More About Recursion

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More About Recursion

• Alice

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Recursion

• Recursion is used when the solution to a problem
  depends on the ability to break a problem down
  into smaller and smaller sub-problems.
• Let's look at an example

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A Towers Problem
Source Spare
Target (coneFrom) (coneSpare) (coneTo)

• The challenge is to move all the disks from the
  source cone to the target cone.
• Move 1 disk at a time
• A larger disk may never be on top of a smaller
  disk

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Initial world

• The disks are instances of the Torus class. (A
  torus is a doughnut shaped object.)
• Each cone is positioned exactly 1 meter from its
  nearest neighbor.
• Other than the bottom disk, each disk is
  positioned exactly 0.1 meter above the disk below.

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Identifying the disks

• To make it easier to describe our solution, we
  give each disk an id number and a name.

id number name 1 disk1
2 disk2 3 disk3 4
disk4
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Solving the problem

• Our solution will use the
• Principle of wishful thinking
• assume we could solve a smaller version of the
  same problem
• if we could solve the smaller version, it would
  make it easier to solve this problem.
• Base case the simplest possible
  version of this problem, one that can obviously
  be solved.

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Wishful thinking in practice
Assume I could move 3 of the disks to the spare
cone.
Then I could move the 4th disk (base case) to the
target cone.
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Storyboard

• To solve the towers problem, we need to know
  howmany disks we have and which cone is the
  source, the target, and the spare

towers Parameters howmany, source, target,
spare If howmany is equal to 1 move it (the
smallest disk) from the source to the
target Else Do in order call towers to
move howmany-1 disks from source to spare
(using target as spare) move it (disk
howmany) from the source to the target
call towers to move howmany-1 disks
from the spare to the target (using
the source as the spare)
base case move 1 disk
a smaller problem -- recursively move the rest of
the disks
Two recursive calls are used in this method.
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Moving a disk

• A challenge in this animation is how to move a
  disk from one tower to another.
• In the storyboard for the towers method, we
  assumed that we had a method named moveIt that
  would accomplish the task.
• To write the moveIt method, we need to know
• What are the parameters to send in to our
  method?
• What steps need to occur?
• How high to raise the disk object?
• How far (forward/back) to move it?

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moveIt Storyboard

• The parameters are
• whichdisk the disk id number
• fromcone the source cone
• tocone the target cone
• A storyboard describing the steps is

moveIt Parameters whichdisk, fromcone,
tocone Do in order Lift the disk
up above the top of the fromcone
Move it (forward or back) to a location above the
tocone Lower the disk down onto the
tocone
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Demo

• Ch8Lec2Towers-V1
• Concepts illustrated in this example
• The base case is written as an If statement
• If the base case is not true, then instructions
  are executed to move 1 step closer to the base
  case and the method calls itself.
• Repeated self-calls eventually work down to the
  base case and the recursion ends.

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Klutzy

• The moveIt method contains three sets of nested
  If statements
• The disk id number is used to determine which
  disk to
• move up
• move over
• move down
• The code is somewhat klutzy.
• In other words, the code is not elegant!

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Using an expression

• We noticed that the distance each disk has to
  move up (and then back down) is 0.3 meters more
  than 0.1 the id number (whichdisk).
• We could use an expression to compute the
  distance for the move up and move down
  instructions.
• move the appropriate disk 0.3 0.1 whichdisk

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Problem

• The problem with trying to use this nifty math
  expression is that we need to have the disk's
  name to write a move instruction.
• For example,
• disk1 move up
• must be an object, cannot use the id number here

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Demo

• Ch08Lec2Towers-V2
• Concepts illustrated in this example
• Condensing the program code to make it more
  elegant may lead to other problems.
• A conversion function is one that has a
  parameter to receive an argument of one data type
  and return some equivalent value, possibly of a
  different data type.
• In this example, we wrote our own function to
  convert the disk id number (i) to the disk name

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Assignment

• Read Alice Chapter 8 Section 2,
• A Second Form of Recursion