Data Structures

Instructor: Sanchita Mal-Sarkar

Course: CIS 265

Data representation

• Computer represents “on” with a 1 and “off” with a 0. These numbers are called binary digits or bits.

• Binary number system: most widely used method for interpreting bit settings as nonnegative integers.

• 38 (decimal) = 21 +22+25 = 00100110

Ones complement notation

• One of the widely used methods for representing negative binary numbers.

• A negative number is represented by changing each bit to the opposite bit setting.

• Example:

• 38 (decimal) = 00100110 (binary)

• -38 (decimal) = 11011001 (binary)

Twos complement notation• Another popular method of representing

negative binary number

• In this method, 1 is added to the ones complement representation of a negative number.

• Example:

-38 (decimal) = 11011001 (ones complement)

-38 (decimal) = 11011010 (twos complement)

Floating-point notation• Usual method to represent real numbers• Real number is represented by a number, called a

mantissa, times a base raised to an integer power, called an exponent.

• Example:

387.53 = 38753 x 10 -2

Other possibilities .38753 x 10 3, 387.53 x100

(we choose mantissa is an integer with no tailing 0s)

Floating-point notation(cont.)• A real number is represented by a 32-bit string.• 24-bit for mantissa and 8-bit for exponent.• Base is fixed to 10.• Example

100 = 0000000000000000000000100000010

-387.53 = 11111111011010001001111111111110

387.53 = 00000000100101110110000111111110

( 24-bit binary representation of 38753 is 000000001001011101100001

8-bit twos complement binary representation of -2 is 11111110)

Modular Programming & Pointers

• Modules in C are called functions.

• Each module or smaller piece is more manageable than the original program.

• This technique is called divide and conquer.

• Arguments make functions more versatile because it enable a function to manipulate different data each time it is called.

• You know, how to pass input parameters into a function.

• I will discuss how to use output parameters to return multiple results from a function.

• Pointer to a variable provides the address to that type of variable.

Example #include <stdio.h> #include <math.h> void determine_sign(double num, char *signp);

int main(void) { double value; char sn; printf(“Enter a value>”); scanf(“%lf”, &value); determine_sign( value, &sn); printf(“The sign of %.4f is: %c\n”, value, sn); return (0); }

Example void determine_sign(double num, char *signp) { if (num < 0) *sign = ‘-’; else if (num == 0) *sign = ‘ ‘; else *sign = ‘+’; }

Pointers• A pointer is a variable that contains

memory address as its value.

• A variable directly contains a specific value.

• A pointer contains an address of a variable that contains a specific value.

• We use pointers extensively to create and manipulate dynamic data structures ( linked lists, queues, stacks, and trees), and to simulate call by reference.

Pointer• Pointers should be initialized either when they

are declared or in an assignment statement.• Pointer can be initialized to 0, NULL, or an

address.• A pointer with the value NULL points to

nothing.• Initializing a pointer to 0 is equivalent to

initializing to NULL (NULL is preferred);• NULL is a symbolic constant defined in

<stdio.h> header file.

Representation of a Pointer int x = 84; // declaration

int *ptr; // declaration

ptr = &x;

84

ptr x

84

x

X directly references avariable whose value is 84

ptr indirectly references a variable whose value is 84

Assigned the address of the variable x to pointervariable ptr. Direct value of ptr is the address of x.

Representation of x and ptr in memory int x = 84; // declaration

int *ptr; // declaration

ptr = &x; // assigned the address of the // variable x to pointer variable

ptr.

60000

ptr

Ptr indirectly references a variable whose value is 84

50000 84

x

X directly references avariable whose value is 84

60000

Different ways to pass arguments to a function

• Two ways to invoke functions:– Call by value– Call by reference

• Call by value: When arguments are passed call by value, – a copy of the argument’s value is made and passed

to the called function. – Changes to the copy do not affect an original

variable’s value in the caller.

