heaps and other data structures

1Liang, Introduction to Java Programming, Sixth Edition, (c) 2007 Pearson Education, Inc. All rights reserved. 0-13-222158-6

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

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

Chapter 21 Generics

Chapter 22 Java Collections Framework

Chapter 23 Algorithm Efficiency and Sorting

Chapter 11 Object-Oriented Design

Chapter 19 Recursion


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).


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:


Limitations of arrays

Once an array is created, its size cannot be altered.

Array provides inadequate support for inserting, deleting, sorting, and searching operations.


Object-Oriented Data StructureIn 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.


Four Classic Data StructuresFour 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.


ListsA 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.


Two Ways to Implement ListsThere 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.


Design of ArrayList and LinkedListFor 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.

MyList MyAbstractList

MyArrayList

MyLinkedList


MyList Interface and MyAbstractList Class MyList

+add(o: Object) : void +add(index: int, o: Object) : void +clear(): void +contains(o: Object): boolean +get(index: int) : Object +indexOf(o: Object) : int +isEmpty(): boolean +lastIndexOf(o: Object) : int +remove(o: Object): boolean +size(): int +remove(index: int) : Object +set(index: int, o: Object) : Object

Appends a new element o at the end of this list. Adds a new element o at the specified index in this list. Removes the element o from this list. Returns true if this list contains the element o. Returns the element from this list at the specified index. Returns the index of the first matching element in this list. Returns true if this list contains no elements. Returns the index of the last matching element in this list. Removes all the elements from this list. Returns the number of elements in this list. Removes the element at the specified index. Sets the element at the specified index.

MyAbstractList

#size: int #MyAbstractList() #MyAbstractList(objects: Object[]) +add(o: Object) : void +isEmpty(): boolean +size(): int +remove(o: Object): boolean

The size of the list. Creates a default list. Creates a list from an array of objects. Implements the add method. Implements the isEmpty method. Implements the size method. Implements the remove method.

MyList

MyAbstractList


Array ListsArray 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.


InsertionBefore 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.

e0

0 1 … i i+1 k-1 Before inserting e at insertion point i e1 … ei ei+1

…

… ek-1

data.length -1 Insertion point e

e0

0 1 … i i+1 After inserting e at insertion point i, list size is incremented by 1

e1 … e ei

…

… ek-1

data.length -1 e inserted here

ek

ek

k

ei-1

ei-1

k+1 k

ei+1

i+2

…shift…


DeletionTo 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.

e0

0 1 … i i+1 k-1 Before deleting the element at index i e1 … ei ei+1

…

… ek-1

data.length -1 Delete this element

e0

0 1 … i After deleting the element, list size is decremented by 1 e1 …

…

… ek

data.length -1

ek

k

ei-1

ei-1 k-1

ei+1

k-2

ek-1

…shift…


Implementing MyArrayList

MyArrayList -data: Object[] +MyArrayList() +MyArrayList(objects: Object[])

MyAbstractList

Creates a default array list. Creates an array list from an array of objects.

MyArrayList RunTestList


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.


Nodes in Linked ListsA linked list consists of nodes. 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; }}

element1

head

next

Node 1

element2

next

Node 2 …

element2

null

Node n tail


Adding Three NodesThe variable head 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 strings in a list, as follows:

Step 1: Declare head and tail:

The list is empty now Node head = null;

Node tail = null;


Adding Three Nodes, cont.Step 2: Create the first node and insert it to the list:

head "Chicago" next: null

tail

After the first node is inserted inserted

head = new Node("Chicago"); last = head;


Adding Three Nodes, cont.Step 3: Create the second node and insert it to the list:

tail.next = new Node("Denver");

head "Chicago" next

"Denver" next: null

tail

tail = tail.next;

head "Chicago" next

"Denver" next: null

tail


Adding Three Nodes, cont.Step 4: Create the third node and insert it to the list:

head "Chicago" next

"Dallas" next: null

tail

"Denver" next

tail.next = new Node("Dallas");

head "Chicago" next

"Dallas" next: null

tail

"Denver" next

tail = tail.next;


Traversing All Elements in the ListEach node contains the element and a data field named next that points to the next element. If the node is the last in the list, its pointer data field next contains the value null. You can use this property to detect the last node. For example, you may write the following loop to traverse all the nodes in the list.

Node current = head;while (current != null) { System.out.println(current.element); current = current.next;}


MyLinkedList

MyLinkedList -first: Node -last: Node +LinkedList() +LinkedList(objects: Object[]) +addFirst(o: Object): void +addLast(o: Object): void +getFirst(): Object +getLast(): Object +removeFirst(): Object +removeLast(): Object

1

m Node element: Object next: Node

Link

1

MyAbstractList

Creates a default linked list. Creates a linked list from an array of objects. Adds the object to the head of the list. Adds the object to the tail of the list. Returns the first object in the list. Returns the last object in the list. Removes the first object from the list. Removes the last object from the list.

