Nested and Excessive Recursion

1
Nested and Excessive Recursion

• Nested recursion
• Excessive Recursion
• Converting a recursive method to iterative.
• Iterative or Recursive?

2
Nested recursion

• When a recursive method has a parameter defined
  in terms of itself, it is said to be nested
  recursive.

3
Nested Recursion Example

• Another example of nested recursion is Ackerman
  formula
• A(m,n) n 1 if m 0
• A(m,n) A(m-1, 1) if m gt 0 n 0
• A(m,n) A(m-1, A(m, n-1)) if m gt 0 n gt 0
• It is interesting for computational reasons due
  to the depth of recursion and due to the nested
  recursion.
• The execution of Ackermann(3,2) produces
  recursion of depth 30.

4
Excessive Recursion

• Some recursive methods repeats the computations
  for some parameters, which results in long
  computation time even for simple cases
• This can be implemented in Java as follows
• int fib(int n)
• if (nlt2)
• return n
• else
• return fib(n-2)Fib(n-1)

5
Excessive Recursion Example

• To show how much this formula is inefficient, let
  us try to see how Fib(6) is evaluated.
• int fib(int n)
• if (nlt2)
• return n
• else
• return fib(n-2)Fib(n-1)

6
Excessive Recursion Example

• For caculating fib(6), the method is called 25
  times.
• The same calculation are repeated again and again
  because the system forgets what have been already
  calculated.
• The method is called 2fib(n1)-1 times to
  compute fib(n)
• 3,000,000 calls are needed to caculate the 31st
  element.

7
Iterative or Recursive?

• Fib can be written iteratively instead of
  recursievely as follows
• int iterativeFib(int n)
• if (nlt2)
• return n
• else
• int i2,tmp, current1, last 0
• for ( iltni)
• tmp current
• currentlast
• lasttmp
• return current
• The method loops (n-1) times making three
  assignments per iteration and only one addition.

8
Iterative or Recursive?
Assignments Assignments Number of Additions n
Recursive Iterative Number of Additions n
25 15 5 6
177 27 9 10
1973 42 14 15
21891 57 19 20
242785 72 24 25
2692537 87 29 30
9
Excessive Recursion Example

• How many combinations of members items can be
  taken from a set of group items, that is how to
  calculate C(group, members)?
• Base Case
• If members1, return group
• If membersgroup, return 1
• General Case
• If (groupgt membersgt1)
• return C(group-1,members-1)C(group-1,members)

10
Combination Example
11
Removing Recursion

• Some times we need to convert a recursive
  algorithm into an iterative one if
• The Language does not support recursion
• The recursive algorithm is expensive
• There are two general techniques for that
• Iteration
• Stacking

12
When to use Recursion

• The main two factors to consider are
• Efficiency
• Clarity
• Recursive methods are less efficient in time and
  space.
• Recursive methods are easier to write and to
  understand
• It is Good to use recursion when
• Relatively shallow
• Same amount of work as the nonrecursive verion
• Shorter and simpler than the nonrecursive solution