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RECURSION

Annie Calpe

11.12.04

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Overview

Introduction Review: the Basics

How it works - Examples- Factorial- Fibonacci Sequence- Sierpinski Curve

Stacks Applications Design Considerations

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Introduction to Recursion

Recursion can be used to manage

repetition.

Recursion is a process in which amodule achieves a repetition of

algorithmic steps by calling itself.

Each recursive call is based on a

different, generally simpler, instance.

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Introduction to Recursion

Recursion is something of a divide andconquer, top-down approach to problem

solving.

- It divides the problem into pieces orselects out one key step, postponing therest.

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4 Fundamental Rules :

1. Base Case: Always have at least one casethat can be solved without recursion.

2. Make Progress: Any recursive call mustprogress towards a base case.

3. Always Believe: Always assume therecursive call works.

4. Compound Interest Rule: Never duplicatework by solving the same instance of aproblem in separate recursive calls.

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Basic Form :

void recurse (){

recurse (); //Function calls itself}

int main ()

{recurse (); //Sets off the recursion

}

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How does it work?

1. The module calls itself.

2. New variables and parameters are allocatedstorage on the stack.

3. Function code is executed with the newvariables from its beginning. It does not makea new copy of the function. Only thearguments and local variables are new.

4. As each call returns, old local variables andparameters are removed from the stack.

5. Then execution resumes at the point of the

recursive call inside the function.

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Recursion Trees

A key tool for analyzing recursive algorithmsis the recursion tree.

The total processing time is related to thetotal # of nodes

The necessary storage space is related to itsheight

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To Build a Recursion Tree:

root = the initial call

Each node = a particular callEach new call becomes a child of the nodethat called it

A tree branch (solid line) = a call-return pathbetween any 2 call instances

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Factorial

Factorial (n):

IF (n = 0)

RETURN 1ELSE

RETURN n * Factorial (n-1)

Calculates n*(n-1)*(n-2)**(1)*(1)

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Fibonacci Numbers

Fibonacci (n)

IF (n

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Recursion Tree showing

Fibonacci calls

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Space Filling Curves

A continuous mapping from a lower-dimensionalspace into a higher-dimensional one, using fractals.

Fractals are shapes that occur inside other, similarshapes.

A useful property of a space-filling curve is that ittends to visit all the points in a region once it hasentered that region.

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The Sierpinski Curve

The limiting curve of an infinite sequenceof curves numbered by an index n=1,2,3

It ends up covering every point in theregion.

Fills 2-D space (fills a plane using lines)

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The Sierpinski Curve

ZIG (n):

if (n = 1)

turn left, advance 1turn left, advance 1

else

ZIG (n/2)ZAG (n/2)

ZIG (n/2)

ZAG (n/2)

ZAG (n):

if (n = 1)

turn right, advance 1turn right, advance 1

turn left

else

ZAG (n/2)ZAG (n/2)

ZIG (n/2)

ZAG (n/2)

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ZIGZAG

ZIG

ZAG

ZIG (4)

ZIG (2)

ZIG

(1)

ZAG(1)

ZIG

(1)

ZAG(1)

ZIG(4) Complete

** End of first ZIG (2) call

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ZIG(4) Complete

ZIG (4)

ZIG (2)

ZIG(1)

ZAG(1)

ZIG(1)

ZAG(1)

ZAG(1)

ZIG(1)

ZAG(1)

ZAG(1)

ZAG (2) ZIG (2) ZAG (2)

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ZIGZAG

ZIG

ZAGZAG

ZAG

ZIGZAG

ZIG(4) Complete

** End of first ZAG (2) call

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ZIG(4) The Rest?

No need to go further. Why?

ANSWER: Because of Rule #3- Weve shown that the base case works as

well as the next case.

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Recursion is controlled in a computer bymeans of a pushdown stack.

A push = a new function callA pop = a completed execution of

a function call

Stack overflow is possible

Run-time Stack Use

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Some Uses For Recursion

Numerical analysis

Graph theory

Symbolic

manipulation

Sorting

List processing

Game playing

General heuristic

problem-solving

Tree traversals

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Why use it?

PROS Clearer logic

Often more compact code

Often easier to modify

Allows for complete analysis of runtime

performance

CONS Overhead costs

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Summary

Recursion can be used as a very powerfulprogramming tool

There is a tradeoff between time spentconstructing and maintaining a program andthe cost in time and memory of execution.

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References

The New Turing Omnibus Dewdney

Data Structures & Problem Solving in Java

Weiss Computing and Algorithm Shackelford

Recursion Tutorial National University ofIreland, Dept of I.T.

http://geminga.it.nuigalway.ie/cai_tutor/recursio.htmhttp://geminga.it.nuigalway.ie/cai_tutor/recursio.htm