1
Iteration
Chapter 4Spring 2007CS 101Aaron Bloomfield
2
Java looping Options
while do-while for
Allow programs to control how many times a statement list is executed
33
Averaging valuesAveraging values
4
Averaging Problem
Extract a list of positive numbers from standard input and produce their average Numbers are one per line A negative number acts as a sentinel to indicate that
there are no more numbers to process
Observations Cannot supply sufficient code using just assignments and
conditional constructs to solve the problem Don’t how big of a list to process
Need ability to repeat code as needed
5
Averaging Algorithm
Prepare for processing Get first input While there is an input to process do {
Process current input Get the next input
} Perform final processing
6
Averaging Problem
Extract a list of positive numbers from standard input and produce their average Numbers are one per line A negative number acts as a sentinel to indicate that
there are no more numbers to process
Sample runEnter positive numbers one per line.Indicate end of list with a negative number.4.50.51.3-1Average 2.1
public class NumberAverage {// main(): application entry pointpublic static void main(String[] args) {
// set up the input
// prompt user for values
// get first value
// process values one-by-onewhile (value >= 0) {
// add value to running total// processed another value// prepare next iteration - get next value
}
// display resultif (valuesProcessed > 0)
// compute and display averageelse
// indicate no average to display}
}
int valuesProcessed = 0;double valueSum = 0;
// set up the input
Scanner stdin = new Scanner (System.in);
// prompt user for values
System.out.println("Enter positive numbers 1 per line.\n" + "Indicate end of the list with a negative number.");
// get first value
double value = stdin.nextDouble();
// process values one-by-one
while (value >= 0) {valueSum += value;++valuesProcessed;value = stdin.nextDouble();
}
// display result
if (valuesProcessed > 0) {double average = valueSum / valuesProcessed;System.out.println("Average: " + average);
} else {System.out.println("No list to average");
}
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Program DemoProgram Demo
NumberAverage.javaNumberAverage.java
10
While syntax and semantics
Logical expression thatdetermines whether Action
is to be executed
while ( Expression ) Action
Action is either a singlestatement or a statement
list within braces
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While semantics for averaging problem
// process values one-by-onewhile ( value >= 0 ) {
// add value to running totalvalueSum += value;
// we processed another value++valueProcessed;
// prepare to iterate – get the next inputvalue = stdin.nextDouble();
}
Test expression is evaluated at thestart of each iteration of the loop.
If test expression is true, these statementsare executed. Afterward, the test expression
is reevaluated and the process repeats
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While Semantics
Expression
Action
true false
Expression isevaluated at the
start of eachiteration of the
loop
If Expression istrue, Action is
executed If Expression isfalse, program
executioncontinues with
next statement
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int valuesProcessed = 0;double valueSum = 0;
double value = stdin.nextDouble();
while (value >= 0) {valueSum += value;++valuesProcessed;value = stdin.nextDouble();
}
if (valuesProcessed > 0) {double average = valueSum / valuesProcessed;System.out.println("Average: " + average);
}else {
System.out.println("No list to average");}
int valuesProcessed = 0;double valueSum = 0;
double value = stdin.nextDouble();
while (value >= 0) {valueSum += value;++valuesProcessed;value = stdin.nextDouble();
if (valuesProcessed > 0) {double average = valueSum / valuesProcessed;System.out.println("Average: " + average);
Execution Trace
Suppose input contains: 4.5 0.5 1.3 -1
0valuesProcessed
valueSum 0
value 4.5
Suppose input contains: 4.5 0.5 1.3 -1
4.5
1
Suppose input contains: 4.5 0.5 1.3 -1
0.5
5.0
2
1.3
6.3
Suppose input contains: 4.5 0.5 1.3 -1
3
-1
Suppose input contains: 4.5 0.5 1.3 -1
average 2.1
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What do these pictures What do these pictures mean?mean?
