Chapter Goals

• To understand the implementation of linked lists and array lists
• To analyze the efficiency of fundamental operations of lists and arrays
• To implement the stack and queue data types
• To implement a hash table and understand the efficiency of its operations

Implementing Linked Lists - The Node Class

• We will implement a simplified, singly-linked list.
• A linked list stores elements in a sequence of nodes.
• A Node object stores an element and a reference to the next node.
  • private inner class
  • public instance variables

public class LinkedList
{
   . . .
   class Node
   {
      public Object data;
      public Node next;
   }
}

Implementing Linked Lists - The Node Class

• A linked list object holds a reference to the first node:
  • each node holds a reference to the next node.

public class LinkedList
{
   private Node first;
   
   public LinkedList() { first = null; }
   
   public Object getFirst()
   {
      if (first == null) { throw new NoSuchElementException(); }
         return first.data;
   }
}

Implementing Linked Lists - Adding and Removing the First Element

• When adding or removing the first element, the reference to the first node must be updated.

  public class LinkedList
  {
     . . .
     public void addFirst(Object element)
     {
        Node newNode = new Node(); 
        newNode.data = element;
        newNode.next = first; 
        first = newNode; 
     }
     . . .
  }

Implementing Linked Lists - Adding the First Element

Figure 1 Adding a Node to the Head of a Linked List

Implementing Linked Lists - Removing the First Element

• The data of the first node are saved and later returned as the method result.
• The successor of the first node becomes the first node of the shorter list.
• The old node is eventually recycled by the garbage collector.

public class LinkedList
{
   . . .
   public Object removeFirst()
   {
      if (first == null) { throw new NoSuchElementException(); }
      Object element = first.data;
      first = first.next; 
      return element;
   }
   . . .
}

Implementing Linked Lists - Removing the First Element

Figure 2 Removing the First Node from a LinkedList

The Iterator Class

• Our simplified ListIterator interface has methods: next, hasNext, remove, add, and set.
• Our LinkedList class declares a private inner class LinkedListIterator.
  • LinkedListIterator implements our simplified ListIterator interface.
  • As an inner class LinkedListIterator has access to
    • The instance variable first
    • The private Node class.
• A list iterator object has:
  • A reference to the the currently visited node, position
  • A reference to the last node before that, previous
  • A isAfterNext flag to track when the next method has been called.

The Iterator Class

• The LinkedListIterator class:

  public class LinkedList
  {
     . . .
     public ListIterator listIterator()
     {
        return new LinkedListIterator();
     }
     class LinkedListIterator implements ListIterator
     {
        private Node position;
        private Node previous;
        private boolean isAfterNext;
        public LinkedListIterator()
        {
           position = null;
           previous = null;
           isAfterNext = false;
        }
        . . .
     }
  }

Advancing an Iterator

• To advance an iterator:
  • Update the position
  • Remember the old position for the remove method.
• The next method:

  class LinkedListIterator implements ListIterator
  {
     . . .
     public Object next()
     {
        if (!hasNext()) { throw new NoSuchElementException(); }
        previous = position; // Remember for remove
        isAfterNext = true;
  
        if (position == null)
        {
           position = first;
        }
        else
        {
           position = position.next;
        }
        return position.data;
     }
     . . .
  }

Advancing an Iterator

• The iterator is at the end if the list is empty (first == null) or if there is no element after the current position (position.next == null).
• The hasNext method:

  class LinkedListIterator implements ListIterator
  {
     . . .
     public boolean hasNext()
     {
        if (position == null)
        {
           return first != null;
        }
        else
        {
           return position.next != null;
        }
     }
     . . .
  }

Removing an Element

• If this is the first element:
  • Call removeFirst
  • Otherwise, update the next reference of the previous node
• Update isAfterNext to disallow another call to remove.
• The remove method:

  class LinkedListIterator implements ListIterator
  {
     . . .
     public void remove()
     {
        if (!isAfterNext) { throw new IllegalStateException(); }
        if (position == first)
        {
           removeFirst();
        }
        else
        {
           previous.next = position.next; 
        }
        position = previous; 
        isAfterNext = false; 
     }
     . . .
  }

