + 1
[End of If.]
4. Set MID = INT((BEG+END)/2)
[End of step2 loop.]
5. If A[MID]=ITEM , then
Set LOC= MID
else
Set LOC=NULL
[End of If]

Example
Consider an array with 5 elements :
Let ITEM= 76
12 , 15 , 20 , 23 , 32, 45 , 54 , 76 , 98 beg=1,end=9 ,mid=5
12 , 15 , 20 , 23 , 32 , 45 ,
12 , 15 , 20 , 23 , 32 , 45 , 54
54 , 76 , 98
, 76 , 98 Beg=6 ,end=9 ,mid=7
12 , 15 , 20 , 23 , 32 , 45 , 54 ,
12 , 15 , 20 , 23 , 32 , 45 , 54 , 76
76 , 98
, 98 Beg=8 ,end= 9 ,mid=8
Left sub array Right sub array
Right sub array

Limitations of Binary Search
 Even if the binary search algorithm is so efficient
it requires two conditions :
The list must be sorted and
One must have direct access to the middle element in
any sub list .
 This means that one must essentially use a
sorted array to hold data.
 But keeping data in sorted array is normally
very expensive when there are many insertions
and deletions.

Multi-Dimensional arrays
Linear arrays in which each element is
referenced by a single subscript is called a
one – dimensional array.
Arrays in which elements are referenced by
more than one subscript is called multi-
dimensional arrays.
A two-dimensional array A is a collection of
m*n elements such that each element is
specified by a pair of subscripts A [i][j]

Representation of 2-D arrays
 A 2-D array is stored in memory by a block of m*n
sequential memory locations.
 The programming language will store the array A
either
1) Column by column (column-major order)
2) Row by row (row-major order)
.
.
.
A(1,1)
A(1,2)
.
.
.
.
A(m,n)

2-D arrays
 For a one-dimensional array computer uses the
formula
LOC(A[K])=Base (A) +w(K-1)
to find the address of A[K].
 For a 2-D array the address of A[J,K] is found
out using the formula
LOC(A[J,K])=Base(A)+w[M(K-1)+(J-1)]
(Column major order)
LOC(A[J,K] = Base (A)+[N(J-1)+(K-1)]

arrays1.ppt python programme arrays insertion

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