Light beerLight beer Dandy lionsDandy lions Assaulted Assaulted
peanutpeanut EggplantEggplant Dr. PepperDr. Pepper Pool tablePool table Tap dancersTap dancers Card sharkCard shark King of popKing of pop I PodI Pod Gator aideGator aide Knight mareKnight mare Hole milkHole milk
1515
Converting text to lower caseConverting text to lower case
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Converting text to strictly lowercasepublic static void main(String[] args) {
Scanner stdin = new Scanner (System.in);
System.out.println("Enter input to be converted:");
String converted = "";
while (stdin.hasNext()) { String currentLine = stdin.nextLine(); String currentConversion =
currentLine.toLowerCase(); converted += (currentConversion + "\n");}
System.out.println("\nConversion is:\n" + converted);
}
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Sample run
A Ctrl+z wasentered. It is theWindows escape
sequence forindicatingend-of-file
An empty linewas entered
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Program DemoProgram Demo
LowerCaseDisplay.javaLowerCaseDisplay.java
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Program trace
public static void main(String[] args) {
Scanner stdin = new Scanner (System.in);
System.out.println("Enter input to be converted:");
String converted = "";
while (stdin.hasNext()) { String currentLine = stdin.nextLine(); String currentConversion =
currentLine.toLowerCase(); converted += (currentConversion + "\n");}
System.out.println("\nConversion is:\n" + converted);
}
public static void main(String[] args) {
Scanner stdin = new Scanner (System.in);
System.out.println("Enter input to be converted:");
String converted = "";
while (stdin.hasNext()) { String currentLine = stdin.nextLine(); String currentConversion =
currentLine.toLowerCase(); converted += (currentConversion + "\n");}
System.out.println("\nConversion is:\n" + converted);
}
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Program trace
Representation of lower caseconversion of current input line
converted += (currentConversion + "\n");
The append assignment operator updates the representationof converted to include the current input line
Newline character is neededbecause method nextLine()
"strips" them from the input
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Another optical illusionAnother optical illusion
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Loop Design & Reading From Loop Design & Reading From a Filea File
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Loop design Questions to consider in loop design and analysis
What initialization is necessary for the loop’s test expression?
What initialization is necessary for the loop’s processing?
What causes the loop to terminate?
What actions should the loop perform?
What actions are necessary to prepare for the next iteration of the loop?
What conditions are true and what conditions are false when the loop is terminated?
When the loop completes what actions are need to prepare for subsequent program processing?
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Reading a file Background
Same Scanner class!
Scanner fileIn = new Scanner (new File (filename) );
The File class allows access to filesIt’s in the java.io package
filename is a String
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Reading a file Class File
Allows access to files (etc.) on a hard drive
Constructor File (String s) Opens the file with name s so that values can be
extracted Name can be either an absolute pathname or a pathname
relative to the current working folder
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Reading a file
Scanner stdin = new Scanner (System.in);
System.out.print("Filename: ");String filename = stdin.nextLine();
Scanner fileIn = new Scanner (new File (filename));
String currentLine = fileIn.nextLine();
while (currentLine != null) {System.out.println(currentLine);
currentLine = fileIn.nextLine();}
Scanner stdin = new Scanner (System.in);
System.out.print("Filename: ");String filename = stdin.nextLine();
Scanner fileIn = new Scanner (new File (filename));
String currentLine = fileIn.nextLine();
while (currentLine != null) {System.out.println(currentLine);
currentLine = fileIn.nextLine();}
Set up standard input streamDetermine file nameSet up file streamProcess lines one by oneGet first lineMake sure got a line to processDisplay current lineGet next lineMake sure got a line to processIf not, loop is doneClose the file stream
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Today’s demotivatorsToday’s demotivators
2929
The For statementThe For statement
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The For Statement
currentTerm = 1;
for ( int i = 0; i < 5; ++i ) {System.out.println(currentTerm);currentTerm *= 2;}
After each iteration of thebody of the loop, the updateexpression is reevaluated
The body of the loop iterateswhile the test expression is
trueint
Initialization stepis performed onlyonce -- just prior
to the firstevaluation of thetest expression
The body of the loop displays thecurrent term in the number series.It then determines what is to be thenew current number in the series
ForExpr
Action
true false
ForInit
ForUpdate
Evaluated onceat the beginning
of the forstatements's
executionThe ForExpr is
evaluated at thestart of each
iteration of theloop
If ForExpr is true,Action isexecuted
After the Actionhas completed,
thePostExpression
is evaluated
If ForExpr isfalse, program
executioncontinues with
next statement
After evaluating thePostExpression, the next
iteration of the loop starts
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for statement syntax
