Download Recursion - WordPress.com

Stacks & Recursion
Slide : 1
CS 201: Data Structures & Algorithms
Mehreen Saeed
Spring 2012
Stack
• Linear data structure
• Can only be accessed from one end
• Very useful when data has to stored and
then retrieved in the reverse order.
Slide : 2
CS 201: Data Structures & Algorithms
Mehreen Saeed
Spring 2012
LIFO : Last In First Out
Stack
10
6
9
Slide : 3
CS 201: Data Structures & Algorithms
Mehreen Saeed
Spring 2012
Stacks
10
Stack
6
9
LIFO : Last In First Out
Slide : 4
CS 201: Data Structures & Algorithms
Mehreen Saeed
Spring 2012
Operations
•
•
•
•
•
Clear()
IsEmpty()
IsFull()
Push(el)
Pop()
• TopEl()
removing it
:Clear the stack
: Check to see if stack is empty
: Check to see if stack is full
: Put an element (el) on the stack
: Remove the topmost element from the stack &
return its value
: Return the topmost element from the stack without
Find out what is overflow and underflow???
Slide : 5
CS 201: Data Structures & Algorithms
Mehreen Saeed
Spring 2012
Operations
By Adam Drozdek
Slide : 6
CS 201: Data Structures & Algorithms
Mehreen Saeed
Spring 2012
RECURSIVE STRUCTURE
Sierpinski_Triangle from www.wikipedia.org
Slide : 7
CS 201: Data Structures & Algorithms
Mehreen Saeed
Spring 2012
Recursive Definition
•
•
•
•
•
An object is defined in terms of smaller version of itself
A problem is decomposed into simpler subproblems of the same kind
Recursion is the process in which repetition in a “self-similar” manner
Generally used to define functions or sequences of numbers
Has two parts
– Base Case / Anchor Case / Ground Case
Define the basic building blocks
– Rules / General Case
Define how objects / elements can be built from basic elements or already
constructed objects
• Serves two purposes
– Generating new elements
– Testing whether an element belongs to a set
Slide : 8
CS 201: Data Structures & Algorithms
Mehreen Saeed
Spring 2012
Examples
Factorials
The set of natural numbers
1.
2.
1.
2.
3.
n! = 1
if n=0
n! = n(n-1)! if n> 0
This definition will evaluate the factorial
of a number
Will generate
numbers
0N
If n  N then (n+1)  N
There are no other natural
numbers
This definition will generate natural
numbers
Will test if a
number is
member of series
Slide : 9
CS 201: Data Structures & Algorithms
Mehreen Saeed
Spring 2012
Recursive Function
Recursion means having the characteristic of coming up again and again
A function that invokes itself by making a call to itself is called a recursive
function
Direct Recursion: When a function calls itself
Indirect Recursion: When A calls B and B calls A
Activation Records/Stack Frame
The state of each function is characterized by an activation record. It is the private
pool of information for a function. It has the following information:
• Parameters and local variables: of the function
• Dynamic link: Link to the calling function’s activation record
• Return Address: Caller function’s return address
Runtime Stack
Runtime stack is a stack whose each element is the activation records of the
different functions being called during the running of the program
Slide : 10
CS 201: Data Structures & Algorithms
Mehreen Saeed
Spring 2012
Example
int factorial(n)
{
int fact = 1;
if (n==0)
return fact;
else
{
fact = n*factorial(n-1);
return fact;
}
}
void main()
{
int answer = factorial(3);
cout << answer;
}
Factorial(0) fact = 1
Factorial(1) fact = 1 * factorial(0)
fact = 1*1=1
Factorial(2) fact = 2 * factorial(1)
fact = 2 * factorial(1)
fact = 2 * 1=2
Factorial(3) fact = 3 * factorial(2)
fact = 3 * factorial(2)
fact = 3 * factorial(2)
fact = 3 * 2 = 6
answer = factorial (3)
answer = factorial (3)
answer = factorial (3)
answer = factorial (3)
Main()
answer = 6
Slide : 11
CS 201: Data Structures & Algorithms
Mehreen Saeed
Spring 2012
Towers of Hanoi
Watch animation with 4 disks!!!
Pictures and animation from: www.wikipedia.org
void Hanoi(int N,char src,char dest,char intermediate)
{
if (N>0)
{
//move top n-1 disks from ‘src’ to ‘intermediate’
//using ‘dest’as intermediate location
Hanoi(N-1,src,intermediate,dest);
//move the bottom disk from ‘src’ to ‘dest’
Move(src,dest);
//now move the rest of N-1 disks from ‘intermediate’
//to ‘dest’ using ‘src’ as intermediate location
Hanoi(N-1,intermediate,dest,src);
}
Slide : 12
CS 201: Data Structures & Algorithms
Mehreen Saeed
}
Spring 2012
How it is working??? Lets trace for
Hanoi(3,’A’,’C’,’B’)
Hanoi(1,A,B,C)
See animation!!!
Hanoi(0,A,C,B)
Move(A,B)
Hanoi(0,C,B,A)
Move(AC)
Move(A,C)
Hanoi(2,A,C,B)
Hanoi(1,B,C,A)
Hanoi(3,A,B,C)
Hanoi(0,B,A,C)
Move(B,C)
Hanoi(0,A,C,B)
Move(B,C)
Move(A,B)
Move(A,B)
Hanoi(1,B,A,C)
Hanoi(2,B,C,A)
Sequence of
moves:
Move(A,B)
Hanoi(0,B,C,A)
Move(B,A)
Hanoi(0,C,A,B)
Move(B,C)
Move(B,C)
Hanoi(1,A,C,B)
Move(B,A)
Hanoi(0,A,B,C)
Move(A,C)
Hanoi(0,B,C,A)
Move(A,C)
Slide : 13
CS 201: Data Structures & Algorithms
Mehreen Saeed
Spring 2012
Tail Recursion
• Use of only one recursive call at the very end of the
function implementation. Useful in logic programming
languages like prolog.
Non-Tail Recursion
• When the recursive call is NOT tail recursion
Slide : 14
CS 201: Data Structures & Algorithms
Mehreen Saeed
Spring 2012
More examples, which you can
try…
• Fibonacci Series
• Linear search
• Binary search
Slide : 15
CS 201: Data Structures & Algorithms
Mehreen Saeed
Spring 2012