Chapter 10 Recursion Dr. Jiung-yao Huang Dept. Comm. Eng. Nat. Chung Cheng Univ. E-mail : [email protected] TA: 鄭筱親陳昱豪

Chapter 10Recursion

Dr. Jiung-yao HuangDept. Comm. Eng.Nat. Chung Cheng Univ.E-mail : [email protected]: 鄭筱親陳昱豪

Copyright ©2004 Pearson Addison-Wesley. All rights reserved. 10-2

Outline• 10.1 THE NATURE OF RECURSION• 10.2 TRACING A RECURSIVE FUNCTION• 10.3 RECURSIVE MATHMETICAL FUNCTIONS• 10.4 RECURSIVE FUNCTIONS WITH ARRAY AND

STRING PARAMETERS– CASE STUDY

• FINDING CAPITAL LETTERS IN A STRING– CASE STUDY

• RECURSIVE SELECTION SORT• 10.5 PROBLEM SOLVING WITH RECURSION

– CASE STUDY• OPERATIONS ON SETS

• 10.6 A CLASSIC CASE STUDY IN RECURSION– TOWERS OF HANOI

• 10.7 COMMON PROGRAMMING ERRORS


10.1 THE NATURE OF RECURSION

• Recursive function– Function that calls itself or that is part of a cycle in the

sequence of function calls• Simple case

– Problem case for which a straightforward solution is known

• Recursive solution characteristics– One or more simple cases of the problem have a

straightforward, nonrecursive solution– The other cases can be redefined in terms of problems

that are closer to the simple cases– By applying this redefinition process every time the

recursive function is called, eventually the problem is reduced entirely to simple cases, which are relatively easy to solve


Recursive Algorithm

if this is a simple case

solve it

else

redefine the problem using recursion


Figure 10.1 Splitting a Problem into Smaller Problems


EXAMPLE 10.1

• Figure 10.2 implements multiplication as the recursive C function multiply that returns the product m x n of its two arguments.

• The body of function multiply implements the general form of a recursive algorithm.

• The simplest case is reached when the condition n==1 is true.


Figure 10.2 Recursive Function multiply


EXAMPLE 10.2

• Develop a function to count the number of times a particular character appears in a string.

• Figure 10.3 shows thought process that fits into our generic else clause.


Figure 10.3 Thought Process of Recursive Algorithm Developer


Figure 10.4 Recursive Function to Count a Character in a String


10.2 TRACING A RECURSIVE FUNCTION

• Two types of tracing the execution of a recursive function– Tracing a recursive function that returns a value– Tracing a void function that is recursive


Tracing a recursive function that returns a value

• Activation frame– Representation of one call to a function

• Figure 10.5 shows three calls to function multiply.


Figure 10.5 Trace of Function multiply


Tracing a void function that is recursive

• Function reverse_input_words in Fig.10.6 is a recursive module that takes n words of input and prints them in reverse order.

• Terminating condition– A condition that is true when a recursive

algorithm is processing a simple case


Figure 10.6 Function reverse_input_words


Figure 10.7 Trace of reverse_input_words(3) When the Words Entered are "bits" "and" "bytes"


Figure 10.8 Sequence of Events for Trace of reverse_input_words(3)


Parameter And Local Variable Stacks (1/5)

• C uses the stack data structure to keep track of the values of n and word at any given point.

After first call to reverse_input_words

n word

3 ?



Before the second call to reverse_input_words

n word

3 bits

After second call to reverse_input_words

n word

3 bits2 ?



Before the third call to reverse_input_words

n word

bits32 and

After third call to reverse_input_words

n word

3 bits2 and1 ?



During this execution of the function, the word “bytes” is scanned and stored in word, and “bytes” is echo printed

immediately because n is 1 (a simple case)

n word

3 bits2 and1 bytes

The function return pops both stacksAfter first return

n word

3 bits2 and



Because control is returned to a printf call, the value of word at the top of the stack is then displayed.

Another return occurs, poping the stacks again.After second return

n word

3 bits


Implementation of Parameter Stacks in C

• System stack– Area of memory where parameters and local variables

are allocated when a function is called and deallocated when the function returns

• When and how to trace recursive functions– During algorithm development, it is best to trace a

specific case simply by trusting any recursive call to return a correct value based on the function purpose.

– Figure 10.9 shows a self-tracing version of function multiply as well as output generated by the call.