Logical test expression that determines whether the action and update step areexecuted
for ( ForInit ; ForExpression ; ForUpdate ) Action
Update step is performed afterthe execution of the loop body
Initialization step prepares for thefirst evaluation of the test
expression
The body of the loop iterates wheneverthe test expression evaluates to true
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for vs. while A for statement is almost like a while statement
for ( ForInit; ForExpression; ForUpdate ) Action
is ALMOST the same as:
ForInit;while ( ForExpression ) {
Action;ForUpdate;
}
This is not an absolute equivalence! We’ll see when they are different in a bit
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Variable declaration You can declare a variable in any block:
while ( true ) {int n = 0;n++;System.out.println (n);
}System.out.println (n);
Variable n gets created (and initialized) each time
Thus, println() always prints out 1
Variable n is not defined once while
loop ends
As n is not defined here, this causes
an error
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Variable declaration You can declare a variable in any block:
if ( true ) {int n = 0;n++;System.out.println (n);
}System.out.println (n);
Only difference from last slide
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System.out.println("i is " + i);}
System.out.println("all done");
System.out.println("i is " + i);}
System.out.println("all done");
i is 0i is 1i is 2all done
Execution Tracei 0int i = 0; i < 3; ++ifor ( ) {int i = 0; i < 3; ++i 123
Variable i has gone out of scope – it
is local to the loop
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for vs. while An example when a for loop can be directly translated into a while
loop:
int count;for ( count = 0; count < 10; count++ ) {
System.out.println (count);}
Translates to:
int count;count = 0;while (count < 10) {
System.out.println (count);count++;
}
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for vs. while An example when a for loop CANNOT be directly translated
into a while loop:
for ( int count = 0; count < 10; count++ ) {System.out.println (count);
}
Would (mostly) translate as:
int count = 0;while (count < 10) {
System.out.println (count);count++;
}count IS defined here
count is NOT defined here
only difference
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for loop indexing Java (and C and C++) indexes everything from zero
Thus, a for loop like this:
for ( int i = 0; i < 10; i++ ) { ... }
Will perform the action with i being value 0 through 9, but not 10
To do a for loop from 1 to 10, it would look like this:
for ( int i = 1; i <= 10; i++ ) { ... }
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Nested loops
int m = 2;int n = 3;for (int i = 0; i < n; ++i) {
System.out.println("i is " + i);for (int j = 0; j < m; ++j) {
System.out.println(" j is " + j);}
}i is 0 j is 0 j is 1i is 1 j is 0 j is 1i is 2 j is 0 j is 1
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Nested loops
int m = 2;int n = 4;for (int i = 0; i < n; ++i) {
System.out.println("i is " + i);for (int j = 0; j < i; ++j) {
System.out.println(" j is " + j);}
}
i is 0i is 1 j is 0i is 2 j is 0 j is 1i is 3
j is 0j is 1j is 2
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How well do you understand How well do you understand for loops?for loops?
20% 20% 20%20%20%1.1. Very well! This Very well! This stuff is easy!stuff is easy!
2.2. Fairly well – with Fairly well – with a little review, a little review, I’ll be goodI’ll be good
3.3. Okay. It’s not Okay. It’s not great, but it’s great, but it’s not horrible, not horrible, eithereither
4.4. Not well. I’m Not well. I’m kinda confusedkinda confused
5.5. Not at all. I’m Not at all. I’m soooooo lostsoooooo lost
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From DubaiFrom Dubai
4444
do-while loopsdo-while loops
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The do-while statement Syntax
do Action while (Expression)
Semantics Execute Action If Expression is true then
execute Action again Repeat this process until
Expression evaluates to false
Action is either a single statement or a group of statements within braces
Action
true
false
Expression
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Picking off digits Consider
System.out.print("Enter a positive number: ");int number = stdin.nextInt();do { int digit = number % 10; System.out.println(digit); number = number / 10;} while (number != 0);
Sample behaviorEnter a positive number: 11299211
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Guessing a number This program will allow the user to guess the number the
computer has “thought” of
Main code block:
do {System.out.print ("Enter your guess:
");guessedNumber = stdin.nextInt();count++;
} while ( guessedNumber != theNumber );
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Program DemoProgram Demo
GuessMyNumber.javaGuessMyNumber.java
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while vs. do-while If the condition is false:
while will not execute the action do-while will execute it once
while ( false ) {System.out.println (“foo”);
}
do {System.out.println (“foo”);
} while ( false );
never executed
executed once
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while vs. do-while A do-while statement can be translated into a while
statement as follows:
do {Action;
} while ( WhileExpression );
can be translated into:
boolean flag = true;while ( WhileExpression || flag ) {
flag = false;Action;
}
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How well do you understand do-How well do you understand do-while loops?while loops?
20% 20% 20%20%20%1.1. Very well! This Very well! This stuff is easy!stuff is easy!
