Data Structures, Algorithms,

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Data Structures, Algorithms,

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Title: Data Structures, Algorithms,

1
Data Structures, Algorithms, Applications

John Barr

2
What The Course Is About

Data structures is concerned with the
representation and manipulation of data.
All programs manipulate data.
So, all programs represent data in some way.
Data manipulation requires an algorithm.

3
What The Course Is About

Algorithm design methods needed to develop
programs that do the data manipulation.

4
Prerequisites

Java

Asymptotic Complexity

Big Oh, Theta, and Omega notations

5
Web Site

www.ithaca.edu/barr/Student/CS311

6
Assignments

Submission procedures
Assignment guidelines

7
Source Codes

Read download and use instructions.
Must have Java 1.2 or higher.
See ProgramIndex.htm, AllNames.html and other
html files produced by Javadoc for Java codes.
I will use the java 2 JDK. You can develop in
anything, but it must run in the JDK.

8
Grades

25 for problem sets/programs (10)
40 for tests (3 tests see web)
30 for projects
5 participation

9
Organization of Text

Three parts
Part I Chapters 1-4, Background
Part 2 Chapters 5-17, Data Structures
Part 3 Chapters 18-22, Algorithms
Each chapter concepts applications

10
Java

Topics
Packages
Inheritance
Method Overriding
Exceptions
Generic Methods
Interfaces, Abstract Classes

11
Java

Exception Classes
EmptyQueueException
MyInputException
UndefinedMethodException

12
MyInputException

Thrown by methods of class MyInputStream, defined
in the book.
Javas standard input methods throw exceptions
that are subclasses of IOException.
Thus must be declared in the throws clause of all
methods that use std input.
MyInputException is a subclass of
RunTimeException. Do not have to catch these
exceptions.

13
MyInputException

/ This exception is thrown when an input
exception occurs. It is used by
MyInputStream
to convert checked exceptions such as
FileNotFoundException and EOFException
into unchecked runtime exceptions. /

14
MyInputException

package exceptions
public class MyInputException
extends RuntimeException
public MyInputException()
super()
public MyInputException(String message)
super(message)

15
MyInputStream

Catch exceptions of type IOException thrown by
Javas input methods
Throws exceptions of type MyInputException

16
Generic Methods

Uses an interface class
Is not an interface itself
Can accept sub-classes that implement an
interface class.

17
Generic Methods

Not possible for Java primitive types.
Use with subclasses of type Object
Must write wrapper classes for primitive types.

18
Generic Methods

/ Swap the integers ai and aj. /
public static void swap(int a, int i, int
j)
// Don't bother to check that indexes i and
j
// are in bounds. Java will do this and
throw
// an ArrayIndexOutOfBoundsException if i
or
// j is out of bounds.
int temp ai
ai aj
aj temp

19
Generic Methods

/ Generic method to swap the object
references ai and aj. /
public static void swap(Object a, int i,
int j)
Object temp ai
ai aj
aj temp

20
Generic Methods

package misc
public class Abc
public static int abc(int a, int b, int c)
return abbc(ab-c)/(ab)4
public static void main(String args)
System.out.println(abc(2,3,4))

21
Generic Methods

package misc
public class Abc
public static float abc(float a, float b,
float c)
return abbc(ab-c)/(ab)4
public static void main(String args)
System.out.println(abc(2,3,4))

22
Generic Methods

Problem with making a generic abc method
Not all objects can or or /
Need to restrict the objects that the generic
method accepts.

23
Interfaces

A list of 0 or more static final data members
(constants)
And 0 or more method headers with no
implementation.
A class can implement 0 or more interfaces, but
can only extend one class (or one abstract class)

24
Interfaces

Can define methods that have formal parameters
whose data type is an interface
Cannot create instances of an interface
Instances of a class that implements an interface
may be cast into the data type of the interface.