Figure 10.9 Recursive Function multiply with Print Statements to Create Trace and Output from multiply(8, 3)


Figure 10.9 Recursive Function multiply with Print Statements to Create Trace and Output from multiply(8, 3) (cont’d)


10.3 RECURSIVE MATHMETICAL FUNCTIONS

• Many mathmatical functions can be defined recursively.

• For example– The factorial of a number n (n!)


Figure 10.10 Recursive factorial Function


EXAMPLE 10.4

• Figure 10.11 shows the trace of fact=factorial(3)

• Figure 10.12 uses iterative version to solve the factorial of the number n


Figure 10.11 Trace of fact = factorial(3);


Figure 10.12 Iterative Function factorial


EXAMPLE 10.5

• The Fibonacci numbers are a sequence of numbers that have many varied uses.

• The Fibonacci sequence is defined as– Fibonacci1 is1

– Fibonacci2 is 1

– Fibonaccin is Fibonaccin-2 +Fibonaccin-1, for n>2


Figure 10.13 Recursive Function fibonacci


EXAMPLE 10.6

• Euclid’s algorithm for finding the gcd can be defined recursively– gcd(m,n) is n if n divides m evently– gcd(m,n) is gcd(n, remainder of m divided by n)

otherwise


Figure 10.14 Program Using Recursive Function gcd


Figure 10.14 Program Using Recursive Function gcd (cont’d)


10.4 RECURSIVE FUNCTIONS WITH ARRAY AND STRING PARAMETERS

CASE STUDY:FINDING CAPITAL LETTERS IN A STRING(1/4)

Step 1: Problem– Form a string containing all the capital letters

found in another string.


CASE STUDY:FINDING CAPITAL LETTERS IN A

STRING(2/4)Step 2: Analysis

– Problem Input• char *str

– Problem Output• char *caps



STRING(3/4)Step 3: Design

– Algorithm1. if str is the empty string

2. Store empty string in caps

(a string with no letters certainly has no caps)

else

3. if initial letter of str is a capital letter

4. Store in caps this letter and the capital letters

from the rest of str

else

5. Store in caps the capital letters from the rest of str



STRING(4/4)Step 4: Implementation (Figure10.15)

Step 5: Testing (Figure 10.16 、 Figure 10.17)


Figure 10.15 Recursive Function to Extract Capital Letters from a String


Figure 10.16 Trace of Call to Recursive Function find_caps


Figure 10.17 Sequence of Events for Trace of Call to find_caps from printf Statements


CASE STUDYRECURSIVE SELECTION SORT(1/4)

Step 1: Problem– Sort an array in ascending order using a

selection sort.



Step 2: Analysis (Figure 10.18)


Figure 10.18 Trace of Selection Sort



Step 3: Design– Recursive algorithm for selection sort

1. if n is 1

2. The array is sorted.

else

3. Place the largest array value in last array

element

4. Sort the subarray which excludes the last

array element (array[0]..array[n-2])



Step 4: Implementation (Figure10.19)

Step 5: Testing


Figure 10.19 Recursive Selection Sort


Figure 10.19 Recursive Selection Sort (cont’d)


10.5 PROBLEM SOLVING WITH RECURSIONCASE STUDY

OPERATIONS ON SETS(1/4)

Step 1: Problem– Develop a group of functions to perform the E

(is an element of), (is a subset of) and ∪(union) operations on sets of characters.

– Also develop functions to check that a certain set is valid (that is, that it contains no duplicate characters), to check for the empty set, and to print a set in standard set notation.

Set: any collection of distinct things considered as a wholeFrom wikipedia


CASE STUDYOPERATIONS ON SETS(2/4)

Step 2: Analysis – Character strings provide a fairly natural representation

of sets of characters.– Like sets, strings can be of varying sizes and can be

empty.– If a character array that is to hold a set is declared to

have one more than the number of characters in the universal set (to allow room for the null character), then set operations should never produce a string that will overflow the array.



