problem solving with loops

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Problem Solving with Loops

Lesson 7

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Overview• Flowchart Symbols • The Loop Logic

Structure• Incrementing/

Decrementing• Accumulating• While/EndWhile• Do Until/Loop

• Repeat/Until• Automatic-

Counter Loop (For)• Nested Loops• Indicators• Algorithm

Instructions • Recursion

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Flowchart Symbols• Decision

– True/False/Else• Process• Assign

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Assign

Process

Decision

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Repetition Structure Logic• Allows the programmer to specify that an

action is to be repeated based on the truth or falsity of some condition.

• Otherwise known as iteration.• Two types of Iteration

– Definite– Indefinite

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Definite Repetition• Known as definite because the

number of iterations to be performed at runtime is known.

• Also known as Counter-controlled Repetition – Uses a variable called a counter to

control the number of times a set of statements should execute.

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Definite ExampleWhile Count < 3

Total = Total + GradeCount = Count + 1

End While

As long as the value in Count is less than 3, the loop will continue to process.

PseudocodeMcManus

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Indefinite Repetition• Known as indefinite because the number

of iterations at runtime is not known before the loop begins executing.– Uses a sentinel value (signal value, a flag

value, or a dummy value) to indicate “end of data entry.”• When using a sentinel

– Choose a sentinel that will not naturally exist within the range of data being used.

– Ex. 999-99-9999 for Social Security Numbers

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Indefinite ExampleWhile TimeCard.SSN <> Sentinel

Obtain time from each card Calculate pay

End While

As long as there are items still remaining on the list the loop will continue.

Pseudocode

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The Looping Logic Structure

• Is one of the three types of program control structures – Sequence– Selection (Decision)– Repetition (Looping or Iteration)

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The Looping Process• Is made up of:

– Initialization– Condition (the test)– Increment (or Decrement)

• The Accumulator, although used in frequently in loops, is NOT part of the looping process

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This is a Test

Question!!!

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Looping Process• The Initialization• set to an initial value

– usually zero but not all the time• Examples:

– Count = 0 – Count = 1– Count = 100

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Looping Process• The Test or Condition

– tested before the start of each loop repetition, called the iteration or pass.

• Examples:– Count < 3 Count = 10– Count <= 3 Count <> 10– Count > 5– Count >= 5

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Looping Process• The Increment (or Decrement)

– updates the variable during each iteration

– must be part of the loop body (usually the last line)

• Examples:– Count = Count + 1 increment– Count = Count – 1 decrement

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The Accumulator• Variables used to store values being

computed in increments during the execution of a loop.

• Not part of the Looping Process– But quite often an integral addition to the process– Very often looks like an increment, which is part of the

looping process• Examples:

– Sum = Sum + 1– TotalGrade = TotalGrade + CurrentGrade

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Common Forms of Loops• The Three Most Common Loop-

control statements: – Pretest Loops:

• the While/End While, • the Do Until/Loop,• the For/Next, and

– Posttest Loops:• the Repeat/Until

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There are lots of other loop variations dependent on the language.

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While/End While • The While/End While statement

– The most versatile of the loops• Can be used for both definite and indefinite

loops– The loop body contains the instructions

to be repeated.– The loop-repetition condition is the

Boolean expression after the reserved word while which is evaluated before each repetition of the loop body.

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While StructureSet counter To 0While counter < final_value

statementsIncrement counter by 1

End While

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?

Init

Stmt

true

false

Stmt

Pseudocode Example

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Three Examples

Using Different Languages

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While ExampleX := 3; How many times does theCount := 0; loop execute?while Count < 3 do begin What is displayed? X := X * 2; 6 Print (X); 12 Count := Count + 1 24 end; {while Count}

Pascal Code Example

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Another While ExampleDim X as IntegerDim Count as IntegerX = 3Count = 0While Count < 3 X = X * 2 Print X Count = Count + 1End While

VB Code ExampleMcManus

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Another While Exampleint X = 3;int Count = 0;While (Count < 3)

{ X = X * 2; cout << X << endl; Print statement Count = Count + 1;}

C++ Code Example

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Looping Examples• Two computer language examples:

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BASIC C++WHILE condition InstructionsEND WHILE

while (condition) { Instructions }

FOR i = Start TO End InstructionsNEXT i

for (initial; conditional; incr) { Instructions }

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Indefinite Examples• A Sentinel-Controlled Loop

–Controlled by a sentinel value.–Examples

end-of-file marker (EOF)end-of-line marker (EOL)999999999 for SSN 999999 for date**

**Be careful with this one!

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Indefinite Examples• A Sentinel-Controlled Loop

Pseudocode TemplateInitialize Sum to 0 Read the first value into counter variable While counter variable is not sentinel do

Add counter variable to SumRead next value into counter variable

End While;

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Another Indefinite Example

• Boolean Flag-Controlled Loops– Executes until the event being

monitored occurs. • A program flag, or flag, is a Boolean variable

whole value (True or False) signals whether a particular event occurs.

• The flag should initially be set to False and reset to True when the event occurs.

• Primarily used when interfacing with devices.

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Another Indefinite Example

• Boolean Flag-Controlled Loops–Pseudocode TemplateInitialize flag to FalseWhile Not flag (While Not False

statements.. Reset flag to True if the event being monitored

occursEnd While; {while}

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Do/Until• Logical Opposite of

the While Loop• Unlike the

While/While-End, the Do/Until tests for falsity.– Should be used when

the question being asked is

more naturally asked in the negative.

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Counter < 5

Init

Stmt

false

true

Stmt

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Do/Until Examplex = 3Count = 3Do Until Count < 1 X = X * 2 Print X Count = Count – 1 ‘DecrementLoop

VB CodeMcManus

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Repeat/Until Structure• Similar to the While/While-End

structure, except:– In the While Loop

• loop-continuation is tested at the beginning of the loop before the body of the loop is performed.

– In the Repeat/Until Loop• loop-continuation is tested after the loop

body is performed, thus executing the loop body at least once.

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Repeat Until Loop• Pseudocode Template

Repeatstatementsincrement counterVariable

Until testCondition• Notice that the

statements are executed before thecondition to end the loop

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counter = 1

counter < 5

True

False

statements

ActionoccursbeforeTest

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The For/Next Loop• Combines all of the Loop Process

components into one statement.• Notes on the Counter:

– Can be used non-destructively but must not be modified (destructively) within the loop body.

– Should be a local variable.• The loop body will not be executed if initial

is greater than the final value, unless the increment value is negative.

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The For/Next Loop• Pseudocode Example

For counter = initialvalue To finalvalue Step 1statements… ‘to increment

Next counter

For counter = finalvalue To initialvalue Step -1statements… ‘to decrement

Next counter

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For Loop Flowchart• Note that the “To” is equivalent

to “while less than or equal to”

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counter = 1

Print counter counter = counter + 1

counter <= 5

True

False

(implicit) (implicit)

(explicit)

For counter = 1 to 5 Step 1 Print counter Next counter

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Nested Loops• Loops can be nested just as IF

statements.– Cannot use the same counter-control

variable for the inner loop as is used for the outer loop.

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x yTrue

False False

True

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Nested Loop TemplateInitialize outer loopWhile outer loop test {while}

statements...

Initialize inner loop While inner loop test {while} Inner loop processing and Update inner loop variable

End While {inner while}

statements... Update outer loop variable

End While {outer while}McManus

Pseudocode Example

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Nested Loop Code Example

Dim outercounter As IntegerDim innercounter As Integeroutercounter = 1

While outercounter <= 3innercounter = 1

While innercounter <= 3 innercounter = innercounter + 1End While ‘innercounter

outercounter = outercounter + 1

End While ‘outercounterMcManus

VB Code

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Nested Loop Code Example

Dim OC As IntegerDim IC As IntegerOC= 1 While OC <= 3

Print OCIC = 1While IC<= 3Print IC IC = IC+ 1End While ‘ICOC = OC + 1

End While ‘OCVB Code

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OC IC

True

False False

True Print ICPrint OC

Outercounter = OCInnercounter = IC

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Loop Invariants• Are statements of the loop’s eventual

goal– Assertions about the characteristics of a

loop that always must be true for a loop to execute properly.

– The assertions are true on loop entry, at the start of each loop iteration, and on exit from the loop.

– They are not necessarily true at each point within the body of the loop.