25
The Computable Interface

package utilities
/ Interface to be implemented by all classes
that permit the standard arithmetic
operations. /
public interface Computable
/ _at_return this x /
public Object add(Object x)
/ _at_return this - x /
public Object subtract(Object x)
/ _at_return this x /
public Object multiply(Object x)

26
The Computable Interface (cont)

/ _at_return quotient of this / x /
public Object divide(Object x)
/ _at_return remainder of this / x /
public Object mod(Object x)
/ _at_return this incremented by x /
public Object increment(Object x)
/ _at_return this decremented by x /
public Object decrement(Object x)
/ _at_return the additive zero element /
public Object zero()
/ _at_return the multiplicative identity
element /
public Object identity()

27
Generic Method abc

package misc
import wrappers.
import utilities.
public class GenericAbc
public static Computable abc(Computable a,
Computable b, Computable c)
Computable t (Computable)
a.add(b.multiply(c))
return (Computable) t.add(b.divide(c))
public static void main(String args)
MyInteger x, y, z
x new MyInteger(2)
y new MyInteger(3)
z new MyInteger(4)
System.out.println(abc(x, y, z))

28
The MyComparable Interface

package utilities
/ Interface to be implemented by all classes
that permit comparison between their objects.
Comparable has only one method.
/
public interface MyComparable extends Comparable
/ _at_return -1 if this lt x,
0 if this x,
1 if this gt x /
public int compareTo(Object x)
/ _at_return true iff this lt x /
public boolean lessThan(Object x)
/ _at_return true iff this lt x /
public boolean lessThanOrEqual(Object x)

29
The MyComparable Interface

/ _at_return true iff this gt x /
public boolean greaterThan(Object x)
/ _at_return true iff this gt x /
public boolean greaterThanOrEqual(Object x)
/ _at_return true iff this x /
public boolean equals(Object x)

30
The Operable Interface

/ Marker interface for classes that are
both Computable and Comparable. /
package utilities
public interface Operable extends Computable,
Comparable

31
The Zero Interface

package utilities
/ Interface to be implemented by all classes
that implement provide a zero value and test
for zero. /
public interface Zero
public Object zero()
public boolean equalsZero()

32
The CloneableObject Interface

Cloning vs copying
X new currancy(3.42)
Y x
y.setValue(6.25)
System.out.println(x)
gtgt 6.25
Why?

33
The CloneableObject Interface
X new currancy(3.42)
x
3.42
34
The CloneableObject Interface
X new currancy(3.42)
x
y
Y x
3.42
35
The CloneableObject Interface
X new currancy(3.42)
x
y
Y x
y.setValue(6.25)
3.42
6.25
36
The CloneableObject Interface
X new currancy(3.42) Y x.clone() y.setValue(
6.25) System.out.println(x)
x
x
x
y
y
3.42
3.42
3.42
6.25
3.42
37
The CloneableObject Interface

package utilities
/ Interface to be implemented by all classes
that implement the method clone. /
public interface CloneableObject extends
Cloneable
public Object clone()

38
Wrapper Classes

Built-in in Java
Redefined in book to use books interfaces.

39
Integer Wrapper Class

/ wrapper class for int /
package wrappers
import utilities.
import exceptions.
public class MyInteger
implements Operable, Zero, CloneableObject
// value of the integer
private int value

40
Integer Wrapper Class

// constructor methods
/ MyInteger initialized to theValue /
public MyInteger(int theValue)
value theValue
/ MyInteger initialized to 0 /
public MyInteger()
this(0)
/ MyInteger initialized to s /
public MyInteger(String s)
throws NumberFormatException
value Integer.parseInt(s)

41
Integer Wrapper Class(partial)

/ input from the given input stream /
public static MyInteger input(MyInputStream
stream)
System.out.println("Enter an integer
value")
return new MyInteger(stream.readInteger())
/ make a clone /
public Object clone()
return new MyInteger(value)

42
Integer Wrapper Class

// Computable interface methods
/ _at_return this x /
public Object add(Object x)
return new MyInteger
(value ((MyInteger) x).value)

43
Integer Wrapper Class

// Comparable interface method
/ _at_return -1 if this lt x,
0 if this x,
1 if this gt x /
public int compareTo(Object x)
int y ((MyInteger) x).value
if (value lt y) return -1
if (value y) return 0
return 1

44
Integer Wrapper Class

// override Object.equals
/ _at_return true iff this x /
public boolean equals(Object x)
return value ((MyInteger) x).value

45
Data Types and Methodsas paramters

Example. Class whose instances are 1-D arrays.
Method InputArray inputs n items, creates array
of size n and stores elements into array.
Problem data type of elements is unknown until
run time.

