This is an advanced programming course in Computer Science that teaches how to design efficient structures and algorithms to process big data and methods to benchmark their performance for large-scale computing. Topics cover data structures such as priority queues, binary trees, tries, and graphs and their applications in constructing algorithms such as sorting, searching, balancing, traversing, and spanning. Advanced topics such as network flow and dynamic programming are also discussed. Throughout this course, students are expected to

• Have a deep conceptual understanding of various data structures and algorithms.

• Implement their conceptual understanding in a programming language.

• Explore the most effective structures and algorithms for given tasks.

• Properly assess the quality of their implementations.

There are topical quizzes and homework assignments that require sufficient skills in Java programming, Git version control, Gradle software project management, and scientific writing. Intermediate-level Java programming is a prerequisite of this course.

Contents

1. Environment Setup

Resources

• Main:

• Test:

Contents

1. Abstraction

Resources

• Main:

• Test:

References

Coding

Right-click on the src/main/java directory under the project and create the package edu.emory.cs.utils. Right-click on the utils package and create the Java class Utils:

• Add Utils.java to git.

• Add the following methods to the Utils class:

Run the program by clicking [Run -> Run]. If you see 5 on the output pane, your program runs successfully.

Testing

Open and add the following configurations (if not already), which would allow you to perform :

Right-click on the directory under the project and create the package . Right-click on the utils package and create the Java class :

• Add UtilsTest.java to Git.

• Add the following method to the UtilsTest class. Make sure to include all imports:

Run the test by clicking [Run -> Run]. If you see the test passed, your unit test runs successfully.

Submission

1. Add the instructors as collaborators in your GitHub repository:

• Jinho Choi: jdchoi77

• Peilin Wu: qualidea1217

• Jeongrok Yu: jeongrok

2. Commit and push the following to your GitHub repository:

3. Submit the URL of your GitHub repository to Canvas.

We are going to create a class called LongInteger inheriting SignedNumeral that can store an indefinite size of an integer value beyond the primitive types such as int and long.

What is so special about primitive data types in Java?

Field

Let us declare the member field digits that is an array of bytes holding the values of this integer:

• L1: LongInteger is passed to specify the generic type T in SignedNumeral.

The i'th dimension of digits is the i'th least significant digit in the integer such that the integer 12345 would be stored as digits = {5, 4, 3, 2, 1}, which makes it convenient to implement the arithmetic methods, add() and multiply().

Is the array of bytes the most efficient way of storing a long integer?

Constructors

Let us define the following three constructors:

• L2: the default constructor that initializes this integer with 0 by calling the constructor in L20.

• L10: a copy constructor that initializes this integer with n.

Do the constructors in L10 and L20 call any constructor in the super class?

Arrays.copyOf() is a referenced by the class type Arrays, not an object. Java provides many classes with static methods that are commonly used (e.g., , ).

Can you call non-static methods or fields in the body of a static method?

Method: set()

Let us define the set() method that takes a string and sets the sign and the value of this integer:

• L1-7: javadoc comments.

  • L4: this method throws if n is null.

When should we use statements over blocks for error handling and vice versa?

Method: add()

Let us override the add() method that calls two helper methods:

• L3-4: adds n to this integer that has the same sign by calling addSameSign().

• L5-6: adds n to this integer that has a different sign by calling addDifferentSign().

The following shows an implementation of addSameSign() based on the simple arithmetic:

• L7-9: creates the byte array result by copying values in this integer.

  • L8: the dimension of result can be 1 more than m after the addition.

What are tradeoffs to make the size of result to be m instead of m+1 and vice versa?

The following shows addDifferentSign() that throws :

The implementation of addDifferentSign() is quite similar to addSameSign() although it involves a few more logics. We will leave this as an .

Method: multiply()

Let us override the multiply() method:

• L4: sets the sign after the multiplication.

• L7-15: multiplies n to this integer:

What is the worst-case complexity of the multiply() method?

Method: main()

Let us create a runnable class called that contains the main method:

Can we define the main method in LongInteger instead without creating LongIntegerRun?