2.2. Fairly well – with Fairly well – with a little review, a little review, I’ll be goodI’ll be good
3.3. Okay. It’s not Okay. It’s not great, but it’s great, but it’s not horrible, not horrible, eithereither
4.4. Not well. I’m Not well. I’m kinda confusedkinda confused
5.5. Not at all. I’m Not at all. I’m soooooo lostsoooooo lost
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Today’s demotivatorsToday’s demotivators
5454
Loop controlsLoop controls
55
The continue keyword The continue keyword will immediately start the next iteration of the
loop The rest of the current loop is not executed
But the ForUpdate part is, if continue is in a for loop
for ( int a = 0; a <= 10; a++ ) {if ( a % 2 == 0 ) {
continue;}System.out.println (a + " is odd");
}
Output: 1 is odd3 is odd5 is odd7 is odd9 is odd
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The break keyword The break keyword will immediately stop the execution of the loop
Execution resumes after the end of the loop
for ( int a = 0; a <= 10; a++ ) {if ( a == 5 ) {
break;}System.out.println (a + " is less than
five");}
Output: 0 is less than five1 is less than five2 is less than five3 is less than five4 is less than five
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Four HobosFour Hobos
58
Four Hobos An example of a program that uses nested for loops
Credited to Will Shortz, crossword puzzle editor of the New York Times And NPR’s Sunday Morning Edition puzzle person
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Problem Four hobos want to split up 200 hours of work The smart hobo suggests that they draw straws with numbers
on it If a straw has the number 3, then they work for 3 hours on 3
days (a total of 9 hours) The smart hobo manages to draw the shortest straw How many ways are there to split up such work? Which one did the smart hobo choose?
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Analysis We are looking for integer solutions to the formula:
a2+b2+c2+d2 = 200 Where a is the number of hours & days the first hobo
worked, b for the second hobo, etc.
We know the following: Each number must be at least 1 No number can be greater than 200 = 14 That order doesn’t matter
The combination (1,2,1,2) is the same as (2,1,2,1) Both combinations have two short and two long
straws
We will implement this with nested for loops
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Implementationpublic class FourHobos {
public static void main (String[] args) {for ( int a = 1; a <= 14; a++ ) { for ( int b = 1; b <= 14; b++ ) {
for ( int c = 1; c <= 14; c++ ) { for ( int d = 1; d <= 14; d++ ) {
if ( (a <= b) && (b <= c) && (c <= d) ) { if ( a*a+b*b+c*c+d*d == 200 ) {
System.out.println ("(" + a + ", " + b
+ ", " + c + ", " + d + ")");
}}
}}
}}
}
}
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Program DemoProgram Demo
FourHobos.javaFourHobos.java
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Results The output:
(2, 4, 6, 12)(6, 6, 8, 8)
Not surprisingly, the smart hobo picks the short straw of the first combination
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Today’s demotivatorsToday’s demotivators
65
Alternate implementation We are going to rewrite the old code in the inner most for
loop:
if ( (a <= b) && (b <= c) && (c <= d) ) { if ( a*a+b*b+c*c+d*d == 200 ) {
System.out.println ("(" + a + ", " + b+ ", " + c + ", " + d + ")");
}}
First, consider the negation of ( (a <= b) && (b <= c) && (c <= d) )
It’s ( !(a <= b) || !(b <= c) || !(c <= d) ) Or ( (a > b) || (b > c) || (c > d) )
66
Alternate implementation This is the new code for the inner-most for loop:
if ( (a > b) || (b > c) || (c > d) ) {continue;
}if ( a*a+b*b+c*c+d*d != 200 ) {
continue;}System.out.println ("(" + a + ", " + b + ", "
+ c + ", " + d + ")");
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How well do you understand four How well do you understand four hobos?hobos?
20% 20% 20%20%20%1.1. Very well! This Very well! This stuff is easy!stuff is easy!
