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Gentle
Introduction to
Programming
Session 5: Sorting, Searching, TimeComplexity Analysis, Memory Model,
Object Oriented Programming
1
Admin.
• Please come on time after the first break
• Who goes to the Mathematics' summer introduction
course?
2
Review
• Recursive vs. Iterative
• Guest lecture: Prof. Benny Chor
• Arrays
• Arrays in memory
• Initialization and usage
• foreach, filter
• Arrays as functions arguments
• Multi-dimensional arrays
• References to array
• Sorting, searching
• Binary search
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Today
• Home work review
• Sorting, searching and time-complexity analysis
• Binary search
• Bubble sort, Merge sort
• Scala memory model
• Object-oriented programming (OOP)
• Classes and Objects
• Functional Objects (Rational Numbers example)
• Home work
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Decimal  Binary
• We want to print the binary representation of a
decimal number
• Examples:
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0 -> 0
8 -> 1000
12 -> 1100
31 -> 11111
Decimal  Binary Recursively
• The conversion of a number d from decimal to
binary:
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If d is 0 or 1 – write d to the left and stop, else:
If d is even, write ‘0’ to the left
If d is odd, write ‘1’ to the left
Repeat recursively with floor(d/2)
Example: d = 14
d
Output at the
end of stage
Action at the end of stage
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0
Insert 0 to left of output; divide d by 2
7
10
Insert 1 to left of output; divide d by 2
and round down
3
110
Insert 1 to left of output; divide d by 2
and round down
1
1110
Insert 1 to left of output; return.
Solution
Dec2Bin.scala
Exercise 1
Write a program that gets 10 numbers from the user.
It then accepts another number and checks to see if
that number was one of the previous ones.
Example 1:
Please enter 10 numbers:
1 2 3 4 5 6 7 8 9 10
Please enter a number to search for: 8
Found it!
Example 2:
Please enter 10 numbers:
1 2 3 4 5 6 7 8 9 10
Please enter a number to search for: 30
Sorry, it’s not there
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Solution
FindNumber.scala
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Exercise 2
• Implement a function that accepts two integer
arrays and returns true if they are equal, false
otherwise. The arrays are of the same size
• Write a program that accepts two arrays of
integers from the user and checks for equality
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Solution
CompareArrays.scala
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Solution (main)
CompareArrays.scala
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Today
• Home work review
• Sorting, searching and time-complexity analysis
• Binary search
• Bubble sort, Merge sort
• Scala memory model
• Object-oriented programming (OOP)
• Classes and Objects
• Functional Objects (Rational Numbers example)
• Home work
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Sort
• We would like to sort the elements in an array
in an ascending order
7 2 8 5 4
sort
2 4 5 7 8
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What is Sorting Good For?
• Finding a number in an array
• Consider a large array (of length n) and
multiple queries on whether a given number
exists in the array (and what is its position in it)
• Naive solution: given a number, traverse the
array and search for it
• Not efficient ~ n/2 steps for each search operation
• Can we do better?
• Sort the array as a preliminary step. Now search can
be performed much faster!
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Binary Search
• Input:
• A sorted array of integers A
• An integer query q
• Output:
• -1 if q is not a member of A
• The index of q in A otherwise
• Algorithm:
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Check the middle element of A
If it is equal to q, return its index
If it is >= q, search for q in A[0,…,middle-1]
If it is < q, search for q in A[middle+1,...,end]
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Example
index
0
1
2
3
4
5
6
7
8
9
value
-5
-3
0
4
8
11
22
56
57
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http://www.youtube.com/watch?v=ZrN6J8No080
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Code – Binary Search
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Code – Usage
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So, How Fast is it?
• Worst case analysis
• Size of the inspected array:
n  n/2  n/4  …..  1
• Each step is very fast (a small constant number
of operations)
• There are log2(n) such steps
• So it takes ~ log2(n) steps per search
• Much faster then ~ n
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Bubble Sort
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Bubble Sort Example
7 2 8 5 4
2 7 5 4 8
2 5 4 7 8
2 4 5 7 8
2 7 8 5 4
2 7 5 4 8
2 5 4 7 8
2 4 5 7 8
2 7 8 5 4
2 5 7 4 8
2 4 5 7 8
2 7 5 8 4
2 5 4 7 8
(done)
2 7 5 4 8
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Another Example
http://www.youtube.com/watch?v=myKlT30nl5Y
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Bubble Sort
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Orders of Growth
• Suppose n is a parameter that measures the size
of a problem (the size of its input)
• R(n) measures the amount of resources needed
to compute a solution procedure of size n
• Two common resources are space, measured
by the number of deferred operations, and time,
measured by the number of primitive steps
The worst-case over all
inputs of size n!
