pseudocode when designing an algorithm to solve a problem, pseudocode, can be used. –artificial,...

Pseudocode

• When designing an ALGORITHM to solve a problem, Pseudocode, can be used.– Artificial, informal language used to develop algorithms

– Similar to everyday English

• Not executed on computers – Used to think out program before coding

• Easy to convert into C++ program

– Only executable statements• No need to declare variables

2.4 Control Structures

• Sequential execution– Statements executed in order

• Transfer of control– Next statement executed not next one in sequence

• 3 control structures (Bohm and Jacopini)– Sequence structure

• Programs executed sequentially by default

– Selection structures• if, if/else, switch

– Repetition structures• while, do/while, for

2.4 Control Structures

• Flowchart– Graphical representation of an algorithm

– Special-purpose symbols connected by arrows (flowlines)

– Rectangle symbol (action symbol)• Any type of action

– Oval symbol• Beginning or end of a program, or a section of code (circles)

• Single-entry/single-exit control structures – Connect exit point of one to entry point of the next

– Control structure stacking

2.5 if Selection Structure

• Selection structure– Choose among alternative courses of action

– Pseudocode example: If student’s grade is greater than or equal to 60

Print “Passed”

– If the condition is true• Print statement executed, program continues to next statement

– If the condition is false• Print statement ignored, program continues

– Indenting makes programs easier to read• C++ ignores whitespace characters (tabs, spaces, etc.)


• Translation into C++If student’s grade is greater than or equal to 60

Print “Passed”

if ( grade >= 60 ) cout << "Passed";

• Diamond symbol (decision symbol)– Indicates decision is to be made

– Contains an expression that can be true or false• Test condition, follow path

• if structure – Single-entry/single-exit


• Flowchart of pseudocode statement

true

false

grade >= 60

print “Passed”

A decision can be made on any expression.

zero - false

nonzero - true

Example:

3 - 4 is true

2.6 if/else Selection Structure

• if– Performs action if condition true

• if/else– Different actions if conditions true or false

• Pseudocodeif student’s grade is greater than or equal to 60

print “Passed”else

print “Failed”

• C++ codeif ( grade >= 60 ) cout << "Passed";else cout << "Failed";


• Ternary conditional operator (?:)– Three arguments (condition, value if true, value if false)

• Code could be written:cout << ( grade >= 60 ? “Passed” : “Failed” );

truefalse

print “Failed” print “Passed”

grade >= 60

Condition Value if true Value if false


• Nested if/else structures– One inside another, test for multiple cases

– Once condition met, other statements skippedif student’s grade is greater than or equal to 90

Print “A”

else if student’s grade is greater than or equal to 80

Print “B”else

if student’s grade is greater than or equal to 70 Print “C” else if student’s grade is greater than or equal to 60 Print “D”

else

Print “F”


• Example

if ( grade >= 90 ) // 90 and above cout << "A";else if ( grade >= 80 ) // 80-89 cout << "B";else if ( grade >= 70 ) // 70-79 cout << "C"; else if ( grade >= 60 ) // 60-69 cout << "D";else // less than 60 cout << "F";


• Compound statement– Set of statements within a pair of braces if ( grade >= 60 )

cout << "Passed.\n";else { cout << "Failed.\n"; cout << "You must take this course again.\n";}

– Without braces,cout << "You must take this course again.\n";

always executed

• Block– Set of statements within braces

1.25 Decision Making: Equality and Relational Operators

Standard algebraic equality operator or relational operator

C++ equality or relational operator

Example of C++ condition

Meaning of C++ condition

Relational operators

> > x > y x is greater than y

< < x < y x is less than y

>= x >= y x is greater than or equal to y

<= x <= y x is less than or equal to y

Equality operators

= == x == y x is equal to y

!= x != y x is not equal to y

Values of Relational Expressions

a - b a < b a > b a <= b a >= b

positive 0 1 0 1

zero 0 0 1 1

negative 1 0 1 0

2003 Prentice Hall, Inc. All rights reserved.

Equality Operators Exmaples

Valid

-----------------------------------------

c == ‘A’

k != -2

y == 2 * z – 5

Not Valid

----------------------------------------

a = b // assignment statement

a = = b – 1 // space not allowed

y =! z // this is equivalent to y = (!z)

Numerical Accuracy

• Many decimal numbers cannot be exactly represented in binary by a finite number of bits. Thus testing for exact equality can fail.

