Data Structure and Algorithms Arrays

 Linear arrays: Memory representation
 Traversal
 Insertion
 Deletion
 Linear Search
 Binary Search
 Merging
 2D Array : Memory representation
1

CONTENTS
2.1 Introductions
2.2 Linear Array
2.2.1 Linear Array Representations in Memory
2.2.2 Traversing Algorithm
2.2.3 Insert Algorithms
2.2.4 Delete Algorithms
2.2.5 Sequential and Binary Search Algorithm
2.2.6 Merging Algorithm
2.3 Multidimensional Array
2.3.1 2-D Array
2.3.2 Representations in Memory
2

2.1 Introduction
Data Structure can be classified as:
linear
non-linear
Linear (elements arranged in sequential in memory
location) i.e. array & linear link-list
Non-linear such as a tree and graph.
Operations:
Traversing, Searching, Inserting, Deleting, Sorting, Merging
Array is used to store a fix size for data and a link-list
the data can be varies in size.
3

2.1 Introduction
Advantages of an Array:
Very simple
Economy – if full use of memory
Random accessed at the same time
Disadvantage of an Array:
wasting memory if not fully used
4

2.2 Linear Array
 Homogeneous data:
a) Elements are represented through indexes.
b) Elements are saved in sequential in memory locations.
 Number of elements, N –> length or size of an array.
If:
UB : upper bound ( the largest index)
LB : lower bound (the smallest index)
Then: N = UB – LB + 1
Length = N = UB when LB = 1
5

2.2 Linear Array
 All elements in A are written symbolically as, 1 .. n is the
subscript.
A1, A2, A3, .... , An
 In FORTRAN and BASIC  A(1), A(2), ..., A(N)
 In Pascal, C/C++ and Java  A[0], A[1], ..., A[N-1]
 subscript starts from 0
LB = 0, UB = N–1
6

2.2.1 Representation of Array in a Memory
 The process to determine the address in a memory:
a) First address – base address.
b) Relative address to base address through index function.
Example: char X[100];
Let char uses 1 location storage.
If the base address is 1200 then the next element is in 1201.
Index Function is written as:
Loc (X[i]) = Loc(X[0]) + i , i is subscript and LB = 0
1200 1201 1202 1203
X[0] X[1] X[2]
7

2.2.1 Representation of Array in a Memory
 In general, index function:
Loc (X[i]) = Loc(X[LB]) + w*(i-LB);
where w is length of memory location required.
For real number: 4 byte, integer: 2 byte and character: 1 byte.
 Example:
If LB = 5, Loc(X[LB]) = 1200, and w = 4, find Loc(X[8]) ?
Loc(X[8])= Loc(X[5]) + 4*(8 – 5)
= 1212
8

2.2.2 Traversing Algorithm
 Traversing operation means visit every element once.
e.g. to print, etc.
 Example algorithm:
9
1. [Assign counter]
K=LB // LB = 0
2. Repeat step 2.1 and 2.2 while K <= UB // If LB = 0
2.1 [visit element]
do PROCESS on LA[K]
2.2 [add counter]
K=K+1
3. end repeat step 2
4. exit

2.2.3 Insertion Algorithm
 Insert item at the back is easy if there is a space. Insert
item in the middle requires the movement of all elements
to the right as in Figure 1.
0 1 2 3 4 k MAX_LIST-1
1 2 3 4 5 k+1 MAX_LIST
10
12 3 44 19 100 … 5 10 18 ? … ?
k+1
size
Array indexes New item
ADT list positions
items
Figure 1: Shifting items for insertion at position 3

2.2.3 Insertion Algorithm
11
INSERT(LA, N, K, ITEM)
//LA is a linear array with N element
//K is integer positive where K < N and LB = 0
//Insert an element, ITEM in index K
1. [Assign counter]
J = N – 1; // LB = 0
2. Repeat step 2.1 and 2.2 while J >= K
2.1 [shift to the right all elements from J]
LA[J+1] = LA[J]
2.2 [decrement counter] J = J – 1
3. [Stop repeat step 2]
4. [Insert element] LA[K] = ITEM
5. [Reset N] N = N + 1
6. Exit

2.2.4 Deletion Algorithm
 Delete item.
(a)
0 1 2 3 4 k-1 k MAX_LIST-1
1 2 3 4 5 k k+1 MAX_LIST
12
12 3 44 100 … 5 10 18 ? … ?
k
size
Array indexes
Delete 19
ADT list positions
items
Figure 2: Deletion causes a gap

