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Lecture DS & Algorithms:09
Abstract Data Types
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Abstract Data Types
Data Type: A data type is a collection of values and a set of operations
on those values.
The collection and operations form a mathematical construct
An ADT refers to the mathematical concept that defines the data type
Each ADT operation is defined by its inputs and outputs.
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Def.
e.g. whole numbers (integers) and arithmetic operators for addition, subtraction, multiplication and division.
e.g. Flight reservation Basic operations: find empty seat, reserve a seat,
cancel a seat assignmentWhy "abstract?" Data, operations, and relations are studied
independent of implementation.
What not how is the focus.
a collection of related data items together with an associated set of operations
Abstract Data Types
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Def.Consists of
storage structures (data structures) to store the data items
andalgorithms for the basic operations.
The storage structures/data structures used in implementations are provided in a language (primitive or built-in) or are built from the language constructs (user-defined).
In either case, successful software design uses data abstraction:
Separating the definition of a data type from its implementation.
Abstract Data Types
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Separation of interface and implementation
• Think of ADT as a black box
• ADT is represented by an interface and implementation is hidden from the user
– This means that the ADT can be implemented in various ways, as long as it adheres to interface
– For example, a ListADT can be represented using an array based implementation or a linked list implementation
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Linear list data structure
• Def: An ordered collection of elements– some examples are an alphabetized list of students, a list of gold
medal winners ordered by year, etc.
• With these examples in mind, we feel the need to perform the following operations on a linear list
– Determine whether the list is empty– Determine the size of list– Find the element with a given index– Find the index of a given element– Delete an element given its index– Insert a new element
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ADT - linearList
• The ADT specification is independent of any representation and programming language
AbstractDataType linearList{
elementsordered finite collection of zero or more elements
operationsempty(): return true if the list is emptysize(): return the list sizeget(index): return the indexed elementindexOf(x): return the index of the first occurance of x in
the list, returns -1, if x is not in the listerase(index): delete the indexth element, element with higher
index have their index reduced by 1insert(index, x): insert x as the indexth elementoutput(): output the list elements from left to right
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Array representation
• [5, 2, 4, 8,1]• Some of the implementations can be
18425location(i) = i
52481location(i) = 9- i
42518location(i) = (7+i)%10
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Simple Data Types
Also known as built-in data types
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Boolean data
Data values: {false, true}
In C/C++: false = 0, true = 1 (or nonzero)
Operations: and &&or ||not !
x !x0 11 0
&& 0 1
0 0 0
1 0 1
| | 0 1
0 0 1
1 1 1
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Character Data
Store numeric codes (ASCII, EBCDIC, Unicode)
1 byte for ASCII and EBCDIC,
2 bytes for Unicode
Basic operation: comparison to determine if Equal, Less than, Greater than, etc. use their numeric value.
,
ASCII/EBCDIC
Unicode
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Integer Data
Non-negative (unsigned) integer:
Store its base-two representation in a fixed number w of bits
(e.g., w = 16 or w = 32)
88 = 00000000010110002
Signed integer: Store in a fixed number w of bits using one of the following
representations:
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Sign-magnitude representation
Save one bit (usually most significant) for sign
(0 = +, 1 = – )
Use base-two representation in the other bits.
88 _000000001011000
0sign bit
–88 _0000000010110001
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Two's complement representation
For negative n (–n):(1) Find w-bit base-2 representation of n (2) Complement each bit.(3) Add 1
Example: –881. 88 as a 16-bit base-two number000000000101100
0
Same as sign mag.
For nonnegative n: Use ordinary base-two representation with leading (sign) bit 0
2. Complement this bit string3. Add 1
11111111101001111111111110101000
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Array ADT
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Linear Arrays
• A linear array is a finite number N homogeneous data elements
– Elements are referenced respectively by an index set of N consecutive numbers
– Elements are stored respectively in successive memory locations
– Fixed number of elements
• Index always integer
• Length=UB-LB+1
• Notation A1 or A(1) or A[1]
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Declaring arrays in C++
whereelement_type is any typearray_name is the name of the array — any valid identifierCAPACITY (a positive integer constant) is the number of
elements in the array
score[0]
score[1]
score[2]
score[3]
score[99]
.
.
....
element_type array_name[CAPACITY];
e.g., double score[100];
The elements (or positions) of the array are indexed 0, 1, 2, . . ., CAPACITY - 1.
Can't input the capacity, Why?
The compiler reserves a block of “consecutive” memory locations, enough to hold CAPACITY values of type element_type.
