Chapter 20 Lists, Stacks, Queues, Trees, and Heaps

1. Chapter 20 Lists, Stacks, Queues, Trees, and Heaps

2. Objectives • To describe what a data structure is (§20.1). • To explain the limitations of arrays (§20.1). • To implement a dynamic list using an array (§20.2.1). • To implement a dynamic list using a linked structure (§20.2.2 Optional). • To implement a stack using an array list (§20.3). • To implement a queue using a linked list (§20.3). • To implement a binary search tree (§20.4).

3. What is a Data Structure? A data structure is a collection of data organized in some fashion. A data structure not only stores data, but also supports the operations for manipulating data in the structure. For example, an array is a data structure that holds a collection of data in sequential order. You can find the size of the array, store, retrieve, and modify data in the array. Array is simple and easy to use, but it has two limitations:

4. Limitations of arrays • Once an array is created, its size cannot be altered. • Array provides inadequate support for inserting, deleting, sorting, and searching operations.

5. Object-Oriented Data Structure In object-oriented thinking, a data structure is an object that stores other objects, referred to as data or elements. So some people refer a data structure as a container object or a collection object. To define a data structure is essentially to declare a class. The class for a data structure should use data fields to store data and provide methods to support operations such as insertion and deletion. To create a data structure is therefore to create an instance from the class. You can then apply the methods on the instance to manipulate the data structure such as inserting an element to the data structure or deleting an element from the data structure.

6. Four Classic Data Structures Four classic dynamic data structures to be introduced in this chapter are lists, stacks, queues, and binary trees. A list is a collection of data stored sequentially. It supports insertion and deletion anywhere in the list. A stack can be perceived as a special type of the list where insertions and deletions take place only at the one end, referred to as the top of a stack. A queue represents a waiting list, where insertions take place at the back (also referred to as the tail of) of a queue and deletions take place from the front (also referred to as the head of) of a queue. A binary tree is a data structure to support searching, sorting, inserting, and deleting data efficiently.

7. Lists A list is a popular data structure to store data in sequential order. For example, a list of students, a list of available rooms, a list of cities, and a list of books, etc. can be stored using lists. The common operations on a list are usually the following: ·Retrieve an element from this list. ·Insert a new element to this list. ·Delete an element from this list. ·Find how many elements are in this list. ·Find if an element is in this list. ·Find if this list is empty.

8. Two Ways to Implement Lists There are two ways to implement a list. One is to use an array to store the elements. The array is dynamically created. If the capacity of the array is exceeded, create a new larger array and copy all the elements from the current array to the new array. The other approach is to use a linked structure. A linked structure consists of nodes. Each node is dynamically created to hold an element. All the nodes are linked together to form a list.

9. Design of ArrayList and LinkedList For convenience, let’s name these two classes: MyArrayList and MyLinkedList. These two classes have common operations, but different data fields. The common operations can be generalized in an interface or an abstract class. A good strategy is to combine the virtues of interfaces and abstract classes by providing both interface and abstract class in the design so the user can use either the interface or the abstract class whichever is convenient. Such an abstract class is known as a convenience class.

10. MyList Interface and MyAbstractList Class MyList MyAbstractList

11. Array Lists Array is a fixed-size data structure. Once an array is created, its size cannot be changed. Nevertheless, you can still use array to implement dynamic data structures. The trick is to create a new larger array to replace the current array if the current array cannot hold new elements in the list. Initially, an array, say data of Object[] type, is created with a default size. When inserting a new element into the array, first ensure there is enough room in the array. If not, create a new array with the size as twice as the current one. Copy the elements from the current array to the new array. The new array now becomes the current array.

12. Insertion Before inserting a new element at a specified index, shift all the elements after the index to the right and increase the list size by 1.

13. Deletion To remove an element at a specified index, shift all the elements after the index to the left by one position and decrease the list size by 1.

14. Implementing MyArrayList MyArrayList TestList Run

15. Linked Lists Since MyArrayList is implemented using an array, the methods get(int index) and set(int index, Object o) for accessing and modifying an element through an index and the add(Object o) for adding an element at the end of the list are efficient. However, the methods add(int index, Object o) and remove(int index) are inefficient because it requires shifting potentially a large number of elements. You can use a linked structure to implement a list to improve efficiency for adding and remove an element anywhere in a list.