Removing an Element

Figure 3 Removing a Node from the Middle of a Linked List

Adding an Element

• After adding the new element
  • set the isAfterNext flag to false to disallow a subsequent call to the remove or set
• The add method:

  class LinkedListIterator implements ListIterator
  {
     . . .
     public void add(Object element)
     {
        if (position == null)
        {
           addFirst(element);
           position = first;
        }
        else
        {
           Node newNode = new Node();
           newNode.data = element;
           newNode.next = position.next; 
           position.next = newNode; 
           position = newNode; 
        }
        isAfterNext = false; 
     }
     . . .
  }

Adding an Element

Figure 4 Add a Node to the Middle of a Linked List

Setting an Element to a Different Value

• Set method changes the data in the previously visited element.
• Must follow a call to next.
• The set method:

  public void set(Object element)
  {
     if (!isAfterNext) { throw new IllegalStateException(); }
     position.data = element;
  }

Efficiency of Linked List Operations

• To get the k^th element of a linked list, you start at the beginning of the list and advance the iterator k times
• To get to the k^th node of a linked list, one must skip over the preceding nodes.
  

Efficiency of Linked List Operations

• When adding or removing an element, we update a couple of references in a constant number of steps.
• Adding and removing an element at the iterator position in a linked list takes O(1) time.

Efficiency of Linked List Operations

• To add an element at the end of the list
  • Must get to the end - an O(n) operation
  • Add the element O(1) operation
• Adding to the end of a linked list in our implementation takes O(n) time
• If the linked list keeps a reference to last as well as first
  • The time is reduced to constant time: O(1)
• We will conclude that adding to the end of a linked list is O(1).

Efficiency of Linked List Operations

• To remove an element from the end of the list:
  • Need a reference to the next-to-last element so that we can set its next reference to null
  • Takes n-1 iterations
• Removing an element from the end of the list is O(n).

Efficiency of Linked List Operations

Figure 5 Removing the Last Element of a Singly-Linked List

Efficiency of Linked List Operations

• In a doubly-linked list, each node has a reference to the previous node in addition to the next one.

public class LinkedList
{
   . . .
   class Node
   {
      public Object data;
      public Node next;
      public Node previous;
   }
}

Efficiency of Linked List Operations

• In a doubly-linked list, removal of the last element takes a constant number of steps.

  last = last.previous; 
  last.next = null; 

Efficiency of Linked List Operations

Figure 6 Removing the Last Element of a Doubly-Linked List

Efficiency of Linked List Operations

section_1/LinkedList.java

section_1/ListIterator.java

Self Check 16.1

Trace through the addFirst method when adding an element to an empty list.

• Answer: When the list is empty, first is null. A new Node is allocated. Its data instance variable is set to the element that is being added. It’s next instance variable is set to null because first is null. The first instance variable is set to the new node. The result is a linked list of length 1.

Self Check 16.2

Conceptually, an iterator is located between two elements (see Figure 9 in Chapter 15). Does the position instance variable refer to the element to the left or the element to the right?

• Answer: It refers to the element to the left. You can see that by tracing out the first call to next. It leaves position to refer to the first node.

Self Check 16.3

Why does the add method have two separate cases?

• Answer: If position is null, we must be at the head of the list, and inserting an element requires updating the first reference. If we are in the middle of the list, the first reference should not be changed.

Self Check 16.4

Assume that a last reference is added to the LinkedList class, as described in Section 16.1.8. How does the add method of the ListIterator need to change?

• Answer: If an element is added after the last one, then the last reference must be updated to point to the new element. After
  position.next = newNode;
  add
  if (position == last) { last = newNode; }

Self Check 16.5

Provide an implementation of an addLast method for the LinkedList class, assuming that there is no last reference.

• Answer:

  public void addLast(Object element)
  {
     if (first == null) { addFirst(element); }
     else
     {
        Node last = first;
        while (last.next != null)
        {
           last = last.next;
        }
        last.next = new Node();
        last.next.data = element;
     }
  }

Self Check 16.6

Expressed in big-Oh notation, what is the efficiency of the addFirst method of the LinkedList class? What is the efficiency of the addLast method of Self Check 5?

• Answer: O(1) and O(n).

Self Check 16.7

How much slower is the binary search algorithm for a linked list compared to the linear search algorithm?

• Answer: To locate the middle element takes n / 2 steps. To locate the middle of the subinterval to the left or right takes another n / 4 steps. The next lookup takes n / 8 steps. Thus, we expect almost n steps to locate an element. At this point, you are better off just making a linear search that, on average, takes n / 2 steps.

