# 3.2 Recursion  (Page 2/2)

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The basic notions of creating a recursive algorithm on a composite design pattern structure are

• The abstract superclass or interface of the data structure has the invariant abstract behavior of being able to perform the algorithm (and thus the computations associated with it).
• Each concrete subclass has its own implementation of that abstract behavior, which is just the variant part of the algorithm that pertains to that particular subclass.

This is the Interpreter Design pattern. Notice that no checks of the type of data being processed (e.g. base case or inductive case) are necessary. Each data object knows intrinsically what it is and thus what it should do. This is called "polymorphic dispatching" when an abstract method is called on an abstract data object, resulting in a particular concrete behavior corresponding to the concrete object used. In other words, we call a method on a list, but get the behavior of an empty list if that what the list is, or we get the behavior of a non-empty list if that is what the list is.

In order to prove that a recursive algorithm will eventually complete, one must show that every time the recursive call is made, the "problem" is getting "smaller". The "problem" is usually the set of possible objects that the recursive call could be called upon. For instance, when recursively processing a list, every call to the rest of the list is calling on a list that is getting progessively shorter. At times, one cannot prove that the problem is definitely getting smaller. This does not mean that the algorithm will never end, it just means that there is a non-zero probability that it will go on forever.

One of the key aspects of a recursive algorithm is that in the inductive case, the inductive method makes the recursive call to another object's method. But in doing so, it has to wait for the called method to return with the needed result. This method that is waiting for the recursive call to return is called a " pending operation ". For instance, at the time the empty list (base case) is reached during a recursive algorithm on a list, every non-empty node in that list has a pending operation.

## Animated recursion demo

Below is an example of generally what is happening in four linked objects during the call to the recursive method of the first object:

## Tail recursion

Consider the problem of finding the last element in a list. Again we need to interpret what it means to be the last element of (a) the empty list and (b) a non-empty list.

• Last element of the empty list: the empty list has no element; so there is no such thing as the last element for the empty list. How do we represent the non-existence of an object? For now, we use a key word in Java called `null` . `null` is a special value in Java that can be assigned to any variable of `Object` type to signify that the variable is not referencing any `Object` at all.
• Last element of a non-empty list with first and rest: it all depends on rest! rest has the capability to tell whether or not first is the last element of the current list. When rest is empty, then first is the last element. When rest is not empty, the first is certainly not the last element. rest has its own fist in this case, and it's up to the rest of rest to determine whether or not this new first is the last element! It's recursion again, isn't it?

To recapitulate, here is how a list can find its own last element.

• empty list case: return null or throw an exception to signify there is no such element.
• non-empty list case: pass first to rest and ask rest for help to find the last element.

How does rest use the first element of the enclosing list to help find the last element of the enclosing list?

• empty list case: the first element of the enclosing list is the last element.
• non-empty list case: recur! Pass its own first to its rest to help find the last element.

Here is the code.

 ```/** * Represents the abstract list structure.*/ public interface IList {/** * Returns the last element in this IList.*/ Object getLast();/*** Given the first of the preceding list, returns the last element of the preceding list. * @param acc the first of the preceding list.*/ Object getLastHelp(Object acc);}```
 ```/** * Represents empty lists.*/ public class MTList implements IList {// Singleton Pattern public static final MTList Singleton= new MTList(); private MTList() {}/** * Returns null to signify there is* no last element in the empty list. */public Object getLast() { return null;} /*** Returns acc, because being the * first element of the preceding* list, it is the last element. */public Object getLastHelp( Object acc) {return acc; }}``` ```/** * Represents non-empty lists.*/ public class NEList implements IList {private Object _first; private IList _rest;public NEList(Object f, IList r) { _first = f;_rest = r; }/** * Passes first to rest and asks for* help to find the last element. */public Object getLast() { return _rest.getLastHelp(_first);} /*** Passes first to rest and asks for * help to find the last element.*/ public Object getLastHelp (Object acc) { return _rest. getLastHelp (_first); }}```

The above algorithm to compute the last element of a list is another example of forward accumulation. Note that in the above, `getLast` is not recursive while `getLastHelp` is recursive. Also note that for the `NEList` , the last computation in `getLastHelp` is a recursive call to `getLastHelp` on `_rest` . There is no other computation after the recursive call returns. This kind of recursion is called tail recursion . Tail recursion is important for program performance. A smart compiler can recognize tail recursion and generate code that speeds up the computation by bypassing unnecessary setup code each time a recursive call is made.

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where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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