Use the technique:|operand1 - operand2| < epsilon

Ex. x/y == 17abs(x/y - 17) < 0.000001

*

Logical Operators

• Negation (unary) !

• Logical and &&

• Logical or ||

*

Logical Operators: Examples

Valid(a < 9) && (b> 7)((a< 6) || (b > 8)) && (c< 7)!(a < b) && c > k3 && (-2 * a + 7)

Not Valida > l && // one operand missinga>9 | | b <m // extra space not alloweda> 9 & b<8 // this is a bitwise operation&b // the address of b

* *

int a = 0, b = 3, c = 1, d =4;

a && !c || d

bF

bF

* * *

Logical Operators: Examples

b T

Logical Operators

Expression Expression Equivalent

!(a == b)

!(a == b || a == c)

!(a == b && c > d)

a != b

a != b && a != c

a != b || c <= d

* * *

Operator Precedence and AssociativityOperators Associativity

() ++(postfix) --(postfix) left to right+ (unary) - (unary) ++ (prefix) -- (prefix) right to left

* / % left to right+ - left to right

< <= > >= left to right== != left to right

&& left to right|| left to right?: right to left

= += -= *= /= etc. right to left, (comma operator) left to right

relational

logical

arithmetic

!not

The Empty Statement

The empty statement is written as a semicolon.

Example:

; // an empty statement

Other statements:

a = b; // an assignment statement

a + b + c; // legal, no useful work done

cout << a() << "\n"; // a function call

Common Errors!

= = means means equalityequality

= used for assignmentused for assignment

FALSE isFALSE is zero

TRUE isTRUE is nonzero

Boolean operators give a Boolean resultBoolean operators give a Boolean result

* ** *

2.16switch Multiple-Selection Structure

• switch– Test variable for multiple values– Series of case labels and optional default caseswitch ( variable ) {

case value1: // taken if variable == value1statementsbreak; // necessary to exit switch

case value2:case value3: // taken if variable == value2 or == valuesstatementsbreak;

default: // taken if variable matches no other cases

statements break;

}


true

false

.

.

.

case a case a action(s) break

case b case b action(s) break

false

false

case z case z action(s) break

true

true

default action(s)

The switchswitch Statement

Syntax switch (expression){case value1:

statement1;break;

case value2:statement2;break;

case valuen:

statementn;break;

default:statement;

}

no no ;;useuse : :

* *


Syntax switchswitch (expression){case value1:

statement1;breakbreak;

case value2:statement2;breakbreak;

case valuen:

statementn;breakbreak;

defaultdefault:statement;

}

no no ;;useuse : :

char let_grd;cout <<“Please type in your grade”<<endl;cin >> let_grd;switch (let_grd){

case ‘A’:cout << “Congratulations!”;break;

case ‘B’:cout << “Good job!”;break;

case ‘C’:cout << “ok, but you can do better!”;break;

cont.


case ‘D’:cout << “Better luck in PMII”;break;

case ‘F’:cout << “ Have fun in summer school!”;break;

default:cout << “You entered an invalid grade.”;

}next statementnext statement

The switch and Breakswitch and Break Statementswitch (let_grd){

case ‘A’:cout << “Congratulations!”;break;

case ‘B’:cout << “Good Job!”;break;

case ‘C’:cout << “OK, but you can do better!”;break;

case ‘D’:cout << “Better luck in PMII!”;break;

case ‘E’:cout << “Have fun in summer school!”;break;

default:cout << “You entered an invalid grade.”;

}

The breakbreak Statement

switch (let_grd){

case ‘A’:case ‘B’: cout << “Good Work”;

break;case ‘C’: cout << “ok!”;

break;case ‘D’:case ‘E’: cout << “Have fun in

summer school!”;}

The breakbreak Statement

switch (let_grd){

case ‘A’:case ‘a’:case ‘B’:case ‘b’: cout << “Good Work”;

break;case ‘C’:case ‘c’: cout << “OK!”;

break;etc.