MyLinkedList


Implementing addFirst(Object o)public void addFirst(Object o) { Node newNode = new Node(o); newNode.next = head; head = newNode; size++; if (tail == null) tail = head;}

head

e0

next

…

A new node to be inserted here

ei

next

ei+1

next

tail

… ek

null

element

next

New node inserted here

(a) Before a new node is inserted.

(b) After a new node is inserted.

e0

next

… ei

next

ei+1

next

tail

… ek

null

element

next

head


Implementing addLast(Object o)public void addLast(Object o) { if (tail == null) { head = tail = new Node(element); } else { tail.next = new Node(element); tail = tail.next; } size++;}

head

e0

next

… ei

next

ei+1

next

tail

… ek

null

o

null




head

e0

next

… ei

next

ei+1

next

tail

… ek

next


o

null


Implementing add(int index, Object o) public void add(int index, Object o) { if (index == 0) addFirst(o); else if (index >= size) addLast(o); else { Node current = head; for (int i = 1; i < index; i++) current = current.next; Node temp = current.next; current.next = new Node(o); (current.next).next = temp; size++; }}

current head

e0

next

…


ei

next

temp

ei+1

next

tail

… ek

null

o

next

current head

ei

next

…


ei

next

temp

ei+1

next

tail

… ek

null

o

next




Implementing removeFirst() public Object removeFirst() { if (size == 0) return null; else { Node temp = head; head = head.next; size--; if (head == null) tail = null; return temp.element; }}

head

e0

next

…

Delete this node

ei

next

ei+1

next

tail

… ek

null

(a) Before the node is deleted.

(b) After the first node is deleted

e1

next

… ei

next

ei+1

next

tail

… ek

null

e1

next

head


Implementing removeLast()

public Object removeLast() { if (size == 0) return null; else if (size == 1) { Node temp = head; head = tail = null; size = 0; return temp.element; } else { Node current = head; for (int i = 0; i < size - 2; i++) current = current.next; Node temp = tail; tail = current; tail.next = null; size--; return temp.element; }}

head

e0

next

…

Delete this node

ek-2

next

ek-1

next

tail

ek

null


(b) After the last node is deleted

e1

next

current

head

e0

next

… ek-2

next

ek-1

null

e1

next

tail


Implementing remove(int index) public Object remove(int index) { if (index < 0 || index >= size) return null; else if (index == 0) return removeFirst(); else if (index == size - 1) return removeLast(); else { Node previous = head; for (int i = 1; i < index; i++) { previous = previous.next; } Node current = previous.next; previous.next = current.next; size--; return current.element; }}

previous head

element

next

…

Node to be deleted

element

next

element

next

tail

… element

null

element

next


current

previous head

element

next

… element

next

element

next

tail

… element

null

(b) After the node is deleted.

current.next

current.next


Circular Linked Lists A circular, singly linked list is like a singly

linked list, except that the pointer of the last node points back to the first node.

element1

head

next

Node 1

element2

next

Node 2 …

element2

next

Node n tail


Doubly Linked Lists A doubly linked list contains the nodes with two

pointers. One points to the next node and the other points to the previous node. These two pointers are conveniently called a forward pointer and a backward pointer. So, a doubly linked list can be traversed forward and backward.

element1

head

next

Node 1

element2

next

Node 2 …

element2

null

Node n tail

null previous previous


Circular Doubly Linked Lists A circular, doubly linked list is doubly linked

list, except that the forward pointer of the last node points to the first node and the backward pointer of the first pointer points to the last node.

element1

head

next

Node 1

element2

next

Node 2 …

element2

next

Node n tail

previous previous previous


StacksA 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.

Data1 Data2

Data1 Data1 Data2 Data3

Data1 Data2 Data3

Data1 Data2

Data3

Data1

Data2 Data1


QueuesA 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.

Data1 Data2

Data1 Data1 Data2 Data3

Data1 Data2 Data3

Data2 Data3

Data1

Data3

Data2 Data3


Implementing Stacks and QueuesUsing an array list to implement StackUse a linked list to implement Queue

Since the insertion and deletion operations on a stack are made only at 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.


Design of the Stack and Queue ClassesThere 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.

MyArrayList MyStack MyLinkedList MyQueue

– 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.

MyStack

MyArrayList MyQueue MyLinkedList


Composition is BetterBoth 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.


MyStack and MyQueue MyStack

-list: MyArrayList

+isEmpty(): boolean +getSize(): int +peek(): Object +pop(): Object +push(o: Object): Object +search(o: Object): int

Returns true if this stack is empty. Returns the number of elements in this stack. Returns the top element in this stack. Returns and removes the top element in this stack. Adds a new element to the top of this stack. Returns the position of the specified element in this stack.