Call by Reference

When an argument is passed by reference,

• the caller actually allows the called function to modify the original variable’s value.

• It is possible to simulate call by reference by using address operators and indirection operators.

Cube a variable using call by value #include <stdio.h>

int cubeByValue(int);

main()

{

int number = 5, number1;

printf(“The original value of number is %d\n”, number);

number1 = cubeByvalue(number));

printf(“The new value of number is %d\n”, number1);

return 0;

}

int cubeByValue(int n)

{

return n * n * n;

}

Cube a variable using call by Reference #include <stdio.h>

void cubeByReference(int *);

int main()

{

int number = 5;

printf(“The original value of number is %d\n”, number);

cubeByReferencee(&number));

printf(“The new value of number is %d\n”, number);

return 0;

}

void cubeBy Reference(int *nPtr)

{

*nPtr = *nPtr ** nPtr * *nPtr;

}

Pointer Arithmetic

• A limited set of arithmetic operators may be performed on pointers.

• A pointer may be incremented (++) or decremented (--).

• An integer may be added to a pointer (+ or +=).

• An integer may be subtracted from a pointer

(+ or +=).

• One pointer may be subtracted from another.

Example• int a[10]= {1,2,3,4,5};• int *ptr;• ptr = a;

• When an integer is added to or subtracted from a pointer, the pointer is not simply incremented or decremented by that integer.

• It depends on the size of the object a pointer point to => machine dependent.

1 5432

a + 1 a + 3

ptr

a

Example (Increment)

• When an integer is added to a pointer, the pointer is incremented by that integer times the size of the object to which the pointer points to.

int *ptr; ptr = a; ptr += 2; will produce: 3008 (3000 + 2 * 4) integer is stored in 4 bytes of memory.

1 5432

a + 1 a + 3ptr

a

30003004

30083012

3016

Example (Decrement)• When an integer is subtracted from a pointer,

decremented by that integer times the size of the object to which the pointer points to.

• int *ptr;

• ptr = a;

• ptr += 2;

• ptr -= 1;

will produce:

3004 (3008 - 1 * 4)

integer is stored in 4 bytes of memory.

1 5432

a + 1 a + 3ptr

a

30003004

30083012

3016

Subtraction• Pointer variables may be subtracted from one

another.• ptr1 contains the location 3000.• ptr2 contains the location 3008. x = ptr2 - ptr1; = 2 will assign to x the number of array elements from ptr1 to ptr2

1 5432

ptr2ptr1

a

30003004

30083012

3016

Relationship between Pointers and Arrays• Arrays and Pointers are intimately related in C.

• Array subscripting notation is converted to pointer notation during compilation.

• Array name is a pointer to the first element of the array.

• ptr = &a[0]• *(ptr + 3) a[3]

• &a[3] (ptr + 3)

• a += 2 invalid => attempts to modify the value of array name with pointer arithmetic.

1 5432

a[1] a[3]

ptr

aOffset to the pointer

Array of Pointers• Array may contain pointers (e.g. string array).

• In C, string is essentially a pointer to its first character.

• Each entry in an array of strings is actually a pointer to the first character of a string.

• Example:

• char *names[4] = {“John”, “Cynthia”, “David”, “Daniel”};

• names[4] => an array of 4 elements.

• Char * => each element of array names is of type “pointer to char.”

• Four array elements are: “John”, “Cynthia”, “David”, “Ruben”.

• Each of these are stored in memory as a NULL-terminated character string that is one character longer than the number of characters between quotes.

• E.g. Four strings are 5, 8, 6, and 7.

Example of array of pointers• It seems that the strings are placed in the names array,

• Actually, only the pointers are stored in the array.

• Each pointer points to the first character of its corresponding string.

• Names is a fixed size array.

• However it provides access to character strings of any length.

• Names could have been placed into a double array of fixed number of columns per row => memory could be wasted.