46
Data Types and Methodsas paramters

Solution make the data type a parameter,
theClass.
Data type of theClass is Class (defined in
java.lang)
Obtain a type by
Class typeOfInt int.class
Class typeofMyInteger MyInteger.class

47
Data Types and Methodsas paramters

Other information
Where to get data from (stnd input, file, etc)
How to input a type (e.g., to input an element of
type Currency must read a sign, a dollar value, a
cents value)
Solution
Pass in input stream as parameter
Require each data type implement a public static
method input(MyInputStream stream)

48
Methods as paramters

package misc
import java.lang.reflect.
import wrappers.
import utilities.
public class Array1D
// instance data member
Object a

49
Methods as paramters

To get a reference to the input method of the
desired class use
Java.lang.Class.getMethod(String name, Class
parameterTypes)
Returns an object of type Method (defined in
java.lang.reflect)
Name name of method we are looking for
parameterTypes signature of method Name.

50
Methods as paramters

To invoke the returned method, must use the class
function
Method.invoke(instance, actual parameters)
Use null for first parameter if method invoking
is static.
Second parameter must be an array of parameters.
getMethod and invoke throw exceptions that are
not of a type that is a subclass of
runtimeException, thus must catch or declare a
throws.

51
/ input objects of type theClass and store in
an array / public void inputArray(Class
theClass, MyInputStream stream) try
// get the proper method to be used
to read in the values Class
parameterTypes MyInputStream.class
Method inputMethod theClass.getMethod("input",
parameterTypes) // input number of
elements and create an array of that size
System.out.println("Enter number of elements")
int n stream.readInteger() a
new Object n
52
// input the elements Object
inputMethodArgs stream for (int i
0 i lt n i)
System.out.println("Enter element " (i1))
ai inputMethod.invoke(null,
inputMethodArgs) catch
(Exception e) System.out.println(
e) throw new IllegalArgumentException("A
rray1D.inputArray")
53
Recursion

Recursive (mathematical) functions
Induction
Recursive methods

54
Recursive functionsfactorial f(n) n!

Base f(n) defined directly for some value(s) of
n
Recursive component right side has a parameter
less than n

55
Recursive functions

f(5) 5f(4) 20f(3) 60f(2) 120f(1)

56
Recursive functionsFibonacci numbers

f(4)

f(3)f(2) f(2)f(1)f(1) f(1)f(0)f(1)f(1)
3
57
Induction

Induction base claim is true for one or more
base values of n
Induction hypothesis assume that claim is true
for values of n from base through m, m is
arbitrary number greater than the base
Induction step prove that claim is true for
next value of n (ie, m1)

58
Induction

Induction base show true for n0.
Induction hypothesis assume that claim is true
for m, 0 lt m lt n
Induction step prove for m1
Do it!

59
Induction
60
Induction
61
Recursive Methods

/ _at_return n! /
public static int factorial(int n)
if (n lt 1)
return 1
else
return n factorial(n - 1)

public static Computable recursiveSum(Computable
a, int n)
// Driver for true recursive method rsum.
count // for if-else statement
if (a.length gt 0)
return rSum(a, n)
else return null // no elements to sum
private static Computable rSum(Computable
a, int n)
if (n 0)
count // for conditional and return
return (Computable) a0.zero()
else
count // for conditional, rSum
invocation,
// add, and return

63
public class Permutation / perm(x, 0, n)
outputs all permutations of x0n / public
static void perm(Object list, int k, int m)
// Generate all permutations of listkm.
int i if (k m) // listkm has
one permutation, output it for (i 0 i
lt m i) System.out.print(listi)
System.out.println()
else // listkm has more than one
permutation // generate these recursively
for (i k i lt m i)
MyMath.swap(list, k, i) perm(list,
k1, m) MyMath.swap(list, k, i)

64
Permutations
a
b
c
a
b
c
c
a
a
b
b
c
a
a
b
b
c
c
a
b
c
a
b
c
a
b
b
c
a
c
65
TestingWhy?

Problem formal proofs not feasible

Problem Number of possible inputs to any
program is generally too large to test all of
them.
Result the objective of testing is not
to establish correctness, but to expose the
presence of errors.
66
Testing

Test set must be chosen to expose any errors in
program
Different test sets used to expose different
errors

67
Testing

Black Box. Consider programs function, not the
actual code.
I/O partitioning
Data divided into classes
Data in same class cause qualitatively same
behavior
Data in different classes cause different
behavior
Test set should include at least one input from
each class
Cause-effect graphing