MyQueue

-list: MyLinkedList +enqueue(element: Object): void +dequeue(): Object +getSize(): int

Adds an element to this queue. Removes an element from this queue. Returns the number of elements from this queue.

MyStack

MyQueue


Example: Using Stacks and Queues

TestStackQueue

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).

Run


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.

60

55 100

57 67 107 45

G

F R

M T A

(A) (B)


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.


Representing Binary TreesA 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.

60

55 100

57 45 67 107

root class TreeNode { Object element; TreeNode left; TreeNode right; public TreeNode(Object o) { element = o; }}


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:


Inserting an Element to a Binary Treeif (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 }

Insert 101 into the following tree.

60

55 100

57 45 67 107

root


Trace Inserting 101 into the following treeif (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);



60

55 100

57 45 67 107

root


Trace Inserting 101 into the following tree, cont.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);



60

55 100

57 45 67 107

root

current





60

55 100

57 45 67 107

root

current





60

55 100

57 45 67 107

root

current

101 < 60?





60

55 100

57 45 67 107

root

current

101 > 60?





60

55 100

57 45 67 107

root

current parent

101 > 60 true





60

55 100

57 45 67 107

root

current

parent

101 > 60 true





60

55 100

57 45 67 107

root

current

parent

101 > 100 false





60

55 100

57 45 67 107

root

current

parent

101 > 100 true





60

55 100

57 45 67 107

root

current

parent

101 < 107 true





60

55 100

57 45 67 107

root

Since current.left is null,current becomes null

parent

101 < 107 true





60

55 100

57 45 67 107

root


parent

current is null now





60

55 100

57 45 67 107

root


parent

101 < 107 true





60

55 100

57 45 67 107

root

parent

101

101 < 107 true


Inserting 59 into the Treeif (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);


60

55 100

57 45 67 107

root

59 101


Tree TraversalTree 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.


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.


The BinaryTree ClassLet’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

-root: TreeNode +BinaryTree() +BinaryTree(objects: Object[]) +insert(o: Object): boolean +inorder(): void +preorder(): void +postorder(): void

1

m TreeNode element: Object left: TreeNode

right: TreeNode

Link

1

Creates a default binary tree. Creates a binary tree from an array of objects. Adds an element to the binary tree. Prints the nodes in inorder traversal. Prints the nodes in preorder traversal. Prints the nodes in postorder traversal.

BinaryTree


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.

TestBinaryTree Run


Tree After Insertions

George

Adam Michael

Daniel Jones Tom

root

Peter

Inorder: Adam, Daniel George, Jones, Michael, Peter, Tom

Postorder: Daniel Adam, Jones, Peter, Tom, Michael, George

Preorder: George, Adam, Daniel, Michael, Jones, Tom, Peter


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.


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).

22 29 14 33

32 39

42

22 29 14

32 42

39

22 14 33

32 39

42

22 29

32

42


Representing a HeapFor 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).

22 29 14 33 30 17 9

32 39 44 13

42 59

62 62 42 59 32 39 44 13 22 29 14 33 30 17 9

[0] [1] [2] [3] [4] [5] [6] [7] [8] [9] [10][11][12][13] [10][11]

parent left

right


Adding Elements to the Heap

Adding 3, 5, 1, 19, 11, and 22 to a heap, initially empty

3

(a) After adding 3 (b) After adding 5 (c) After adding 1

(d) After adding 19

3

19

5 1

(e) After adding 11

3 5

19

11 1

(f) After adding 22

3 5 1

22

11 19

5

3

5

3 1


Rebuild the heap after adding a new node

Adding 88 to the heap

(a) Add 88 to a heap

3 5 1 88

22

11 19

(b) After swapping 88 with 19 3 5 1 19

22

11 88

(b) After swapping 88 with 22 3 5 1 19

88

11 22


Removing the Root and Rebuild the Tree

22 29 14 33 30 17 9

32 39 44 13

42 59

62

Removing root 62 from the heap



22 29 14 33 30 17

32 39 44 13

42 59

9

Move 9 to root



22 29 14 33 30 17

32 39 44 13

42 9

59

Swap 9 with 59



22 29 14 33 30 17

32 39 9 13

42 44

59

Swap 9 with 44



22 29 14 33 9 17

32 39 30 13

42 44

59

Swap 9 with 30


The Heap Class

Heap RunTestHeap

Heap -list: java.util.ArrayList +Heap() +Heap(objects: Object[]) +add(newObject: Object): void +remove(): Object +getSize(): int

Creates a default heap. Creates a heap with the specified objects. Adds a new object to the heap. Removes the root from the heap and returns it. Returns the size of the heap.


Priority QueueA 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

-heap: Heap +enqueue(element: Object): void +dequeue(): Object +getSize(): int

Adds an element to this queue. Removes an element from this queue. Returns the number of elements from this queue.

MyPriorityQueue RunTestPriorityQueue

heaps and other data structures

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