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indices numbered 0, 1, 2, . . ., CAPACITY - 1
How well does C/C++ implement an array ADT?
As an ADT In C++
ordered
fixed size
same type elements
direct access
element_type is the type of elements
CAPACITY specifies the capacity of the array
subscript operator []
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an array literal
Array Initialization
Example:double rate[5] = {0.11, 0.13, 0.16, 0.18, 0.21};
Note 1: If fewer values supplied than array's capacity, remaining elements assigned 0.
double rate[5] = {0.11, 0.13, 0.16};
rate
0 1 2 3 4
0.11 0.13 0.16 0 0
rate
0 1 2 3 4
0.11 0.13 0.16 0.18 0.21
In C++, arrays can be initialized when they are declared.Numeric arrays:element_type num_array[CAPACITY] = {list_of_initial_values};
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Note 1: If fewer values are supplied than the declared size of the array, the zeroes used to fill un-initialized elements are interpreted as the null character '\0' whose ASCII code is 0.
const int NAME_LENGTH = 10;char collegeName[NAME_LENGTH]={'C', 'a', 'l', 'v', 'i', 'n'};
vowel
0 1 2 3 4
A E I O U
char vowel[5] = {'A', 'E', 'I', 'O', 'U'};
Character Arrays
Character arrays may be initialized in the same manner as numeric arrays.
declares vowel to be an array of 5 characters and initializes it as
follows:
collegeName
0 1 2 3 4 5 6 7 8 9
C a l v i n \0 \0 \0 \0
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The contents of the memory word with this address 0x13BA can then be retrieved and displayed.
An address translation like this is carried out each time an array element is accessed.
What will be the time complexity
AddressesWhen an array is declared, the address of the first byte (or word) in the block of memory associated with the array is called the base address of the array.
Each array reference must be translated into an offset from this base address.
For example, if each element of array score will be stored in 8 bytes and the base address of score is 0x1396. A statement such as
cout << score[3] << endl;requires that array reference score[3] be translated into a memory address: 0x1396 + 3 * sizeof (double)
= 0x1396 + 3 * 8 = 0x13BA
score[3]
[0]
[1]
[2]
[3]
[99]
.
.
....
score 0x1396
0x13ae
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The value of array_name is actually the base address of array_name
array_name + index is the address of array_name[index].
An array reference array_name[index]
is equivalent to
For example, the following statements of pseudocode are equivalent:
print score[3]print *(score + 3)
* is the dereferencing operator*ref returns the contents of the memory location with address ref
*(array_name + index)
Character arrays
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Operations
• Traverse• Insert• Delete• Search
– Linear Search– Binary Search
• Sorting
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Traversing Linear Arrays
Repeat for K=LB to UB
Apply PROCESS to Array[K]
[End of Loop]
Exit
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Sorting
BUBBLE(DATA,N)1. Repeat Steps 2 and 3 for K=1 to N-1
2. Set PTR:=13. Repeat while PTR<=N-K
a) if DATA[PTR]>DATA[PTR+1] then:Swap DATA[PTR] and DATA [PTR+1]
b) Set PTR:=PTR+14. Exit
Complexity of Bubble Sort?n(n-1)/2 + O(n)=?Can you see any problem?How to make bubble sort efficient?
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Bubble Sort Source Code
for(x = 0; x < n; x++)
{ for(y = 0; y < n-1; y++)
{
if(array[y] > array[y+1])
{
temp = array[y+1];
array[y+1] = array[y];
array[y] = temp;
}
}
}
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• void bubbleSort(int numbers[], int array_size) • {• int i, j, temp; • for (i = (n - 1); i >= 0; i--) • { • for (j = 1; j <= i; j++) • { • if (numbers[j-1] > numbers[j]) • { • temp = numbers[j-1]; • numbers[j-1] = numbers[j]; • numbers[j] = temp; • } • }• }• }
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Bubble Sort: Example
• Consider the unsorted array to the right
• We start with the element in the first location, and move forward:– if the current and next items are in
order, continue with the next item, otherwise
– swap the two entries
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Bubble Sort: Example
• After one loop, the largest element is in the last location
• Repeat the procedure
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Bubble Sort: Example
• Now the two largest elements are at the end
• Repeat again
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Bubble Sort: Example
• With this loop, 12 is brought to its appropriate location
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Bubble Sort: Example
• Finally, we swap the last two entries to order them
• At this point, we have a sorted array
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Problems with arrays
• Capacity can not be changed during program execution– Memory wastage– Out of range errors