2.2. Fairly well – with Fairly well – with a little review, a little review, I’ll be goodI’ll be good
3.3. Okay. It’s not Okay. It’s not great, but it’s great, but it’s not horrible, not horrible, eithereither
4.4. Not well. I’m Not well. I’m kinda confusedkinda confused
5.5. Not at all. I’m Not at all. I’m soooooo lostsoooooo lost
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OrnithologyOrnithology NutritionNutrition PeacePeace
AcousticsAcoustics
MathematicsMathematics
LiteratureLiterature
MedicineMedicine
PhysicsPhysics ChemistryChemistry
BiologyBiology
The 2006 Ig Nobel PrizesThe 2006 Ig Nobel PrizesFor explaining why woodpeckers don’t get headachesFor explaining why woodpeckers don’t get headachesFor showing that Kuwaiti dung beetles are finicky eatersFor showing that Kuwaiti dung beetles are finicky eatersFor development of a high-pitched electronic teen-ager For development of a high-pitched electronic teen-ager
repellent (and, later, ring tones)repellent (and, later, ring tones)For experiments to determine why people don’t like the For experiments to determine why people don’t like the
sound of fingernails scraping on a blackboardsound of fingernails scraping on a blackboardFor calculating the number of photos you must take to For calculating the number of photos you must take to
ensure that (almost) nobody in a group will have their ensure that (almost) nobody in a group will have their eyes closedeyes closed
For a report entitled, “Consequences of Erudite Vernacular For a report entitled, “Consequences of Erudite Vernacular Utilized Irrespective of Necessity: Problems with Using Utilized Irrespective of Necessity: Problems with Using Long Words Needlessly.“Long Words Needlessly.“
For a medical case report titled, “"Termination of Intractable For a medical case report titled, “"Termination of Intractable Hiccups with Digital Rectal Massage“Hiccups with Digital Rectal Massage“
For studying why dry spaghetti breaks into multiple piecesFor studying why dry spaghetti breaks into multiple piecesFor a study entitled, “Ultrasonic Velocity in Cheddar Cheese For a study entitled, “Ultrasonic Velocity in Cheddar Cheese
as Affected by Temperature," as Affected by Temperature," For showing that the female malaria mosquito is equally For showing that the female malaria mosquito is equally
attracted to the smells of limburger cheese and human attracted to the smells of limburger cheese and human feetfeet
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3 card poker3 card poker
70
3 Card Poker This is the looping HW from a previous fall The problem: count how many of each type of hand in a 3
card poker game
Standard deck of 52 cards (no jokers) Four suits: spades, clubs, diamonds, hearts 13 Faces: Ace, 2 through 10, Jack, Queen, King
Possible 3-card poker hands Pair: two of the cards have the same face value Flush: all the cards have the same suit Straight: the face values of the cards are in succession Three of a kind: all three cards have the same face value Straight flush: both a flush and a straight
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The Card class A Card class was provided
Represents a single card in the deck
Constructor: Card(int i) If i is in the inclusive interval 1 ... 52 then a card is
configured in the following manner If 1 <= i <= 13 then the card is a club If 14 <= i <= 26 then the card is a diamond If 27 <= i <= 39 then the card is a heart If 40 <= i <= 52 then the card is a spade If i % 13 is 1 then the card is an Ace; If i % 13 is 2, then the card is a 2, and so on.
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Card class methods String getFace()
Returns the face of the card as a String String getSuit()
Returns the suit of the card as a String int getValue()
Returns the value of the card boolean equals(Object c)
Returns whether c is a card that has the same face and suit as the invoking card
String toString() Returns a text representation of the card. You may find
this method useful during debugging.
73
The Hand class A Hand class was (partially) provided
Represents the three cards the player is holding
Constuctor: Hand(Card c1, Card c2, Card c3) Takes those cards and puts them in sorted order
74
Provided Hand methods public Card getLow()
Gets the low card in the hand public Card getMiddle()
Gets the middle card in the hand public Card getHigh()
Gets the high card in the hand public String toString()
We’ll see the use of the toString() method later public boolean isValid()
Returns if the hand is a valid hand (no two cards that are the same)
public boolean isNothing() Returns if the hand is not one of the “winning” hands
described before
75
Hand Methods to Implement The assignment required the students to implement the other
methods of the Hand class We haven’t seen this yet
The methods returned true if the Hand contained a “winning” combination of cards public boolean isPair() public boolean isThree() public boolean isStraight() public boolean isFlush() public boolean isStraightFlush()
76
Class HandEvaluation Required nested for loops to count the total number of each
hand
Note that the code for this part may not appear on the website
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Program DemoProgram Demo
HandEvaluation.javaHandEvaluation.java
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How well do you understand 3-How well do you understand 3-card poker?card poker?
20% 20% 20%20%20%1.1. Very well! This Very well! This stuff is easy!stuff is easy!
