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Gentle Introduction to Programming

Session 5: Sorting, Searching, Time-Complexity Analysis, Memory Model,

Object Oriented Programming

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Admin.

• Please come on time after the first break• Who goes to the Mathematics' summer introduction course?

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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

Decimal Binary

• We want to print the binary representation of a decimal number

• Examples:• 0 -> 0• 8 -> 1000• 12 -> 1100• 31 -> 11111

Decimal Binary Recursively

• The conversion of a number d from decimal to binary:• 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

14 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

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Exercise 1

Write a program that gets 10 numbers from the user.It then accepts another number and checks to see ifthat number was one of the previous ones.

Example 1:Please enter 10 numbers:1 2 3 4 5 6 7 8 9 10Please enter a number to search for: 8Found it!

Example 2:Please enter 10 numbers:1 2 3 4 5 6 7 8 9 10Please enter a number to search for: 30Sorry, 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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SolutionCompareArrays.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 2 4 5 7 8sort

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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:• 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

0 1 2 3 4 5 6 7 8 9-5 -3 0 4 8 11 22 56 57 97

index

value

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 8 5 4

2 7 8 5 4

2 7 5 8 4

2 7 5 4 8

2 7 5 4 8

2 5 7 4 8

2 5 4 7 8

2 7 5 4 8

2 5 4 7 8

2 4 5 7 8

2 5 4 7 8

2 4 5 7 8

2 4 5 7 8

(done)

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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) = n2Are 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 ComplexityArray of size n

n iterationsi 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 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

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Merge Sort Example

http://en.wikipedia.org/wiki/Merge_sort

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Marge Sort Time Complexity• 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

• A reference is also passed by value• Example: arrays• Objects (?)• This explains why we can change an array’s

content within a function

4 5 6 7 8 9a

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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)

• 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 5y

6

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Memory Image 2

val ar1 = Array(1,2,3)

val ar2 = ar1

ar2(0) = 4

ar1 1 2 3

ar2

4

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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 BlueprintA class is a blueprint of objects

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Class as a BlueprintA 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

• 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 aelse 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• 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

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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