• L2: the parameter args is passed from the command line.

Why does the main method need to be static?

This prints something like the following:

• [: one-dimensional array.

• L: the element of this array is an object.

• java.lang.String

What is the hash code of an object?

Every object implicitly inherits that defines a few member methods including , which gets called automatically by the println() method to retrieve the string representation of this object. We can use the helper method that gives a more readable representation:

How is the Arrays.toString() method implemented?

Since no argument is passed to the main method at the moment, this prints an empty array:

If you set the arguments to 123 -456 using the [Run - Edit Configurations - Program arguments]setting, it prints the following array:

Given those two arguments, we can create two integers:

This prints something like the following, which are returned by a.toString():

How is the toString() method implemented in the Object class?

Method: toString()

To print a more readable representation, we need to override the toString() method in LongInteger:

• L5: provides an efficient way of concatenating different data types into one string.

What are the advantages of using StringBuilder instead of concatenating values with the + operator as follows:

Given the overridden method, the above main method now prints the following:

What are the advantages of overriding toString() instead of creating a new method with the same code, and calling the new method to get the string representation of LongInteger?

Method: compareTo()

Java does not allow , so it is not possible to use logical operators to compare the two integers above, a and b:

In fact, any object that is comparable must inherit the interface as follows:

• L2: LongInteger is passed to Comparable as a generic type.

Is extends always used to inherit a class whereas implements is used to inherit an interface?

The Comparable interface contains one abstract method called that returns a negative value if this object is smaller than n, a positive value if this object is greater than n, and zero if this object equals to n. The compareTo() method must be overridden by the LongInteger class:

The compareAbs() method compares the absolute values of this and n:

• L7: if digits has more dimensions, its absolute value is greater.

• L10-13: compares the significant digits iteratively.

Is it safe to use the same variable i to iterate both digits and n.digits?

Once LongInteger properly inherits Comparable by overriding compareTo(), objects instantiated by this class can be compared using many built-in methods.

• L2: is a specific implementation of the interface .

  • All collections in Java inheriting uses generics.

What is the advantage of declaring list as List instead of ArrayList? What kind of sorting algorithm does Collections.sort() use?

The above code prints the following sorted lists:

What would be the case that needs to distinguish -0 from 0?

Coding

• Create a class called LongIntegerQuiz under the main algebraic package that extends the LongInteger class.

• Override the method so that the add() method in can handle integers with different signs.

Testing

• Create the class under the test package.

• Test the correctness of your LongIntegerQuiz using the unit tests.

• Add more tests for a more thorough assessment if necessary.

Quizzes

1. What is the advantage of using Generics?

2. How do you make the class you define comparable?

3. What is the advantage of overriding member methods in the Object class?

Submission

1. Commit and push everything under the following packages to your GitHub repository:

• Main:

• Test:

2. Submit answers to the above to Canvas.

Test: LongInteger()

Let us create a testing class called LongIntegerTest and make a unit test for the constructors:

• L6: the annotation indicates that this method is used for unit testing.

• L7: methods used for unit testing must be public.

• L8: tests the default constructor

  • : compares two parameters where the left parameter is the expected value and the right parameter is the actual value.

• L12-15: tests the constructor with a string parameter.

• L18-19: tests the copy constructor.

When should we use import static instead of import?

When you run this test, you see a prompt similar to the followings:

Test: multiply()

Let us define another method for testing the multiply() method:

• L11-12: a can hold the integer 152415787517146788750190521 (), which is much larger than the maximum value of long that is .

Test: compareTo()

Let us define a method for testing the compareTo() method:

• : passes if the parameter returns true.

Class

A class is a template to instantiate an object.

What is the relationship between a class and an object?

Let us create a class called Numeral that we want to be a super class of all numeral types:

• L1: packageindicates the name of the package that this class belongs to in a hierarchy.

• L3: public is an .

What are the acceptable access-level modifiers to declare a top-level class?

Let us declare a method, add() , that is an operation expected by all numeral types:

• L4: @param adds a comment about the parameter.