2.2. Fairly well – with Fairly well – with a little review, a little review, I’ll be goodI’ll be good
3.3. Okay. It’s not Okay. It’s not great, but it’s great, but it’s not horrible, not horrible, eithereither
4.4. Not well. I’m Not well. I’m kinda confusedkinda confused
5.5. Not at all. I’m Not at all. I’m soooooo lostsoooooo lost
7979
All your base are belong to All your base are belong to usus
Flash animationFlash animation Reference: http://en.wikipedia.org/wiki/All_your_base_are_belong_to_usReference: http://en.wikipedia.org/wiki/All_your_base_are_belong_to_us
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The Halting ProblemThe Halting Problem
82
What’s wrong with this program?
public class LoopsForever {public static void main (String args[]) {
while ( true ) {System.out.println ();
}}
}
Given a more complicated program, how do we tell if it gets stuck in an infinite loop? Such as when an application “hangs”?
83
84
The Halting problem Given a Java program P, and input I
Let P be a filename for a program file on a disk somewhere
Let I be a filename for a file that contains all the input the program takes in
Will the program P with input I ever terminate? Meaning will program P with input I loop forever or halt?
Can a computer program determine this? Can a human?
First shown by Alan Turing in 1936 Before digital computers existed! (I’m ignoring which way he showed it for now)
85
A few notes To “solve” the halting problem means we have a method
Oracle.CheckHalt (String P, String I) Let Oracle be a class that can give lots of (truthful)
answers Oracle.PredictFuture(),
Oracle.GetNextLotteryNumbers(), etc. P is the (filename of the) program we are checking for
halting I is the (filename of the) input to that program
And it will return “loops forever” or “halts” As a boolean: true means “loops forever”, false means
“halts” Note it must work for any (Java) program, not just some
programs Or simple programs
86
Take your best guess – do you Take your best guess – do you think it’s possible to solve the think it’s possible to solve the
halting problem?halting problem?
33% 33%33%1.1. YesYes
2.2. NoNo
3.3. I don’t I don’t understand understand what the what the halting problem halting problem isis
87
Can a human determine if a program halts?
Given a program of 10 lines or less, can a human determine if it halts? Assuming no tricks – the program is completely
understandable And assuming the computer works properly, of course
And we ignore the fact that an int will max out at 4 billion As there are ways we can get around this…
For the sample programs on the next page: Assume that the code is in a proper main() method in a
proper class Assume “…print” stands for “System.out.print”
Likewise for “…println”
88
Halting problem examples: will they halt? First sample program:
...println (“Alan Turing”); ...println (“was a genius”); System.exit();
Second sample program:
for (int n = 0; n < 10; n++) ...println (n); System.exit();
Third sample program
while (true) ...println (“hello world”); System.exit();
Fourth sample program:
int x = 10; while ( x > 0 ) { ...println (“hello world”); x = x + 1; } System.exit();
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Take your best guess – do you Take your best guess – do you think it’s possible to solve the think it’s possible to solve the
halting problem?halting problem?
33% 33%33%1.1. YesYes
2.2. NoNo
3.3. I don’t I don’t understand understand what the what the halting problem halting problem isis
90
Perfect numbers Numbers whose divisors (not including the number) add
up to the number 6 = 1 + 2 + 3 28 = 1 + 2 + 4 + 7 + 14
The list of the first 10 perfect numbers:6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216 The last one was 54 digits!
All known perfect numbers are even; it’s an open (i.e. unsolved) problem if odd perfect numbers exist
Sequence A000396 in OEIS
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Odd perfect number search Will this program ever halt?
int n = 1; // arbitrary-precision integer while (true) { int sumOfFactors = 0; for ( int factor = 1; factor < n; factor++ ) if ( n % factor == 0 ) // factor is a factor of n sumOfFactors = sumOfFactors + factor; if (sumOfFactors == n) then break; n = n + 2; } System.out.exit();
Adapted from http://en.wikipedia.org/wiki/Halting_problem
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Take your best guess – do you Take your best guess – do you think it’s possible to solve the think it’s possible to solve the
halting problem?halting problem?
33% 33%33%1.1. YesYes
2.2. NoNo
3.3. I don’t I don’t understand understand what the what the halting problem halting problem isis
93
Where does that leave us?
If a human can’t figure out how to do the halting problem, we can’t make a computer do it for us
It turns out that it is impossible to write such a CheckHalt() method But how to prove this?
94
CheckHalt()’s non-existence Consider a program P with input I Suppose that a method Oracle.CheckHalt(P,I) exists
Tests if P(I) will either “loop forever” or “halt” A program is a series of bits
And thus can be considered data as well Thus, we can call CheckHalt(P,P)
It’s using the bytes of program P as the input to program P
95
CheckHalt()’s non-existence Consider a new program:
public class Test {public static void main (String args[]) {
if ( Oracle.CheckHalt(“Test.java”, “Test.java”) )// if Test.java loops
foreverSystem.exit(); // then halt
else // else if Test.java haltswhile (true) { } // then loop forever
}}
Do we agree that class Test is a valid program?
