Transcript What is Computer Science

Algorithmic Recursion
Recursion
Alongside the algorithm, recursion is one of the most
important and fundamental concepts in computer
science as well as many areas of mathematics.
Recursive solutions to problems tend be

simple

eloquent

concise
Definition:
A recursive algorithm is one that calls or refers to
itself within its algorithmic body.
Like other iterative constructs such as the FOR
loop and the WHILE loop a particular
condition needs to be met that prevents
the algorithm from executing indefinitely.
In recursive algorithms this condition is called
the base or end case.
The base case usually represents a lower
bound on the number of times the
recursive algorithm will call itself.
A recursive algorithm normally expects INPUT
parameters and returns a single RETURN
value.
Each subsequent recursive call of the algorithm
should pass INPUT parameters that are closer to
the base case
Recursion Example
The Fibonacci Sequence is powerful integer number
sequence which has many applications in
mathematics, computer science and even
nature ( the arrangements of flower petals,
spirals in pine cones all follow the Fibonacci
sequence ).
The Fibonacci sequence is defined as follows:
Fib(n) = Fib(n-1) + Fib(n-2) where N > 1
Fib(0) = Fib(1) = 1
The first ten terms in the sequence are
1,1,2,3,5,8,13,21,33,54
Fibonacci Sequence Module
{Module to compute the nth term in the Fibonacci sequence}
Module Name: Fib
Input:
n
Output:
nth term in sequence
Process:
IF (n<=1)
{base condition}
THEN
RETURN 1
ELSE
{recursive call}
RETURN Fib(n-1) + Fib(n-2)
ENDIF
Compute
fib_5
Fib(5)
Fib(5)
Fib(4)
Fib(3)
+
Fib(2) + Fib(1)
Fib(2)
+
Fib(3)
Fib(2)
Fib(1) + Fib(0)
+
Fib(1)
Fib(1) + Fib(0)
Fib(1) + Fib(0)
= 1+1+1+1+1+1+1+1 = 8
When looking at recursive solution to a
problem we try to identify two main
characteristics:
1.
Can we identify a solution where the
solution is based upon solving a simpler
problem of itself each time.
2.
Can we identify a base case where the
simpler solutions stop.
Most recursive modules (functions) take the
following form:
IF ( base case )
THEN
RETURN base_case solution
ELSE
RETURN recursive solution
ENDIF
Example:
Evaluate xy recursively
Firstly we need to answer the two questions.
(1)
(2)
Answer yes
xy = x * xy-1
Answer yes
x0 = 1
xy Recursive Module
{ Module to compute xy recursively }
Module Name: power
Input:
x,y
Output:
x to the power of y
Process:
IF (y=0)
{base condition}
THEN
RETURN 1
ELSE
{recursive call}
RETURN x*power(x,y-1)
ENDIF
Example:
Triangular values
Design an iterative algorithm to read a number from a user
and print it’s triangular value.
( if the number 5 is input its triangular value is
5+4+3+2+1 = 15 )
{Print triangular value of input number}
PRINT “Enter a number “
READ number
triangular_value  0
FOR ( value FROM 1 TO number )
triangular_value  triangular_value + value
ENDFOR
PRINT triangular_value
Now design a recursive module to solve the same
problem.
Step 1 – Answer questions
(1) Yes - tri(n) = n + tri(n-1)
(2) Yes - n = 1
Algorithm:
Module Name: tri
Inputs:
n
Outputs:
triangular value of n
Process:
IF ( n = 1 )
THEN
/* Base Case */
RETURN 1
ELSE
/* Recursive solution */
RETURN n + tri(n-1)
ENDIF