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1

Chapter 2

Objectives

Upon completion you will be able to:

• Explain the difference between iteration and recursion

• Design a recursive algorithm

• Determine when an recursion is an appropriate solution

• Write simple recursive functions

Recursion

Data Structures: A Pseudocode Approach with C , Second Edition

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How to writing repetitive algorithms

Iteration

Recursion

Is a repetitive process in which an algorithm calls itself.


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2-1 Factorial - A Case Study

We begin the discussion of recursion with a case

study and use it to define the concept.

This section also presents an iterative and a

recursive solution to the factorial algorithm.

• Recursive Defined

• Recursive Solution


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Iterative

The definition involves only the algorithm parameter(s) and not the algorithm itself.


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Recursive

A repetitive algorithm use recursion whenever the algorithm appears within the definition itself.


6Data Structures: A Pseudocode Approach with C , Second Edition

7

Note that

Recursion is a repetitive process in which an algorithm called itself.


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Which code is simpler?

Which one has not a loop?


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Calling a recursive algorithm


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2-2 Designing Recursive Algorithms

In this section we present an analytical

approach to designing recursive algorithms.

We also discuss algorithm designs that are not

well suited to recursion.

• The Design Methodology

• Limitation of Recusion

• Design Implemenation


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The Design Methodology

Every recursive call either solves a part of the problem or it reduce the size of the problem.


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The Design Methodology

Base case

The statement that “solves” the problem.

Every recursive algorithm must have a base case.

General case

The rest of the algorithm

Contains the logic needed to reduce the size of the problem.


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Rules for designing a recursive algorithm

1. Determine the base case.

2. Determine the general case.

3. Combine the base case and the general cases into an algorithm.


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Combine the base case and the general cases into an algorithm

Each call must reduce the size of the problem and move it toward the base case.

The base case, when reached, must terminate without a call to the recursive algorithms; that is, it must execute a return.


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Limitations of recursion

You should not use recursion if the answer to any of the following questions is no:

1. Is the algorithm or data structure naturally suited to recursion?

2. Is the recursive solution shorter and more understandable?

3. Does the recursive solution run within acceptable time and space limits?


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Design implementation – reverse keyboard input


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data=6

data=20

data=14

data=5

※ 請注意 print data

在什麼時候執行


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2-3 Recursive Examples

Slide 05

Four recursive programs are developed and

analyzed. Only one, the Towers of Hanoi, turns

out to be a good application for recursion.

• Greatest Common Divisor

• Fiboncci Numbers

• Prefix to Postfix Conversion

• The Towers of Honoi


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GCD design

otherwise

aif

bif

bab

b

a

ba 0

0

)mod,gcd(

),gcd(

Greatest Common Divisor Recursive Definition


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Pseudocode


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GCD C implementation


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gcd(10, 25)

gcd(25, 10)

gcd(10, 5)

gcd(5, 0)


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Another G.C.D. Recursive Definition


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Fibonacci numbers

Mentioned a problem FIBONACCI:A pair of rabbits, a month later able to produce a pair of rabbits, and newborn rabbit have in a month after fertility, but also gave birth to rabbits.Start from a pair of rabbits, a year later there will be a number of rabbits?


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Fibonacci numbers


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Fibonacci numbers -

Each number is the sum of the previous two numbers.

The first few numbers in the Fibonacci series are

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …


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Fibonacci numbers


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Fibonacci numbers


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Fibonacci numbers – an example


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0 represents .T.


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(Continued)


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Analysis


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We omit

Prefix to Postfix Conversion

The Towers of Honoi


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HW2

Write a recursive algorithm to calculate the combination of n objects taken k at a time.

Due date : (甲：941012、乙：941012)


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2-3 Recursive Examples

Slide 05

Four recursive programs are developed and

analyzed. Only one, the Towers of Hanoi, turns

out to be a good application for recursion.

•The Towers of Honoi


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Towers of Hanoi Problem:

Invented by French mathematician Lucas in 1880s.

Original problem set in India in a holy place called Benares.

There are 3 diamond needles fixed on a brass plate. One needle contains 64 pure gold disks. Largest resting on the brass plate, other disks of decreasing diameters.

Called tower of Brahma.


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Priests are supposed to transfer the disks from one needle to the other such that at

no time a disk of larger diameter should sit on a disk of smaller diameter.

Only one disk can be moved at a time.

Later setting shifted to Honoi, but the puzzle and legend remain the same.

How much time would it take? Estimate…..


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Today we know that we need to have

264-1


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Recursive Towers of Hanoi Design

Find a pattern of moves.

Case 1:

move one disk from source to destination needle.


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Move two disks

Case 2:

Move one disk to auxiliary needle.

Move one disk to destination needle.

Move one disk to from auxiliary to destination needle.


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Move three disks

Case 3:

Move two disks from source to auxiliaryneedle.

Move one disk from source to destinationneedle.

Move two disks from auxiliary to destination needle.


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(Continued)

Move two disks from source to auxiliary needle.

Move two disks from auxiliary to destination needle.


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Algorithm Tower of Hanoi

Towers (numDisks, source, dest, auxiliary)

numDisks is number of disks to be moved

source is the source tower

dest is the destination tower

auxiliary is the auxiliary tower


Data Structures: A Pseudocode Approach with C 49

Generalize Tower of Hanoi

General case

Move n-1 disks from source to auxiliary.

Base case

Move one disk from source to destination.

General case

Move n-1 disks from auxiliary to destination.

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