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1 CompSci 105 SS 2005 Principles of Computer Science Lecture 6: Recursion Lecturer: Santokh Singh Assignme nt 1 due tomorrow . Should have started working on it.
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1CompSci 105 SS 2005
Principles of Computer Science
Lecture 6: Recursion
Lecturer: Santokh Singh
Assignment 1 due
tomorrow. Should
have started
working on it.
2
Recursion:Compute using self-reference
Iteration: Compute using loops (for, while, etc.)
3
Definition: “A phone directory is a list of names and telephone numbers in alphabetical order by surname”
Searching a Phone Directory
“It’s easy if there’s only one name in the book!”
“But it’s hard if the book is any larger.”
4
Factorial
Factorial(n) = 1 x 2 x 3 x … n
Factorial(0) = 1
Iterative Definition
Factorial(n) = n * Factorial(n-1)
Factorial(0) = 1
Recursive Definition
Textbook, pp. 51-52
5
In Java
public static int fact( int n ) {
if ( n == 0 ) {
return 1;
}
else {
return n * fact( n-1 );
}
}Textbook, p. 53
6
Tracing Recursive Programs
Recursion for void methods
The Towers of Hanoi
7public static int fact( int n ) {
if ( n == 0 ) {
return 1;
}
else {
return n * fact( n-1 );
}
}
Textbook, p. 57-58
n = ?A: fact(n-1) = ?return = ?
A
For each method call, make a box with space for:
Values of method’s parameters and local variables
Return values of all called methods
Return value for this method
Box Tracing
8public static int fact( int n ) {
if ( n == 0 ) {
return 1;
}
else {
return n * fact( n-1 );
}
}
Textbook, p. 57-58
n = 3A: fact(n-1) = ?return = ?
A
9public static int fact( int n ) {
if ( n == 0 ) {
return 1;
}
else {
return n * fact( n-1 );
}
}
Textbook, p. 57-58
n = 3A: fact(n-1) = ?return = ?
n = 2A: fact(n-1) = ?return = ?
AA
10public static int fact( int n ) {
if ( n == 0 ) {
return 1;
}
else {
return n * fact( n-1 );
}
}
Textbook, p. 57-58
n = 1A: fact(n-1) = ?return = ?
n = 3A: fact(n-1) = ?return = ?
n = 2A: fact(n-1) = ?return = ?
A
A
A
11public static int fact( int n ) {
if ( n == 0 ) {
return 1;
}
else {
return n * fact( n-1 );
}
}
Textbook, p. 57-58
n = 0 A: fact(n-1) = ?return = ?
n = 1A: fact(n-1) = ?return = ?
n = 3A: fact(n-1) = ?return = ?
n = 2A: fact(n-1) = ?return = ?
A
A
A
A
12public static int fact( int n ) {
if ( n == 0 ) {
return 1;
}
else {
return n * fact( n-1 );
}
}
Textbook, p. 57-58
n = 0 A: fact(n-1) = ?return = 1
n = 1A: fact(n-1) = 1return = ?
n = 3A: fact(n-1) = ?return = ?
n = 2A: fact(n-1) = ?return = ?
A
A
A
A
13public static int fact( int n ) {
if ( n == 0 ) {
return 1;
}
else {
return n * fact( n-1 );
}
}
Textbook, p. 57-58
n = 0 A: fact(n-1) = ?return = 1
n = 1A: fact(n-1) = 1return = 1
n = 3A: fact(n-1) = ?return = ?
n = 2A: fact(n-1) = 1return = ?
A
A
A
A
14public static int fact( int n ) {
if ( n == 0 ) {
return 1;
}
else {
return n * fact( n-1 );
}
}
Textbook, p. 57-58
n = 0 A: fact(n-1) = ?return = 1
n = 1A: fact(n-1) = 1return = 1
n = 3A: fact(n-1) = 2return = ?
n = 2A: fact(n-1) = 1return = 2
A
A
A
A
15
Proving recursive algorithms
1. Prove each base cases works
2. Prove each recursive case works*
3. Prove all recursive calls meet a base casepublic static int fact( int n ) {
if ( n == 0 ) {
return 1;
}
else {
return n * fact( n-1 );
}
} Proof by inductionTextbook, p. 751-5
16
Tracing Recursive Programs
Recursion for void methods
The Towers of Hanoi
17
Reversing a String
Textbook, p. 59ff
god“ ”
18
Reversing a String
Textbook, p. 59ff
god“ ”• If string is empty
do nothing
19
Reversing a String
Textbook, p. 59ff
god“ ”• If string is empty
do nothing• Otherwise,
Output last character
Reverse substring of first (n-1) characters
20
Reversing a String
Formal Box TraceTextbook, p. 62
godreverse “ ”• If string is empty
do nothing• Otherwise,
Output last character
Reverse substring of first (n-1) characters
21
More than one way ….
Textbook, p. 63ff
• If string is empty
do nothing• Otherwise,
Reverse substring of last (n-1) characters
Output first character
22
Invariants
public static int fact( int n ) {
if ( n == 0 ) {
return 1;
}
else {
// Invariant:
return n * fact( n-1 );
}
}Textbook, p. 56
23
Loop invariants
Is there something that we want to be true every time the while test is executed?
// Computes the sum of
// the first n items.
int sum = 0;
int j = 0;
while (j < n) {
sum += item[j];
j++;
}
24
Loop invariants
sum contains the sum of the first j items
j is in the range 0..n-1
Textbook, p. 12
// Computes the sum of
// the first n items.
int sum = 0;
int j = 0;
while (j < n) {
sum += item[j];
j++;
}
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