68
Black Box Testing

Example find the roots of a quadratic function
(ie where equation is 0)
Solution

public class QuadraticRoots
/ The quadratic is ax2 bx c. a must be
nonzero.
_at_exception IllegalArgumentException
thrown when coefficient of x2 is zero /
public static void outputRoots(double a,
double b, double c)
if (a 0)
throw new IllegalArgumentException
("Coefficient of x2 must be
nonzero")
double d b b - 4 a c

if (d gt 0) // two real roots
double sqrtd Math.sqrt(d)
System.out.println
("There are two real roots "
(-b sqrtd) / (2 a) " and
"
(-b - sqrtd) / (2 a))
else if (d 0)
// both roots are the same
System.out.println
("There is only one distinct root "
-b / (2 a))
else // complex conjugate roots
System.out.println("The roots
are complex")
System.out.println("The real
part is " -b / (2 a))
System.out.println("The
imaginary part is "
Math.sqrt(-d) / (2 a))

71
Black Box Testing

Three different qualitative behaviors
The roots are complex
The roots are real and distinct
The roots are real and the same
So partition the input into three classes.
Design test data so that each of the three
classes are represented.

72
White Box Testing

Create test set based on an examination of the
code.
Different levels of testing coverage.
Statement coverage. Weakest condition The test
set results in each program statement being
executed at least once.

73
White Box Testing

Statement coverage.
Example quadratic roots program
Test set (0,1,2), (1, -5,6),(1,-8,16),(1,2,5)
causes all statements to be executed.
Test set (0,1,2),(1,-5,6),(1,3,2),(2,5,2) does
not provide statement coverage.

74
White Box Testing

decision coverage. Test set causes each
conditional in the program to take on both true
and false values.
Example roots program

75
White Box Testing

outputRoots program. Three conditionals
a 0
d gt 0
d 0
for decision coverage require at least one set of
test data
for which a 0 is true
and one for which it is false.
for which d gt 0 is true
and one for which it is false
for which d 0 is true
and one for which it is false.

76
White Box Testing

Example max program.
Test set (a,-1), (a,4) with a04
2,4,6,8,9 provides statement coverage, not
decision coverage.
Why?

apositionOfCurrentMax.compareTo(ai) lt 0
never
becomes false

When a04 4,2,6,8,9 get both decision and
statement coverage.

77
/ generic method to find maximum object in
a0n _at_throws IllegalArgumentException
when n lt 0 _at_return position of max element
in a0n / public static int max(Comparable
a, int n) if (n lt 0) throw
new IllegalArgumentException
("MyMath.max Cannot find max of zero elements
") int positionOfCurrentMax 0
for (int i 1 i lt n i) if
(apositionOfCurrentMax.compareTo(ai) lt 0)
positionOfCurrentMax i return
positionOfCurrentMax
78
White Box Testing

Clause coverage. Each clause of each conditional
must take both true and false values.
Clause a Boolean expression that contains no
boolean connector (eg, , ).

79
White Box Testing

Example
if ((C1 C2) (C3 C4) S1
else S2
Decision coverage use one test set that causes
((C1 C2) (C3 C4) to be true, one that
causes it to be false
Clause coverage
Must use a test set that causes each of the Ci to
evaluate to true and each to evaluate to false.

80
White Box Testing

Clause coverage (Strong)
Must use a test set that causes each combination
of clause values to be used
In example requires 16 sets of test data
Note that some combinations may not be possible.

81
White Box Testing

Execution Path Coverage.
Execution path a sequence of statements of a
program in order of their execution.
Example outputRoots has only 4 execution paths.
Lines 1 - 3
Lines 1, 4 - 10
Lines 1, 4, 5, 11 - 14
Lines 1, 4, 5, 11, 15 - 20
Example max has varying number of execution
paths
Depends on n.
When n lt 0, one execution path 1, 2, 3
When n 0, one path 1, 4, 5, 8
When n 1, 2 paths
1, 4, 5, 6, 5, 8
1, 4, 5, 6, 7, 5, 8
When n 3, 4 paths
for n, n gt0 there are 2n paths.

82
White Box Testing

Execution Path Coverage.
Requires test set that causes all execution paths
to be executed.
For outputRoots program statement coverage,
decision coverage, clause coverage, and execution
path coverage are equivalent requirements.
For max program statement coverage, decision
coverage, execution path coverage are different
Decision and clause coverage are same.

83
White Box Testing

Execution Path Coverage. Most demanding.
Test set that results in total execution path
coverage also results in statement and decision
coverage.
My not result in clause coverage.
Often requires an prohibitively large number of
test data.
Not practical.
Minimum requirement for this class
Statement coverage
Must also test for special cases (ie boundary
cases).

84
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