The issue is that we cannot define the methods unless we know what specific numeral type this class should implement; in other words, it is too abstract to define those methods. Thus, we need to declare Numeral as a type of abstract class.

What are the advantages of havingNumeral as a super class of all numeral types?

There are two types of abstract classes in Java, abstract class and interface.

Can an object be instantiated by an abstract class or an interface?

Interface

Let us define Numeral as an :

• L2: abstract method

  • All methods in an interface are public that does not need to be explicitly coded.

Who defines the bodies of the abstract methods?

Let us create a new interface called SignedNumeral that inherits Numeral and adds two methods, flipSign() and subtract():

Can an interface inherit either an abstract class or a regular class?

• L1: extends inherits exactly one class or interface.

• L9: default allows an interface to define a (introduced in Java 8).

Can we call add() that is an abstract method without a body in the default method subtract()?

Although the logic of subtract() seems to be correct, n.flipSign() gives a compile error because n is a type of Numeral that does not include flipSign(), which is defined in SignedNumeral that is a subclass of Numeral.

What kind of a compile error does n.flipSign() cause?

There are three ways of handling this error: , , and .

Casting

The first way is to downcast the type of n to SignedNumeral, which forces the compiler to think that n can invoke the flipSign() method:

This removes the compile error; however, it will likely cause a worse kind, a .

Why is a runtime error worse than a compile error?

, although allowed in Java, is generally not recommended unless there is no other way of accomplishing the job without using it.

How can downcasting cause a runtime error in the above case?

Polymorphism

The second way is to change the type of n to SignedNumeral in the parameter setting:

This seems to solve the issue. Then, what about add() defined in Numeral? Should we change its parameter type to SignedNumeral as well?

It is often the case that you do not have access to change the code in a super class unless you are the author of it. Even if you are the author, changing the code in a super class is not recommended.

Why is it not recommended to change the code in a super class?

How about we the add() method as follows?

• L2: @Override is a predefined type to indicate the method is overridden.

The annotation @Override gives an error in this case because it is not considered an overriding.

What are the criteria to override a method?

When @Override is discarded, the error goes away and everything seems to be fine:

However, this is considered an , which defines two separate methods for add(), one taking n as Numeral and the other taking it as SignedNumeral. Unfortunately, this would decrease the level of abstraction that we originally desired.

What are good use cases of method overriding and overloading?

Generics

The third way is to use , introduced in Java 5:

• L1: T is a generic type that is a subtype of Numeral.

  • A generic type can be recursively defined as T extends Numeral<T>.

Can we define more than one generic type per interface or class?

The generic type T can be specified in a subclass of Numeral:

• L1: T is specified as SignedNumeral.

This would implicitly assign the parameter type of add() as follows:

The issue is that the implementation of add() may require specific features defined in the subclass that is not available in SignedNumeral. Consider the following subclass inheriting SignedNumeral:

• L1: implements inherits .

• L2-6: LongInteger is a regular class, so all declared in the super classes must be defined in this class.

Since the n is typed to SignedNumeral in L6, it cannot call any method defined in LongInteger, which leads to the same issue addressed in the section.

Would the type of n being SignedNumeral an issue for the subtract() method as well?

Thus, SignedNumeral needs to define its own generic type and pass it onto Numeral:

• L1: T is a generic type inheriting SignedNumeral, that implies all subclasses of SignedNumeral.

T can be safely passed onto Numeral because if it is a subclass of SignedNumeral, it must be a subclass of Numeral, which is how T is defined in the Numeral class.

Enum

Let us create an class called to represent the "sign" of the numeral:

• All items in an enum have the scope of and the access-level of public.

• Items must be delimited by , and ends with ;.

The items in the enum can be assigned with specific values to make them more indicative (e.g., +, -):

• L5: final makes this field a constant, not a , such that the value cannot be updated later.

• L8: this points to the created by this constructor.

Why should the member field value be private in the above example?

Note that value in L8 indicates the local parameter declared in the constructor whereas value in L13 indicates the member field declared in L5.

