i1101 algorithm & imperative programmingantoun.me/teaching/2018-2019/i1101/ch6.pdf ·...

I1101Algorithm & Imperative Programming

Semester: 2

Academic Year: 2018/2019

Credits: 5 (60 hours)

Dr. Antoun Yaacoub

Lebanese University

Faculty of Science

Computer Science BS Degree

Syllabus1. Algorithmics

2. Fundamental elements of a C program

3. Basic types, operators and expressions

4. Read and write data

5. Alternative structure

6. Repetitive structure

7. Arrays

8. Strings

9. Functions

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Arrays

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Introduction

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• Arrays are certainly the most popular structured variables

One-dimensional arrays• A (uni-dimensional) array A is a structured variable formed of an integer N of simple variables of the same type, which are called the components of the array. The number of components N is then the dimension of the array.

• In making the connection with mathematics, we still say that "A is a vector of dimension N"

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One-dimensional arraysExample• For example, the declaration:

int DAYS [12] = {31,28,31,30,31,30,31,31,30,31,30,31};

defines an array of type int of dimension 12.

The 12 components are initialized by the respective values 31, 28, 31, ..., 31.

• The first component of the array can be accessed by DAYS [0], the second component by DAYS [1],. . . , at the last component by DAYS [11].

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One-dimensional arraysDeclaration and Storage• <SimpleType> <ArrayName> [<Dimension>];

• Array names are identifiers that must match the known restrictions.

• Examples:

� int A[25]; // array containing 25 integers

� double B[100]; // array containing 100 doubles

� int C[10]; // array containing 10 characters

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One-dimensional arraysDeclaration and Storage• In C, the name of an array is the representative of the address of the first element of the array. The addresses of the other components are calculated (automatically) relative to this address.

• Example:short A [5] = {1200, 2300, 3400, 4500, 5600};

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One-dimensional arraysDeclaration and Storage• If an array consists of N components and if a component needs M bytes in memory, then the array will occupy N * M bytes.

• Example

• Assuming that a variable of the long type occupies 4 bytes (ie, sizeof (long) = 4), for the array T declared by: long T [15];C will reserve N * M = 15 * 4 = 60 bytes in memory.

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One-dimensional arraysInitialization and automatic reservation• When declaring an array, we can initialize the components of the table, by indicating the list of the respective values between braces.

• Examples

� int A[5] = {10, 20, 30, 40, 50};

� float B[4] = {-1.05, 3.33, 87e-5, -12.3E4};

� int C[10] = {1, 0, 0, 1, 1, 1, 0, 1, 0, 1};

• Of course, you must ensure that the number of values in the list matches the size of the array. If the list does not contain enough values for all components, the remaining components are initialized with zero.

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One-dimensional arraysInitialization and automatic reservation• If the dimension is not specified explicitly during initialization, then the computer automatically reserves the necessary number of bytes.

• Examples

� int A[] = {10, 20, 30, 40, 50};==> reservation of 5 * sizeof (int) bytes (in our case: 20 bytes)

� float B[] = {-1.05, 3.33, 87e-5, -12.3E4};==> reservation of 4 * sizeof (float) bytes (in our case: 32 bytes)

� int C[] = {1, 0, 0, 1, 1, 1, 0, 1, 0, 1};==> reservation of 10 * sizeof (int) bytes (in our case: 40 bytes)

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One-dimensional arraysInitialization and automatic reservation• Examples

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One-dimensional arraysAccess to components• By declaring an array by:int A[5];we have defined a table A with five components, which can be accessed by:A[0], A[1], ..., A[4]

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One-dimensional arraysAccess to components• Examples

� MAX = (A[0]> A[1])? A[0]: A[1];

� A[4] * = 2;

• Consider an array T of dimension N:

• In C,

� - access to the first element of the array is done by T[0]

� - access to the last element of the array is done by T[N-1]

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One-dimensional arraysDisplay and assignment• The for structure is particularly suitable for working with arrays. Most

applications can be implemented simply by modifying the standard examples of display and assignment.

• Display the contents of an array

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void main(){

int A[5];int I; /* counter */// assign values here by the userfor (I=0; I<5; I++)

printf("%d ", A[I]);

printf("\n");}

One-dimensional arraysDisplay and assignment

• Assignment with values from the user

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void main(){

int A[5];int I; /* counter */for (I=0; I<5; I++)

scanf("%d", &A[I]);

for (I=0; I<5; I++)printf("%d ", A[I]);

printf("\n");}

Exercise• Write a program that reads the dimension N of an integer array T (maximum dimension: 50 components), fills the array with values entered on the keyboard and displays the array.