LOOPING

do {

} while ( )

for (I = 0; I < 5; I++) {

}

while ( ) {

}

2.7 while Repetition Structure

• Repetition structure– Action repeated while some condition remains true

– Psuedocodewhile there are more items on my shopping list

Purchase next item and cross it off my list

– while loop repeated until condition becomes false

• Exampleint product = 2;

while ( product <= 1000 )

product = 2 * product;

2.7 The while Repetition Structure

• Flowchart of while loop

product <= 1000 product = 2 * producttrue

false

Taste of Algorithm Analysis

How many times does this loop run?

int product = 2;

while ( product <= 1000 )

product = 2 * product;

2.8 Formulating Algorithms (Counter-Controlled Repetition)

• Counter-controlled repetition– Loop repeated until counter reaches certain value

• Definite repetition– Number of repetitions known

• Example A class of ten students took a quiz. The grades (integers in

the range 0 to 100) for this quiz are available to you. Determine the class average on the quiz.

2.8 Formulating Algorithms (Counter-Controlled Repetition)

• Pseudocode for example:Set total to zero

Set grade counter to one

While grade counter is less than or equal to tenInput the next gradeAdd the grade into the totalAdd one to the grade counter

Set the class average to the total divided by tenPrint the class average

• Next: C++ code for this example

fig02_07.cpp(1 of 2)

1 // Fig. 2.7: fig02_07.cpp2 // Class average program with counter-controlled repetition.3 #include <iostream>4 5 using std::cout;6 using std::cin;7 using std::endl;8 9 // function main begins program execution10 int main()11 {12 int total; // sum of grades input by user13 int gradeCounter; // number of grade to be entered next14 int grade; // grade value15 int average; // average of grades16 17 // initialization phase18 total = 0; // initialize total19 gradeCounter = 1; // initialize loop counter20


fig02_07.cppoutput (1 of 1)

21 // processing phase22 while ( gradeCounter <= 10 ) { // loop 10 times23 cout << "Enter grade: "; // prompt for input24 cin >> grade; // read grade from user25 total = total + grade; // add grade to total26 gradeCounter = gradeCounter + 1; // increment counter27 }28 29 // termination phase30 average = total / 10; // integer division31 32 // display result33 cout << "Class average is " << average << endl; 34 35 return 0; // indicate program ended successfully36 37 } // end function main

Enter grade: 98

Enter grade: 76

Enter grade: 71

Enter grade: 87

Enter grade: 83

Enter grade: 90

Enter grade: 57

Enter grade: 79

Enter grade: 82

Enter grade: 94

Class average is 81

The counter gets incremented each time the loop executes. Eventually, the counter causes the loop to end.

2.9 Formulating Algorithms (Sentinel-Controlled Repetition)

• Suppose problem becomes: Develop a class-averaging program that will process an arbitrary number of grades each time the program is run

– Unknown number of students

– How will program know when to end?

• Sentinel value– Indicates “end of data entry”

– Loop ends when sentinel input

– Sentinel chosen so it cannot be confused with regular input • -1 in this case


• Top-down, stepwise refinement– Begin with pseudocode representation of top

Determine the class average for the quiz

– Divide top into smaller tasks, list in orderInitialize variables

Input, sum and count the quiz grades

Calculate and print the class average


• Many programs have three phases– Initialization

• Initializes the program variables

– Processing• Input data, adjusts program variables

– Termination• Calculate and print the final results

– Helps break up programs for top-down refinement


• Refine the initialization phaseInitialize variables

goes to

Initialize total to zero

Initialize counter to zero

• ProcessingInput, sum and count the quiz grades

goes to

Input the first grade (possibly the sentinel)

While the user has not as yet entered the sentinel

Add this grade into the running total

Add one to the grade counter

Input the next grade (possibly the sentinel)


• TerminationCalculate and print the class average

goes to

If the counter is not equal to zero

Set the average to the total divided by the counter

Print the average

Else

Print “No grades were entered”

• Next: C++ program


1 // Fig. 2.9: fig02_09.cpp2 // Class average program with sentinel-controlled repetition.3 #include <iostream>4 5 using std::cout;6 using std::cin;7 using std::endl;8 using std::fixed;9 10 #include <iomanip> // parameterized stream manipulators11 12 using std::setprecision; // sets numeric output precision 13 14 // function main begins program execution15 int main()16 {17 int total; // sum of grades18 int gradeCounter; // number of grades entered19 int grade; // grade value20 21 double average; // number with decimal point for average22 23 // initialization phase24 total = 0; // initialize total25 gradeCounter = 0; // initialize loop counter

Data type double used to represent decimal numbers.