Limit of Interface

In SignedNumeral, it would be convenient to have a member field that indicates the sign of the numeral:

• L2: All member fields of an interface are static and public.

Can you declare a member field in an interface without assigning a value?

Given the sign field, it may seem intuitive to define flipSign() as a default method:

• L3: condition ? A : B is a that returns A if the condition is true; otherwise, it returns B.

Is there any advantage of using a ternary operator instead of using a regular if statement?

Unfortunately, this gives a compile error because sign is a constant whose value cannot be reassigned. An interface is not meant to define so many default methods, which were not even allowed before Java 8. For such explicit implementations, it is better to declare SignedNumeral as an abstract class instead.

Abstract Class

Let us turn into an :

• L9: the default constructor with no parameter.

• L17: another constructor with the sign parameter.

Why calling this(Sign.POSITIVE) in L10 instead of stating this.sign = Sign.POSITIVE?

• L29: abstract indicates that this is an abstract method.

Member fields and methods in an abstract class can be decorated by any modifiers, which need to be explicitly coded.

Is there anything that is not allowed in an abstract class but allowed in a regular class?

In summary, SignedNumeral includes 2 abstract methods, add() inherited from Numeral, and multiply() declared in this class.

Can you define an abstract class or an interface without declaring an abstract method?

Auxiliary Method

Let us create PriorityQueueTest and define testRobustnessAux() that checks the robustness of any PQ inheriting AbstractPriorityQueue:

• L6: the generic type T is defined specifically for this method such that the priority queue pq, the list keys, and the comparator comp take the same type T that is comparable.

• L7: any collection that inherits the interface has the member method that takes a and applies each key in the collection to the consumer.

• L8: any collection has the member method that returns the of the collection.

  • : creates a stream of the collection whose keys are sorted

  • : creates a collection specified by the

• L9: iterates each of the sorted keys and compares it to the returned value of pq.remove().

What are the advantages of defining a method-level generic type?

Functional Programming

The following code shows a traditional way of iterating over a list (that is equivalent to L7 above):

Java 5 introduced an enhanced for loop that simplified the traditional for loop:

Java 8 introduced that enabled functional programming in Java. The following code takes each key (as a variable) in keys and applies it to pq.add():

The syntax of the above code can be simplified as follows (as shown in L7):

What are the main differences between object-oriented programming and functional programming?

Test: Robustness

Let us define the testRobustness() method that calls the auxiliary method testRobustnessAux():

• L7-9: tests different types of max-PQs. The keys need to be sorted in reverse order for it to test the remove() method.

• L11-13: tests different types of min-PQs. The keys need to be sorted in the natural order for it to test the remove() method.

The generic types of the PQs in L4-5 are explicitly coded as <Integer>, whereas they are simplified to <> in L7-13. When do generic types need to be explicitly coded?

Contents

1. Simple Priority Queues

Resources

• Main:

• Test:

References

Development Kit

Install the latest version of the Java SE Development Kit (JDK) on your local machine:

• The required version: 17.x.x (or higher)

Version Control

Install Git using any of the following instructions:

Run the following commands on a terminal by replacing user.email and user.name with your email address and name:

Integrated Development Environment

Install the latest version of on your local machine:

• The recommended version: 2022.3.x (Ultimate Edition)

• Apply for the with your school email address to use the ultimate version.

Project Management

Launch IntelliJ and create a project by clicking the [New Project] button at the top:

• Name: dsa-java

• Location: local_path/dsa-java

• Check "Create Git repository"

Open and make sure distributionUrl indicates the latest version of Gradle:

Click [Settings - Build, Execution, Deployment] on the menu:

• Click [Build Tools - Gradle] and set Gradle JVM to 17.

• Click [Compiler - Java Compiler] and set Project bytecode version to 17.

Click [File - Project Structure] and select [Project Settings]:

• Click [Project Settings - Project] and set SDK to 17 and Project language level to SDK default.

• Click [Project Settings - Modules - Dependencies] and set Module SDK to 17.

