Recursion

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Recursion

• COMP 114
• Tuesday February 26

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Announcements

• Program 2 due, 1100 AM Thursday
• Dont forget
• Folder, envelope
• Clean, working code
• No bugs
• Good documentation!!!

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Topics

• Last Time
• File I/O

• This Class
• GUI Message Box
• Recursion

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Message Box

• Used to signal errors
• Several types, but they all work the same.
• Demo Message Box Project

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Recursion

• In computer science, recursion is when a method
  calls itself.
• Not as crazy as it seems.
• First, lets look at a non-programming example

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Recursive Definition

• Define a list of number separated by commas
• Example 23, 27, 31, 1
• like this
• a LIST is a number
• or a LIST is a number comma LIST

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Example

• Try to see whether 23, 27, 31, 1 is a list
• Apply definition

LIST number or LIST number , LIST
23, 27, 31, 1
23, LIST
27, LIST
31, LIST
1
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Erroneous Examples

• 2 . 5 , 6
• 5, A

LIST number or LIST number , LIST
Compilers work like this!
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Base Case

• Notice that we had a case (LIST number) that
  does no recursion
• Thats called the base case
• Otherwise, wed have infinite recursion
• Go back to example we stopped at base case

LIST number or LIST number , LIST
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Recursion Used in Math

• Induction
• Recursive proofs
• Also have to provide a base case
• The one thats trivial to prove

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What About Programs?

• We can solve a lot of math problems recursively.
• Example

public static int sum(int n) if(n lt
1) return 0 return( sum(n 1) n )
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Example
public static int sum(int n) if(n lt
1) return 0 return( sum(n 1) n )
sum(4)
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Progress

• Notice that we must make progress toward base
  case
• There is steady reduction in size of problem
• Otherwise, trouble.

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Iterative Solution

• Of course, for this problem a recursive solution
  doesnt make sense.
• Elegant but slower than

public static int sum(int n) int result
0 for(int i 1 i lt n i) result
i return(n)
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How Does This Work?

• When method is called (recursive or not), new
  space is created for local variables
• In data structure called a stack
• Every time method is called, local variables are
  different
• Cant get to values in old local variables. They
  are out of scope

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Costs of Recursion

• Method call overhead
• Stack overhead
• Basically, space needs to be allocated for method
  variables
• Instructions to invoke method
• Recursive solutions can be made Iterative
• Sometimes hard to follow, though

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Indirect Recursion

• Method A can call method B
• Then method B calls method A
• Still recursive

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Tail Recursion

• When recursion happens just before the method
  exits, its called tail recursion
• Its simplest
• Recursion can be anywhere in method, though

public static int sum(int n) if(n lt
1) return 0 return( sum(n 1) n )
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Towers of Hanoi

• Game on a board with three pegs
• Three disks with holes (torii or donuts)
• Goal move disks to 3rd peg
• Rules
• Move one disk at a time
• Bigger disk cant be on top of smaller disk

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Towers of Hanoi Applet

• http//www.mazeworks.com/hanoi/
• Lets try it!

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Recursive Algorithm

• Simple Algorithm for N rings
• Move top N 1 rings to extra peg
• Move big ring to destination
• Move N 1 rings to destination

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Whats the Base Case?

• When theres only one ring to move, just move it.

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Code

• void moveTwr (int disks, int start, int end, int
  temp)
• if (disks 1)
• moveOneDisk (start, end)
• else
• moveTwr (disks-1, start, temp, end)
• moveOneDisk (start, end)
• moveTwr (disks-1, temp, end, start)
• Called as
• moveTwr (totalDisks, 1, 3, 2)

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Recursion Step 1

• Move top N 1 rings to extra peg
• Move big ring (N) to destination
• Move N 1 rings to destination

Move top 2 rings
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Recursion Step 1

• Move top N 1 rings to extra peg
• Move big ring (N) to destination
• Move N 1 rings to destination

Move top 2 rings
Move big ring (base)
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Recursion Step 1

• Move top N 1 rings to extra peg
• Move big ring (N) to destination
• Move N 1 rings to destination

Move top 2 rings
Move big ring (base)
Move 2 rings
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Recursion Step 1

• Move top N 1 rings to extra peg
• Move big ring (N) to destination
• Move N 1 rings to destination

Move top 2 rings
Move big ring (base)
Move 2 rings
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The Legend

• Order of monks, the priests of Brahma, in Benares
  are working on this puzzle

• Using 64 gold disks

• When they finish, the world will end.

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Dont Worry!

• To solve puzzle with N rings takes 2N 1 steps
• This is called exponential time complexity (well
  talk about this soon)

• Even if the monks move a disk every second
• Itll take them 584 billion years

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The Truth

• This puzzle was invented by French mathematician
  Edouard Lucas
• Sold as toy
• No monks working on it

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Another Example Koch Snowflake

• Take a line

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Divide in Three
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Replace Middle Part
with two lines of same length
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Keep Doing This

• Show Example
• Project Koch App
• From Lewis and Loftus book

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Fractal

• Koch snowflake is an example of a fractal
• Fractal is geometric shape made of
• same patterns
• at different scales
• and sometimes different orientations
• Many are recursively defined
• Benoit Mandelbrot of IBM Research popularized
  fractal research

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Fractal Images
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Next Time

• Running Time of Algorithms
• Big O notation