MEIN 50010: Python Recursive Functions

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1 : Python Recursive Functions Fabian Sievers Higgins Lab, Conway Institute University College Dublin Wednesday,

2 Recursion / Timing

3 On the N-th Day of Christmas... On the 1st day of Christmas my true love gave to me A partridge in a Pear Tree On the 2nd day of Christmas my true love gave to me 2 Turtle Doves... and a partridge in a Pear Tree On the 3rd day of Christmas my true love gave to me 3 French Hens... and all the stuff I got on day 2 = day(3-1)

4 Factorial (Infinite Loop) 1/2 Factorial most famous example Needed in combinatorics: n! ways to arrange n objects n n! = n (n 1) 2 1 = i Iteration i=1 n! = n (n 1)! Recursion! (not quite right) Example 3! = 3 2! = 3 (3 1)! 3! = 3 2 1! = 3 2 (2 1)! 3! = ! = (1 1)! 3!? = (0 1)! 3!? = ( 1) ( 1 1)!..??? Need or exit condition!

5 Factorial () 2/2 Recursive Definition of Factorial n (n 1)! n > 0 n! = 1 n = 0 not defined n < 0 usually means a certain number attains biggest/smallest value but can be any condition that extracts you from infinite loop

6 Fibonacci () 1/3 Fibonacci Number F n is the sum of the two previous Fibonacci Numbers F n 1 + F n 2 Recursion (s): F 0 = 0, F 1 = 1 Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34,... Fibonacci Numbers occur in CS (compression, sorting, pseudo-random), Nature (rabbits, leaf arrangement) and in Dan Browne s da Vinci Code Example F 5 F 5 = F 4 + F 3 F 5 = (F 3 + F 2 ) + (F 2 + F 1 ) F 5 = ((F 2 + F 1 ) + (F 1 + F 0 )) + ((F 1 + F 0 ) + (F 1 )) F 5 = (((F 1 + F 0 ) + F 1 ) + (F 1 + F 0 )) + ((F 1 + F 0 ) + (F 1 ))! have to calculate F 2 three times!

7 Who Cares? 2/3 Operations take time If you can find a closed expression, go for it Timing: >>>import time >>>start = time.time() >>>Do Stuff >>>end = time.time() >>>elapsed = end - start addition additions N (N 1) additions N = N 2 (N + 1) 1 Addition, 1 Multiplication, 1 Division regardless of N Additions/subtractions are cheap, multiplications are the same or slightly more expensive, divisions are more expensive, function calls are really expensive

8 Pros/Cons 3/3 Advantages Decompose large task into smaller tasks, e.g., tree traversal, compression, nested loops, ray-tracing, XML, compiling, sorting Hide implementation details from user Disadvantages Time required to launch a function Stack space required for the function Stack space in Python much less than heap Depth of recursion limited (4096?)

9 Tower of Hanoi 1/2 Three rods, N disks of varying sizes Start with all disks on one rod, sorted by size Objective: move tower from one rod to another Move only one disk (top of stack) at a time No disk may be placed on top of smaller disk To move bottom disk must remove all other disks first

10 Tower of Hanoi 2/2 : move single disk from A to B Move pile of n disks from A to B move (n 1) (all but the bottom disk) to C ( A,B) move the bottom disk to B move (n 1) disks from C to B def move(n, from, to, via): if (n == 1): print (from, to), else: move(n-1, from, via, to) move( 1, from, to, via) move(n-1, via, to, from) move(3, 0, 1, 2)

11 Take-Home Message Recursive function calls incur greater over-head Recursion tends to be slower than iteration or closed expression Recursion needs extra work/implementation to speed up Recursion is elegant to design but difficult to debug Avoid recursion if iteration or closed expression possible! Some Problems can/should/must be solved using recursion if recursive formulation is clearer In order to understand recursion you first have to understand recursion!

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APCS-AB: Java. Recursion in Java December 12, week14 1

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