Open and make sure sourceCompatibility and targetCompatibility are set to java version 17 (add the following configurations if they do not exist already):

Lastly, check mavenCentral() is configured as a repository in your build.gradle:

GitHub Integration

To integrate the project with your repository, click [Settings]:

• Choose [Version Control - Github] on the left pane.

• Click [+] and login with your GitHub ID and password.

• If you use two-factor authentication, log in with your .

Create under the project and add the following contents:

Click [Git - GitHub - Share Project on Github] and create a repository:

• Make sure to check private.

• Repository name: dsa-java

• Remote: origin

Add all files and make the initial commit. Check if the repository is created under your GitHub account: https://github.com/your_id/dsa-java.

General

• Book: https://emory.gitbook.io/dsa-java/

• GitHub:

• Time: MW 11:30AM - 12:45PM

• Location: Atwood 240

Instructors

• Associate Professor of Computer Science Office Hours → MW 4PM - 5:30PM, MSC W302F

• BS in Computer Science; BA in Physics and Astronomy Office Hours → TuTh 10:30AM - 12PM, MSC E308

• BS in Computer Science and Economics Office Hours -> MW 2:30PM - 4PM, MSC E308

Grading

• 1 + 10 topical quizzes: 70%

• 3 homework assignments: 30%

• Your work is governed by the . Honor code violations (e.g., copies from any source, including colleagues and internet sites) will be referred to the .

Notes

• For every topic, one quiz will be assigned to check if you keep up with the materials.

• Homework assignments assess conceptual understanding, programming ability, and analytical writing skills relevant to this course.

• All quizzes and assignments must be submitted individually. Discussions are allowed; however, your work must be original.

A binary heap is a PQ that takes$O(\log n)$for both add() and remove(), where keys are stored in a balanced binary tree in which every node has a higher priority than its children (when used as a max-PQ).

When should we use a binary heap over a lazy PQ or an eager PQ?

Balanced Binary Tree

A balanced binary tree can be implemented by a list such that given the 'th key in the list:

• The index of its parent:

• The index of its left child: .

• The index of its right child: .

Is the tree represented by the list [1, 2, 3, 4, 5] balanced?

Implementation

Let us create the class inheriting AbstractPriorityQueue:

• L11: adding null as the first item makes it easier to calculate the indices of parents and children.

• L16: keys has the extra item of null from L11 that is not an input key.

How does adding null make the calculation of the indices easier?

Additionally, let us define the helper method compare() that compares two keys in the list:

• L8: calls the compare() method in the member field priority declared in the super class.

Add() with Swim

The add() method in a heap uses the operation called swim as follows:

• L3: appends the new key to the list.

• L7-10: keeps swimming until the list becomes a heap:

How many keys are compared at most for the `add operation?

Remove() with Sink

The remove() method in a heap uses the operation called sink as follows:

• L3: checks if the heap is empty.

• L4: swaps the root and the last key in the list.

• L3

How many keys are compared at most for the remove operation?

A priority queue (PQ) is a data structure that supports the following two operations:

• add(): adds a comparable key to the PQ.

• remove(): removes the key with the highest (or lowest) priority in the PQ.

A PQ that removes the key with the highest priority is called a maximum PQ (max-PQ), and with the lowest priority is called a minimum PQ (min-PQ).

Does a priority queue need to be sorted at all time to support those two operations? What are the use cases of priority queues?

Abstract Priority Queue

Let us define that is an abstract class to be inherited by all priority queues:

• L1: declares the type T that is the type of input keys to be stored in this PQ.

  • T must be by its priority.

What are comparable data types in Java? Can you define your own comparator?

Let us define three abstract methods, add(), remove(), and size() in AbstractPriorityQueue:

Given the abstract methods, we can define the regular method isEmpty():

Lazy Priority Queue

Let us define whose core methods satisfy the following conditions:

• add(): takesto add a key to the PQ.

• remove(): takes to remove the key with the highest/lowest priority from the PQ.

In other words, all the hard work is done at the last minute when it needs to remove the key.

