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1 DATA STRUCTURES LECTURE NOTES INFO1105 Work Street Work City, Work State Work ZIP, T Work Phone, F Work Fax Phone, M Mobile Phone, E Work , Work URL 2015 Julian O.

2 TABLE OF CONTENTS WEEK 1 Data Structures Analysis Tools Java Review Recursion WEEK 2 Java Generics Array Lists Linked Lists Iterators WEEK 3 Stacks Queues Deques WEEK 4 General Trees Binary Trees Tree Traversal Algorithms WEEK 5 Priority Queues Heaps WEEK 6 Binary Search Trees AVL Trees WEEK 7 Maps and Hash Tables Ordered Maps and Sets INFO1105 NOTES 2

3 WEEK 8 Graph Structures Graph Traversals WEEK 9 Directed Graphs Weighted Graphs Shortest Paths WEEK 10 Adaptable Priority Queues Minimum Spanning Trees Pattern Matching and Text Compression WEEK 11 Sorting Algorithms Merge-Sort and Quick-Sort Bucket-Sort and Radix-Sort INFO1105 NOTES 3

4 Week 1 Data Structures: A data structure is a systematic method of organising and accessing data. The Java collections library has many data structures built-in including ArrayList, HashMap, LinkedList, and more. An Abstract Data Type (ADT) is a mathematical model of a data structure that specifies the type of data stored, the methods usable on them, and the parameters of these methods. ADTs are expressed in Java as interfaces. There can be several different data structures that use different implementations of the same ADT. Analysis Tools: Data structures are analysed and evaluated using analysis tools. There are seven functions that we use to analyse data structures which are ordered below by growth rate (where b is a positive constant). Linear Function f(x) = c Logarithmic Function f(x) = log bx Linear Function f(x) = x n- log- n Function f(x) = xlog 2x Quadratic Function f(x) = x 2 Cubic Function f(x) = x 3 Exponential Function f(x) = b x Methods of Analysis: Our aim is to design data structures with smallest running time. Generally, running time is directly proportional to the size of the input, and so we use the seven functions above to study the relationship between the running time and the size of the input for an data structure. Experimental analysis is one method of analysing data structures that involves using the Java method System.currentTimeMillis() to test the running time when the input size is at different values. This way we can create a graph of the running time in milliseconds against the input size. There are limitations to using this method though as not all input sizes can be tested, the data structure must be fully implemented, and we often cannot compare the results of two different data structures. INFO1105 NOTES 4

5 Another method of analysing data structures is to count the number of primitive operations the data structure performs. A primitive operation is a simple low-level instruction such as assigning a value to a variable, or calling a method. All primitive operations take approximately the same amount of time to perform, and so the amount performed is directly proportional to the running time for that input. Since data structures may run faster on some inputs than others of the same size, it is too difficult to define an accurate average running time, and so we tend to focus on the worst case. Asymptotic Notation: Since the running time of a data structure is dependent on the size of the input, we describe the running time using special notation. We use an O to describe that the running time as a function of input size f(x) grows at a rate that is less than or equal to that of another function in an asymptotic sense like so: f(n) is O(g(n)). Another way to think of this is that we the maximum amount of operations a method would perform in the parentheses following the O. For example, a method that returns a single value from an array would have a running time of since it only performs a single operation regardless of the size of the array. However, a method that searches an array element by element may have to perform operations on every single element, so this method has a running time of O(n). Methods where the choice of the next element to perform an operation is one of several possibilities have a running time of O(log(n)), for example, a binary search. Additionally, we can use an Ω to denote that a function grows at a rate that is greater than or equal to that of another functions, and a Ө to denote if two functions grow at the same rate. We can use running times in asymptotic notation to analyse the running times of data structures in the long run. Consider the following table illustrating the growth rates of different functions. Java Review: Java is an object-oriented programming language. This means that a program is seen as a collection of objects (or instances) with different structures and behaviours that interact with each other. The structure and behaviour of an object is dependent on which class it belongs to. Classes are part of a program that hold a set of instance variables and methods. Objects can be created using a special method called the constructor. The class essentially defines the type of an object. INFO1105 NOTES 5

6 Instance variables are a means of storing data of different types. These types include integers, floating-point numbers, characters, strings, boolean values and more. There are many operations we can perform on instance variables. Arrays can be used to store large amounts of instance variables of the same type. Methods are a set of instructions that are stored in a class. Rather than writing out the same instructions over and over, we can write them once in a method and invoke or call the method whenever necessary to execute these instructions. The main method is a special method written in a class that is invoked by the computer when the program is run. In this method, we specify instructions that program will perform at runtime. We manipulate the behaviour of a program depending on certain circumstances using the concept of control flow, which involves using if-else statements, switch statements, and loops. One method of creating a set of classes involves defining a very general class, then creating subclasses of this class that inherit all the methods and variables of the general class (called a superclass). The subclasses are said to extend the superclass. This concept is called inheritance. Another method of creating a set of classes involves creating an interface, which is a set of only constants and method headings and not their implementations. From here, we can create classes that implement this interface, which means that they contain all the constants and implementations for all the methods. To account for any unexpected errors, like accidentally dividing by zero or storing a data value in an instance variable of the wrong type, we use exceptions, which are a mechanism of handling an error. Recursion: Recursion occurs when a method calls itself. To avoid an infinite loop occurring, recursive methods have a base case, which are an instance where the method stops calling itself. Recursion can be very useful but is not always ideal, and as such it is often replaced with loops. Linear recursion is the simplest form of recursion. Methods of linear recursion make one recursive call each time it is invoked until the base case. This type of recursion is useful when we are performing an algorithm on a set of elements. We say that tail recursion occurs when a linearly recursive method s last operation is a recursive call. This can occur when we run out of elements in an array to perform recursive calls on. Binary recursion is when a method makes two recursive calls. A method used to calculate the k th Fibonacci number returns a sum of two recursive calls, the first for the k-1 th Fibonacci number and the second for the k-2 th Fibonacci number. If a method makes n recursive calls, then it runs in O(n) time. Similarly, if a method makes log(n) recursive calls, then it runs in O(log(n)) time. INFO1105 NOTES 6

11 for (list_type variable_name : list){ statements } Here the statements will be repeated for each element in the list (where the list implements the Iterable ADT). This syntax is a shorthand Java provides us with for an actual for-loop that uses iterators. The listiterator method that is implemented in many lists Java provides us with produces an iterator that can traverse forward and backward. This iterator is can be thought of as a list cursor. We can use methods like add, previous, next, set, and more to use and manipulate lists based on the position of the list cursor. Lists in the Java Collections Framework: The Java Collections Framework is a set of classes and interfaces that Java provides us with that allow us to define and implement various data structures for storing collections of data. The root interface of these data structures is java.util.collection. The Collection interface extends the Iterable interface, which means that it contains the method iterator. Also, the Collection interface is extended by an interface called List which is implemented by the classes ArrayList and LinkedList. INFO1105 NOTES 11

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