96
CheckHalt()’s non-existence A (somewhat condensed)
version of class Test:
public class Test {… main … (String args[]) {
if ( Oracle.CheckHalt (“Test.java”,
“Test.java”) )
System.exit(); elsewhile (true) { } }
}
Two possibilities:
Either class Test halts… Then CheckHalt(Test,Test)
returns true (“loops forever”)…
Which means that class Test loops forever
Contradiction!
Or class Test loops forever… Then CheckHalt(Test,Test)
returns false (“halts”)… Which means that class
Test halts Contradiction!
97
How well do you understand How well do you understand the halting problem?the halting problem?
20% 20% 20%20%20%1.1. Very well! This Very well! This stuff is easy!stuff is easy!
2.2. Fairly well – with Fairly well – with a little review, a little review, I’ll be goodI’ll be good
3.3. Okay. It’s not Okay. It’s not great, but it’s great, but it’s not horrible, not horrible, eithereither
4.4. Not well. I’m Not well. I’m kinda confusedkinda confused
5.5. Not at all. I’m Not at all. I’m soooooo lostsoooooo lost
98
Why do we care about the halting problem?
It was the first algorithm that was shown to not be able to exist by a computer You can prove something exists by showing an example (a
correct program) But it’s much harder to prove that a program can never
exist
First shown by Alan Turing in 1936 Before digital computers existed!
9999
New 2005 demotivatiors!New 2005 demotivatiors!
100100
Not going over any more Not going over any more slides in this slide setslides in this slide set
101101
Triangle countingTriangle counting
102
The programming assignment This was the looping HW from two springs ago
List all the possible triangles from (1,1,1) to (n,n,n) Where n is an inputted number In particular, list their triangle type
Types are: equilateral, isosceles, right, and scalene
103
Sample execution
Enter n: 5
(1,1,1) isosceles equilateral(1,2,2) isosceles(1,3,3) isosceles(1,4,4) isosceles(1,5,5) isosceles(2,2,2) isosceles equilateral(2,2,3) isosceles(2,3,3) isosceles(2,3,4) scalene(2,4,4) isosceles(2,4,5) scalene
(2,5,5) isosceles(3,3,3) isosceles equilateral(3,3,4) isosceles(3,3,5) isosceles(3,4,4) isosceles(3,4,5) right scalene(3,5,5) isosceles(4,4,4) isosceles equilateral(4,4,5) isosceles(4,5,5) isosceles(5,5,5) isosceles equilateral
104104
Program DemoProgram Demo
TriangleDemo.javaTriangleDemo.java
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The Triangle class That semester we went over classes by this homework
So they had to finish the class We will be seeing class creation after spring break
Methods in the class: public Triangle() public Triangle (int x, int y, int z) public boolean isTriangle() public boolean isRight() public boolean isIsosceles() public boolean isScalene() public boolean isEquilateral() public String toString()
106
The TriangleDemo class Contained a main() method that tested all the triangles
Steps required: Check if the sides are in sorted order (i.e. x < y < z)
If not, then no output should be provided for that collection of side lengths
Create a new Triangle object using the current side lengths Check if it is a valid triangle
If it is not, then no output should be provided for that collection of side lengths
Otherwise, indicate which properties the triangle possesses Some side length values will correspond to more than 1
triangle e.g., (3, 3, 3) is both isosceles and equilateral Thus, we can’t assume that once a property is present, the
others are not.
107107
Look at that them there code…Look at that them there code…
TriangleDemo.javaTriangleDemo.java
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How well do you understand How well do you understand triangle counting?triangle counting?
20% 20% 20%20%20%1.1. Very well! This Very well! This stuff is easy!stuff is easy!
2.2. Fairly well – with Fairly well – with a little review, a little review, I’ll be goodI’ll be good
3.3. Okay. It’s not Okay. It’s not great, but it’s great, but it’s not horrible, not horrible, eithereither
4.4. Not well. I’m Not well. I’m kinda confusedkinda confused
5.5. Not at all. I’m Not at all. I’m soooooo lostsoooooo lost
109109
Fibonacci numbersFibonacci numbers
110
Fibonacci sequence Sequences can be neither geometric or arithmetic
Fn = Fn-1 + Fn-2, where the first two terms are 1 Alternative, F(n) = F(n-1) + F(n-2)
Each term is the sum of the previous two terms Sequence: { 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … } This is the Fibonacci sequence
Full formula:
n
nn
nF25
5151)(
111
Fibonacci sequence in nature
1385321
112
Reproducing rabbits You have one pair of rabbits on an island
The rabbits repeat the following: Get pregnant one month Give birth (to another pair) the next month
This process repeats indefinitely (no deaths) Rabbits get pregnant the month they are born
How many rabbits are there after 10 months?