• L1: declares T and passes it to its super class, AbstractPriorityQueue.

• L2: defines a list to store input keys.

Can you add keys to the member field keys when it is declared as final (a constant)? Why does all constructors in LazyPriorityQueue need to call the super constructor?

We then override the core methods, add() and remove():

• L6-8: appends a key to the list in .

• L15-20: removes a key to the list in .

Is ArrayList the best implementation of List for LazyPriorityQueue? Why does remove() in L18 cost ?

Eager Priority Queues

Let us define whose core methods satisfy the following conditions:

• add(): takesto add a key to the PQ.

• remove(): takes to remove the key with the highest/lowest priority from the PQ.

In other words, all the hard work is done as soon as a key is added.

What are the situations that LazyPQ is preferred over EagerPQ and vice versa?

• The implementations of the two constructors and the size() method are identical to the ones in LazyPriorityQueue.

Should we create an abstract class that implements the above code and make it as a super class of LazyPQ and EagerPQ? What level of abstraction is appropriate in object-oriented programming?

We then override the core methods, add() and remove():

• L6-12: inserts a key to the list in .

  • L8: finds the index of the key to be inserted in the list using binary search in .

What are the worst-case complexities of add() and remove() in LazyPQ and EagerPQ in terms of assignments and comparison?

Contents

1. Abstraction

Resources

• Main:

• Test:

References

• Comparison-based Algorithms

Contents

1. Binary Search Trees

References

Abstract Sort

Let us create an abstract class AbstractSort:

• L2-4: member fields:

  • comparator: specifies the precedence of comparable keys.

  • comparisons: counts the number of comparisons performed during sorting.

  • assignments

Let us define three helper methods, compareTo(), assign(), and swap():

• L7: compares two keys in the array and increments the count.

• L18: assigns the value to the specific position in the array and increments the count.

• L29

These helper methods allow us to analyze the exact counts of those operations performed by different sorting algorithms.

How is it different to benchmark runtime speeds from counting these two operations?

Finally, let us define the default sort() that calls an overwritten abstract method, sort():

• L5: calls the abstract method sort() that overwrites it.

• L15: sorts the array in the range of (beginIndex, endIndex) as specified in comparator

When would it be useful to sort a specific range in the input array?

Time

Let us define a static nested class Time under PriorityQueueTest that stores runtimes for the add() and remove() methods in any PQ:

What is the advantage of defining static nested class over non-static nested class?

Runtime Estimation

Let us define addRuntime() that estimates the runtime for add() and remove()for a particular PQ:

• L5-8: estimates the runtime for add():

  • L5: gets the snapshot of the starting time using .

Is the approach using currentTimeMillis() a proper way of benchmarking the speed?

We then define benchmark() that estimates the speeds of multiple PQs using the same lists of keys by calling addRuntime():

• L1: method declaration:

  • AbstractPriorityQueue[]: It is unusual to declare an array of objects involving generics in Java (you will see the unique use case of it in the testSpeedAux() method).

Why do we need to benchmark the priority queues multiple times?

Benchmark

Let us define testSpeedAux() that takes multiple PQs and prints out the benchmarking results:

• L1: annotates indicating that this method takes vararg parameters.

• L2: declares a final method that cannot be overridden by its subclasses.

Why is it recommended to declare the method final if it takes vararg parameters? Why does JVM need to be warmed-up before the actual benchmarking? Why should you use StringJoiner instead of concatenating strings with the + operator?

Speed Comparison

The followings show runtime comparisons between LazyPQ, EagerPQ, and BinaryHeap:

Coding

• Create a class called TernaryHeapQuiz under the main queue package that extends the abstract class AbstractPriorityQueue.

• Each node in TernaryHeapQuiz takes up to 3 children, so it becomes a ternary instead of a binary tree.

• Override the required abstract methods, , , and , such that both add() and remove() give .

• Feel free to use the code in .

Testing

• Create the class under the test package.

• Test the correctness of your TernaryHeapQuiz using the method.