16. Nodes in Linked Lists A linked list consists of nodes, as shown in Figure 20.7. Each node contains an element, and each node is linked to its next neighbor. Thus a node can be defined as a class, as follows: class Node { Object element; Node next; public Node(Object o) { element = o; } }

17. Nodes in a Linked List The variable first refers to the first node in the list, and the variable last refers to the last node in the list. If the list is empty, both are null. For example, you can create three nodes to store three circle objects (radius 1, 2, and 3) in a list: Node first, last; // Create a node to store the first circle object first = new Node(new Circle(1)); last = first; // Create a node to store the second circle object last.next = new Node(new Circle(2)); last = last.next; // Create a node to store the third circle object last.next = new Node(new Circle(3)); last = last.next;

18. MyLinkedList My LinkedList

19. The addFirst(Object o) Method Since variable size is defined as protected in MyAbstractList, it can be accessed in MyLinkedList. When a new element is added to the list, size is incremented by 1, and when an element is removed from the list, size is decremented by 1. The addFirst(Object o) method (Line 20-28) creates a new node to store the element and insert the node to the beginning of the list. After the insertion, first should refer to this new element node.

20. The addLast(Object o) Method The addLast(Object o) method (Lines 31-41) creates a node to hold element o and insert the node to the end of the list. After the insertion, last should refer to this new element node.

21. The add(int index, Object o) Method The add(int index, Object o) method (Lines 45-57) adds an element o to the list at the specified index. Consider three cases: (1) if index is 0, invoke addFirst(o) to insert the element to the beginning of the list; (2) if index is greater than or equal to size, invoke addLast(o) to insert the element to the end of the list; (3) create a new node to store the new element and locate where to insert the new element. As shown in Figure 20.12, the new node is to be inserted between the nodes current and temp. The method assigns the new node to current.next and assigns temp to the new node’s next.

22. The removeFirst() Method The removeFirst() method (Lines 61-69) removes the first node in the list by pointing first to the second node, as shown in Figure 20.13. The removeLast() method (Lines 73-88) removes the last node from the list. Afterwards, last should refer to the former second-last node.

23. Stacks and Queues A stack can be viewed as a special type of list, where the elements are accessed, inserted, and deleted only from the end, called the top, of the stack. A queue represents a waiting list. A queue can be viewed as a special type of list, where the elements are inserted into the end (tail) of the queue, and are accessed and deleted from the beginning (head) of the queue. Since the insertion and deletion operations on a stack are made only the end of the stack, using an array list to implement a stack is more efficient than a linked list. Since deletions are made at the beginning of the list, it is more efficient to implement a queue using a linked list than an array list. This section implements a stack class using an array list and a queue using a linked list.

24. Design of the Stack and Queue Classes There are two ways to design the stack and queue classes: ·Using inheritance: You can declare the stack class by extending the array list class, and the queue class by extending the linked list class. ·Using composition: You can declare an array list as a data field in the stack class, and a linked list as a data field in the queue class. Both designs are fine, but using composition is better because it enables you to declare a complete new stack class and queue class without inheriting the unnecessary and inappropriate methods from the array list and linked list.

25. MyStack and MyQueue MyStack MyQueue

26. Example: Using Stacks and Queues Write a program that creates a stack using MyStack and a queue using MyQueue. It then uses the push (enqueu) method to add strings to the stack (queue) and the pop (dequeue) method to remove strings from the stack (queue). TestStackQueue Run

27. Binary Trees A list, stack, or queue is a linear structure that consists of a sequence of elements. A binary tree is a hierarchical structure. It is either empty or consists of an element, called the root, and two distinct binary trees, called the left subtree and right subtree. Examples of binary trees are shown in Figure 20.18.

28. Binary Tree Terms The root of left (right) subtree of a node is called a left (right) child of the node. A node without children is called a leaf. A special type of binary tree called a binary search tree is often useful. A binary search tree (with no duplicate elements) has the property that for every node in the tree the value of any node in its left subtree is less than the value of the node and the value of any node in its right subtree is greater than the value of the node. The binary trees in Figure 20.18 are all binary search trees. This section is concerned with binary search trees.

