Recursion

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Recursion!
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Question

• Stephen Kings books
• J. R. R. Tolkiens books
• Recursion
• Do they have something in common?

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Answer

• J. R. R. Tolkien has written Two towers
• Stephen King has written Dark Tower books
• The Towers of Hanoi!

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The Towers of Hanoi

• The Legend. In an ancient city in India, so the
  legend goes, monks in a temple have to move a
  pile of 64 sacred disks from one location to
  another. The disks are fragile only one can be
  carried at a time. A disk may not be placed on
  top of a smaller, less valuable disk. And, there
  is only one other location in the temple (besides
  the original and destination locations) sacred
  enough that a pile of disks can be placed there.
  

.jedi
.jedi
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Source http//www.math.toronto.edu/mathnet/games/
towers.html
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The Towers of Hanoi
Source http//www.mathematik.uni-muenchen.de/hin
z/tower.jpg
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The Towers of Hanoi

• How should the monks proceed?
• Will they make it?

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The Towers of Hanoi

• Recursive solution!
• For N 0 do nothing
• Move the top N-1 disks from Src to Aux (using Dst
  as an intermediary peg)
• Move the bottom disks from Src to Dst
• Move N-1 disks from Aux to Dst (using Src as an
  intermediary peg)
• The first call
• Solve(3, 1, 2, 3)

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The Towers of Hanoi

• Move from Src to Dst
• Move from Src to Aux
• Move from Dst to Aux
• Move from Src to Dst
• Move from Aux to Src
• Move from Aux to Dst
• Move from Src to Dst

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The Towers of Hanoi

• How much time will monks spend? ¹
• For one disk, only one move is necessary
• For two disks, we need three moves
• For n disks???

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The Towers of Hanoi

• Let Tn denote the number of moves needed to move
  n disks.
• T1 1
• T2 3
• Tn 2Tn-1 1

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The Towers of Hanoi

• Let Tn denote the number of moves needed to move
  n disks.
• T1 1
• T2 3
• Tn 2Tn-1 1
• Tn 2n - 1
• How to prove it?

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Induction!

• Base case
• T1 1 21 - 1. OK!
• Inductive hypothesis
• Tn 2n -1
• Inductive step we show that
• Tn12n1-1.
• Tn12Tn12(2n-1)1
• 2n1-2 1 2n1-1.
• QED!

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Towers of Hanoi

• Suppose it takes one minute for a monk to move a
  disk. The whole task hence would take 264-1
  minutes
• (210)624-1 minutes
• (103)615 minutes
• 251016 hours
• 1016 days
• 1000000000000000 days
• the age of universe

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Recursion

• Sierpinski triangle
• Waclaw Sierpinski Polish mathematician
  1882-1969

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Sierpinski Triangle

• Draw a black triangle
• Draw a white triangle in the middle of the
  triangle.
• Call the procedure for three left black triangles

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Tail recursion

• Consider the following method
• int last(int n, int arr)
• // returns the last element in arr
• // starting from n
• // -1 if n is too large
• if (n gt arr.length - 1)
• return 1
• if (n arr.length 1)
• return arrn
• return last(n1, arr)

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Tail recursion

• The recursive function call is the last command
  in the method
• We can transform then recursion into iteration
• Some compilers can do it!

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Tail recursion

• int last(int n, int arr)
• // returns the last element in arr
• // starting from n
• while (n lt arr.length 1)
• n
• if (n arr.length 1)
• return arrn
• else
• return 1

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Tail recursion

• The transformed program doesnt use any extra
  memory
• The transformation can save space
• Exercise Transform Towers of Hanoi.