Recursive Linked Lists

An alternative way to define a linked list is to do so recursively .  This is the approach taken by some programming languages, such as Lisp and Scheme.  It can also be done in C++ or Java.

In this view, a list can be defined as follows.

definition  A list is either

a.  an empty list, containing no data items, or
b.  a first data item followed by a list containing the rest of its data items.


For example, in a list written

    (  x  y  z  w  )

first is x, and rest is (  y  z  w  ).

A class definition for a list defined in this way might start like this:

class RecursiveList implements List {
    Object  first;
    RecursiveList rest;
    .
    .
    .


Note that RecursiveList is defined recursively; that is, the definition contains a reference to the thing being defined.  That's frowned upon in English, but it is a powerful tool used by mathematicians and computer scientists.

This definition lends itself to writing list operations as recursive functions.

example  Write a method for RecursiveList which counts the number of elements in the list.

int size() {
    if ( isEmpty() )
        return 0;
    else
        return 1 + rest.size();
}

example  Write a method to print the contents of a RecursiveList.

void display(){
    if( !isEmpty() ){
        System.out.println(first.toString());
        rest.display();
    }
}

Q:  In what order will the items be displayed?  Would it be possible to display them in the reverse order?

example  Suppose that a RecursiveList is being used to hold a list of Integers.  Write a method which finds the sum of the elements in the list.

example  Given a list of strings, and a particular string to search for, search the list for occurrences of the string.  Return a count of the number of occurrences found.

Implementation

How can a recursively defined linked list be implemented in Java?

We start by defining a class with two instance variables:

class RecursiveList {
    Object  first;
    RecursiveList rest;


Add get and set methods for first and rest:

public Object getFirst(){ return first; }, etc.


When we write constructors for the list class, we have to decide some implementation details:  What does an empty list look like?  What does a list with one element look like?  In general, what is the physical structure of a valid linked list?  We can answer these questions as follows:

• The empty list is represented by a node whose first and rest are both null.
• In a list with one element, "rest" should be an empty list.
• In a list with n elements, "rest" should be a list of (n-1) elements.
• A list with n data items is represented by a linked list of n+1 nodes.  The first n nodes contain the n data items as their "first"s.  In the (n+1)-th node, both first and rest are null.

So we can write the following constructors:

RecursiveList() {        //  initializes an empty list
    first = rest = null;
}

RecursiveList(Object item){        //  initializes a list with one element
    first = item;
    rest = new RecursiveList();
}

RecursiveList(Object item, RecursiveList rest){  //  initializes a list with a given first element and rest-of list
    first = item;
    this.rest = rest;
}


It is possible to implement any list operation with these basic ones.  However, it is also useful to define a few additional operations

private void prepend(Object item){
    rest = new RecursiveList(first,rest);
    first = item;
}

private Object removeFirst(){
    if(isEmpty())
        return null;
    else {
        Object temp = first;
        first = rest.first;
        rest = rest.rest;
        return temp;
        }
}


We can then implement the methods of the list interface.  Here are a few examples:


public boolean add(Object item){
    if(isEmpty())
        prepend(item);
    else
        rest.add(item);
    return true;
}

public Object get(int index){
    if(isEmpty())
        throw new ArrayIndexOutOfBoundsException();
    else if(index==0)
        return first;
    else
        return rest.get(index-1);
}

public Object remove(int index){
    if(isEmpty())
        throw new ArrayIndexOutOfBoundsException();
    else if(index==0)
        return removeFirst();
    else
        return rest.remove(index-1);
}

public void add(int index, Object item){
    if(index==0)
        prepend(item);
    else if(isEmpty())
        throw new ArrayIndexOutOfBoundsException();
    else
        rest.add(index-1, item);
}