Step 3: Design– Algorithm for is_empty(set)

1. Is initial character ‘\0’ ?– Algorithm for is_element(ele, set)

1. if is_empty(set)2. Answer is false

else if initial character of set matches ele3. Answer is true

else4. Answer depends on whether ele is

in the rest of set


Step 3: Design (cont’s)

– Algorithm for is_set(set)1. if is_empty(set)

2. Answer is trueelse if is_element(initial set character, rest of set)

3. Answer is falseelse

4. Answer depends on whether rest of set is valid set

– Algorithm for is_subset(sub, set)1. if is_empty(sub)

2. Answer is trueelse if initial character of sub is not an element of set

3. Answer is falseelse

4. Answer depends on whether rest of sub is a subset of set



– Algorithm for union of set1 and set21. if is_empty(set1)

2. Result is set2

else if initial character of sset1 is also an element of set2

3. Result is union of the rest of set1 with set2

else

4. Result includes initial character of set1 and the

union of the rest of set1 with set2



– Algorithm for print_set(set)1. Output a { .

2. if set is not empty, print elements seperated by commas.

3. Output a } .

– Algorithm for print_with_commas(set)1. If set has exactly one element

2. Print it

else3. Print initial element and a comma

4. print_with_commas the rest of set




Step 5: Testing


Figure 10.20 Recursive Set Operations on Sets Represented as Character Strings


Figure 10.20 Recursive Set Operations on Sets Represented as Character Strings (cont’d)










10.6 A CLASSIC CASE STUDY IN RECURSIONTOWERS OF HANOI (1/)

Step 1: Problem– Move n disks from peg A to peg C using peg B

as needed.– The following conditions apply

• 1. Only one disk at a time may be moved, and this disk must be the top disk on a peg

• 2. A larger disk can never be placed on top of a smaller disk


TOWERS OF HANOI (2/)

Step 2: Analysis – Problem inputs

• int n• char from_peg• char to_peg• char aux_peg

– Problem output• A list of individual disk moves


Figure 10.21 Towers of Hanoi


Figure 10.22 Towers of Hanoi After Steps 1 and 2


Figure 10.23 Towers of Hanoi After Steps 1, 2, 3.1, and 3.2


TOWERS OF HANOI (3/)

Step 3: Design– Algorithm

1. if n is 1 then2. Move disk 1 from the from peg to the to peg

else3. Move n-1 disks from the from peg to the

auxiliary peg using the to peg4. Move disk n from the from peg to the to peg5. Move n-1 disks from the auxiliary peg to the

to peg using the from peg


TOWERS OF HANOI (4/4)


Step 5: Testing


Figure 10.24 Recursive Function tower


Figure 10.25 Trace of tower ('A', 'C', 'B', 3);


Figure 10.26 Output Generated by tower('A', 'C', 'B', 3);


Comparison of Iterative and Recursive Functions

• Recursive– Requires more time and space because of extra

function calls– Is much easier to read and understand– To researchers developing solutions to the

complex problems that are at the frontiers of their research areas, the benefits gained from increased clarity far outweigh the extra cost in time and memory of running a recursive program


10.7 COMMON PROGRAMMING ERRORS

• A recursive function may not be terminate properly.– A run-time error message noting stack overflow or an

access violation is an indicator that a recursive function is not terminating

• Be aware that it is critical that every path through a nonvoid function leads to a return statement

• The recopying of large arrays or other data structures can quickly consume all available memory

• Introduce a nonrecursive function to handle preliminaries and call the recursive function when there is error checking


Chapter Review

• A recursive function either calls itself or initiates a sequence of function calls in which it may be called again

• Designing a recursive solution involves identifying simple cases that have straightforward solutions and then redefining more complex cases in terms of problems that are closer to simple cases

• Recursive functions depend on the fact that for each cal to a function, space is allocated on the stack for the function’s parameters and local variables


Question?


Programming Projects 1

• Develop a program to count pixels (picture elements) belonging to an object in a photograph.

• The data are in a two-dimensional grid of cells, each of which may be empty (value 0) or filled (value 1).

• The filled cells that are connected form a blob (an object).

• Figure 10.27 shows a grid with three blobs.• Include in your program a function blob_check that

takes as parameters the grid and the x-y coordinates of a cell and returns as its value the number of cells in the blob to which the indicated cell belongs.


Figure 10.27 Grid with Three Blobs

Documents

Chapter 10 Recursion Dr. Jiung-yao Huang Dept. Comm. Eng. Nat. Chung Cheng Univ. E-mail : [email protected] TA: 鄭筱親 陳昱豪

Chapter 10 Recursion Dr. Jiung-yao Huang Dept. Comm. Eng. Nat. Chung Cheng Univ. E-mail : [email protected] TA: 鄭筱親陳昱豪