fundamentals of programming (python)...

Fundamentals of Programming(Python)

Recursion

Ali TaheriSharif University of Technology

Spring 2018

Outline1. Recursive Functions

2. Example: Fibonacci

3. Memoization

4. Example: Binomial Coefficient

5. Example: Towers of Hanoi

6. Example: Fast Exponentiation

2ALI TAHERI - FUNDAMENTALS OF PROGRAMMING [PYTHON]Spring 2018

Recursive Function

A function which calls itself is referred to as a Recursive function◦ Example: the factorial function

𝑛! = 𝑛 ∗ 𝑛 − 1 ! 𝑖𝑓 𝑛 > 01 𝑜𝑡ℎ𝑒𝑟𝑤ℎ𝑖𝑠𝑒

A recursive function always consists of two parts◦ Base Case: one or more cases for which no recursion is applied

◦ Recursive Step: all chains of recursion eventually end up in one of the base cases

The concept is very similar to mathematical induction


Example: Fibonacci

The Fibonacci sequence:◦ 0,1,1,2,3,5,8,13,21,34,55,89,134

𝐹𝑖𝑏 𝑛 =

0 𝑖𝑓 𝑛 = 01 𝑖𝑓 𝑛 = 1

𝐹𝑖𝑏 𝑛 − 1 + 𝐹𝑖𝑏 𝑛 − 2 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒


def fib(n):

if n == 0: # Base Case

return 0


return 1

return fib(n-1) + fib(n-2) # Recursion Step

Example: Binomial Coefficient

The binomial coefficient counts the number of ways to select 𝑘 items out of 𝑛:

𝑛

𝑘=

𝑛

𝑘 − 1+

𝑛 − 1

𝑘 − 1𝑛

0=

𝑛

𝑛= 1


def bin(n,k):

if k == 0 or k == n: # Base Cases

return 1

return bin(n-1,k) + bin(n-1,k-1) # Recursion Step

Example: Towers of Hanoi

An elegant application of recursion is to the Towers of Hanoi problem:◦ There are three pegs and a pyramid-shaped stack of 𝑛 disks

◦ The goal is to move all disks from the source peg into the target peg with the help of the third peg under some constraints



Constraints:◦ Only one disk may be moved at a time

◦ A disk may not be “set aside”. It may only be stacked on one of the three pegs

◦ A larger disk may never be placed on top of a smaller one.




def hanoi(n, source, target, third):

if n == 1: # Base case (we have only one disk)

print('Moving disk %d from %s to %s'

% (n, source, target))

else: # Recursive Step

# Move n-1 disk from source to third

hanoi(n - 1, source, third, target)

# Move nth disk from source to target

print('Moving disk %d from %s to %s'

% (n, source, target))

# Move n-1 disk from third to target

hanoi(n - 1, third, target, source)

Some Problems


Problem #1 Assume a table 𝑇 having following value in cell (𝑖, 𝑗):

𝑇(𝑖, 𝑗) =

𝑗 𝑖 = 1

𝑘=1

𝑗

𝑇(𝑖 − 1, 𝑘) 𝑖 > 1

Write a function to calculate 𝑇(𝑖, 𝑗) recursively.

Problem #2We have a game board having ∞ rows and 𝑁 columns. Having a marble in row 1 and column 𝑐 (cell (1, 𝑐)).

In each phase, we can move the marble to the next row of current column or one of the columns beside.

Write a recursive function to compute the number of ways we can move the marble to cell (𝑖, 𝑗).

Memoization

Recursive step often computes similar results multiple times which is inefficient

We can save partial results for further use◦ This technique is called memoization


memo = {} # Partial results

def fib(n):


return 0


return 1

if n in memo: # Return the result if

return memo[n] # it has already computed

memo[n] = fib(n-1) + fib(n-2) # Save the result

return memo[n]

Example: Fast Exponentiation

Given a matrix 𝐴, we want to calculate 𝐴𝑛

◦ Suppose we have written a function mat_prod to calculate the product of two matrices

◦ We can use mat_prod 𝑛 times to calculate 𝐴𝑛, but it is inefficient for large 𝑛

◦ Divide and conquer technique can lead to a more efficient solution

𝐴𝑛 =

𝐴𝑛/2 × 𝐴𝑛/2 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛

𝐴 𝑛/2 × 𝐴 𝑛/2 × 𝐴 𝑖𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑𝐼 𝑖𝑓 𝑛 = 0


Example: Fast Exponentiation


def exp(A, n):


return [[1, 0],

[0, 1]]

temp = exp(A, n // 2) # A**(n/2)

prod = mat_prod(temp, temp) # A**n

if n % 2 == 0:

return prod

else:

return mat_prod(prod, A)

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