1 programming for engineers in python autumn 2011-12 lecture 8: recursion

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Programming for Engineers in

Python

Autumn 2011-12

Lecture 8: Recursion

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Lecture 7: Highlights

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Lecture 6 (OOP): Highlights• Representation design options and implications (Rectangle)• Inheritance and Polymorphism• Rational - customize classes so that they are natural to use

• Attributes, methods, constructor• Method overriding (e.g., __str__)• Operators overloading (e.g., __add__)

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Variables in Functions• Defined only in the function’s scope• Are not available for other functions/shell (even function’s parameters)• Upon completion of execution:

•Variables are not defined anymore•Variables values are not kept for future invocations of the function

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The Call Stack

call stack

resultmain

base

exponent

res

ipower

= 2

= 20

= 3

= 15

= 5

= 17

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Towers of Hanoi

• Objective:• Move the entire stack of disks from rode S

(source) to T (target) using A (auxiliary)

S TA

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The Rules

• Only one disk may be moved at a time• Each move consists of taking the upper disk from one

of the rods and sliding it onto another rod• No disk may be placed on top of a smaller disk

S TA

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Examples

S TA

1 disk – very easy!

S TA

2 disks – very easy!

S TA3 disks – easy

S TA4 disks?

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Solution for 4 Disks

S TA

S TA

S TA

S TA

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For n Disks?

S TA

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Assume We Have a Solution for n-1 Disks

S TA

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Solution: Step 1

S TA

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Solution: Step 2

S TA

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Solution: Step 3

S TA

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How to Solve for n-1 Disks?

Easy!

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Algorithm for n Disks

S TA

1. If there is only a single disk – easy!2. Move n-1 disks from S to A (T is the auxiliary rode)3. Move disk from S to T4. Move n-1 disks from A to T (S is the auxiliary rode)

Let us write it in Python…

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A Python Program for Towers of Hanoi

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Remember the Calling Stack?

hanoi(3,'S','T','A')main

hanoi(2,'S',‘A',‘T')

hanoi(1,‘S',‘T',‘A')

And so on…

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Recursion

• We use Recursion to solve the Towers of Hanoi problem

• Recursion: a function whose implementation references itself

• Recursion enables to solve a “large” problem using solutions to “small” problems that assemble it

• In every recursive call the problem is reduced. When the problem is small enough -solve directly (base case, מקרה קצה)

• Divide and conquer

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Iterative Versus Recursive

Step by step (iteratively): 4! = 1*2*3*4 = 2*3*4 = 6*4 = 2424 = 2*2*2*2 = 4*2*2 = 8*2 = 16

n! = 1*2*3*….*nan = a*a*…..*a

n iterations

n! = n*(n-1)!0! = 1

Recursively: 4! = 4*3! = 4*(3*2!) = 4*(3*(2*1!)) = 4*(3*(2*(1*0!))) = 4*(3*(2*(1*1))) = 4*(3*(2*1)) = 4*(3*2) = 4*6 = 24

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Recursive Definition

n! = n*(n-1)!0! = 1

Base condition

קצה מקרה

Recursive call

Smaller instance

Factorial

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Pros and Cons

• Pros: short and natural for many problems• Cons: Computational inefficient, sometimes hard to

understand or inconvenient

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Recursion Example in Python

Halting condition (and infinite loops)

Advance towards base case

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Recursive factorial – step by step

