Paradigms

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Paradigms
Lect20

• Define the Problem.
• Gather Information.
• Generate a Possible Solution.
• Implement the Solution.
• Test the Solution.

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(No Transcript)
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Nine Dots
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Nine Dots Solution
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Six Matches
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6 Matches - 2 Triangles
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Six Matches Solution
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Bottles Solution

• All cases except 8 can be solved by
• B-A-2C B contains the required amount.
• 6 can also be solved by A-C
• 7 can also be solved by AC
• 8 can only be solved by A-C

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Recursion

• Sometimes it is possible and useful to offer an
  algorithm that calls itself.
• This calling of itself is termed recursion.

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Possible

• It is possible because the structure of the
  algorithm naturally breaks down into
• solution of a large problem
• solution of the same problem of a smaller size
• a small step.

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Useful

• It is useful because recursion may be the natural
  way to express the solution to a problem.

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Factorial

• The factorial is a classic example of recursion.
• 5! 5 x 4 x 3 x 2 x1
• 4! 4 x 3 x 2 x 1
• 5! 5 x 4!
• Generally n! n (n-1)!

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Factorial Program

• include ltstdio.hgt
• long fact(int)
• int main()
• int n5
• printf("Factorial d is ld\n", n, fact(n))
• system("PAUSE") return 0
• long fact(int n)
• if(n1) return 1
• else return ((long) n)fact(n-1)

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Diagramatic Expansion
main
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Diagramatic Expansion
main
fact(5)
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Diagramatic Expansion
main
fact(5)
fact(4)
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Diagramatic Expansion
main
fact(5)
fact(4)
fact(3)
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Diagramatic Expansion
main
fact(5)
fact(4)
fact(3)
fact(2)
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Diagramatic Expansion
main
fact(5)
fact(4)
fact(3)
fact(2)
fact(1)
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Diagramatic Expansion
main
fact(5)
fact(4)
fact(3)
1
fact(2)
fact(1)
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Diagramatic Expansion
main
fact(5)
fact(4)
2
fact(3)
1
fact(2)
fact(1)
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Diagramatic Expansion
main
fact(5)
6
fact(4)
2
fact(3)
1
fact(2)
fact(1)
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Diagramatic Expansion
main
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fact(5)
6
fact(4)
2
fact(3)
1
fact(2)
fact(1)
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Diagramatic Expansion
120
main
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fact(5)
6
fact(4)
2
fact(3)
1
fact(2)
fact(1)
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End Condition

• Recursion can be dangerous.
• fact() will continue to call itself unless we
  program the end condition
• if(n1) return 1
• An explicit end condition is essential.

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Fibonacci Series
Nautilus Shell
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Mathematics

• fibonacci(n) fibonacci(n-1) fibonacci(n-2)
• Ex 1,2,3,5,8,13,21, ....
• long fibonacci(int n)
• if(n0) return 1
• if(n1) return 2
• return fibonacci(n-1)fibonacci(n-2)

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Towers of Hanoi

• The purpose of the game is to move the tower from
  one post to another with the following rules
• Only one disk can be moved at a time
• no disk can be placed on one of smaller diameter
  and
• the third post can be used for temporary storage.

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Exhaustive Search
Disk1 to Peg3
Disk1 to Peg2
Disk2 to Peg3
Disk2 to Peg2
Disk1 to Peg1
Disk1 to Peg3
Disk1 to Peg2
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A Careful Review
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A Careful Review
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Solution

• To move a tower of n disks from the start peg to
  the destination peg using the intermediate peg
• To perform
• movetower(n, start, dest, inter)
• Do
• movetower(n-1, start, inter, dest)
• movedisk(n, start, dest)
• movetower(n-1, inter, dest, start)

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End Condition

• The remaining question is how to complete the
  recursion? The movement of a tower containing
  one disk is just the movement of the disk
• if(n1) movedisk(1, start, dest)

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movetower(4, 1, 3, 2)
movetower(3, 1, 2, 3) movedisk(4, 3)
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movetower(3, 1, 2, 3)
movetower(2, 1, 3, 2) movedisk(3, 2)
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movetower(2, 1, 3, 2)
movetower(1, 1, 2, 3) movedisk(2, 3)