26 27 // processing phase28 // get first grade from user29 cout << "Enter grade, -1 to end: "; // prompt for input30 cin >> grade; // read grade from user31 32 // loop until sentinel value read from user33 while ( grade != -1 ) { 34 total = total + grade; // add grade to total35 gradeCounter = gradeCounter + 1; // increment counter36 37 cout << "Enter grade, -1 to end: "; // prompt for input38 cin >> grade; // read next grade39 40 } // end while41 42 // termination phase43 // if user entered at least one grade ...44 if ( gradeCounter != 0 ) {45 46 // calculate average of all grades entered47 average = static_cast< double >( total ) / gradeCounter;48

static_cast<double>() treats total as a double temporarily (casting).

Required because dividing two integers truncates the remainder.

gradeCounter is an int, but it gets promoted to double.



49 // display average with two digits of precision50 cout << "Class average is " << setprecision( 2 )51 << fixed << average << endl;52 53 } // end if part of if/else54 55 else // if no grades were entered, output appropriate message56 cout << "No grades were entered" << endl;57 58 return 0; // indicate program ended successfully59 60 } // end function main

Enter grade, -1 to end: 75








Enter grade, -1 to end: -1

Class average is 82.50

setprecision(2)prints two digits past decimal point (rounded to fit precision).

Programs that use this must include <iomanip>

fixed forces output to print in fixed point format (not scientific notation). Also, forces trailing zeros and decimal point to print.

Include <iostream>

2.10 Nested Control Structures

• Problem statement We are taking a vote amongst 10 students to see if there are

more Giants fans or Patriots fans.

• Notice that– Program processes 10 results

• Fixed number, use counter-controlled loop

– Two counters can be used

– Each test result is 1 or 2• If not 1, assume 2

1

3 #include <iostream>4 5 using std::cout;6 using std::cin;7 using std::endl;8 9 // SUPERBOWL 2008 -- function main begins program execution10 int main()11 {12 // initialize variables in declarations13 int giants = 0; // number of votes for giants14 int patriots = 0; // number of patriots15 int studentCounter = 1; // student counter16 int result; // one exam result17 18 // process 10 students using counter-controlled loop19 while (studentCounter <= 10 ) {20 21 // prompt user for input and obtain value from user22 cout << "Enter result (1 = giants, 2 = patriots): ";23 cin >> result;24

25 // if result 1, increment passes; if/else nested in while26 if ( result == 1 ) // if/else nested in while27 giants++; 28 29 else // if result not 1, increment failures 30 patriots++; 31 32 // increment studentCounter so loop eventually terminates33 studentCounter = studentCounter + 1; 34 35 } // end while36 37 // termination phase; display number of votes38 cout << “Giants " << giants << endl; 39 cout << “Patriots " << patriots << endl;40 41 // print team cheer42 if ( giants > patriots )43 cout << “GO GIANTS " << endl;44 else cout <<“GO PATRIOTS” << endl;45 return 0; // successful termination46 47 } // end function main

2.13Essentials of Counter-Controlled Repetition

• Counter-controlled repetition requires– Name of control variable/loop counter

– Initial value of control variable

– Condition to test for final value

– Increment/decrement to modify control variable when looping


1 // Fig. 2.16: fig02_16.cpp2 // Counter-controlled repetition.3 #include <iostream>4 5 using std::cout;6 using std::endl;7 8 // function main begins program execution9 int main()10 {11 int counter = 1; // initialization12 13 while ( counter <= 10 ) { // repetition condition14 cout << counter << endl; // display counter15 ++counter; // increment16 17 } // end while 18 19 return 0; // indicate successful termination20 21 } // end function main


1

2

3

4

5

6

7

8

9

10

2.13Essentials of Counter-Controlled Repetition

• The declarationint counter = 1;

– Names counter– Declares counter to be an integer

– Reserves space for counter in memory

– Sets counter to an initial value of 1

2.14for Repetition Structure

• General format when using for loopsfor ( initialization; LoopContinuationTest;

increment )

statement

• Examplefor( int counter = 1; counter <= 10; counter++ )

cout << counter << endl;