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Exercise - continued• Calculate and then display the sum of the elements of the array.

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Exercise - continued• Then delete all occurrences of the value 0 in the table T and tamp the remaining elements. Display the resulting array.

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• Then store the elements of T in the reverse order without using an auxiliary array. Display the resulting array.

• Idea: Swap the elements of the array using two indices that run through the array, starting at the beginning and at the end of the array respectively and meeting in the middle.

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• Then copy all strictly positive components into a second TPOS array and all strictly negative values into a third TNEG array. Display TPOS and TNEG arrays.

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Two-dimensional arrays• In C, a two-dimensional array A is to be interpreted as an array (uni-dimensional) of dimension R, each component of which is a (one-dimensional) array of dimension C.

• We call R the number of rows in the array and C the number of columns in the array. R and C are then the two dimensions of the array. A two-dimensional array therefore contains R * C components.

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Two-dimensional arrays• A two-dimensional array is said to be square, if R is equal to C.

• Comparing with mathematics, we can say that "A is a vector of R vectors of dimension C", or better:

• "A is a matrix of dimensions R and C".

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Two-dimensional arraysExample• Consider a one-dimensional array GRADE to memorize the grades of 20 students in a class in a assignment:int GRADE[20] = {45, 34, ..., 50, 48};

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Two-dimensional arraysExample• To memorize students' grades in 10 assignments, we can collect several of these one-dimensional tables in a two-dimensional NOTES table:int GRADES[10] [20] = {{45, 34, ..., 50, 48},

{39, 24, ..., 49, 45},... ... ...

{40, 40, ..., 54, 44}};

• In one line we find the notes of all students in a homework assignment. In a column, we find all the notes of a student.

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Two-dimensional arrays Declaration and Storage• <SimpleType> <TableName> [<DimRow>] [<DimCol>];

• Examples

� int A[10][10];

� double B[2][20];

� chariot D[15][40];

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Two-dimensional arrays Declaration and Storage• As with one-dimensional arrays, the name of a array is the representative of the address of the first array element (ie the address of the first row of the array). The components of a two-dimensional array are stored line by line in memory.

• Example: Storing a two-dimensional arrayshort A [3] [2] = {{1, 2},

{10, 20},{100, 200}};

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27An array of R and C dimensions, consisting of components each of which

needs M bytes, will occupy R * C * M bytes in memory.

Two-dimensional arrays Initialization and automatic reservation• When declaring an array, we can initialize the components of the array, by indicating the list of the respective values between braces. Inside the list, the components of each row of the table are once again enclosed in braces. To improve the readability of the programs, we can indicate the components in several lines.

• Examples

int A[3][10] ={{ 0,10,20,30,40,50,60,70,80,90},

{10,11,12,13,14,15,16,17,18,19},

{ 1,12,23,34,45,56,67,78,89,90}};

float B[3][2] = {{-1.05, -1.10 },

{86e-5, 87e-5 },

{-12.5E4, -12.3E4}}; Alg

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Two-dimensional arrays Initialization and automatic reservation

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Two-dimensional arrays Initialization and automatic reservation• If the number of rows R is not specified explicitly during initialization, the computer automatically reserves the necessary number of bytes.

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Two-dimensional arrays Access to components• The elements of an array of dimensions R and C are as follows:

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/ \| A[0][0] A[0][1] A[0][2] . . . A[0][C-1] || A[1][0] A[1][1] A[1][2] . . . A[1][C-1] || A[2][0] A[2][1] A[2][2] . . . A[2][C-1] || . . . . . . . . . . . . . . . || A[R-1][0] A[R-1][1] A[R-1][2] . . . A[R-1][C-1] |\ /

Two-dimensional arraysDisplay and assignment

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void main(){

int A[5][10];int I,J;/* for each row... */for (I=0; I<5; I++) {

/* ... Consider each component*/for (J=0; J<10; J++)

printf("%7d", A[I][J]);printf("\n");

}}

Two-dimensional arraysDisplay and assignment

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void main(){

int A[5][10];int I,J;/* for each row... */for (I=0; I<5; I++) {

/* ... Consider each component*/for (J=0; J<10; J++)

scanf("%d", &A[I][J]);printf("\n");

}}

Exercise• Write a program that reads the R and C dimensions of a two-dimensional array T of type int (maximum dimensions: 50 rows and 50 columns). Fill in the array with values entered on the keyboard and display the table and the sum of all its elements.