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Reproducing rabbits First month: 1 pair
The original pair Second month: 1 pair
The original (and now pregnant) pair Third month: 2 pairs
The child pair (which is pregnant) and the parent pair (recovering)
Fourth month: 3 pairs “Grandchildren”: Children from the baby pair (now
pregnant) Child pair (recovering) Parent pair (pregnant)
Fifth month: 5 pairs Both the grandchildren and the parents reproduced 3 pairs are pregnant (child and the two new born rabbits)
114
Reproducing rabbits Sixth month: 8 pairs
All 3 new rabbit pairs are pregnant, as well as those not pregnant in the last month (2)
Seventh month: 13 pairs All 5 new rabbit pairs are pregnant, as well as those not
pregnant in the last month (3) Eighth month: 21 pairs
All 8 new rabbit pairs are pregnant, as well as those not pregnant in the last month (5)
Ninth month: 34 pairs All 13 new rabbit pairs are pregnant, as well as those not
pregnant in the last month (8) Tenth month: 55 pairs
All 21 new rabbit pairs are pregnant, as well as those not pregnant in the last month (13)
115
Reproducing rabbits Note the sequence:
{ 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … }
The Fibonacci sequence again
116
Fibonacci sequence Another application:
Fibonacci references from http://en.wikipedia.org/wiki/Fibonacci_sequence
117
Fibonacci sequence As the terms increase, the ratio between successive terms
approaches 1.618
This is called the “golden ratio” Ratio of human leg length to arm length Ratio of successive layers in a conch shell
Reference: http://en.wikipedia.org/wiki/Golden_ratio
618933989.12
15
)(
)1(lim
nF
nFn
118
The Golden Ratio
119
120120
Number countingNumber counting
121
The programming assignment This was the looping HW from last fall
Get an integer i from the user The homework had four parts
Print all the Fibonacci numbers up to i Print all the powers of 2 up to i Print all the prime numbers up to i Time the previous three parts of the code
122
Sample executionInput an integer i: 10
The 10th Fibonacci number is 55Computation took 1 ms
2 3 5 7 11 13 17 19 23 29The 10th prime is 29Computation took 0 ms
The 10th power of 2 is 1024Computation took 6 ms
2 4 8 16 32 64 128 256 512 1024BigInteger: The 10th power of 2 is 1024Computation took 2 ms
123
Background: Prime numbers Remember that a prime number is a number that is ONLY
divisible by itself and 1
Note that 1 is not a prime number! Thus, 2 is the first prime number
The first 10 prime numbers: 2 3 5 7 11 13 17 19 23 29
The easiest way to determine prime numbers is with nested loops
124
How to time your code Is actually pretty easy:
long start = System.currentTimeMillis();// do the computationlong stop = System.currentTimeMillis();long timeTakenMS = stop-start;
This is in milliseconds, so to do the number of actual seconds:
double timeTakenSec = timeTakenMS / 1000.0;
125125
Program DemoProgram Demo
NumberGames.javaNumberGames.java
Note what happens when you enter 100Note what happens when you enter 100 With the Fibonacci numbersWith the Fibonacci numbers With the powers of 2With the powers of 2
126
BigIntegers An int can only go up to 2^31 or about 2*109
A long can only go up to 2^63, or about 9*1018
What if we want to go higher?
2100 = 1267650600228229401496703205376
To do this, we can use the BigInteger class It can represent integers of any size
This is called “arbitrary precision” Not surprisingly, it’s much slower than using ints and
longs
The Fibonacci number part didn’t use BigIntegers That’s why we got -980107325 for the 100th term It “flowed over” the limit for ints – called “overflow”
127
BigInteger usage BigIntegers are in the java.math library
import java.math.*;
To get nn:
BigInteger bigN = new BigInteger (String.valueOf(n));BigInteger biggie = new BigInteger
(String.valueOf(1));for ( int i = 0; i < n; i++ )
biggie = biggie.multiply (bigN);System.out.println (biggie);
128128
Look at that them there code…Look at that them there code…
NumberGames.javaNumberGames.java