6-Python-Recursion.pdf

Recursive Functions
Recall factorial function:
Iterative Algorithm
Loop construct (while)
can capture computation in a
set of state variables that
update on each iteration
through loop

Recursive Functions
Alternatively:
Consider
5! = 5x4x3x2x1
can be re-written as 5!=5x4!
In general n! = nx(n-1)!
factorial(n) = n * factorial(n-1)

Recursive Functions
Alternatively:
Consider
5! = 5x4x3x2x1
can be re-written as 5!=5x4!
In general n! = nx(n-1)!
factorial(n) = n * factorial(n-1)
Recursive Algorithm
function calling itself

Recursive Functions
Known
Base case

Recursive Functions
Base case
Recursive step

Recursive Functions
•  No computation in first
phase, only function calls
•  Deferred/Postponed
computation
– after function calls
terminate, computation
starts
•  Sequence of calls have
to be remembered Execution trace for n = 4
fact (4)
4 * fact (3)
4 * (3 * fact (2))
4 * (3 * (2 * fact (1)))
4 * (3 * (2 * (1 * fact (0))))
4 * (3 * (2 * (1 * 1)))
4 * (3 * (2 * 1))
4 * (3 * 2)
4 * 6
24
Courtesy Prof PR Panda CSE Department IIT Dehi

Another Example (Iterative)
Source:https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-0001-introduction-to-computer-science-and-
programming-in-python-fall-2016/lecture-slides-code/
Iterative

Another Example (Recursive)
Iterative Algorithm
Recursive

Recursive Functions
•  Size of the problem reduces at each step
•  The nature of the problem remains the same
•  There must be at least one terminating condition
•  Simpler, more intuitive
–  For inductively defined computation, recursive algorithm
may be natural
•  close to mathematical specification
•  Easy from programing point of view
•  May not efficient computation point of view

GCD Algorithm
gcd (a, b) =
b if a mod b = 0
gcd (b, a mod b) otherwise
Iterative
Algorithm
Recursive Algorithm

GCD Algorithm
gcd (a, b) =
b if a mod b = 0
gcd (b, a mod b) otherwise
gcd (6, 10)
gcd (10, 6)
gcd (6, 4)
gcd (4, 2)
2
2
2
2

Fibonacci Numbers
fib (n) =
0 n = 1
1 n = 2
fib (n-1) + fib (n-2) n > 2
Recursive Algorithm

Fibonacci Numbers
fib (6)
fib (5)
fib (4)
fib (3) fib (2)
fib (2) fib (1)
fib (3)
fib (2) fib (1)
fib (4)
fib (3) fib (2)
fib (2) fib (1)

Power Function
•  Write a function power (x,n) to compute the
nth power of x
power(x,n) =
1  if n = 0
x * power(x,n-1) otherwise

Power Function
•  Efficient power function
•  Fast Power
–  fpower(x,n) = 1 for n = 0
–  fpower(x,n) = x * (fpower(x, n/2))2 if n is odd
–  fpower(x,n) = (fpower(x, n/2))2 if n is even

Power Function
•  Efficient power function

Towers of Hanoi Problem
•  64 gold discs with different
diameters
•  Three poles: (Origin, Spare, Final)
•  Transfer all discs to final pole
from origin
–  one at a time
–  spare can be used to temporarily
store discs
–  no disk should be placed on a
smaller disk
•  Intial Arrangement:
–  all on origin pole, largest at bottom,
next above it, etc.
Origin Spare Final

3-Step Strategy
Origin Spare Final Origin Spare Final
Origin Spare Final Origin Spare Final
Solve for (n-1) disks. Move to Spare.
Use Final as Spare.
Move bottom disk to Final
Move (n-1) disks to Final.
Use Origin as Spare.

Recursive Solution
•  Use algorithm for (n-1) disks to solve n-disk problem
•  Use algorithm for (n-2) disks to solve (n-1) disk problem
•  Use algorithm for (n-3) disks to solve (n-2) disk problem
•  ...
•  Finally, solve 1-disk problem:
–  Just move the disk!

Recursive Solution

6-Python-Recursion.pdf

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