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Four Special Cases of Loops

• When the loop is skipped entirely (zero iteration loop)

Ex. If Balance <> -100Write Check

End If ???• When the loop body is executed just

once• When the loop executes some

normal number of times• When the loop fails to exit (infinite

loop)McManus

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Watch out for…• Beware of infinite loops• Beware of off-by-one Loop Errors

– These will cause a logic error, but not a run-time error• Executes the loop one too many times• Executes the loop one too few times

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Loop Testing Strategy• Verify the Algorithm

– Test the value of the algorithm • before the loop• during the loop, and • after the loop.

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Recursion

Or a case of twisted tails

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Recursion• Is used in computer science and software

engineering as a:– Means of describing, defining, or specifying

things– Means of designing solutions to problems

(using divide & conquer)

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More info in Notes Section

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Recursion• Occurs when a function calls itself from

within the body of the function• Also known as “procedural iteration”• Classic Examples of Recursion

– Factorial– Fibonacci

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Recursive Definitions• A recursive definition is one where

something is defined in terms of itself

• Almost all algorithms that require looping can be defined iteratively or recursively

• Requires two parts:– Base case (or stopping case)– Recursive step

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Factorial x!• How it works:

If X = 3, the chain of recursive calls would be as follows:

Factorial(3) 3 * Factorial(2) 3 * (2 * Factorial(1) )

Precondition x > 0Postcondition

Returns the product 1 * 2 * 3 * …* x for x > 1

Returns 1 when X is 0 or 1.

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Breaking Recursion Down

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Factorial(3) = 3 * Factorial(2)

Factorial(2) = 2 * Factorial(1)

Factorial(1) = 1 * 1 = 1Factorial(1) = 1 * Factorial(0)

Factorial(2) = 2 * 1 = 2

Factorial(3) = 3 * 2 = 6

Factorial(0) = 1


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VB Factorial ExamplePrivate Function Factorial(ByRef y As Double) _

As Double ‘2nd double defines Factorial If y <= 1 Then Factorial = 1 ' Base case Else Factorial = y * Factorial(y - 1) ' Recursive step End IfEnd FunctionThe Base Case is also known as the Stopping Case.

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Call Pop5 1204 243 62 21


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The Original Fibonacci Problem

• Investigated (in the year 1202) was about how fast rabbits could breed in ideal circumstances.

• Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on.

• The puzzle that Fibonacci posed was...

• How many pairs will there be in one year?

• At the end of the first month, they mate, but there is still one only 1 pair.

• At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.

• At the end of the third month, the original female produces a second pair, making 3 pairs.

• At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs.

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Fibonacci Tree of Rabbits

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FemaleMale


Fibonacci Curve or Golden Spiral

• We can make another picture showing the Fibonacci numbers 1,1,2,3,5,8,13

• If we start with two small squares of size 1 next to each other.

• On top of both of these draw a square of size 2 (=1+1).

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Golden Rectangle & Golden Spiral

• Corners of the Golden Rectangle are joined to form a Golden Spiral, which gets 1.6180339887 times wider with each 90 turn.

• The Spiral occurs frequently in Nature!

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The Fibonacci Sequence• The number of clockwise spirals and

the number of counterclockwise spirals formed by the seeds of certain varieties of flowers

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Sunflower Spirals

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Fibonacci Sequence• Fibonacci Numbers

Each Fibonacci number is the sum of the two preceding Fibonacci numbers. • The Fibonacci series defined recursively:

Fibonacci(0) = 0Fibonacci(1) = 1Fibonacci(2) = 1Fibonacci(n) = Fibonacci(n-1) + Fibonacci(n - 2)

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A Shortened Form:Fib (1) = 1 Fib (2) = 1Fib (n) = Fib (n - 1) + Fib (n - 2)

Fib(3) = Fib(2) + Fib(1) = 1 + 1 = 2Fib(4) = Fib(3) + Fib(2) = 2 + 1 = 3Fib(5) = Fib(4) + Fib(3) = 3 + 2 = 5Fib(6) = Fib(5) + Fib(4) = 5 + 3 = 8

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Golden Ratio• Is an Irrational Number• 1.618039887• Symbol: • Also: • Used in Stock Market manipulations • Used in Nature

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problem solving with loops

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