Recursion

• This is a programming technique
• I should have covered this in 230, but ...
• A recursive function is a function that calls itself.
• Sometimes we do this just for algorithm development
• Other times, it is for implementation.
• Our goal is to understand recursion and the cost of recursion.
• First some guidance
  • To be successful a recursive algorithm or function must
    • Have a base case where recursion stops.
    • Have a recursive call, where the function calls itself.
    • Make progress towards a base case.
  • When doing a recursive design the following is true
    • Solve part of the problem then use recursion to solve the rest.
    • Or, I could do this if I only had my current routine implemented to solve this sub case.
• A first easy problem computing n!
  • The mathematical definition for n! is sometimes given as
  • This is only defined for positive integer n.
  • $$ n! = \left\{ \begin{array}{lr} n*(n-1)! & n > 1\\ 1 & n =0,1 \\ \end{array} \right. $$
  • Note this matches both cases above.
    • The base case is when n =0 or 1, the answer is 1.
    • The recursive case is n * (n-1)!
    • Progress is n-1 which gets us closer to 0 or 1.
    • And I could find n! if I just had a way to find (n-1)!
  • The implementation
  • Trace this.
• Another easy problem is integer power. ($a^b$) where b is an integer
  • Can you give a recursive definition?
  • Can you give a recursive function?
• How does this work?
  • Each time a function is called, a activation record or frame is placed on the stack.
  • In recursion this happens.
  • This means, there is a unique copy of each local variable, paramter, and other information.
  • This is slower (a hand full of instructions) than a loop.
  • And it takes more memroy.
• Let's look at a classical recursive algorithms
  • Binary Search
    • I know you did an iterative version of this.
    • What is the non-recursive algorithm?
    • What recursive algorithm can we build?
      • What would the recursive case be? (This has two)
      • What wouuld the base case be? (This has two).
• Linked list functions are naturally recursive.
  • Design a function to print a linked list.
  • Design a function to copy a linked list.
• Flood Fill is natural, multi recursive function.
  • See this javascript demo
• We will introduce many recursive algorithms in 385.