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Orders of Growth
• Want to estimate the “order of growth” of R(n):
R1(n)=100n2
R2(n)=2n2+10n+2
R3(n) = n2
Are all the same in the sense that if we
multiply the input by a factor of 2, the
resource consumption increases by a
factor of 4
Order of growth is proportional to n2
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Summary
• Trying to capture the nature of processes
• Quantify various properties of processes:
• Number of steps a process takes (Time Complexity)
• Amount of Space a process uses (Space Complexity)
• You will encounter these issues in many courses
throughout your studies
• We shall focus on the (intuitive) time complexity
analysis of various sorting algorithms
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Examples
• Find a maximum in a general array
• Find the maximum in a sorted array
• Find the 5th largest element in a sorted array
• Answer n Fibonacci quarries, each limited
by MAX
• Find an element in a general array
• Find an element in a sorted array
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Bubble Sort Time Complexity
Array of size n
n iterations
i iterations
constant
(n-1 + n-2 + n-3 + …. + 1) * const ~ ½ * n2
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The Idea Behind Marge Sort
• A small list will take fewer steps to sort than
a large list
• Fewer steps are required to construct a
sorted list from two sorted lists than two
unsorted lists
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Marge Sort Algorithm
• If the array is of length 0 or 1, then it is
already sorted. Otherwise:
• Divide the unsorted array into two subarrays of about half the size
• Sort each sub-array recursively by reapplying merge sort
• Merge the two sub-arrays back into one
sorted array
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Merge Sort Example
http://en.wikipedia.org/wiki/Merge_sort
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Marge Sort Time Complexity
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If the array is of length 0 or 1, then it is already sorted. Otherwise:
Divide the unsorted array into two sub-arrays of about half the size
Sort each sub-array recursively by re-applying merge sort
Merge the two sub-arrays back into one sorted array
n + 2 * (n/2) + 22 * n/22 + 23 * n/23 + … + 2log(n) * n/2log(n) =
n + n + … + n = n * log(n)
log(n)
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Today
• Home work review
• Sorting, searching and time-complexity analysis
• Binary search
• Bubble sort, Merge sort
• Scala memory model
• Object-oriented programming (OOP)
• Classes and Objects
• Functional Objects (Rational Numbers example)
• Home work
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Passing Arguments to Functions
• When a function is called, arguments’ values
are attached to function’s formal parameters
by order, and an assignment occurs before
execution
• Values are copied to formal parameters
• “Call by value”
• Function’s parameters are defined as vals
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Passing Arguments to Functions
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A reference is also passed by value
Example: arrays
Objects (?)
This explains why we can change an array’s
content within a function
a
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Local Names
• Arguments names do not matter!
• Local variable name hides in-scope variables
with the same name
Different x!
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Everything is an Object (in Scala)
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In Java: primitives vs. objects
In Scala everything is an Object
But: special treatment for primitives
Why do we care?
val x = 5
var y = x
y = 6
?
val ar1 = Array(1,2,3)
val ar2 = ar1
ar2(0) = 4
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?
val x = 5
var y = x
y = 6
val ar1 = Array(1,2,3)
val ar2 = ar1
ar2(0) = 4
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Memory Image 1
val x = 5
var y = x
y = 6
x
5
y
5
6
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Memory Image 2
val ar1 = Array(1,2,3)
val ar2 = ar1
ar2(0) = 4
ar1
1 2 3
4
ar2
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Scala Memory Model
• Based on Java…
• Stack: local variables and arguments, every
function uses a certain part of the stack
• Stack variables “disappear” when scope ends
• Heap: global variables and object, scope
independent
• Garbage Collector
• Partial description
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How to Change a Variable via
Functions?
• The arguments are passed as vals thus can not
be changed
• So how can a method change an outer
variable?
• By its return value
• By accessing heap-based memory (e.g., arrays)
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Today
• Home work review
• Sorting, searching and time-complexity analysis
• Binary search
• Bubble sort, Merge sort
• Scala memory model
• Object-oriented programming (OOP)
• Classes and Objects
• Functional Objects (Rational Numbers example)
• Home work
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Programming in Scala
Chapter 4: Classes and Objects
Chapter 6: Functional Objects
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Singletone Objects
• All programs written so far in this course are
Signletone objects
• File start with the reserved word object
• Contain functions that can be used elsewhere
• Application: singeltone object with a main
function
• (Actually singeltone objects are more then that)
• (Java programmers: think of it as a holder of static
methods)
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A Car
• How would you represent a car?
• Parts / features: 4 wheels, steering wheel, horn,
color,…
• Functionality: drive, turn left, honk, repaint,…
• In Scala???
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Object-Oriented Programming
(OOP)
• Represent problem-domain entities using a
computer language
• When building a software in a specific domain,
describe the different components of the domain
as types and variables
• Thus we can take another step up in abstraction
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Class as a Blueprint
A class is a blueprint of objects
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Class as a Blueprint
A class is a blueprint of objects
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Classes as Data Types
• Classes define types that are a composition of
other types and have unique functionality
• An instance of a class is named an object
• Every instance may contain:
• Data members / fields
• Methods
• Constructors
• Instances are accessed only through reference
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Examples
• String
• Members: all private
• Methods: length, replace, startsWith, substring,…
• Constructors: String(), String(String),…
• http://java.sun.com/j2se/1.5.0/docs/api/java/lang/String.html
• Array
• Members: all private
• Methods: length, filter, update,…
• Constructors: initiate with 1-9 dimensions
• http://www.scala-lang.org/docu/files/api/scala/Array.html
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Car Example
• Members: 4 wheels, steering wheel, horn, color,…
• Every car instance has its own
• Methods: drive, turn left, honk, repaint,…
• Constructors: Car(String color),
Car(Array[Wheels], Engine,…), …
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Today
• Home work review
• Sorting, searching and time-complexity analysis
• Binary search
• Bubble sort, Merge sort
• Scala memory model
• Object-oriented programming (OOP)
• Classes and Objects
• Functional Objects (Rational Numbers example)
• Home work
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Rational Numbers
• A rational number is a number that can be
expressed as a ration n/d (n,d integers, d not 0)
• Examples: 1/2, 2/3, 112/239, 2/1
• Not an approximation
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Specification
• Add, subtract, multiply, divide
• println should work smoothly
• Immutable (result of an operation is a new rational number)
• It should feel like native language support
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Constructing a Rational
• How client programmer will create a new
Rational object?
