WEEK10: recursion
---------------------------------------------------------------
 M: recursive functions

recursion is when a function calls itself to get it's work done

recursion is most useful when the algorithm is one that repeats
something over and over, each time on a smaller set of the initial
data (ex: binary search)

MERGE SORT:

 given a list...
 if it has more than 1 item in it:
    split it into to lists
    sort each list
    merge them back together

 last week we wrote the merge(L1,L2,L3) function.
 this week we are writing the mergeSort() function.

 How should we do the "sort each list" step above?

 IDEA: let's call mergeSort() to do that step!!!

 So here it is...

def mergeSort(L):

  n = len(L)
  if n > 1:
    half = n/2
    L1 = L[0:half]    # use slicing to split list into two
    L2 = L[half:]
    mergeSort(L1)     # use another call to this function to
    mergeSort(L2)     # sort each sub-list!!
    merge(L1,L2,L)

 Think about the stack and how each called function is placed
 on top of the stack. There is nothing wrong with a function
 calling itself, as long as it doesn't repeat forever.


FACTORIAL EXAMPLE:

  5! = 5*4*3*2*1
  6! = 6*5*4*3*2*1 or 6*5!
  7! = 7*6!

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

math definition:

 n! = 1 if n is 0, n*(n-1)! if n is greater than 0

**easy** to express this using recursion:

def factorial(n):
  if n == 0:
    return 1
  else:
    return n * factorial(n-1)

 - try running /home/jk/inclass/factorialstack.py to see the stack
   for the recursive factorial function:

$ python factorialstack.py

please enter a positive integer...
           n: 4
-------------------------------
  in factorial...stacknum = 1, n = 4
  calling 4 * factorial(3)
    in factorial...stacknum = 2, n = 3
    calling 3 * factorial(2)
      in factorial...stacknum = 3, n = 2
      calling 2 * factorial(1)
        in factorial...stacknum = 4, n = 1
        calling 1 * factorial(0)
          in factorial...stacknum = 5, n = 0
          returning 1
        returning 1 
      returning 2 
    returning 6 
  returning 24 
factorial(n): 24

 - try running a recursive program in the python tutor:

      http://www.pythontutor.com


REVERSE A STRING:

 - can you write a recursive function to reverse a string?

$ python revstr.py 
  string: we love computer science
  reversed: ecneics retupmoc evol ew

 Hint: if you have a string, s, and a working recursive revstr 
 function, what will revstr(s[1:]) + s[0] give you??

 What is the base case/stop-the-recursion condition for this one??

def revstr(str):
  """ Use recursion to reverse a given string.  """
  if len(str) <= 1:
    return str
  else:
    return revstr(str[1:]) + str[0]

 and just for fun, here is the non-recursive version:

def NRrevstr(str):
  """ Non-Recursive version """
  rs = ""
  for i in range(len(str)-1, -1, -1):
    rs = rs + str[i]
  return rs


GRAPHICS:

 - can you write a recursive function to draw concentric circles?



   here's how it would be called from main: 

    reccircle(cenpt, size, color, win)

   your function should draw a circle at the given center point,
   with the given size and color, then call itself again with a
   smaller size.

   when should the recursion end? 


def reccircle(p, radius, color, w):
  """ draw recursive circles """
  if radius > 1:
    c = Circle(p, radius)
    c.setOutline(color)
    c.draw(w)
    reccircle(p, radius*0.45, color, w)

So far all of the recursive functions we have looked at could easily be written without recursion. How about the following image? Can you think of ways to create it with and without recursion?