Traversals of Linked Lists Preorder BST Traversal Postorder BST Traversal Miscellaneous BST Traversals Depth First vs Breadth First Array Traversals

1
Traversals of Linked ListsPreorder BST
TraversalPostorder BST TraversalMiscellaneous
BST TraversalsDepth First vs Breadth FirstArray
Traversals
Lecture 14
2
Traversals of Linked Lists
3
The Scenario

• We need to visit each element in the linked list.
• At each element, we do some work.
• Well stop when we reach the end of the list.
• Examples
• Printing all elements
• Updating/changing the data of all elements

4
Traversals

• A traversal visits every element in a collection.
• Two forms
• Recursive
• Iterative (linear structures)

5
Visiting Every Element
//
head
48
17
142

• Regardless of whether we use iteration or
  recursion, we continue until we reach the end
  (NIL).

6
Dont De-reference NIL

• We must make sure we dont de-reference (follow)
  the NIL pointer.
• To do this, always test to see if the pointer is
  NIL.
• If so, we cant use the operator

head
//
Cant follow head! (i.e. head doesnt work!)
7
Testing for NIL

• Always test for NIL at the beginning of your
  loop.
• Your exitif statement must appear at the top
• Always test for NIL at the beginning of your
  recursive module.
• Your terminating condition must appear at the
  top of your module

8
Iterative Traversal Template

• procedure Traverse (cur iot in Ptr toa Node)
• // Purpose call Do_Something on each element
• // Precondition none
• // Postcondition Do_Something on each element
• loop
• exitif( cur NIL )
• Do_Something( cur.data )
• cur lt- cur.next
• endloop
• endprocedure // Traverse

9
An Iterative Traversal Example

• procedure Print_List(cur iot in Ptr toa Node)
• loop
• exitif (cur NIL)
• print(cur.data)
• cur lt- cur.next
• endloop
• endprocedure //Print_List

Called via Print_List(list_head)
cur
list_head
17
42
4
10
An Iterative Traversal Example

• procedure Print_List(cur iot in Ptr toa Node)
• loop
• exitif (cur NIL)
• print(cur.data)
• cur lt- cur.next
• endloop
• endprocedure //Print_List

cur
list_head
17
42
4
11
An Iterative Traversal Example

• procedure Print_List(cur iot in Ptr toa Node)
• loop
• exitif (cur NIL)
• print(cur.data)
• cur lt- cur.next
• endloop
• endprocedure //Print_List

4
cur
list_head
17
42
4
12
An Iterative Traversal Example

• procedure Print_List(cur iot in Ptr toa Node)
• loop
• exitif (cur NIL)
• print(cur.data)
• cur lt- cur.next
• endloop
• endprocedure //Print_List

4
cur
list_head
17
42
4
13
An Iterative Traversal Example

• procedure Print_List(cur iot in Ptr toa Node)
• loop
• exitif (cur NIL)
• print(cur.data)
• cur lt- cur.next
• endloop
• endprocedure //Print_List

4
cur
list_head
17
42
4
14
An Iterative Traversal Example

• procedure Print_List(cur iot in Ptr toa Node)
• loop
• exitif (cur NIL)
• print(cur.data)
• cur lt- cur.next
• endloop
• endprocedure //Print_List

4
cur
list_head
17
42
4
15
An Iterative Traversal Example

• procedure Print_List(cur iot in Ptr toa Node)
• loop
• exitif (cur NIL)
• print(cur.data)
• cur lt- cur.next
• endloop
• endprocedure //Print_List

4 17
cur
list_head
17
42
4
16
An Iterative Traversal Example

• procedure Print_List(cur iot in Ptr toa Node)
• loop
• exitif (cur NIL)
• print(cur.data)
• cur lt- cur.next
• endloop
• endprocedure //Print_List

4 17
cur
list_head
17
42
4
17
An Iterative Traversal Example

• procedure Print_List(cur iot in Ptr toa Node)
• loop
• exitif (cur NIL)
• print(cur.data)
• cur lt- cur.next
• endloop
• endprocedure //Print_List

4 17
cur
list_head
17
42
4
18
An Iterative Traversal Example

• procedure Print_List(cur iot in Ptr toa Node)
• loop
• exitif (cur NIL)
• print(cur.data)
• cur lt- cur.next
• endloop
• endprocedure //Print_List

4 17
cur
list_head
17
42
4
19
An Iterative Traversal Example

• procedure Print_List(cur iot in Ptr toa Node)
• loop
• exitif (cur NIL)
• print(cur.data)
• cur lt- cur.next
• endloop
• endprocedure //Print_List

4 17 42
cur
list_head
17
42
4
20
An Iterative Traversal Example

• procedure Print_List(cur iot in Ptr toa Node)
• loop
• exitif (cur NIL)
• print(cur.data)
• cur lt- cur.next
• endloop
• endprocedure //Print_List

4 17 42
cur
list_head
17
42
4
21
An Iterative Traversal Example

• procedure Print_List(cur iot in Ptr toa Node)
• loop
• exitif (cur NIL)
• print(cur.data)
• cur lt- cur.next
• endloop
• endprocedure //Print_List

4 17 42
cur
list_head
17
42
4
22
An Iterative Traversal Example

• procedure Print_List(cur iot in Ptr toa Node)
• loop
• exitif (cur NIL)
• print(cur.data)
• cur lt- cur.next
• endloop
• endprocedure //Print_List

4 17 42
cur
list_head
17
42
4
23
Recursive Traversal Template

• procedure Traverse (cur iot in Ptr toa Node)
• // Purpose call Do_Something on each element
• // Precondition none
• // Postcondition Do_Something on each element
• if (cur ltgt NIL) then
• Do_Something( cur.data )
• Traverse( cur.next )
• endif
• endprocedure // Traverse

24
A Recursive Traversal Example

• procedure Print_List(cur iot in Ptr toa Node)
• if (cur ltgt NIL) then
• print(cur.data)
• Print_List(cur.next)
• endif
• endprocedure //Print_List

Called via Print_List(list_head)
cur
list_head
17
42
4
You already know how to do this...
25
A Loophole in Parameter Protection

• If a pointer is an in parameter, the pointer
  cannot be changed.
• But, anything to which the pointer points may be
  changed.
• Thus, giving access to a list means giving
  ability to make changes to it you cannot protect
  against this.

26
An Example

• procedure DestroyList (cur iot in Ptr toa Node)
• if (cur ltgt nil) then
• cur.data lt- -1
• DestroyList(cur.next)
• endif
• endprocedure // DestroyList

//
head
48
17
142
27
Another Example

• procedure DestroyList (cur iot in Ptr toa Node)
• cur.next lt- NIL
• endprocedure // DestroyList

//
head
48
17
142
28
Summary

• Traversals involve visiting every element
• Recursion or iteration
• Stop when you reach NIL
• Make sure to not de-reference NIL
• Be aware that passing a pointer to a module via
  an IN parameter allows the module to modify the
  entire list.

29
Questions?
30
Traversing a Binary Search Tree (BST)Pre-Order
31
Outline of Pre-Order Traversal

• Three principle steps
• Do work (Current)
• Traverse Left
• Traverse Right
• Work can be anything
• Separate work from traversal

32

• Traverse the tree Pre-order
• Visit the current and do work
• Visit the trees left sub-tree
• Visit right sub-tree

33
Pre-Order Traversal Procedure

• procedure Pre_Order(cur iot in Ptr toa Tree_Node)
• // Purpose perform pre-order traversal, call
• // DoWhatever for each node
• // Preconditions cur points to a binary tree
• // Postcondition DoWhatever will be performed
• // on each tree node in pre-order order
• if( cur ltgt NIL ) then
• Do_Whatever( cur.data )
• Pre_Order( cur.left_child )
• Pre_Order( cur.right_child )
• endif
• endprocedure // Pre_Order

34
Proc PreOrderPrint(pointer) pointer NOT NIL?
print(data) PreOrderPrint(left child)
PreOrderPrint(right child)
root
P
L
R
35
Questions?
36
Traversing a Binary Search Tree (BST)Post-Order
37
Outline of Post-Order Traversal

• Three principle steps
• Traverse Left
• Traverse Right
• Do work (Current)
• Work can be anything
• Separate work from traversal

38

• Traverse the tree Post order
• Visit the trees left sub-tree
• Visit right sub-tree
• Visit the current and do work

39
Post-Order Traversal Procedure

• procedure Post_Order(cur iot in Ptr toa
  Tree_Node)
• // Purpose perform post-order traversal, call
• // DoWhatever for each node
• // Preconditions cur points to a binary tree
• // Postcondition DoWhatever will be performed
• // on each tree node in post-order order
• if( cur ltgt NIL ) then
• Post_Order( cur.left_child )
• Post_Order( cur.right_child )
• Do_Whatever( cur.data )
• endif
• endprocedure // Post_Order

40
Proc PostOrderPrint(pointer) pointer NOT NIL?
PostOrderPrint(left child) PostOrderPrint(right
child) print(data)
root
L
R
P
41
Questions?
42
Miscellaneous BST Traversals
43
Miscellaneous Traversals

• Defined Traversals
• In-order L C R
• Pre-order C L R
• Post-order L R C
• Other Possibilities
• R C L
• C R L
• R L C

44
Proc MiscTraverse(pointer) pointer NOT NIL?
print(data) MiscTraverse(right child)
MiscTraverse(left child)
root
P
R
L
45
Proc MiscTraverse(pointer) pointer NOT NIL?
print(data) MiscTraverse(right child)
MiscTraverse(left child)
root
P
R
L
46
Proc MiscTraverse(pointer) pointer NOT NIL?
print(data) MiscTraverse(right child)
MiscTraverse(left child)
root
P
P
R
L
Output 22
47
Proc MiscTraverse(pointer) pointer NOT NIL?
print(data) MiscTraverse(right child)
MiscTraverse(left child)
root
P
P
R
R
L
Output 22
48
Proc MiscTraverse(pointer) pointer NOT NIL?
print(data) MiscTraverse(right child)
MiscTraverse(left child)
root
P
P
R
R
L
P
Output 22 67
49
Proc MiscTraverse(pointer) pointer NOT NIL?
print(data) MiscTraverse(right child)
MiscTraverse(left child)
root
P
P
R
R
L
P
Output 22 67
R
50
Proc MiscTraverse(pointer) pointer NOT NIL?
print(data) MiscTraverse(right child)
MiscTraverse(left child)
root
P
P
R
R
L
P
Output 22 67 94
R
P
51
Proc MiscTraverse(pointer) pointer NOT NIL?
print(data) MiscTraverse(right child)
MiscTraverse(left child)
root
P
P
R
R
L
P
Output 22 67 94
R
P
R
52
Proc MiscTraverse(pointer) pointer NOT NIL?
print(data) MiscTraverse(right child)
MiscTraverse(left child)
root
P
P
R
R
L
P
Output 22 67 94 97
R
P
R
P
53
Proc MiscTraverse(pointer) pointer NOT NIL?
print(data) MiscTraverse(right child)
MiscTraverse(left child)
root
P
P
R
R
L
P
Output 22 67 94 97
R
P
R
P
R
54
Proc MiscTraverse(pointer) pointer NOT NIL?
print(data) MiscTraverse(right child)
MiscTraverse(left child)
root
P
P
R
R
L
L
P
P
Output 22 67 94 97 36 44 9 14 3 7 1
R
R
L
L
P
P
P
P
R
R
R
R
L
L
L
L
P
P
P
P
R
R
R
R
L
L
L
L
55
Questions?
56
Depth First vs. Breadth First Traversal
57
Depth vs. Breadth First Traversals

• Depth First Traversals
• Go down (deep)
• In-, Pre-, Post-order
• Breadth First Traversals
• Go across (shallow)
• Require a queue to help

58
Depth-First Traversal
root
59
Breadth-First Traversal
root
60
Depth First Traversal (In-Order)

• Procedure DFT
• (current isoftype in Ptr toa Node)
• // In order DFT
• if(current ltgt NIL) then
• DFT(current.left)
• print(current.data)
• DFT(Current.right)
• endif
• endprocedure

61
Depth First Traversal (Pre-Order)

• Procedure DFT
• (current isoftype in Ptr toa Node)
• // In order DFT
• if(current ltgt NIL) then
• DFT(current.left)
• print(current.data)
• DFT(Current.right)
• endif
• endprocedure

Move this here to make a Preorder DFT
62
Depth First Traversal (Post-Order)

• Procedure DFT
• (current isoftype in Ptr toa Node)
• // In order DFT
• if(current ltgt NIL) then
• DFT(current.left)
• print(current.data)
• DFT(Current.right)
• endif
• endprocedure

Move this here to make a Postorder DFT
63
Proc DFT(pointer) pointer NOT NIL? DFT(left
child) print(data) DFT(right child)
root
We have already seen these!
64
Breadth-First Traversal

• Requires a queue to maintain which nodes to visit
  next.
• Enqueue the root pointer
• Loop until no elements in the queue
• Dequeue(current)
• Enqueue currents left and right children
• Do work at current

65
Breadth-First Traversal
LB

• Procedure BFT(root isoftype in Ptr toa Node)
• Q isoftype Queue
• Initialize(Q)
• temp isoftype Ptr toa Node
• OK isoftype Boolean
• enqueue(Q, root)
• // continued

66
Breadth-First Traversal
LB

• loop
• dequeue(temp, OK, Q)
• exitif(NOT OK)
• if(temp.left ltgt NIL) then
• enqueue(temp.left, Q))
• endif
• if(temp.right ltgt NIL) then
• enqueue(temp.right, Q)
• endif
• print(temp.data) // processing node
• endloop
• endprocedure

67

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(data)
• endloop

root
Q
aNode
68

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(data)
• endloop

root
Q22
aNode
69

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(data)
• endloop

root
Q22
aNode
70

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q
aNode22
71

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q9 67
aNode22
72

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q9 67
aNode22
73

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q9 67
aNode22
74

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q67
aNode9
75

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q67 3 14
aNode9
76

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q67 3 14
aNode9
77

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q67 3 14
aNode9
78

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q3 14
aNode67
79

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q3 14 36 94
aNode67
80

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q3 14 36 94
aNode67
81

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q3 14 36 94
aNode67
82

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q14 36 94
aNode3
83

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q14 36 94 1 7
aNode3
84

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q14 36 94 1 7
aNode3
85

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q14 36 94 1 7
aNode3
86

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q36 94 1 7
aNode14
87

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q36 94 1 7
aNode14
88

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q36 94 1 7
aNode14
89

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q36 94 1 7
aNode14
90

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q94 1 7
aNode36
91

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q94 1 7 44
aNode36
92

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q94 1 7 44
aNode36
93

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q94 1 7 44
aNode36
94

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q1 7 44
aNode94
95

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q1 7 44 97
aNode94
96

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q1 7 44 97
aNode94
97

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q1 7 44 97
aNode94
98

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q7 44 97
aNode1
99

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q7 44 97
aNode1
100

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q7 44 97
aNode1
101

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q7 44 97
aNode1
102

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q44 97
aNode7
103

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q44 97
aNode7
104

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q44 97
aNode7
105

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q44 97
aNode7
106

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q97
aNode44
107

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q97
aNode44
108

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q97
aNode44
109

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q97
aNode44
110

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q
aNode97
111

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q
aNode97
112

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q
aNode97
113

• Enqueue(root)
• loop
• exitif Q empty
• dequeue(aNode)
• enqueue(children)
• print(aNode.data)
• endloop

root
Q
aNode97
114
Queuestions?
115
Traversals of Arrays
116
The Scenario

• We need to visit each element in the array.
• At each element, we do some work.
• Well stop when we reach the end of the array
  (when we reach MAX).
• Examples
• Printing all elements
• Updating/changing the data of all elements

117
Traversals

• A traversal visits every element in a collection.
• Two forms
• Recursive
• Iterative (linear structures)

118
Recursive Array Traversal Template

• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Purpose call Do_Something on each element
• // Precondition i 1 on first call
• // Postcondition Do_Something on each element
• if (i lt MAX) then
• Do_Something(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

119
Recursive Array Traversal Example

• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

120

• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

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• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

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• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

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• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

procedure RecursiveTraverse (i iot in num,
NumArray iot in NumArrayType) // Pre i 1 on
first call if (i lt MAX) then
print(NumArrayi) RecursiveTraverse (i1,
NumArray) endif endprocedure
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• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

procedure RecursiveTraverse (i iot in num,
NumArray iot in NumArrayType) // Pre i 1 on
first call if (i lt MAX) then
print(NumArrayi) RecursiveTraverse (i1,
NumArray) endif endprocedure
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• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

procedure RecursiveTraverse (i iot in num,
NumArray iot in NumArrayType) // Pre i 1 on
first call if (i lt MAX) then
print(NumArrayi) RecursiveTraverse (i1,
NumArray) endif endprocedure
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• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

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• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

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• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

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• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

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• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

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• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

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• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

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• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

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• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

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• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

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• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

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• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

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• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

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• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

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• procedure RecursiveTraverse (i iot in num,
  NumArray iot in NumArrayType)
• // Pre i 1 on first call
• if (i lt MAX) then
• print(NumArrayi)
• RecursiveTraverse (i1, NumArray)
• endif
• endprocedure

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Iterative Traversal Template

• procedure Traverse (my_array iot in/out
  ArrayType)
• // Purpose call Do_Something on each element
• // Precondition none
• // Postcondition Do_Something on each element
• i isoftype Num
• i lt- 1
• loop
• exitif( i gt MAX )
• Do_Something( my_arrayi )
• i lt- i 1
• endloop
• endprocedure // Traverse

142
An Iterative Traversal Example
Traverse the array and fill in every elementso
that the ith element contains the value i.
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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10

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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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MAX is 10 Vec_max definesa Array1..MAX of
Num data isoftype Vec_max i isoftype Num i lt-
1 loop exitif(i gt MAX) datai lt- i i lt- i
1 endloop
data
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A More Complex Problem

• Write a module which returns an array with the
  following characteristics
• Every odd element has the first half of the
  alphabet in ascending order every even element
  has the second half of the alphabet in descending
  order.

A1 A A2 Z A3 B A4 Y A5
C A6 X . . . A23 L A24
O A25 M A26 N
181
A
A
Z
B
Y
C
L
O
M
N
Every odd element has the first half of the
alphabet in ascending order every even element
has the second half of the alphabet in descending
order.
182
A First Try

• ALPHAS is 26
• A isoftype CharArrayType
• i isoftype Num
• up isoftype char
• i lt- 1
• up lt- A
• loop
• exitif( i gt ALPHAS )
• Ai lt- up
• up lt- next_character(up)
• i lt- i 2
• endloop

183
A
A
B
C
L
M
Every odd element has the first half of the
alphabet in ascending order every even element
has the second half of the alphabet in descending
order.
184

• ALPHAS is 26
• A isoftype Array_Type
• i isoftype Num
• up, down isoftype char
• i lt- 1
• up lt- A
• down lt- Z
• loop
• exitif( i gt ALPHAS )
• Ai lt- up
• Ai 1 lt- down
• up lt- next_character(up)
• down lt- previous_character(down)
• i lt- i 2
• endloop

185
A
A
Z
B
Y
C
L
O
M
N
Every odd element has the first half of the
alphabet in ascending order every even element
has the second half of the alphabet in descending
order.
186
A 2-D Iterative Traversal Example

• We have a 2-D array
• We want to traverse the array and set each
  element to the product of its row position and
  column position
• (i.e. the element at position i,j should
  contain the value ij)

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MAX_COLS is 4 Row_Type definesa
Array1..MAX_COLS of Num Sample isoftype
Row_type
Sample
1 2 3 4
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MAX_ROWS is 3 MAX_COLS is 4 Row_Type definesa
Array1..MAX_COLS of Num Matrix definesa
Array1..MAX_ROWS of Row_Type Data isoftype
Matrix
1 2 3
1 2 3 4
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MAX_ROWS is 3 MAX_COLS is 4 Row_Type definesa
Array1..MAX_COLS of Num Matrix definesa
Array1..MAX_ROWS of Row_Type Data isoftype
Matrix Row, Col isoftype Num Row lt- 1 loop
exitif(Row gt MAX_ROWS) Col lt- 1 loop
exitif(Col gt MAX_COLS) DataRowCol lt- Row
Col Col lt- Col 1 endloop Row lt- Row
1 endloop
190
MAX_ROWS is 3 MAX_COLS is 4 Row_Type definesa
Array1..MAX_COLS of Num Matrix definesa
Array1..MAX_ROWS of Row_Type Data isoftype
Matrix Row, Col isoftype Num
1 2 3
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Row lt- 1 loop exitif(Row gt MAX_ROWS) Col lt-
1 loop exitif(Col gt MAX_COLS)
DataRowCol lt- Row Col Col lt- Col 1
endloop Row lt- Row 1 endloop
MAX_ROWS is 3 MAX_COLS is 4
1 2 3
1 2 3 4
192
Row lt- 1 loop exitif(Row gt MAX_ROWS) Col lt-
1 loop exitif(Col gt MAX_COLS)
DataRowCol lt- Row Col Col lt- Col 1
endloop Row lt- Row 1 endloop
MAX_ROWS is 3 MAX_COLS is 4
1 2 3
1 2 3 4
193
Row lt- 1 loop exitif(Row gt MAX_ROWS) Col lt-
1 loop exitif(Col gt MAX_COLS)
DataRowCol lt- Row Col Col lt- Col 1
endloop Row lt- Row 1 endloop
MAX_ROWS is 3 MAX_COLS is 4
1 2 3
1 2 3 4
194
Row lt- 1 loop exitif(Row gt MAX_ROWS) Col lt-
1 loop exitif(Col gt MAX_COLS)
DataRowCol lt- Row Col Col lt- Col 1
endloop Row lt- Row 1 endloop
MAX_ROWS is 3 MAX_COLS is 4
1 2 3
1 2 3 4
195
Row lt- 1 loop exitif(Row gt MAX_ROWS) Col lt-
1 loop exitif(Col gt MAX_COLS)
DataRowCol lt- Row Col Col lt- Col 1
endloop Row lt- Row 1 endloop
MAX_ROWS is 3 MAX_COLS is 4
1 2 3
1 2 3 4
196
Row lt- 1 loop exitif(Row gt MAX_ROWS) Col lt-
1 loop exitif(Col gt MAX_COLS)
DataRowCol lt- Row Col Col lt- Col 1
endloop Row lt- Row 1 endloop
MAX_ROWS is 3 MAX_COLS is 4
1 2 3
1 2 3 4
197
Row lt- 1 loop exitif(Row gt MAX_ROWS) Col lt-
1 loop exitif(Col gt MAX_COLS)
DataRowCol lt- Row Col Col lt- Col 1
endloop Row lt- Row 1 endloop
MAX_ROWS is 3 MAX_COLS is 4
1 2 3
1 2 3 4
198
Row lt- 1 loop exitif(Row gt MAX_ROWS) Col lt-
1 loop exitif(Col gt MAX_COLS)
DataRowCol lt- Row Col Col lt- Col 1
endloop Row lt- Row 1 endloop
MAX_ROWS is 3 MAX_COLS is 4
1 2 3
1 2 3 4
199
Row lt- 1 loop exitif(Row gt MAX_ROWS) Col lt-
1 loop exitif(Col gt MAX_COLS)
DataRowCol lt- Row Col Col lt- Col 1
endloop Row lt- Row 1 endloop
MAX_ROWS is 3 MAX_COLS is 4
1 2 3
1 2 3 4
200
Row lt- 1 loop exitif(Row gt MAX_ROWS) Col lt-
1 loop exitif(Col gt MAX_COLS)
DataRowCol lt- Row Col Col lt- Col 1
endloop Row lt- Row 1 endloop
MAX_ROWS is 3 MAX_COLS is 4
1 2 3
1 2 3 4
201
Row lt- 1 loop exitif(Row gt MAX_ROWS) Col lt-
1 loop exitif(Col gt MAX_COLS)
DataRowCol lt- Row Col Col lt- Col 1
endloop Row lt- Row 1 endloop
MAX_ROWS is 3 MAX_COLS is 4
1 2 3
1 2 3 4
202
Row lt- 1 loop exitif(Row gt MAX_ROWS) Col lt-
1 loop exitif(Col gt MAX_COLS)
DataRowCol lt- Row Col Col lt- Col 1
endloop Row lt- Row 1 endloop
MAX_ROWS is 3 MAX_COLS is 4
1 2 3
1 2 3 4
203
Row lt- 1 loop exitif(Row gt MAX_ROWS) Col lt-
1 loop exitif(Col gt MAX_COLS)
DataRowCol lt- Row Col Col lt- Col 1
endloop Row lt- Row 1 endloop
MAX_ROWS is 3 MAX_COLS is 4
1 2 3
1 2 3 4
204
Row lt- 1 loop exitif(Row gt MAX_ROWS) Col lt-
1 loop exitif(Col gt MAX_COLS)
DataRowCol lt- Row Col Col lt- Col 1
endloop Row lt- Row 1 endloop
MAX_ROWS is 3 MAX_COLS is 4
1 2 3
1 2 3 4
205
Row lt- 1 loop exitif(Row gt MAX_ROWS) Col lt-
1 loop exitif(Col gt MAX_COLS)
DataRowCol lt- Row Col Col lt- Col 1
endloop Row lt- Row 1 endloop
MAX_ROWS is 3 MAX_COLS is 4
1 2 3
1 2 3 4
206
Row lt- 1 loop exitif(Row gt MAX_ROWS) Col lt-
1 loop exitif(Col gt MAX_COLS)
DataRowCol lt- Row Col Col lt- Col 1
endloop Row lt- Row 1 endloop
MAX_ROWS is 3 MAX_COLS is 4
1 2 3
1 2 3 4
207
Row lt- 1 loop exitif(Row gt MAX_ROWS) Col lt-
1 loop exitif(Col gt MAX_COLS)
DataRowCol lt- Row Col Col l