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Exercise• Write a program that reads the R and C dimensions of a two-dimensional array T of type int (maximum dimensions: 50 rows and 50 columns). Fill in the array with values entered on the keyboard and display the table and the sum of each row and column using only one auxiliary variable for the sum.

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Exercise• Write a program that transfers a two-dimensional array R and C (maximum dimensions: 10 rows and 10 columns) into a one-dimensional array V of size R * C.

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/ \| a b c d | / \| e f g h | ==> | a b c d e f g h i j k l || i j k l | \ /\ /

AttentionThe following exercises are to be treated according to the same scheme:

• Variable declaration

• Reading the dimensions of the arrays

• Reading array data

• Treatments

• Display results

Choose the appropriate maximum dimensions for the tables.

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ExerciseScalar product of two vectors• Write a program that calculates the scalar product of two integer vectors U and V (of the same dimension).

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/ \ / \| 3 2 -4 | * | 2 -3 5 | = 3*2+2*(-3)+(-4)*5 = -20\ / \ /

ExerciseCalculation of a polynomial of degree N• Calculate, for a given float value X, the numeric value of a polynomial of degree n: � � = �� + �� + ⋯ + �� + �

• The values of coefficients An, ..., A0 will be entered by keyboard and stored in a float array A of dimension n + 1.

� Use the pow() function for calculation.

� Use the Horner scheme that avoids the exponentiation operations:

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ExerciseMaximum and minimum values of an array• Write a program that determines the largest and the smallest value in an array of integers A.

• Next, display the value and the position of the maximum and minimum values.

• If the array contains several maxima or minima, the program will retain the position of the first maximum or minimum encountered.

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ExerciseInsert a value into a sorted array• An array A of dimension N + 1 contains N integer values sorted in ascending order; the (N + 1) th value is undefined.

• Insert a given VAL value read from the keyboard in the array A to obtain a sorted array of N + 1 values.

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ExerciseFinding a value in an array• Search an array of integers for a VAL value entered on the keyboard. Display the position of VAL if it is in the array, otherwise display a corresponding message. The POS value that is used to memorize the position of the value in the array, will have the value -1 as long as VAL was not found.

• Implement two versions:

• a) Sequential research: Successively compare the values of the array with the given value.

• b) Dichotomous search ('binary search')� Condition: Table A must be sorted

� Compare the searched number to the value in the middle of the table,

� if there is a tie or if the table is exhausted, stop the treatment with a corresponding message.

� If the value sought precedes the current value of the table, continue the search in the half-table to the left of the current position.

� if the desired value follows the current value of the table, continue the search in the half-table to the right of the current position.

� Write the program for the case where the array A is sorted in ascending order.

• Question: What is the advantage of dichotomous research? Explain briefly.

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ExerciseMerge two sorted arrays• We have two arrays A and B (of respective dimensions N and M), sorted in ascending order. Merge the elements of A and B into a third MERGE array sorted in ascending order.

• Method: Use three indexes IA, IB and IMERGE. Compare A[IA] and B[IB]; replace FUS[IMERGE] with the smaller of the two elements; advance in the table MERGE and in the table that contributed its element. When one of the two arrays A or B is exhausted, simply copy the remaining elements of the other array into the MERGE array.

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ExerciseSort by maximum selection• Sort the elements of an array A in descending order.

• Method: Traverse the array from left to right using the index I. For each element A[I] of the array, determine the PMAX position of the (first) maximum to the right of A[I] and exchange A[I] ] and A[PMAX].

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ExerciseSort by propagation (bubble sort)• Sort the elements of an array A in ascending order.

• Method: Starting again each time at the beginning of the table, we carry out several times the following treatment: We propagate, by successive permutations, the largest element of the array towards the end of the array (like a bubble that goes back to the surface of a liquid).

• Implement the algorithm considering that:

� The part of the table (on the right) where there were no permutations is sorted.

� If no swapping has occurred, the array is sorted.

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ExerciseDistinct elements in an array• Let X be an array containing n positive integers. Write a program that

creates:

� an array Dist containing the distinct elements of X

� An array Effec containing the number of occurrence of each distinct element of X

� An integer k containing the number of distinct elements of X.

