Lesson 21: Recursion vs' Iteration

1
Lesson 21 Recursion vs. Iteration

• Objectives
• Distinguish between an iterative algorithm and a
  recursive algorithm
• Write recursive algorithms for performing
  mathematical tasks

• Outline
• Recursive algorithms
• Iterative algorithms

2
Recursive Functions

• An algorithm is called recursive if it solves a
  problem by reducing it to an instance of the same
  problem with a lower input size.
• Such an algorithm must include a BASE CASE
• Such an algorithm must include a rule for taking
  a standard (non-base case) input and breaking it
  down into a smaller problem.

3
Recursive Functions

• //computes an recursively
• int pow (int a, int n)

4
Greatest Common Divisor

• GCD(a,b) a lt b
• Standard List all of the factors of each a, b.
• Recursive algorithm

5
Recursive calculation of factorial

• factorial n! n(n-1)(n-2)...1
• int rfact (int x)

6
Iteration

• Iteration involves sequentially calculating from
  the ground up and usually involves for loops.
• int ifact (int x)

7
Counting (chapter 4)

• Importance of counting
• Allocation of Resources
• Probability
• Security
• Collisions

8
Counting

• Product Rule
• If there are n ways to make choice 1 and m ways
  to make choice 2, then the total number of
  choices is
• mn

9
Computer passwords

• If lowercase alphabetic characters are allowed,
  how many 8-character passwords are there?
• If upper- and lower-case and numeric characters
  are allowed, how many 10-character passwords are
  there?

10
MAC Addresses

• Ethernet addresses consist of a 48-bit MAC
  address hard-coded into each network card. How
  many possible unique cards can be produced with
  this system?
• If the first 24 bits uniquely identifies the
  vendor of the card (which it does), how many
  cards can be associated with each vendors ID?
• How should 3Com respond to this?

11
License Plates

• Assume each state follows this format for their
  license plates
• AIB 346
• how many license plates could be generated
  throughout the US?

12
Sum Rule

• If a first task can be done n ways and a second
  task m ways, but the tasks cannot be done at the
  same time, there are nm choices.

13
Sum Rule

• A character used in a certain font can either be
  an upper-case Latin character, lower-case English
  character, upper-case Greek character, or a
  Hiragana character.
• How many possible characters are there?