CS 112 Introduction to Programming

1
CS 112 Introduction to Programming

• Lecture 32
• Recursive ProgrammingHttp//zoo.cs.yale.edu/clas
  ses/cs112/

2
Outline

• Recap
• Recursive programming
• Palindrome
• Change Maker
• Towers of Hanoi

3
Recap What is Recursion?

• A method in C can invoke itself if set up that
  way, it is called a recursive method
• Recursion follows the mathematical notion of
  induction
• there is a base case
• there is an inductive (recursion) step
• solve a smaller problem

4
Recap Recursive Sum
int Sum( int n ) if ( n 0 )
// base case return 0 else
// recursive case
return Sum( n-1 ) n
Sum(4)
Sum(0)
4
0
10
0
5
Recap Local and Formal Variables

• Like a normal method call, each invocation of a
  recursive method has its own set of method
  variables, i.e., formal parameters and local
  variables

void LotsOfPrints(int num) int local
2num if (num 0)
else
LotsOfPrints( num 1 )
Console.WriteLine( local )
LotsOfPrints(3) 2 4
6
6
Control Flow
void LotsOfPrints( int num ) int local
2num if (num 0) // base case
else // recursion
LotsOfPrints( num 1 )
Console.WriteLine( local )
LotsOfPrints(3)
2
4
6






LotsOfPrints(2)
LotsOfPrints(1)
LotsOfPrints(0)
WriteLine(local)
WriteLine(local)
WriteLine(local)




7
Exercise

• Without using any loop statement, write a method
  Power(int x, uint n) that raises integer x to the
  power of n, where n is a non-negative integer?
• Examples
• power(2, 1) 2
• power(2, 2) 4
• power(2, 10) 1024

8
Recursive Exponentiation
int Power( int x, uint n ) if ( n 0 )
// base case return 1 else
// recursive case
int temp Power( x, n/2 ) temp
temp if ( n 2 1 )
temp x return temp
See Power.cs
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Writing Recursive Programs

• Determine the structure of your method
• think of the sub-problem(s) that you may solve to
  solve the current problem
• make the definition of your method general enough
  to handle them so that you can use recursion
• i.e., the sub-problem can be solved using the
  same method with different parameters
• Determine the recursive case
• Determine the base case

10
Outline

• Review and admin.
• Palindrome
• Change maker
• Towers of Hanoi

11
Palindrome Problem

• A palindrome is a string that reads the same
  backward or forward
• Question how do you check a string to see
  whether or not it is a Palindrome?
• what is a sub-problem you can reduce your current
  problem to
• and what is the signature of the method to allow
  recursion?
• what is the recursive case?
• what is the base case?

x
y
z
a
z
y
x
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Palindrome

• CheckPalindrome(s, left, right)
• Recursive case
• CheckPalindrome (s, left 1, right - 1)
• Base case
• if left gt right, then true
• if sleft ! sright, then false

13
Palindrome Check
bool CheckPalindrome( string s, int left, int
right ) if ( left gt right ) //
base case 1 return true else if (
sleft ! sright ) // base case 2
return false else
// recursion return CheckPalindrome(
s, left 1, right 1 )
See Palindrome.cs
14
Outline

• Review and admin.
• Palindrome
• Change maker
• Towers of Hanoi

15
Change Maker

• Recursion is useful for trying all possibilities
• Consider the change making problem
• Count the number of ways to make change for a
  given amount with pennies, nickels, dimes,
  quarters
• Questions
• what are sub-problems you can reduce your current
  problem to
• and what is the signature of the method to allow
  recursion?
• what is the recursive case?
• what is the base case?

16
Change Maker Structure

• Order coin types in increasing order and number
  them
• Count( amount, numCoinTypes )
• number of ways to make amount cents using
  numCoinTypes coins where numCoinTypes could be
  0, 1, 2, 3, 4

17
Change Maker Recursive Case

• Question Count( 11, 2 ) ?
• Count( 11, 2 ) sum of
• Count( 6, 2 ) // use one nickel
• Count( 11, 1 ) // discard the nickel type
• Two paths
• use one more coin of the highest denomination,
  reduce amount and try to make change for
  remaining money
• or discard the highest denomination and try to
  make changes with remaining denominations

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Change Maker Recursive Case
3
11, 2
2
1
1
1
1
0
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Change Maker Recursive Case
4
15, 2
3
1
2
1
1
1
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Change Maker Base Case

• amount lt 0 numCoinTypes 0 0
• amount 0 numCoinTypes 1 1

21
Change Maker Code
int coins 1, 5, 10, 25 int Count( int
amount, int numCoinTypes ) if ( amount lt 0
numCoinTypes 0 ) // base case
1 return 0 else if ( amount 0
numCoinTypes 1 ) // base case 2
return 1 else
// recursive
case return Count( amount -
coinsnumCoinTypes - 1, numCoinTypes )
Count( amount,
numCoinTypes 1 )
See ChangeMaker.cs
22
Outline

• Review and admin.
• Palindrome
• Change maker
• Towers of Hanoi

23
Towers of Hanoi

• The Towers of Hanoi is a puzzle made up of three
  vertical pegs and several disks that slide on
  the pegs
• The disks are of varying size, initially placed
  on one peg with the largest disk on the bottom
  with increasingly smaller ones on top
• The goal is to move all of the disks from peg 1
  to peg 3 under the following rules
• we can move only one disk at a time
• we cannot move a larger disk on top of a smaller
  one

24
Questions

• What are sub-problems you can reduce your current
  problem to
• and what is the signature of the method to allow
  recursion?
• What is the recursive case?
• What is the base case?

25
Tower of Hanoi Code
void MoveTower ( int numDisks, int start, int
end, int temp ) if ( NumDisks 1 ) //
base case Console.WriteLine( move
disk from start to end ) else
// recursion
MoveTower ( numDisks-1, start, temp, end
) Console.WriteLine( move disk from
start to end ) MoveTower
( numDisks-1, temp, end, start )
public static void Main( string args )
MoveTower(4, 1, 3, 2)
See HanoiTowers.cs