29. Representing Binary Trees A binary tree can be represented using a set of linked nodes. Each node contains a value and two links named left and right that reference the left child and right child, respectively, as shown in Figure 20.19. class TreeNode { Object element; TreeNode left; TreeNode right; public TreeNode(Object o) { element = o; }

30. Inserting an Element to a Binary Tree If a binary tree is empty, create a root node with the new element. Otherwise, locate the parent node for the new element node. If the new element is less than the parent element, the node for the new element becomes the left child of the parent. If the new element is greater than the parent element, the node for the new element becomes the right child of the parent. Here is the algorithm:

31. Inserting an Element to a Binary Tree For example, to insert 101 into the tree in Figure 20.19, the parent is the node for 107. The new node for 101 becomes the left child of the parent. To insert 59 into the tree, the parent is the node for 57. The new node for 59 becomes the right child of the parent, as shown in Figure 20.20. if (root == null) root = new TreeNode(element); else { // Locate the parent node current = root; while (current != null) if (element value < the value in current.element) { parent = current; current = current.left; } else if (element value > the value in current.element) { parent = current; current = current.right; } else return false; // Duplicate node not inserted // Create the new node and attach it to the parent node if (element < parent.element) parent.left = new TreeNode(elemenet); else parent.right = new TreeNode(elemenet); return true; // Element inserted }

32. Tree Traversal Tree traversal is the process of visiting each node in the tree exactly once. There are several ways to traverse a tree. This section presents inorder, preorder, postorder, depth-first, and breadth-first traversals. The inorder traversal is to visit the left subtree of the current node first, then the current node itself, and finally the right subtree of the current node. The postorder traversal is to visit the left subtree of the current node first, then the right subtree of the current node, and finally the current node itself.

33. Tree Traversal, cont. The breadth-first traversal is to visit the nodes level by level. First visit the root, then all children of the root from left to right, then grandchildren of the root from left to right, and so on. For example, in the tree in Figure 20.20, the inorder is 45 55 57 59 60 67 100 101 107. The postorder is 45 59 57 55 67 101 107 100 60. The preorder is 60 55 45 57 59 100 67 107 101. The breadth-first traversal is 60 55 100 45 57 67 107 59 101.

34. The BinaryTree Class Let’s define the binary tree class, named BinaryTree with the insert, inorder traversal, postorder traversal, and preorder traversal, as shown in Figure 20.21. Its implementation is given as follows: BinaryTree

35. Example: Using Binary Trees Write a program that creates a binary tree using BinaryTree. Add strings into the binary tree and traverse the tree in inorder, postorder, and preorder. BinaryTree Run

36. Heap Heap is a useful data structure for designing efficient sorting algorithms and priority queues. A heap is a binary tree with the following properties: • It is a complete binary tree. • Each node is greater than or equal to any of its children.

37. Complete Binary Tree A binary tree is complete if every level of the tree is full except that the last level may not be full and all the leaves on the last level are placed left-most. For example, in Figure 20.23, the binary trees in (a) and (b) are complete, but the binary trees in (c) and (d) are not complete. Further, the binary tree in (a) is a heap, but the binary tree in (b) is not a heap, because the root (39) is less than its right child (42).

38. Representing a Heap For a node at position i, its left child is at position 2i+1 and its right child is at position 2i+2, and its parent is (i-1)/2. For example, the node for element 39 is at position 4, so its left child (element 14) is at 9 (2*4+1), its right child (element 33) is at 10 (2*4+2), and its parent (element 42) is at 1 ((4-1)/2).

39. Rebuilding a Heap

40. Removing the Root

41. Adding a New Node

42. The Heap Class Heap TestHeap Run

43. Priority Queue A regular queue is a first-in and first-out data structure. Elements are appended to the end of the queue and are removed from the beginning of the queue. In a priority queue, elements are assigned with priorities. When accessing elements, the element with the highest priority is removed first. A priority queue has a largest-in, first-out behavior. For example, the emergency room in a hospital assigns patients with priority numbers; the patient with the highest priority is treated first. MyPriorityQueue TestPriorityQueue Run