– Prints integers from one to ten

No semicolon after last statement


1 // Fig. 2.17: fig02_17.cpp2 // Counter-controlled repetition with the for structure.3 #include <iostream>4 5 using std::cout;6 using std::endl;7 8 // function main begins program execution9 int main()10 {11 // Initialization, repetition condition and incrementing 12 // are all included in the for structure header. 13 14 for ( int counter = 1; counter <= 10; counter++ )15 cout << counter << endl; 16 17 return 0; // indicate successful termination18 19 } // end function main


1

2

3

4

5

6

7

8

9

10

2.14for Repetition Structure

• for loops can usually be rewritten as while loopsinitialization;

while ( loopContinuationTest){

statement

increment;

}

• Initialization and increment– For multiple variables, use comma-separated lists

for (int i = 0, j = 0; j + i <= 10; j++, i++)

cout << j + i << endl;



1 // Fig. 2.20: fig02_20.cpp2 // Summation with for.3 #include <iostream>4 5 using std::cout;6 using std::endl;7 8 // function main begins program execution9 int main()10 {11 int sum = 0; // initialize sum12 13 // sum even integers from 2 through 10014 for ( int number = 2; number <= 100; number += 2 ) 15 sum += number; // add number to sum16 17 cout << "Sum is " << sum << endl; // output sum18 return 0; // successful termination19 20 } // end function main

Sum is 2550


• Example upcoming– Program to read grades (A-F)

– Display number of each grade entered

• Details about characters– Single characters typically stored in a char data type

• char a 1-byte integer, so chars can be stored as ints

– Can treat character as int or char• 97 is the numerical representation of lowercase ‘a’ (ASCII)

• Use single quotes to get numerical representation of charactercout << "The character (" << 'a' << ") has the value "

<< static_cast< int > ( 'a' ) << endl;

PrintsThe character (a) has the value 97


1 // Fig. 2.22: fig02_22.cpp2 // Counting letter grades.3 #include <iostream>4 5 using std::cout;6 using std::cin;7 using std::endl;8 9 // function main begins program execution10 int main()11 {12 int grade; // one grade13 int aCount = 0; // number of As14 int bCount = 0; // number of Bs15 int cCount = 0; // number of Cs16 int dCount = 0; // number of Ds17 int fCount = 0; // number of Fs18 19 cout << "Enter the letter grades." << endl20 << "Enter the EOF character to end input." << endl;21


22 // loop until user types end-of-file key sequence23 while ( ( grade = cin.get() ) != EOF ) {24 25 // determine which grade was input26 switch ( grade ) { // switch structure nested in while27 28 case 'A': // grade was uppercase A29 case 'a': // or lowercase a30 ++aCount; // increment aCount31 break; // necessary to exit switch32 33 case 'B': // grade was uppercase B34 case 'b': // or lowercase b35 ++bCount; // increment bCount 36 break; // exit switch37 38 case 'C': // grade was uppercase C39 case 'c': // or lowercase c40 ++cCount; // increment cCount 41 break; // exit switch42

cin.get() uses dot notation (explained chapter 6). This function gets 1 character from the keyboard (after Enter pressed), and it is assigned to grade.

cin.get() returns EOF (end-of-file) after the EOF character is input, to indicate the end of data. EOF may be ctrl-d or ctrl-z, depending on your OS.

Compares grade (an int) to the numerical representations of A and a.

break causes switch to end and the program continues with the first statement after the switch structure.

Assignment statements have a value, which is the same as the variable on the left of the =. The value of this statement is the same as the value returned by cin.get().

This can also be used to initialize multiple variables:a = b = c = 0;


43 case 'D': // grade was uppercase D44 case 'd': // or lowercase d45 ++dCount; // increment dCount 46 break; // exit switch47 48 case 'F': // grade was uppercase F49 case 'f': // or lowercase f50 ++fCount; // increment fCount 51 break; // exit switch52 53 case '\n': // ignore newlines, 54 case '\t': // tabs, 55 case ' ': // and spaces in input56 break; // exit switch57 58 default: // catch all other characters59 cout << "Incorrect letter grade entered."60 << " Enter a new grade." << endl;61 break; // optional; will exit switch anyway62 63 } // end switch64 65 } // end while66

Notice the default statement, which catches all other cases.

This test is necessary because Enter is pressed after each letter grade is input. This adds a newline character that must be removed. Likewise, we want to ignore any whitespace.


