Introducing Recursion

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Introducing Recursion

• Date June 7th, 2002
• Author Steve Engels,
• University of Waterloo

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Introduction

• Towers of Hanoi
• Moving small towers is easy
• 1 disc 1 move
• 2 discs 3 moves
• 3 discs 7 moves
• 64 discs 1.8 x 1019
• Ouch. Is there a nice solution for large cases?

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Induction

• First, look at induction
• Inductive vs deductive proofs
• Deductive proof based on a series of logical
  steps
• Inductive proof of base case, then proof of
  general case (i.e. all cases bigger than base
  case)
• Intuition behind induction
• Show that the smallest case is true
• Show that each big case is true if the case
  smaller than it is true
• Therefore, if the smallest case is true, all
  cases bigger than it must also be true

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Induction Example

• Thm Given N people in a room, where each person
  shakes hands with every other person exactly
  once, show that the number of handshakes is
  n(n-1)/2
• Proof
• Induction involves a different kind of thinking
  breaking up large problems into smaller
  problems to solve them

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Recursive Definitions

• A recursive definition defines an object in terms
  of smaller objects of the same type.
• Includes base (degenerate) cases and recursive
  cases
• Example 1 Factorial function
• n! 1 if n0 (base case)
• n! n(n-1)! if ngt0 (recursive case)
• Example 2 Fibonacci numbers
• f0 f1 1 (base case)
• fn fn-1 fn-2 if n2 (recursive case)
• Example 3 Binary trees

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Recursive Programs

• Recall Fibonacci numbers. How would you
  calculate fn?
• Solution is defined in terms of solutions to
  smaller problems of the same type
• Example
• Recursive and iterative programs are related in
  the same way as inductive and deductive reasoning
  ? different kind of thinking

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Recursive Programs

• How do you build a recursive method?
• 5 steps
• Step 0 Analyse problem. See how to break up
  large problem into smaller problems
• Step 1 Base case
• Step 2 Break large case into smaller case(s)
• Step 3 Call method recursively on smaller
  case(s)
• Step 4 Use result of recursive call(s) to
  produce result for large case
• Step 5 Return result for large case

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Hanoi Revisited

• So how do we solve the Towers of Hanoi elegantly?
• Note Some solutions can be implemented both
  recursively and iteratively.

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Working the CrowdThe Need for Audience
Participation in University Teaching

• Date June 7th, 2002
• Author Steve Engels,
• University of Waterloo

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How to Keep Your Audience Awake

• Important factors in students
• Interest
• Awareness
• Consciousness
• Techniques for stimulating student interest
• Props
• Demonstrations
• Exercise
• Other techniques

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Using Props

• Replace verbal descriptions with tangible
  examples
• Useful when describing objects, structures, data,
  etc.
• Examples
• Arrays
• Stacks
• Queues
• Linked lists
• Circular
• Sublist
• Trees

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Using Demonstrations

• Replace descriptions of an action or process with
  a visual demonstration
• Illustrates motion, action, operation of
  props/objects
• Examples
• Linked list operations
• Activation stacks
• Binary search
• Tree traversals
• Sorting

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Using Exercises

• Test understanding of material through practice
• Ensure that material is understood, and reinforce
  applications of lesson
• Examples
• Reading parts of lecture material
• Polling for solution to problem
• Random questions
• Group work

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Other Techniques

• Every little bit helps
• Examples
• Names
• Feedback
• Sugar
• Projectiles