• Example:

� X={4, 5, 5, 7, 4, 2, 3, 2}

� Dist={4, 5, 7, 2, 3}

� Effec={2, 2, 1, 2, 1}

� K=5

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ExerciseDelete an element from an array• Write a program that deletes the first occurrence of an element in an array

• Example

� A={0, 1, 0, 1, 2, 3, 1}

� Delete the first occurrence of 0

� A={1, 0, 1, 2, 3, 1}

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ExerciseZeroing the main diagonal of a matrix• Write a program that zeros the elements of the main diagonal of a given square matrix A.

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ExerciseUnit Matrix• Write a program that constructs and displays a unitary square matrix U of dimension N. A unitary matrix is a matrix, such that:

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/ 1 if i=j

uij = |

\ 0 if i≠j

ExerciseTransposition of a matrix• Write a program that performs the transposition tA of a matrix A of dimensions N and M into a matrix of dimensions M and N.

• a) The transposed matrix will be stored in a second matrix B which will then be displayed.

• b) The matrix A will be transposed by permutation of the elements.

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/ \ / \tA = t | a b c d | = | a e i |

| e f g h | | b f j || i j k l | | c g k |\ / | d h l |

\ /

ExerciseMultiplication of a matrix by a real• Write a program that multiplies an A matrix by a real X.

• a) The result of the multiplication will be memorized in a second matrix B which will then be displayed.

• b) The elements of matrix A will be multiplied by X.

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/ \ / \| a b c d | | X*a X*b X*c X*d |

X * | e f g h | = | X*e X*f X*g X*h || i j k l | | X*i X*j X*k X*l |\ / \ /

ExerciseAddition of two matrices• Write a program that performs the addition of two matrices A and B of the same dimensions N and M.

• a) The result of the addition will be stored in a third matrix C which will then be displayed.

• b) Matrix B is added to A.

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/ \ / \ / \| a b c d | | a' b' c' d' | | a+a' b+b' c+c' d+d' || e f g h | + | e' f' g' h' | = | e+e' f+f' g+g' h+h' || i j k l | | i' j' k' l' | | i+i' j+j' k+k' l+l' |\ / \ / \ /

ExerciseMultiplication of two matrices• By multiplying a matrix A of dimensions N and M with a matrix B of dimensions M and P, a matrix C of dimensions N and P is obtained:A (N, M) * B (M, P) = C (N, P)

• The multiplication of two matrices is done by multiplying the components of the two matrices rows by columns:

• Write a program that performs the multiplication of two matrices A and B. The result of the multiplication will be stored in a third matrix C which will then be displayed.

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/ \ / \ / \| a b c | | p q | | a*p + b*r + c*t a*q + b*s + c*u || e f g | * | r s | = | e*p + f*r + g*t e*q + f*s + g*u || h i j | | t u | | h*p + i*r + j*t h*q + i*s + j*u || k l m | \ / | k*p + l*r + m*t k*q + l*s + m*u |\ / \ /

ExercisePascal's triangle• Write a program that builds the PASCAL triangle of degree N and stores it in a square matrix P of dimension N + 1.

• Method:

� Calculate and display only the values up to the main diagonal (included). Limit the degree to enter by the user to 13.

� Construct the triangle line by line:

� Initialize the first element and the element of the diagonal to 1.

� Calculate the values between the initialized elements from left to right using the relation: Pi, j = Pi-1, j + Pi-1, j-1

• Example: Pascal's triangle of degree 6:

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n=0 1

n=1 1 1

n=2 1 2 1

n=3 1 3 3 1

n=4 1 4 6 4 1

n=5 1 5 10 10 5 1

n=6 1 6 15 20 15 6 1

ExerciseSearch for saddle point

• Search in a given matrix A for elements that are both a maximum on their row and a minimum on their column. These elements are called saddle points. Display positions and values of all found saddle points.

• Examples: The underlined elements are saddle points:

• Method: Create two auxiliary MAX and MIN matrices of the same dimensions as A, such as:

• �� , � = �1 �� , � �� 0 ��ℎ��• �� , � = �1 �� , � �� !�� 0 ��ℎ��

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/ \ / \ / \ / \| 1 8 3 4 0 | | 4 5 8 9 | | 3 5 6 7 7 | | 1 2 3 || | | 3 8 9 3 | | 4 2 2 8 9 | | 4 5 6 || 6 7 2 7 0 | | 3 4 9 3 | | 6 3 2 9 7 | | 7 8 9 |\ / \ / \ / \ /

i1101 algorithm & imperative programmingantoun.me/teaching/2018-2019/i1101/ch6.pdf ·...

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