University of Virginia Computer Science
CS216: Program and Data Representation, Spring 2006
 (none)
06 February 2006

CS216 Notes Section 3 (6 February 2006)

Recursive Definitions

To solve a problem recursively:

  1. Be optimistic.
  2. Think of the simplest version of the problem, something you can already solve. (This is the base case.)
  3. Combine them to solve the problem.
Recusive Definitions on Lists

To solve a problem involving list data recursively:

  1. Be very optimistic. Since lists themselves are recursive structures, we should be able to come up with a recursive definition for almost anything we can think of doing with a list.
    1. Assume we can solve a smaller version of the problem.
    2. Break the problem into the first element of the list and the rest of the list.
  2. Think of the simplest version of the problem, something you can solve already. For lists, this is usually the empty list. (Sometimes, it might be the length 1 list.)
  3. Combine them to solve the problem.
The PS2 LinkedList.py implmentation is more complex than we'd like, because we want to deal with empty lists instead of just using None to represent them (this would break Python's type checking). So, we'll consider procedures for the ListNode class. The LinkedList procedure would just need to call these for the self.__node.

Many recursive list procedures can be defined with this template: 

def listproc (lst):
  if lst.__next == None:
      what to do for a one element list
  else:
      f (g(lst[0]), lst.__next.listproc ()
For example: 
    def length (self):
        if self.__next == None:
            return 1
        else:
            return 1 + self.__next.length ()
What are f and g to match the template?

Finish the sumlist procedure defintion below. It should take a ListNode object as its parameter and evaluate the the sum of the values of all elements in the list. 
    def sumlist (self):
        if self.__next == None:

            return ________________________________________
        else:

            return ______________________ + self.__next.________________________
Define the filterMatches procedure that takes a ListNode and an element, and evaluates to the list containing all elements from the original list that do not match el. 
    def filterMatches (self, el):    
       if self.__next == None:
            if self.__info == el:
 
                  __________________________________
            else:
 
                  return self
       else:
           if self.__info == el:
                return filterMatches (self.__next, el)
           else:
Recusive Definitions on Trees

To solve a problem involving trees recursively:

  1. Be extremely optimistic. Since trees themselves are recursive structures, we should be able to come up with a recursive definition for almost anything we can think of doing with a tree.
    1. Assume we can solve a smaller version of the problem.
    2. Break the problem into the root node and the children of that node.
  2. Think of the simplest version of the problem, something you can solve already. For trees, this is usually a leaf or the empty tree.
  3. Combine them to solve the problem.
Finish this definition that counts the number of nodes in a tree: 
   def size(self):
       count = 1  # count this node
       for child in children(self):

           count += child.____________________

       return count

Finish this definition that finds the maximal value in a tree: 
   def largest(self):

       __________________________________
       for child in children(self):

           if child.____________________ > max:
                max = child.largest()

       return max
What is the running time of largest defined above?




Define a procedure that implements largest whose running time is in O(n): 

   def largest (self):

Finding All Possible Lists

Finding all possible lists for a set of elements is much easier than finding all possible trees, but many of the same ideas apply.

What are the possible lists that can be created using the elements {1, 2, 3}?




Define a generator that generates all possible lists (here we use the Python list datatype, not the list types from PS2). (If you are stuck, see a partial solution on the next page.) 

def allLists (elements):
    if len(elements) == 1:

        yield ___________________
    else:
        for index in range(len(elements)):
            # pick this element first
            first = elements[index]

	    # rest is a copy of the original list with first removed
            rest = elements[:]
            del rest[index]

            assert (len(rest) == len(elements) - 1)

            for rlist in allLists (rest):
                if not rlist == None:

		    ____________________________
                    yield rlist
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