Class parameters
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Constructing a Rational
• The Scala compiler will compile any code placed in
the class body, which isn’t part of a field or a method
definition, into the primary constructor
?
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Reimplementing toString
• toString method
• A more useful implementation of toString would
print out the values of the Rational’s numerator and
denominator
• override the default implementation
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Usage
• Now we can remove the debug println…
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Checking Preconditions
• Ensure the data is valid when the object is
constructed
• Use require
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Define “add” Method
• Immutable
• Define add:
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Add Fields
• n, d are in scope in the add method
• Access then only on the object on which add
was invoked 
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Test Add, Access Fields
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Self Reference (this)
• Define method lessThan:
• Define method max:
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Auxiliary Constructors
• Constructors other then the primary
• Example: a rational number with a denominator
of 1 (e.g., 5/1  5)
• We would like to do: new Rational(5)
• Auxiliary constructor first action: invoke
another constructor of the same class
• The primary constructor is thus the single point
of entry of a class
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Revised Rational
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Private Fields and Methods
• 66/42 = 11/7
• To normalize divide the numerator and
denominator by their greatest common divisor (gcd)
• gcd(66,42) = 6  (66/6)/(42/6) = 11/7
• No need for Rational clients to be aware of this
• Encapsulation
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Off Topic: Calculate gcd
• gcd(a,b) = g
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a = n * g
b = m * g
gcd(n,m)=1(otherwise g is not the gcd)
a = t * b + r = t * m * g + r  g is a
divisor of r
• gcd(a,b) = gcd(b,a%b)
• The Euclidean algorithm: repeat iteratively:
if (b == 0) return a
else repeat using a  b, b  a%b
• http://en.wikipedia.org/wiki/Euclidean_algorithm
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Correctness
• Example:
gcd(40,24)  gcd(24,16)  gcd(16,8) 
gcd(8,0)  8
• Prove: g
= gcd(a,b) = gcd(b,a%b)= g1
• g1 is a divisor of a ( g1 ≤ g)
• There is no larger divisor of a (
g1 ≥ g)
• ≤ : a = t * b + r  a = t * h * g1 + v * g1
 g1 is a divisor of a
• ≥ : assume g > g1  a = t * b + r  g is a
divisor of b and r  contradiction
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Implementation
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Revised Rational
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Defining Operators
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Why not use natural arithmetic operators?
Replace add by the usual mathematical symbol
Operator precedence will be kept
All operations are method calls
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Revised Rational
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Usage
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Method Overloading
• Now we can add and multiply rational numbers!
• What about mixed arithmetic?
• r * 2 won’t work 
• r * new Rational(2) is not nice 
• Add new methods for mixed addition and
multiplication
• Method overloading
• The compiler picks the correct overloaded
method
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Usage
• The * method invoked is determined in each
case by the type of the right operand
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Revised Rational
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Implicit Conversions
• 2 * r  2.*(r)  method call on 2 (Int) 
Int class contains no multiplication method
that takes a Rational argument 
• Create an implicit conversion that
automatically converts integers to rational
numbers when needed
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Companion Object
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Revised Rational
• Define implicit conversion in Rational.scala,
after defining object Rational
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In Eclipse
• In Rational.scala:
• Companion object
(object Rational)
• Rational class (class
Rational)
• Place the main method
in another file
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Summary
• Customize classes so that they are natural
to use
• fields, methods, primary constructor
• Method overriding
• Self reference (this)
• Define several constructors
• Encapsulation
• Define operators as method
• Method overloading
• Implicit conversions, companion object
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Today
• Home work review
• Sorting, searching and time-complexity analysis
• Binary search
• Bubble sort, Merge sort
• Scala memory model
• Object-oriented programming (OOP)
• Classes and Objects
• Functional Objects (Rational Numbers example)
• Home work
85
Exercise 1
• Read, understand and implement
selection/insertion sort algorithm
• http://en.wikipedia.org/wiki/Selection_sort
• http://en.wikipedia.org/wiki/Insertion_sort
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Exercise 2
• Implement class Complex so it is natural to use
complex numbers
• Examples:
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Want More Exercises?
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Exercise (tough!)
• Read, understand and implement quick sort
algorithm
• http://en.wikipedia.org/wiki/Quicksort
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