• Add more tests for a more thorough assessment if necessary.

Benchmarking

• Compare runtime speeds between BinaryHeap and TernaryHeapQuiz for add() and remove() using the method.

• Create a PDF file quiz2.pdf and write a report that includes the following:

Submission

1. Commit and push everything under the following packages to your GitHub repository:

• Main:

• Test:

2. Submit quiz2.pdf to Canvas.

Unlike comparison-based algorithms in Sections 3.2 and 3.3, distribution-based sorting algorithms do not make comparisons between keys in the input array; instead, they distribute keys into ordered buckets and merge keys in the buckets.

Bucket Sort

Let us create an abstract class, AbstractBucketSort inheriting AbstractSort:

• L2: declares a list of buckets where each bucket is represented by a .

• L6: does not pass any comparator to AbstractSort since no comparison is needed.

What kind of input keys can work with distribution-based sorting algorithms?

We then define a helper method sort():

• L7: bucketIndex is a function that takes a comparable key and returns the appropriate bucket index of the key.

• L9-10: adds the keys within the range in the input array to the buckets.

How many assignments are made by the sort() method in BucketSort?

Integer Bucket Sort

Let us create the IntegerBucketSort class inheriting BucketSort that sorts integers within a specific range in ascending order:

• L2: takes the smallest integer in the range.

• L8: passes the number of buckets, max - min, to be initialized to the super constructor.\

We then override the sort() method:

• L3: passes a lambda expression that takes key and returns key - MIN indicating the index of the bucket that key should be assigned to.

Is it possible to implement DoubleBucketSort that can handle double instead of integer keys?

Radix Sort

Let us create an abstract class, RadixSort, inheriting BucketSort<Integer>:

• L3: creates 10 buckets to sort any range of integers.

• L6: returns the order of the most significant digit in the input array.

LSD Radix Sort

Let us create the LSDRadixSort class inheriting RadixSort that sorts the integer keys from the least significant digit (LSD) to the most significant digit (MSD):

• L5: iterates from LSD to MSD.

• L7: the lambda expression returns the bucket index, the value in the 'th significant digit.

Benchmarks

The followings compare runtime speeds between advanced sorting algorithms where the input arrays consist of random integer keys:

Selection-based Sort

Selection-based sorting algorithms take the following steps:

• For each key $A_i$ where $|A| = n$ and :

  • Search the maximum key where .

  • Swap and

The complexities differ by different search algorithms:

Selection Sort uses linear search to find the minimum (or maximum) key, whereas Heap Sort turns the input array into a heap, so the search complexity becomes instead of .

Can the search be faster than ?

Selection Sort

Let us create the class inheriting AbstractSort:

Let us then override the sort() method:

• L3-12:

  • L3: iterates all keys within the range .

How does the sort() method work with Comparator.reverseOrder()?

Heap Sort

Let us create the class inheriting AbstractSort:

Before we override the sort() method, let us define the following helper methods:

• L1-7: finds the right position of the k'th key by using the sink operation.

• L9-11: finds the parent index of the k'th key given the beginning index.

How is this sink() method different from the one in the class?

Finally, we override the sort() method:

• L4-5: turns the input array into a heap :

  • L4: iterates from the parent of the key in the ending index .

What is the worst-case scenario for Selection Sort and Heap Sort?

Insertion-based Sort

Insertion-based sorting algorithms take the following steps:

• For each key where and :

  • Keep swapping and until .

The complexities differ by different sequences:

Insertion Sort

Let us create the class inheriting AbstractSort:

Let us then define an auxiliary method, sort():

• L4: iterates keys in the input array .

• L5: compares keys in the sublist of the input array .

• L6

Given the auxiliary method, the sort() method can be defined as follows where h = 1:

How many swaps does Insertion Sort make for the following array? [7, 1, 2, 3, 4, 5, 6, 14, 8, 9, 10, 11, 12, 13, 0]

Shell Sort

Gap Sequence

Shell Sort groups keys whose gap is defined by a particular :

• Knuth:

• Hibbard:

• Pratt:

For the above example, by using the Hibbard sequence, it first groups keys whose gap is 7:[7, 14, 0], [1, 8], [2, 9], [3, 10], [4, 11], [5, 12], [6, 13]

It then sorts each group using Insertion Sort, which results in the following array: [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]

The above procedure is repeated for gaps 3 and 1; however, the array is already sorted, so no more swapping is necessary.