‘J’ ‘\0’

‘D’

‘h’

‘C’

‘n’

‘y’ ‘n’

‘o’

‘\0’‘a’‘i’‘h’

‘n’

‘i’‘v’‘a’

‘t’

‘l’‘a’‘D’

‘\0’‘d’

‘e’‘i’ ‘\0’

names[0]

names[1]

names[2]

names[3]

Pointer to Functions• Pointer to a function contains the address of the function in

memory.

• A function name is really the starting address in memory of code that performs the function’s task.

• Pointers to functions can be passed to functions, returned from functions, stored in arrays, and assigned to another function pointers.

• int (*compare)(int num[1], int num[4])

Parenthesis s are needed around *compare because * has lower precedence than parenthesesenclosing the function parameters.

Representation and conversion of Numeric Types

• Why we need more than one numeric type ?• We could use data type double for all numbers.

However, they are represented in the computer’s memory in different ways. Here are some advantages of integer data type:– On many computers, operation will be faster if we use

integer instead of type double.

– We need less space to store integer number .

– Operations with integer are always precise. But in operations with double numbers can result in some loss of accuracy or round-off error.

Advantages of Type double Format

• A real number has an integral part and a fractional part. But an integer number can not have a fractional part. So real number can not be represented by an integer, it can be represented by double data type.

• Another advantage of the data type double is that much larger range of numbers can be represented as compared to type int.

• Actual ranges vary from one implementation to another.

ANSI standard for C

min range of positive values of type int: 1- 32, 767

min range of positive values of type double: 10 -37 to 10 37

Internal formats of Type int & Type double• All data are represented in memory as binary strings (strings

of 0s and 1s).

• However, the binary string stored for type int value 24 is not the same as the binary string stored for the type double number 24.0.

• The actual internal representation is computer dependent.

• Positive integers are represented by standard binary numbers. Integer 24 is represented by binary number 11000.

• The format of type double is analogous to scientific notation.

mantissa exponent

type double format

binary number

type int format

Scientific Notation A real number has an integral part and a fractional part

which are separated by a decimal point.

• Very large or very small values can be represented by scientific notation.

Example: for the value 123000.0 => scientific notation: 1.23

• In C scientific notation: 1.23e5 or 1.23E5

• Read the letter e or E as "times 10 to the power".

• If the exponent has a minus sign, the decimal point is moved to the left.

Example: 0.34e-4 == 0.000034

Numerical Inaccuracies• Sometimes an error occurs in representing real numbers.

• Analogy: in decimal system, 1/3 is 0.333333…..

• Some fractions canot be represented exactly as binary numbers in the mantissa of the type double format.

• This inaccuracy is called representational error.

• It depends on the number of binary digits used in the mantissa.

• The more bits, the smaller the error.

• An equality comparison of two type double values can lead to unexpected results.

Example

for (count = 0.0; count != 10.0; count = count + 0.1)

{

} Infinite loop

Cancellation Error

• When you add a very large and a very small real numbers, the larger number may cancel out the smaller number.

• Results in cancellation error.

Example:

1000.0 + 0.000000231 = 1000.0

(On some computers)

Arithmetic Underflow

• If we multiply two very small numbers, the result may be too small to be represented accurately.

• The computational result will be represented by zero.

• This phenomenon is called arithmetic underflow.

Arithmetic Overflow

• If we multiply two very large numbers, the result may be too large to be represented.

• This phenomenon is called arithmetic overflow.

• Different C compilers handle it in different ways.

Automatic Conversion of Data Types• In several cases, data of one numeric type are automatically

converted to another numeric type.

• Some examples:

int k = 5, m = 4, n;

double x = 1.5, y = 2.1, z;

A Expression of different numeric types:

k + x Value of int variable k is converted

Value is 6.5 to type double format before operation is performed.