Static Classes

• Every object of an inner class has a reference to the outer class.
  • It can access the instance variables and methods of the outer class
• If an inner class does not need to access the data of the outer class,
  • It does not need a reference
  • Declare it static to save the cost of the reference
• Example: Declare the Node class of the LinkedList class as static:

  public class LinkedList
  {
     . . .
     static class Node
     {
        . . .
     }
  }

Implementing Array Lists

• An array list maintains a reference to an array of elements.
• The array is large enough to hold all elements in the collection.
• When the array gets full, it is replaced by a larger one.
• An array list has an instance field that stores the current number of elements.


Figure 7 An Array List Stores its Elements in an Array

Implementing Array Lists

• Our ArrayList implementation will manage elements of type Object:

  public class ArrayList
  {
     private Object[] elements;
     private int currentSize;
  
     public ArrayList()
     {
        final int INITIAL_SIZE = 10;
        elements = new Object[INITIAL_SIZE];
        currentSize = 0;
     }
  
     public int size() { return currentSize; }
     . . .
  }

Implementing Array Lists - Getting and Setting Elements

• Providing get and set methods:
  • Check for valid positions
  • Access the internal array at the given position
• Helper method to check bounds:

  private void checkBounds(int n)
  {
     if (n < 0 || n >= currentSize)
     {
        throw new IndexOutOfBoundsException();
     }
  }

Implementing Array Lists - Getting and Setting Elements

• The get method:

  public Object get(int pos)
  {
     checkBounds(pos);
     return element[pos];
  }

• The set method:
  

  public void set(int pos, Object element)
  {
     checkBounds(pos);
     elements[pos] = element;
  }

• Getting and setting an element can be carried out with a bounded set of instructions, independent of the size of the array list.
• These are O(1) operations.

Removing or Adding Elements

• To remove an element at position k, move the elements with higher index values.
• The remove method:

  public Object remove(int pos)
  {
     checkBounds(pos);
     Object removed = elements[pos];
     for (int i = pos + 1; i < currentSize; i++)
     {
        elements[i - 1] = elements[i];
     }
     currentSize--;
     return removed;
  }

• On average, n / 2 elements need to move.
• Inserting a element also requires moving, on average, n /2 elements.
• Inserting or removing an array list element is an O(n) operation.

Removing or Adding Elements

• Exception: adding an element after the last element
  • Store the element in the array
  • Increment size
• An O(1) operation
• A the addLast method:

  public boolean addLast(Object newElement)
  {
     growIfNecessary();
     currentSize++;
     elements[currentSize - 1] = newElement;
     return true;
  }

Removing or Adding Elements

Figure 8 Removing and Adding Elements

Growing the Internal Array

 

 

When an array list is completely full, we must move the contents to a larger array.

 

Growing the Internal Array

• When the array is full:
  • Create a bigger array
  • Copy the elements to the new array
  • New array replaces old
• Reallocation is O(n).
• The growIfNecessary method:

  private void growIfNecessary()
  {
     if (currentSize == elements.length)
     {
        Object[] newElements = new Object[2 * elements.length]; 
        for (int i = 0; i < elements.length; i++)
        {
           newElements[i] = elements[i]; 
        }
        elements = newElements; 
     }
  }

Growing the Internal Array

Growing the Internal Array

• Reallocation seldom happens.
• We amortize the cost of the reallocation over all the insertion or removals.
• Adding or removing the last element in an array list takes amortized O(1) time.
  • Written O(1)+

Efficiency of Array List and Linked List Operations

Self Check 16.8

Why is it much more expensive to get the kth element in a linked list than in an array list?

• Answer: In a linked list, one must follow k links to get to the kth elements. In an array list, one can reach the kth element directly as elements[k].

Self Check 16.9

Why is it much more expensive to insert an element at the beginning of an array list than at the beginning of a linked list?

• Answer: In a linked list, one merely updates references to the first and second node––a constant cost that is independent of the number of elements that follow. In an array list of size n, inserting an element at the beginning requires us to move all n elements.

Self Check 16.10

What is the efficiency of adding an element exactly in the middle of a linked list? An array list?

• Answer: It is O(n) in both cases. In the case of the linked list, it costs O(n) steps to move an iterator to the middle.

Self Check 16.11

Suppose we insert an element at the beginning of an array list, and the internal array must be grown to hold the new element. What is the efficiency of the add operation in this situation?

• Answer: It is still O(n). Reallocating the array is an O(n) operation, and moving the array elements also requires O(n) time.

Self Check 16.12

Using big-Oh notation, what is the cost of adding an element to an array list as the second-to-last element?

• Answer: O(1)+. The cost of moving one element is O(1), but every so often one has to pay for a reallocation.

Implementing Stacks and Queues

• Stacks and queues are abstract data types.
• We specify how operations must behave.
• We do not specify the implementation.
• Many different implementations are possible.

Stacks as Linked Lists

• A stack can be implemented as a sequence of nodes.
• New elements are “pushed” to one end of the sequence, and they are “popped” from the same end.
• Push and pop from the least expensive end - the front.
• The push and pop operations are identical to the addFirst and removeFirst operations of the linked list.

Stacks as Linked Lists

Figure 10 Push and Pop for a Stack Implemented as a Linked List

section_3_1/LinkedListStack.java

Stacks as Arrays

• A stack can be implemented as an array.
• Push and pop from the least expensive end - the back.
• The array must grow when it gets full.
• The push and pop operations are identical to the addLast and removeLast operations of an array list.
• push and pop are O(1)+ operations.


Figure 11 A Stack Implemented as an Array

Queues as Linked Lists

• A queue can be implemented as a linked list:
  • Add elements at the back
  • Remove elements at the front.
  • Keep a reference to last element
• The add and remove operations are O(1) operations.

Queues as Linked Lists

Figure 12 A Queue Implemented as a Linked List

Queues as Circular Arrays

• In a circular array, we wrap around to the beginning after the last element.
  
• When removing elements of a circular array,
  • increment the index at which the head of the queue is locate
• When the last element of the array is filled,
  • Wrap around and start storing at index 0
  • If elements have been removed there is room
  • else reallocate
• All operations except reallocating are independent of the queue size
  • O(1)
• Reallocation is amortized constant time
  • O(1)+

Queues as Circular Arrays

Self Check 16.13

Add a method peek to the Stack implementation in Section 16.3.1 that returns the top of the stack without removing it.

• Answer:

  public Object peek()
  {
     if (first == null)
     {
        throw new NoSuchElementException();
     }
     return first.data;
  }

Self Check 16.14

When implementing a stack as a sequence of nodes, why isn't it a good idea to push and pop elements at the back end?

• Answer: Removing an element from a singly-linked list is O(n).

Self Check 16.15

When implementing a stack as an array, why isn’t it a good idea to push and pop elements at index 0?

• Answer: Adding and removing an element at index 0 is O(n).

Self Check 16.16

What is wrong with this implementation of the empty method for the circular array queue?

public boolean empty()
{
   return head == 0 && tail == 0;
}

• Answer: The queue can be empty when the head and tail are at a position other than zero. For example, after the calls q.add(obj) and q.remove(), the queue is empty, but head and tail are 1.

Self Check 16.17

What is wrong with this implementation of the empty method for the circular array queue?

public boolean empty()
{
   return head == tail;
}

• Answer: Indeed, if the queue is empty, then the head and tail are equal. But that situation also occurs when the array is completely full.

Self Check 16.18

Have a look at the growIfNecessary method of the CircularArrayQueue class. Why isn’t the loop simply:

for (int i = 0; i < elements.length; i++)
{
   newElements[i] = elements[i];
}

• Answer: Then the circular wrapping wouldn't work. If we simply added new elements without reordering the existing ones, the new array layout would be
  

Implementing a Hash Table

• In the Java library sets are implemented as hash sets and tree sets.
• Hashing: place items into an array at an index determined from the element.
• Hash code: an integer value that is computed from an object,
  • in such a way that different objects are likely to yield different hash codes.
• Collision: when two or more distinct objects have the same hash code.
• A good hash function minimizes collisions.
• A hash table uses the hash code to determine where to store each element.

Implementing a Hash Table

Hash Tables

• Hash table: An array that stores the set elements.
• Hash code: used as an array index into a hash table.
• Simplistic implementation
• Very large array
• Each object at its hashcode location
  
• Simple to locate an element
• But not practical


Figure 14 A Simplistic Implementation of a Hash Table

Hash Tables

• Realistic implementation:
  • A reasonable size array.
  • Use the remainder operator to calculate the position.

  int h = x.hashCode();
  if (h < 0) { h = -h; }
  position = h % arrayLength;

  • Use separate chaining to handle collisions:
    • All colliding elements are collected in a linked list of elements with the same position value
    • The lists are called buckets
  • Each entry of the hash table points to a sequence of nodes containing elements with the same hash code.
  • A hash table can be implemented as an array of buckets— sequences of nodes that hold elements with the same hash code.


Figure 15 A Hash Table with Buckets to Store Elements with the Same Hash Code

Hash Tables

 

 

Elements with the same hash code are placed in the same bucket.

Implementing a Hash Table - Finding an Element

• Algorithm to find an element, obj
  • Compute the hash code and compress it.
    • gives an index h into the hash table.
  • Iterate through the elements of the bucket at position h.
    • Check element is equal to obj.
  • If a match is found among the elements of that bucket ,
    • obj is in the set.
    • Otherwise, it is not
• If there are no or only a few collision:
  • adding, locating, and removing hash table elements takes O(1) time.

Adding and Removing Elements

• Algorithm to add an element:
  • Compute the compressed hash code h.
  • Iterate through the elements of the bucket at position h.
    • For each element of the bucket, check whether it is equal to obj
  • If a match is found among the elements of that bucket, then exit.
  • Otherwise, add a node containing obj to the beginning of the node sequence.
  • If the load factor exceeds a fixed threshold, reallocate the table.
• Load factor: a measure of how full the table is.
  • The number of elements in the table divided by the table length
• Adding an element to a hash table is O(1)+

Adding and Removing Elements

• Algorithm to remove an element:
  • Compute the hash code to find the bucket that should contain the object
  • Try to find the element
  • If it is present:
    • remove it
    • otherwise, do nothing
  • Shrink the table if it becomes to sparse
• Removing an element from a hash table is O(1)+

Iterating over a Hash Table

• When iterator points to the middle of a node chain,
  • easy to get the next element
• When the iterator is at the end of a node chain,
  • Skip over empty buckets
  • Advance the iterator to the first node of the first non-empty bucket
• Iterator needs to store the bucket number and a reference to the current node in the node chain.

if (current != null && current.next != null)
{
   current = current.next; // Move to next element in bucket
}
else // Move to next bucket
{
   do
   {
      bucketIndex++;
      if (bucketIndex == buckets.length)
      {
         throw new NoSuchElementException();
      }
      current = buckets[bucketIndex];
   }
   while (current == null);
}

Iterating over a Hash Table

Figure 16 An Iterator to a Hash Table

Hash Table Efficiency

• The cost of iterating over all elements of a hash table:
  • Is proportional to the table length
  • Not the number of elements in the table
• Shrink the table when the load factor gets too small.
• One iteration is O(1).
• Iterating over the entire table is O(n).
  

section_4/HashSet.java

section_4/HashSetDemo.java

Program Run:

• Harry
  Sue
  Nina
  Susannah
  Larry
  Eve
  Sarah
  Adam
  Juliet
  Katherine
  Tony
  

Self Check 16.19

If a hash function returns 0 for all values, will the hash table work correctly?

• Answer: Yes, the hash set will work correctly. All elements will be inserted into a single bucket.

Self Check 16.20

If a hash table has size 1, will it work correctly?

• Answer: Yes, but there will be a single bucket containing all elements. Finding, adding, and removing elements is O(n).

Self Check 16.21

Suppose you have two hash tables, each with n elements. To find the elements that are in both tables, you iterate over the first table, and for each element, check whether it is contained in the second table. What is the big-Oh efficiency of this algorithm?

• Answer: The iteration takes O(n) steps. Each step makes an O(1) containment check. Therefore, the total cost is O(n).

Self Check 16.22

In which order does the iterator visit the elements of the hash table?

• Answer: Elements are visited by increasing (compressed) hash code. This ordering will appear random to users of the hash table.

Self Check 16.23

What does the hasNext method of the HashSetIterator do when it has reached the end of a bucket?

• Answer: It locates the next bucket in the bucket array and points to its first element.

Self Check 16.24

Why doesn’t the iterator have an add method?

• Answer: In a set, it doesn’t make sense to add an element at a specific position.