Implementation

Let us create an abstract class, , inheriting AbstractSort:

• L2: stores a particular gap sequence.

• L7: pre-populate the gap sequence that can handle the input size up to 10000.

Then, let us define two abstract methods, populateSequence() and getSequenceStartIndex():

• L5: populates a particular sequence for the input size n.

• L11: returns the index of the first gap to be used given the input size n.

Let us then override the sort() method:

• L4: should not re-populate the sequence unless it has to.

• L6: iterates the sequence where is the number of gaps in the sequence.

• L7

Knuth Sequence

Let us create the class that populates the Knuth Sequence for ShellSort:

The two abstract methods can be overridden as follows:

• L2: populates the Knuth sequence up to the gap .

• L13: returns the index of the first key .

Why should we use as the largest gap in the sequence?

Demonstration

Unit Tests & Benchmarks

The codes for unit testing and benchmarking are available in :

Coding

Shell Sort: Hibbard

• Create the class under the package that inherits .

• Override the populateSequence() and getSequenceStartIndex() methods such that it performs Shell Sort by using the Hibbard sequence: .

• Feel free to use the code in .

Radix Sort: MSD

• Create the class under the package that inherits .

• Override the sort() method such that it performs Radix Sort from the most significant digit (MSD) to the least significant digit (LSD).

• Feel free to use the code in

Testing

• Create the class under the test package.

• Test the correctness of your TernaryHeapQuiz using the method.

• Add more tests for a more thorough assessment if necessary.

Benchmarking

• Compare runtime speeds between ShellSortKnuth and ShellSortQuiz for random, ascending, and descending cases using the method.

• Compare runtime speeds between LSDRadixSort and RadixSortQuiz for random cases.

Submission

• Push everything under the sort package to your GitHub repository:

  • Main:

  • Test:

Divide & Conquer

• Divide the problem into sub-problems (recursively).

• Conquer sub-problems, which effectively solve the super problem.

Why do people ever want to use QuickSort?

Merge Sort

• Divide the input array into two sub-arrays.

• Sort each of the sub-arrays and merge them into the back.

Let us create the class inheriting AbstractSort:

• L2: holds the copy of the input array.

Let us then override the sort() method that calls the helper method:

• L4-5: increases the size of the temp array if necessary.

  • L5: unchecked type Object to T[].

What is the advantage of declaring the member field temp?

The helper method can be defined as follows:

• L11: sorts the left sub-array.

• L12: sorts the right sub-array.

• L13: merges the left and right sub-arrays.

Finally, the merge() method can be defined as follows:

• L10-11: copies the input array to the temporary array and counts the assignments.

• L14-15: no key left in the 1st half.

• L16-17

How many assignments are made for the number of keys by the above version of MergeSort?

Quick Sort

• Pick a pivot key in the input array.

• Exchange keys between the left and right partitions such that all keys in the left and right partitions are smaller or bigger than the pivot key, respectively.

• Repeat this procedure in each partition, recursively.

Let us create the class inheriting AbstractSort:

Let us then override the sort() method:

• L3: stops when the pointers are crossed.

• L6: sorts the left partition.

• L7: sorts the right partition.

The partition() method can be defined as follows:

• L5: finds the left pointer where endIndex > fst > pivot.

• L6: finds the right pointer where beginIndex < snd

Intro Sort

• Although Quick Sort is the fastest on average, the worst-case complexity is .

• sorting algorithms that guarantee faster worst-case complexities than Quick Sort: Quick Sort for random cases and a different algorithm for the worst case.

How can we determine if Quick Sort is meeting the worse-case?

Let us define the IntroSoft class inheriting QuickSort: