Recursion

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Recursion

• CSC 172
• SPRING 2002
• LECTURE 8

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Visual Proof
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Visual Proof
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Visual Proof
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Visual Proof
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Visual Proof
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Visual Proof
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Visual Proof
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Visual Proof
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Visual Proof
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Visual Proof
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Recursion Basis Induction

• Usually thought of as a programming technique
• Can also serve as a method of definition

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Recursive Definition of Expressions

• Example expressions with binary operators
• Basis
• An operand (variable or constant) is an
  expression
• - x1, y, 3, 57, 21.3

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Recursive Definition of Expressions

• Induction
• If E is an expression, then (E) is an expression
• If E1 and E2 are expressions, and is a binary
  operator (e.g. , ), then E1 E2 is an
  expression
• Thus we can build,
• x, y, 3, xy, (xy), (xy)3

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An Interesting Proof

• S(n) An expression E with binary operators of
  length n has one more operand than operators.
• Proof is by complete induction on the length
  (number of operators, operands, and parentheses)
  of the expression

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S(n) An expression E with binary operators of
length n has one more operand than operators.

• Basis n1
• E must be a single operand
• Zero operators

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S(n) An expression E with binary operators of
length n has one more operand than operators.

• Induction (complete)
• Assume S(1), S(2), . . . S(n).
• Let E have length n1 gt 1
• How was E constructed? Rule 1, or Rule 2?
• If by rule 1, then E (E1)
• E1 has length n-1
• BTIH S(n-1) has one more operand than operators
• E and E1 have the same number of operands
  operators
• So, S(n1) will hold under construction by rule 1

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S(n) An expression E with binary operators of
length n has one more operand than operators.

• If by rule 2, then E E1 E2
• Both E1 and E2 have length lt n,
• because
• is one symbol
• length(E1 ) length(E2 ) n
• Let E1 and E2 have a and b operators
• BTIH E1 and E2 have a1 and b1 operands
• Thus, E has (a1)(b1) ab2 operands
• E has ab1 operators (the 1 is for the )

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S(n) An expression E with binary operators of
length n has one more operand than operators.

• E has (a1)(b1) ab2 operands
• E has ab1 operators
• Note we used all of S(1),S(2),. . .,S(n) in the
  inductive step
• The fact that expression was defined
  recursively let us break expressions apart.
• We dont know how it broke up, but we have all
  the cases covered.

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Recursion an organization of process

• A style of programming and problem-solving where
  we express a solution in terms of smaller
  instances of itself.
• Uses basis/induction just like inductive proofs
• Basis needs no smaller instances
• Induction solution in terms of smaller instances

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Sorting Lists with MergeSort

• Note a list of length 1 is sorted
• If two lists are sorted, how can you combine them
  into one sorted list?
• How can you divide one list into two?

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Recursive Mergesort

• Basis A list of length 1 is sorted
• Induction for lists of gt 1 element
• Split the list into two equal (as possible) parts
• Recursively sort each part
• Merge the result of the two recursive sorts
• One at a time, select the smaller element from
  the two fronts of each list
• Physical demo

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Simplified MergeSort

• A linked list is a list of nodes with
• Node getNext(), setNext() methods.
• and getData(), setData methods.

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Node split

• public static Node split(Node head)
• Node secondNode
• if (head null) return null
• else if (head.getNext() null) return null
• else
• secondNode head.getNext()
• head.setNext(secondNode.getNext())
• secondNode.setNext(split(secondNode.getNext())
• return secondNode

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A linked List
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A linked List in Split
Split head
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A linked List in Split
secondNode head.getNext()
Split 0 head secondNode
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A linked List in Split
head.setNext(secondNode.getNext())
Split 0 head secondNode
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First recursive call in Split
secondNode.setNext(split(secondNode.getNext())
Split 0 head secondNode
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New Split
split(secondNode.getNex
t())
Split 1 head secondNode
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New Split
head.setNext(secondNode.getNext())
Split 1 head secondNode
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Second recursive Split
secondNode.setNext(split(secondNode.getNext())
Split 1 head secondNode
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Second recursive Split
secondNode.setNext(split(secondNode.getNext())
Split 2 head secondNode
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Second recursive Split
head.setNext(secondNode.getNext())
Split 2 head secondNode
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secondNode.setNext(split(secondNode.getNext()) ?
return secondNode
What gets returned?
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Backing up
secondNode.setNext(split(secondNode.getNext())
Split 1 head secondNode
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Backing up
secondNode.setNext(split(secondNode.getNext())
Split 1 head secondNode
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return secondNode
What gets returned?
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Backing up
secondNode.setNext(split(secondNode.getNext())
Split 1 head secondNode
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return secondNode
What gets returned?
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Backing up
secondNode.setNext(split(secondNode.getNext())
Split 1 head secondNode
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return secondNode
What gets returned?
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Node Merge

• public static Node merge(Node list1, Node list2)
• if (list1 null) return list2
• else if (list2 null) return list1
• else if (list1.getData.compareTo(list2.getData())
  lt 1)
• list1.setNext(merge(list1.getNext(),list2)
• return list1
• else
• list2.setNext(merge(list1,list2.getNext())
• return list2

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Two linked Lists
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Two linked Lists in merge
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list1 merge 0 list2
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Two linked Lists in merge
0th
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list1 merge 1 list2
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Two linked Lists in merge
0th
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list1 merge 2 list2
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Two linked Lists in merge
0th
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list1 merge 3 list2
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Two linked Lists in merge
0th
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list1 merge 4 list2
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Two linked Lists in merge
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list1 merge 3 list2
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Two linked Lists in merge
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list1 merge 2 list2
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Two linked Lists in merge
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list1 merge 2 list2
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Two linked Lists in merge
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list1 merge 0 list2
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Node MergeSort

• public static Node mergeSort(Node list)
• Node secondList
• if (list null) return null
• else if (list.getNext() null) return list
• else
• secondList split(list)
• return merge(mergeSort(list),mergeSort(secondLis
  t))

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Mergesort
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Mergesort Split
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Mergesort Split
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Mergesort Split
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Mergesort Split
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Mergesort Split
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Mergesort Split
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Mergesort Split
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Mergesort Split
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Mergesort Split
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Mergesort Split
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Mergesort Split
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Mergesort Split
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Mergesort Split
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Mergesort Split
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Mergesort Merge
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Mergesort Merge
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Mergesort Merge
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Mergesort Split
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Mergesort Split
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Mergesort Split
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Mergesort Merge
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Mergesort Merge
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Mergesort Merge
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Mergesort Merge
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Mergesort Merge
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Mergesort Merge
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Mergesort Merge
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Mergesort Split
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Mergesort Split
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Mergesort Split
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Mergesort Split
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Mergesort Split
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Mergesort Split
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Mergesort Split
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Mergesort Merge
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Mergesort Merge
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Mergesort Merge
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Mergesort Split
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Mergesort Split
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Mergesort Split
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Mergesort Merge
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Mergesort Merge
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Mergesort Merge
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Mergesort Merge
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Mergesort Merge
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Mergesort Merge
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Mergesort Merge
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Mergesort Merge
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Mergesort Merge
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Mergesort Merge
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Mergesort Merge
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Mergesort Merge
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Mergesort Merge
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Mergesort Merge
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