recursion colin capham recursion – an introduction to the concept

RecursionColin Capham

Recursion – an introductionto the concept


Aim: • To understand the use of a simple recursion technique

Learning Objectives:

• To be able to explain simple recursion principles• Describe the process of “Divide and Conquer”• To understand basic “sub-problems”• Be able to solve an example of the “Towers of Hanoi”


What is Recursion?

Recursion is a common form of the general-purpose problem-solving technique called “divide and conquer”'.

1. divide P into several sub-problems, P1,P2,...Pn.

The principle of divide-and-conquer, is that you solve agiven problem P in 3 steps:

2. solve, however you can, each of the subproblems to get solutions S1...Sn.

3. Use S1...Sn to construct a solution to the original problem, P.


Lets look at an everyday example:

Suppose you want to carry a large pile of books to your car!

P = carry a pile of 50 books from your office to your car

Well, you can't even lift such a big pile, never mind carry it.


P1 contains 10 books;

So you split the original pile into 3 piles:

P2 contains 15 books;

P3 contains 25 books.

Solution…


Your plan, of course, is to carry each pile to the car separately,and then put them back together into one big pile once theyare all there.

This is a recursive “divide-and-conquer” strategy;you have divided the original problem P into 3 problemsof the same kind.

Now suppose you can directly solve P1 and P2 - they are small enough that you can carry them to the car.

So, P1 and P2 are now sitting out there by the car. Once you get P3 there, you will pile them up again!!


But P3 is too big to be carried.

You apply the same strategy recursively:divide P3 into smaller problems,solve them, and combine the resultsto make the solution for P3.

You divide P3 into two piles,P3a has 11 books, P3b has 14,and carry these separately to the car…


Then you put P3a back onto P3b to form the pile P3 there at the car.

P1 P2 P3

P3a

P3b

Now you have solved all the subproblems: P1, P2, P3 are sitting at the car.


You stack them into one big pile, and presto,you have solved the original problem.


Quick recap:

The principle of divide-and-conquer, is that you solve agiven problem P in 3 steps:

1. divide P into several sub-problems, P1,P2,...Pn.

2. solve, however you can, each of the subproblems to get solutions S1...Sn.

3. Use S1...Sn to construct a solution to the original problem, P.

Recursion is a common form of the general-purpose problem-solving technique called “divide and conquer”'.


Lets now look at another example of using recursion to solve a problem…

The Towers of Hanoi

An old legend tells of a Hindu templewhere the pyramid puzzle might havebeen used for the mental disciplineof young priests using recursion tosolve a problem.


The Towers of Hanoi is a puzzle, consisting of a collection of ringsof different sizes, and three posts mounted on a base.

At the beginning all the rings are on the left most post as shown,and the goal is to move them all to the rightmost post, by movingone ring at a time from one post to another.

The Towers of Hanoi

But, at no time may a larger ring be placed on top of a smaller one!


Student Activity:

Solve the Towers of Hanoi puzzle by breaking down the problem intosub-problems, and transfer your 4 disks from point A to point C.

Time limit: 5 mins

Who can solve the problem in the minimum amount of moves?

The objective of the puzzle is to move the entire stack at A to C,

obeying the following rules:

1. Only one disk may be moved at a time.

2. Each move consists of taking the upper disk from one of the pegs and placing it onto another peg, on top of the other disks that may already be present on that peg.

3. No disk may be placed on top of a smaller disk


Solution

A B C

15 Moves


Questions?

recursion colin capham recursion – an introduction to the concept

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