factorial(4)n4

Returns…

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factorial(4)n4

Returns…4*…

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factorial(4)n4

Returns…

factorial(3)n3

Returns…

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factorial(4)n4

Returns…

factorial(3)n3

Returns…

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factorial(4)n4

Returns…

factorial(3)n3

Returns…3*…

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factorial(4)n4

Returns…

factorial(3)n3

Returns…

factorial(2)n2

Returns…

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factorial(4)n4

Returns…

factorial(3)n3

Returns…

factorial(2)n2

Returns…

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factorial(4)n4

Returns…

factorial(3)n3

Returns…

factorial(2)n2

Returns…2*…

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factorial(4)n4

Returns…

factorial(3)n3

Returns…

factorial(2)n2

Returns…

factorial(1)n1

Returns…

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factorial(4)n4

Returns…

factorial(3)n3

Returns…

factorial(2)n2

Returns…

factorial(1)n1

Returns…

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factorial(4)n4

Returns…

factorial(3)n3

Returns…

factorial(2)n2

Returns…

factorial(1)n1

Returns…1*…

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factorial(4)n4

Returns…

factorial(3)n3

Returns…

factorial(2)n2

Returns…

factorial(1)n1

Returns…

factorial(0)n0

Returns…

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factorial(4)n4

Returns…

factorial(3)n3

Returns…

factorial(2)n2

Returns…

factorial(1)n1

Returns…

factorial(0)n0

Returns…

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factorial(4)n4

Returns…

factorial(3)n3

Returns…

factorial(2)n2

Returns…

factorial(1)n1

Returns…

factorial(0)n0

Returns…

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factorial(4)n4

Returns…

factorial(3)n3

Returns…

factorial(2)n2

Returns…

factorial(1)n1

Returns…

factorial(0)n0

Returns… 1

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factorial(4)n4

Returns…

factorial(3)n3

Returns…

factorial(2)n2

Returns…

factorial(1)n1

Returns…1*1

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factorial(4)n4

Returns…

factorial(3)n3

Returns…

factorial(2)n2

Returns…2*1

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factorial(4)n4

Returns…

factorial(3)n3

Returns…3*2

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factorial(4)n4

Returns…4*6

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General Form of Recursive Algorithms

• Base case: small (non-decomposable) problem• Recursive case: larger (decomposable) problem• At least one base case, and at least one recursive

case.

test + base case

recursive case

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Short Summary

• Design a recursive algorithm by1. Solving big instances using the solution to smaller

instances2. Solving directly the base cases

• Recursive algorithms have1. Stopping criteria2. Recursive case(s)3. Construction of a solution using solution to smaller

instances

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Point for Attention: Memory

• Variables of a called function are kept in memory until the function returns

• Many recursive calls might fill the computers memory

• Even if we minimize the number of variables, recursion has its price

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Example: Fibonacci Series

• Fibonacci series0, 1, 1, 2, 3, 5, 8, 13, 21,

34• Definition:

• fib(0) = 0• fib(1) = 1• fib(n) = fib(n-1) + fib(n-2)

en.wikipedia.org/wiki/Fibonacci_number

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סלט פיבונאצ'י

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Recursive Fibonacci Series

Every call with n > 1 invokes 2 function calls, and so on…

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Redundant Calls

Fib(4)

Fib(3) Fib(2)

Fib(2) Fib(1) Fib(1) Fib(0)

Fib(1)

Fib(0)

Fib(5)

Fib(3)

Fib(2) Fib(1)

Fib(1)

Fib(0)

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Number of Calls to Fibonacci

n value Number of calls

1 1 1

2 1 3

3 2 5

23 28657 92735

24 46368 150049

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Demonstration: Iterative Versus Recursive Fibonacci

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Demonstration: Iterative Versus Recursive Fibonacci

Output (shell):

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Fibonacci: Recursion and Efficiency

• If a recursive function calls itself more than once it will be extremely inefficient

• Define it iteratively

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So Do We Really Need Recursion?

Yes!

(specific examples next week)

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Odd-Even (if time allows)

• Given a function ‘odd(n)’ • Odd n - return True, Even n – return False

• Write a function ‘even(n)’ that:• Even n - return True, Odd n – return False

• This is easy!

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Odd-Even

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Odd-Even

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Odd-Even

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Odd-Even

Questions?

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Off Topics• Documenting your programs• Find the error

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Documenting Your Programs

• An essential part of writing computer code• Understand what was your intension when

writing it• For others to be able to coordinate their

code with yours• For the grader/lecturer to try and understand

your code and grade it accordingly…

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Documenting Your Programs with Comments

• Comments: pieces of code not interpreted or executed by the computer

• One line comments start with the hash character (#)

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Meaningful Variables Names

• Variables names• Meaningful names

Off Topic: Find the Error

And Now?

INDENTATION IS REALLY, REALLY IMPORTANT!

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