(b)
0 1 2 3 k-1 MAX_LIST-1
1 2 3 4 k MAX_LIST
13
12 3 44 100 … 5 10 18 ? … ?
k
size
Array indexes
ADT list positions
items
Figure 3: Fill gap by shifting

14
DELETE(LA, N, K, ITEM)
1. ITEM = LA[K]
2. Repeat for I = K to N–2 // If LB = 0
2.1 [Shift element, forward]
LA[I] = LA[I+1]
3. [end of loop]
4. [Reset N in LA]
N = N – 1
5. Exit

2.2.5 Sequential Search
Compare successive elements of a given list with a search ITEM
until
1. either a match is encountered
2. or the list is exhausted without a match.
0 1 N-1
Algorithm:
SequentialSearch(LA, N, ITEM, LOC)
1. I = 0 // If LB = 0
2. Repeat step 2.1 while (i<N and LA[I] != ITEM )
2.1 I=I+1
3. If LA[I]==ITEM then
Return found at LOC=I
If not
Return not found 15

2.2.5 Binary Search Algorithm
 Binary search algorithm is efficient if the array is sorted.
 A binary search is used whenever the list starts to become large.
 Consider to use binary searches whenever the list contains more
than 16 elements.
 The binary search starts by testing the data in the element at the
middle of the array to determine if the target is in the first or
second half of the list.
 If it is in the first half, we do not need to check the second half. If
it is in the second half, we do not need to test the first half. In other
words we eliminate half the list from further consideration. We
repeat this process until we find the target or determine that it is
not in the list.
16

 To find the middle of the list, we need three variables, one to
identify the beginning of the list, one to identify the middle
of the list, and one to identify the end of the list.
 We analyze two cases here: the target is in the list (target
found) and the target is not in the list (target not found).
17

 Target found case: Assume we want to find 22 in a sorted
list as follows:
 The three indexes are first, mid and last. Given first as 0 and
last as 11, mid is calculated as follows:
mid = (first + last) / 2
mid = (0 + 11) / 2 = 11 / 2 = 5
18
4 7 8 10 14 21 22 36 62 77 81 91
a[0] a[1] a[2] a[3] a[4] a[5] a[6] a[7] a[8] a[9] a[10] a[11]

 At index location 5, the target is greater than the list value (22 > 21).
Therefore, eliminate the array locations 0 through 5 (mid is automatically
eliminated). To narrow our search, we assign mid + 1 to first and repeat
the search.
19
4 7 8 10 14 21 22 36 62 77 81 91
a[0] a[1] a[2] a[3] a[4] a[5] a[6] a[7] a[8] a[9] a[10] a[11]
0 5 11
first mid last
Target: 22Target: 22
22 > 21

 The next loop calculates mid with the new value for first and
determines that the midpoint is now 8 as follows:
mid = (6 + 11) / 2 = 17 / 2 = 8
20
4 7 8 10 14 21 22 36 62 77 81 91
a[0] a[1] a[2] a[3] a[4] a[5] a[6] a[7] a[8] a[9] a[10] a[11]
6 8 11
first mid last
22 < 62

 When we test the target to the value at mid a second time, we discover that the
target is less than the list value (22 < 62). This time we adjust the end of the list
by setting last to mid – 1 and recalculate mid. This step effectively eliminates
elements 8 through 11 from consideration. We have now arrived at index location
6, whose value matches our target. This stops the search.
21
4 7 8 10 14 21 22 36 62 77 81 91
a[0] a[1] a[2] a[3] a[4] a[5] a[6] a[7] a[8] a[9] a[10] a[11]
6 6 7
first mid last
22 equals 228 6 7
function terminates
first mid last

 Target not found case: This is done by testing for first and last crossing:
that is, we are done when first becomes greater than last. Two conditions
terminate the binary search algorithm when (a) the target is found or (b)
first becomes larger than last. Assume we want to find 11 in our binary
search array.
22
4 7 8 10 14 21 22 36 62 77 81 91
a[0] a[1] a[2] a[3] a[4] a[5] a[6] a[7] a[8] a[9] a[10] a[11]
0 5 11
first mid last Target: 11Target: 11
11 < 21

 The loop continues to narrow the range as we saw in the
successful search until we are examining the data at index
locations 3 and 4.
23
4 7 8 10 14 21 22 36 62 77 81 91
a[0] a[1] a[2] a[3] a[4] a[5] a[6] a[7] a[8] a[9] a[10] a[11]
0 2 4
first mid last
11 > 8