B RHS is of type int and LHS is of type double:

z = k / m; Expression is evaluated first. Then, the result is converted to type

double format for assignment.z = 1.0

Expression value = 1

Automatic Conversion of Data Types

• Example:

int k = 5, m = 4, n;

double x = 1.5, y = 2.1, z;

C RHS is of type double and LHS is of type int:

n = x * y; Expression is evaluated first. Then, the result is converted to type int format for assignment, and

fractional part is lost.Expression value= 3.15;

n = 3

Explicit Conversion of Data Types• C also provides an explicit type conversion operation

called a cast.

• Examples:– Using cast operation int n1, n2;

double frac;

frac = (double)n1 / (double)n2;

2.0 / 4.0

0.5

– Without cast operation frac = n1 / n2;

2 / 4 integer division

0

Result:frac = 0.5

Result:frac = 0.0

Explicit Conversion of Data Types A. frac = (double)(n1 / n2); and B. frac = (double)n1 / (double)n2; C. frac = (double)n1 /n2;A The quotient n1/n2 is computed first, results in

loss of fractional part. The cast to double convertsthe whole number

quotient tp type dpouble.B frac = (double)n1 / (double)n2; 2.0 / 4.0 0.5C frac = (double)n1 / n2; n2 will automatically be

converted to double.

Difference

Explicit Conversion of Data Types• Sometimes we include casts that do not affect the result

but simply make clear to the reader the conversions that would occur automatically.

Example: int x, sqrt_x;

sqrt_x = (int)sqrt((double)x);

• The formal parameter of sqrt function is of type double.

• The actual argument x is of type int.

• x will automatically be converted to type double.

• sqrt function returns a value of type double.

• This value will automatically be converted to type int since sqrt_x is of type int.

Note: When a cast operation is applied to a variable, the conversion changes the value of the expression, but it does not change what is stored in the variable.

Representation & Conversion of Type char• We can declare variable of type char to represent a single character• A character can be a letter, digit, or punctuation mark.• A character is enclosed in apostrophes.(e.g. ‘A’, ‘3’).• We can compare character values using equality operators (== and !

=) and relational operators (<,<=, >, >=).• In ASCII (American Standard Code for Information Interchange): digit character ‘0’ = 48 (decimal code value) ‘9’ = 57 (‘0’<‘1’<‘2’<‘3’<‘4’…….<‘9’) ‘A’ = 65 ‘Z’ = 90 (‘A’< ‘B’<‘C’……..<‘X’<‘Y’<‘Z’) ‘a’ = 97 ‘z’ = 122 (‘a’< ‘b’<‘c’……..<‘x’<‘y’<‘z’)• C allows conversion of type char to type int and vice versa.

Enumerated Types• ANSI C allows us to associate a numeric code with each

category by creating an enumerated type that has its own list of meaningful values.

• typedef can be used to name user-defined types.

Example:

typedef enum

{ sunday, monday, tuesday, wednesday, thursday, friday,

saturday}

day_t;

enumerated type day_t has seven possible values (enumeration constants).

sunday will be represented as the integer 0, monday as integer 1, and so on.

day_t today; // declaration of variable today.

Examples of Enumerated Types• Class_t is an enumerated data type:

typedef enum

{fresh, soph, jr, sr}

class_t;

• What is the value of each of the following?

(int) sr , (class_t) 0, (class_t)((int)soph + 1)

• What is displayed by this code fragment?

for (class = fresh; class <= sr; ++class)

printf(“%d “, class);

3 fresh jr

0 1 2 3

Example of Enumerated Type Example:

typedef enum

{ sunday, monday, tuesday, wednesday, thursday, friday,

saturday}

day_t;

day_t next(day_t today)

{

day_t next;

if (today == saturday)

next = sunday;

else

next = (day_t)((int)today + 1);

return (next);

}

Another Example of Enumerated Type typedef enum

{ sunday, monday, tuesday, wednesday, thursday, friday,

saturday}

day_t;

void print_day (day_t day)

{

switch (day)

{

case sunday:

printf(“sunday”);

break;

case monday:

printf(“monday”);

break;

default: printf(“Invalid Code”);

}

}