67 // output summary of results68 cout << "\n\nTotals for each letter grade are:" 69 << "\nA: " << aCount // display number of A grades70 << "\nB: " << bCount // display number of B grades71 << "\nC: " << cCount // display number of C grades 72 << "\nD: " << dCount // display number of D grades73 << "\nF: " << fCount // display number of F grades74 << endl;75 76 return 0; // indicate successful termination77 78 } // end function main


Enter the letter grades.

Enter the EOF character to end input.

a

B

c

C

A

d

f

C

E

Incorrect letter grade entered. Enter a new grade.

D

A

b

^Z

Totals for each letter grade are:

A: 3

B: 2

C: 3

D: 2

F: 1

2.17do/while Repetition Structure

• Similar to while structure– Makes loop continuation test at end, not beginning

– Loop body executes at least once

• Formatdo {

statement

} while ( condition );

true

false

action(s)

condition



1 // Fig. 2.24: fig02_24.cpp2 // Using the do/while repetition structure.3 #include <iostream>4 5 using std::cout;6 using std::endl;7 8 // function main begins program execution9 int main()10 {11 int counter = 1; // initialize counter12 13 do { 14 cout << counter << " "; // display counter15 } while ( ++counter <= 10 ); // end do/while 16 17 cout << endl;18 19 return 0; // indicate successful termination20 21 } // end function main

1 2 3 4 5 6 7 8 9 10

Notice the preincrement in loop-continuation test.

2.18break and continue Statements

• break statement– Immediate exit from while, for, do/while, switch– Program continues with first statement after structure

• Common uses– Escape early from a loop

– Skip the remainder of switch


1 // Fig. 2.26: fig02_26.cpp2 // Using the break statement in a for structure.3 #include <iostream>4 5 using std::cout;6 using std::endl;7 8 // function main begins program execution9 int main()10 {11 12 int x; // x declared here so it can be used after the loop13 14 // loop 10 times15 for ( x = 1; x <= 10; x++ ) {16 17 // if x is 5, terminate loop18 if ( x == 5 )19 break; // break loop only if x is 520 21 cout << x << " "; // display value of x22 23 } // end for 24 25 cout << "\nBroke out of loop when x became " << x << endl;

Exits for structure when break executed.



26 27 return 0; // indicate successful termination28 29 } // end function main

1 2 3 4

Broke out of loop when x became 5

2.18break and continue Statements

• continue statement– Used in while, for, do/while– Skips remainder of loop body

– Proceeds with next iteration of loop

• while and do/while structure– Loop-continuation test evaluated immediately after the continue statement

• for structure– Increment expression executed

– Next, loop-continuation test evaluated


1 // Fig. 2.27: fig02_27.cpp2 // Using the continue statement in a for structure.3 #include <iostream>4 5 using std::cout;6 using std::endl;7 8 // function main begins program execution9 int main()10 {11 // loop 10 times12 for ( int x = 1; x <= 10; x++ ) {13 14 // if x is 5, continue with next iteration of loop15 if ( x == 5 )16 continue; // skip remaining code in loop body17 18 cout << x << " "; // display value of x19 20 } // end for structure21 22 cout << "\nUsed continue to skip printing the value 5" 23 << endl;24 25 return 0; // indicate successful termination

Skips to next iteration of the loop.



26 27 } // end function main

1 2 3 4 6 7 8 9 10

Used continue to skip printing the value 5

do while for

sum = 0;cnt = 1;cnt = 1;dodo{

sum += cnt;cnt++;

} while (cnt <=n);

sum = 0;cnt = 1;cnt = 1;while (cnt <=n){

sum += cnt;cnt++;

}

sum = 0;for (cnt = 1cnt = 1; cnt <= n; cnt++)

sum += cnt;

2.21Structured-Programming Summary

• Structured programming– Programs easier to understand, test, debug and modify

• Rules for structured programming– Only use single-entry/single-exit control structures

– Rules1) Begin with the “simplest flowchart”

2) Any rectangle (action) can be replaced by two rectangles (actions) in sequence

3) Any rectangle (action) can be replaced by any control structure (sequence, if, if/else, switch, while, do/while or for)

