CS141: Topics for Week 13: Nov 27 & 29, Dec 1 2017

Computer Science Headlines: ACM Tech News

Reading:

• Chp 12 Object Oriented Design
• Chap 13 Recursion

Lab and Other Activities

• Lab 10: Final Project

Recursion

• What is recursion?
  • A recursive program is a program which calls itself, usually it calls itself on a problem with smaller size.
  • Be careful - if not done correctly, this can lead to infinite recursion!

• Related to mathematical induction:

  • To prove a statement for a problem of size N:
    • Base case: Prove the problem for small N (usually 0 or 1)
    • Induction step:
      • Assume true for size N-1
      • Prove it is true for size N
  • Example: use method to prove T(N)= 1 + 2 + 3 + ... + N = N (N+1)/2 for all N
    • Base case: T(1) = 1 = 1 (1+1)/2
    • Induction step: T(N) = N + T(N-1) = N + (N-1)(N)/2 = ... = N (N+1)/2
      • Assume true for T(N-1)
      • Prove it is true for size N-1

• Problems which lend themselves to recursive solutions are generally problems with the properties:

  • A solution for the problem of a given size (e.g. n) can be expressed simply in terms of the same problem with smaller size, e.g. n-1
  • There is a size, called the "base case", which is trivial to solve. This serves as the condition for stopping the recursion.
  • Program solutions usually have the form

    myFunc(n)
    if n is the base case, return the answer
    else apply the recursion
    

• Examples: Printing an array, calculating factorial or the Fibonacci Series

  • Factorial n! = 1*2*3* ... *(n-1)*n
    What is the base case and what is the recursion?
  • Other examples can be seen on lab for CS-145.
• Fractals and recursion
  • fractals have the property of self-similarity.
  • Recursive images can generate images with this property. For example, see the sierpinski triangle