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
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
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10-2
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
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Recursive Algorithm
if this is a simple case
solve it
else
redefine the problem using recursion
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Figure 10.1 Splitting a Problem into Smaller
Problems
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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.
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Figure 10.2 Recursive Function multiply
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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.
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Figure 10.3 Thought Process of Recursive
Algorithm Developer
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Figure 10.4 Recursive Function to Count a
Character in a String
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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
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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.
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Figure 10.5 Trace of Function multiply
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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
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Figure 10.6 Function reverse_input_words
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Figure 10.7 Trace of reverse_input_words(3) When
the Words Entered are "bits" "and" "bytes"
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Figure 10.8 Sequence of Events for Trace
of reverse_input_words(3)
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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
?
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Parameter And Local Variable Stacks (2/5)
Before the second call to reverse_input_words
n
word
3
bits
After second call to reverse_input_words
n
word
2
3
?
bits
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10-19
Parameter And Local Variable Stacks (3/5)
Before the third call to reverse_input_words
n
word
2
3
and
bits
After third call to reverse_input_words
n
1
2
3
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word
?
and
bits
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Parameter And Local Variable Stacks (4/5)
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
1
2
3
bytes
and
bits
The function return pops both stacks
After first return
n
word
2
3
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and
bits
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Parameter And Local Variable Stacks (5/5)
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
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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.
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Figure 10.9 Recursive Function multiply with Print
Statements to Create Trace and Output from
multiply(8, 3)
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Figure 10.9 Recursive Function multiply with Print
Statements to Create Trace and Output from
multiply(8, 3) (cont’d)
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10.3 RECURSIVE MATHMETICAL
FUNCTIONS
• Many mathmatical functions can be defined
recursively.
• For example
– The factorial of a number n (n!)
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Figure 10.10 Recursive factorial Function
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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
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Figure 10.11 Trace of fact = factorial(3);
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Figure 10.12 Iterative Function factorial
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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
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Figure 10.13 Recursive Function fibonacci
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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
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Figure 10.14 Program Using Recursive
Function gcd
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Figure 10.14 Program Using Recursive
Function gcd (cont’d)
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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.
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CASE STUDY:
FINDING CAPITAL LETTERS IN A STRING(2/4)
Step 2: Analysis
– Problem Input
• char *str
– Problem Output
• char *caps
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CASE STUDY:
FINDING CAPITAL LETTERS IN A STRING(3/4)
Step 3: Design
– Algorithm
1. 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
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CASE STUDY:
FINDING CAPITAL LETTERS IN A STRING(4/4)
Step 4: Implementation (Figure10.15)
Step 5: Testing (Figure 10.16、Figure 10.17)
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Figure 10.15 Recursive Function to Extract
Capital Letters from a String
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Figure 10.16 Trace of Call to Recursive
Function find_caps
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Figure 10.17 Sequence of Events for Trace
of Call to find_caps from printf Statements
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CASE STUDY
RECURSIVE SELECTION SORT(1/4)
Step 1: Problem
– Sort an array in ascending order using a
selection sort.
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CASE STUDY
RECURSIVE SELECTION SORT(2/4)
Step 2: Analysis (Figure 10.18)
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Figure 10.18 Trace of Selection Sort
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CASE STUDY
RECURSIVE SELECTION SORT(3/4)
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])
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CASE STUDY
RECURSIVE SELECTION SORT(4/4)
Step 4: Implementation (Figure10.19)
Step 5: Testing
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Figure 10.19
Recursive
Selection
Sort
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Figure 10.19 Recursive Selection Sort (cont’d)
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10.5 PROBLEM SOLVING WITH RECURSION
CASE 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.
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CASE STUDY
OPERATIONS 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.
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CASE STUDY
OPERATIONS ON SETS(3/4)
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 ele
3. Answer is true
else
4. Answer depends on whether ele is in the rest
of set
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Step 3: Design (cont’s)
–
Algorithm for is_set(set)
1. if is_empty(set)
2. Answer is true
else if is_element(initial set character, rest of set)
3. Answer is false
else
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 true
else if initial character of sub is not an element of set
3. Answer is false
else
4. Answer depends on whether rest of sub is a subset of set
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Step 3: Design (cont’s)
– Algorithm for union of set1 and set2
1. 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
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Step 3: Design (cont’s)
– 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
else
3. Print initial element and a comma
4. print_with_commas the rest of set
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CASE STUDY
OPERATIONS ON SETS(4/4)
Step 4: Implementation (Figure10.20)
Step 5: Testing
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Figure 10.20 Recursive Set Operations on
Sets Represented as Character Strings
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Figure 10.20
Recursive
Set Operations
on Sets
Represented
as Character
Strings (cont’d)
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Figure 10.20
Recursive
Set Operations
on Sets
Represented
as Character
Strings (cont’d)
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Figure 10.20
Recursive
Set Operations
on Sets
Represented
as Character
Strings (cont’d)
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Figure 10.20
Recursive
Set Operations
on Sets
Represented
as Character
Strings (cont’d)
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Figure 10.20 Recursive Set Operations on
Sets Represented as Character Strings (cont’d)
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10.6 A CLASSIC CASE STUDY IN RECURSION
TOWERS 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
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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
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Figure 10.21 Towers of Hanoi
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Figure 10.22 Towers of Hanoi After Steps 1
and 2
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Figure 10.23 Towers of Hanoi After Steps 1,
2, 3.1, and 3.2
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TOWERS OF HANOI (3/)
Step 3: Design
– Algorithm
1. if n is 1 then
2. Move disk 1 from the from peg to the to peg
else
3. Move n-1 disks from the from peg to the
auxiliary peg using the to peg
4. Move disk n from the from peg to the to peg
5. Move n-1 disks from the auxiliary peg to the
to peg using the from peg
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TOWERS OF HANOI (4/4)
Step 4: Implementation (Figure10.24)
Step 5: Testing
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Figure 10.24 Recursive Function tower
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Figure 10.25 Trace of tower ('A', 'C', 'B', 3);
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Figure 10.26 Output Generated by tower
('A', 'C', 'B', 3);
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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
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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
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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
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Question?
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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.
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Figure 10.27 Grid with Three Blobs
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