4) Rules 2 and 3 can be applied in any order and multiple times

2.21Structured-Programming Summary

• All programs broken down into– Sequence

– Selection• if, if/else, or switch

• Any selection can be rewritten as an if statement

– Repetition• while, do/while or for• Any repetition structure can be rewritten as a while

statement

Loop Counting

cin >> n1; cin >> n2;For (I=0; I<n1; I++)

for (j=0; j< n2; j++)cout <<“The Scarlet Knights are #1”;

How many times does “The Scarlet Knights are #1” get printed?

n2 * n1

Self review

What does this program do?…int k;cout <<“ Please type in a number”;cin << k;do { cout << “Wake UP in the back!” cout << “\n”; k = k/2; } while (k > 1);

32768 16384 8192 4096 2048 1024 512 256 128 64 32 16 8 4 2 1

Binary: 215 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20

Useful summation formulas and rulesliu1 = 1+1+…+1 = u - l + 1

In particular, lin1 = n - 1 + 1 = n

1in i = 1+2+…+n = n(n+1)/2 n2/2

1in i2 = 12+22+…+n2 = n(n+1)(2n+1)/6 n3/3

0in ai = 1 + a +…+ an = (an+1 - 1)/(a - 1) for any a 1

In particular, 0in 2i = 20 + 21 +…+ 2n = 2n+1 – 1

(ai ± bi ) = ai ± bi cai = cai liuai = limai + m+1iuai

Verification of programs

• Assertion: A statement about a particular condition at a certain point in an algorithm– Preconditions and postconditions are examples of assertions

• Invariant: A condition that is always true at a certain point in an algorithm

• Loop invariant: A condition that is true before, during and after each execution of an algorithm’s loop– Can be used to detect errors before coding is started

It is hard to keep track of what is happening with loops. Loops which don't terminate or terminate without achieving their goal behavior is a common problem in computer programming. Loop invariants help. A loop invariant is a formal statement about the relationship between variables in your program which holds true just before the loop is ever run (establishing the invariant) and is true again at the bottom of the loop, each time through the loop (maintaining the invariant). Here is the general pattern of the use of Loop Invariants in your code:

... // the Loop Invariant must be true here while ( TEST CONDITION ) { // top of the loop ... // bottom of the loop // the Loop Invariant must be true here } // Termination + Loop Invariant = Goal ...

invariant + termination => goal

Called the loop guard

Verification

• Loop invariant (continued)– The invariant for a correct loop is true:

• Initially, after any initialization steps, but before the loop begins execution

• Before every iteration of the loop

• After every iteration of the loop

• After the loop terminates

For (i=0; i<5, i++)

cout <<“hello there” <<endl;

The loop invariant: i >= 0 and i <= 5.

Loop invariant

Example 2

• Write a program to compute the factorial of N (N! = N* N-1 * N-2 ….

Factorial N!

• Break down the problem:– Have a counter J that iterates between 1 and N and a running total

Fact that stores the intermediate values

• What do we want after the loop terminates?– Fact = N!

– Expressing this in terms of our variables: Fact = J! and J = N

– Since our iteration is over J<N, we can see that Fact = J! and J<=N - this will be the invariant for our loop.

Invariant: Fact = J! and J<=N

Does the following code match the invariant?

J=Fact=1;

While (J <= N) {

J++;

Fact = Fact*J;

}

NO!!! Since J = N+1 after. The loop guard has to be changed to while (J<N).

Constructing loops with loop invariants

The loop in the previous slide was developedaccording to the following steps:

1. Come up with a loop strategy for solving the problem.2. Determine the set of variables that is needed for the loop.3. Express the required condition (desired result) at the exit of the4. Use the condition from 3 and property 1 above to determine the

reason for looping and the loop invariant.5. Use the loop invariant from 4 to construct the loop body and the

initialization.

Example 2

IN CLASS ASSIGNMENT:

Problem statement: Given an integer dividend

X > 0 and an integer divisor Y > 0, find the

quotient and the remainder of the integer division of X by Y without using division.

HINT

• Use two extra variables called Remainder and Quotient.

• Think about subtraction

Algorithm

Condition to terminate loop

What do we want at the end?

Is this correct?

Q=1;

R = X;

While (R>=Y) {R = R-Y;Q++;}

NO!! Because at the beginning of the loop X != Q*Y + R but equals:X == 1*Y + X. So we must set Q to Zero!!

Q=0

pseudocode when designing an algorithm to solve a problem, pseudocode, can be used. –artificial,...

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