 These settings of first and last set the mid index to 3 as follows:
mid = (3 + 4) / 2 = 7 / 2 = 3
24
4 7 8 10 14 21 22 36 62 77 81 91
a[0] a[1] a[2] a[3] a[4] a[5] a[6] a[7] a[8] a[9] a[10] a[11]
3 3 4
first mid last Target: 11Target: 11
11 > 10

 The test at index 3indicates that the target is greater than the list value, so we set first to mid
+ 1, or 4. We now test the data at location 4 and discover that 11 < 14. The mid is as
calculated as follows:
 At this point, we have discovered that the target should be between two adjacent values; in
other words, it is not in the list. We see this algorithmically because last is set to mid – 1,
which makes first greater than last, the signal that the value we are looking for is not in the
list.
25
4 7 8 10 14 21 22 36 62 77 81 91
a[0] a[1] a[2] a[3] a[4] a[5] a[6] a[7] a[8] a[9] a[10] a[11]
4 4 4
first mid last
11 < 14 4 4 3
first mid last
Function terminates

 Example algorithm:
DATA – sorted array
ITEM – Info
LB – lower bound
UB – upper bound
ST – start Location
MID – middle Location
LAST – last Location
26

27
1. [Define variables]
ST = LB, LAST= UB;
MID = (ST+LAST)/2;
2. Repeat 3 and 4 WHILE (ST <= LAST & DATA[MID] !=
ITEM)
3. If ITEM < DATA[MID] then
LAST = MID-1
If not
ST = MID+1
4. Set MID = INT((ST + LAST)/2)
[LAST repeat to 2]
5. If DATA[MID] == ITEM then
LOK = MID
If not
LOK = NULL
6. Stop

Suppose A is a sorted list with r elements and B is a
sorted list with s elements. The operation that
combines the element of A and B into a single sorted
list C with n=r + s elements is called merging.
28

Algorithm: Merging (A, R,B,S,C)
Here A and B be sorted arrays with R and S elements
respectively. This algorithm merges A and B into an array
C with N=R+ S elements
Step 1: Set NA=1, NB=1 and NC=1
Step 2: Repeat while NA ≤ R and NB ≤ S:
if A[NA] ≤ B[NB], then:
Set C[NC] = A[NA]
Set NA = NA +1
else
Set C[NC] = B[NB]
Set NB = NB +1
[End of if structure]
Set NC= NC +1
[End of Loop]
29

Step 3: If NA >R, then:
Repeat while NB ≤ S:
Set C[NC] = B[NB]
Set NB = NB+1
Set NC = NC +1
[End of Loop]
else
Repeat while NA ≤ R:
Set C[NC] = A[NA]
Set NC = NC + 1
Set NA = NA +1
[End of loop]
[End of if structure]
Step 4: Return C[NC]
30

Complexity of merging: The input consists of the
total number n=r+s elements in A and B. Each
comparison assigns an element to the array C, which
eventually has n elements. Accordingly, the number
f(n) of comparisons cannot exceed n:
f(n) ≤ n = O(n)
31

Exercises
Find where the indicated elements of an array a
are stored, if the base address of a is 200* and
LB = 0
a) double a[10]; a[3]?
b) int a[26]; a[2]?
*(assume that int(s) are stored in 4 bytes and
double(s) in 8 bytes).
32

2.3 MULTIDIMENSIONAL ARRAY
 Two or more subscripts.
33

2-D ARRAY
 A 2-D array, A with m X n elements.
 In math application it is called matrix.
 In business application – table.
 Example:
Assume 25 students had taken 4 tests.
The marks are stored in 25 X 4 array locations:
34
U0 U1 U2 U3
Stud 0 88 78 66 89
Stud 1 60 70 88 90
Stud 2 62 45 78 88
.. .. .. .. ..
.. .. .. .. ..
Stud 24 78 88 98 67
n
m

2-D ARRAY
 Multidimensional array declaration in C++:-
int StudentMarks [25][4];
StudentMarks[0][0] = 88;
StudentMarks[0][1] = 78;…..
OR
int StudentMarks [25][4] = {{88, 78, 66, 89},
{60, 70, 88, 90},…}
35

2.3.1 2-D ARRAY
 In C++ the 2-D array is visualized as follows:
36
…
[0]
[1]
[2]
[3]
[4]
[5]
[6]
[24]
StudentMarks
88 78 66 89
60 70 88 90
62 45 78 88
[0] [1] [2] [3]

2.3.2 Representation of
2D arrays in Memory
Column Major Order:
LOC(A[j, k])=Base(A)+w[m*k + j]
Row Major order:
LOC(A[j, k])=Base(A)+w[n*j + k]
Given: A 2-D array, A with m X n elements.

Data Structure and Algorithms Arrays

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