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Lecture 14. Traversals of Linked Lists Preorder BST Traversal Postorder BST Traversal Miscellaneous BST Traversals Depth First vs Breadth First Array Traversals. Traversals of Linked Lists. The Scenario. We need to visit each element in the linked list. At each element, we do some work. - PowerPoint PPT Presentation
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Traversals of Linked ListsPreorder BST Traversal
Postorder BST TraversalMiscellaneous BST Traversals
Depth First vs Breadth FirstArray Traversals
Traversals of Linked Lists
The Scenario• We need to visit each element in the
linked list.• At each element, we do some work.• We’ll stop when we reach the end of
the list.
• Examples:– Printing all elements– Updating/changing the data of all
elements
Traversals
• A traversal visits every element in a collection.
• Two forms:– Recursive– Iterative (linear structures)
Visiting Every Element
• Regardless of whether we use iteration or recursion, we continue until we reach the end (NIL).
48 17 142head //
Don’t De-reference NIL
• We must make sure we don’t de-reference (follow) the NIL pointer.
• To do this, always test to see if the pointer is NIL.– If so, we can’t use the ‘^’ operator
head //
Can’t follow head! (i.e. head^ doesn’t work!)
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
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 <- cur^.next endloopendprocedure // Traverse
An Iterative Traversal Exampleprocedure Print_List(cur iot in Ptr toa Node) loop exitif (cur = NIL) print(cur^.data) cur <- cur^.next endloopendprocedure //Print_List
4 17
list_head
42
Called via Print_List(list_head)
cur
An Iterative Traversal Exampleprocedure Print_List(cur iot in Ptr toa Node) loop exitif (cur = NIL) print(cur^.data) cur <- cur^.next endloopendprocedure //Print_List
4 17
list_head
42
cur
An Iterative Traversal Exampleprocedure Print_List(cur iot in Ptr toa Node) loop exitif (cur = NIL) print(cur^.data) cur <- cur^.next endloopendprocedure //Print_List
4 17
list_head
42
4
cur
An Iterative Traversal Exampleprocedure Print_List(cur iot in Ptr toa Node) loop exitif (cur = NIL) print(cur^.data) cur <- cur^.next endloopendprocedure //Print_List
4 17
list_head
42
4
cur
An Iterative Traversal Exampleprocedure Print_List(cur iot in Ptr toa Node) loop exitif (cur = NIL) print(cur^.data) cur <- cur^.next endloopendprocedure //Print_List
4 17
list_head
42
4
cur
An Iterative Traversal Exampleprocedure Print_List(cur iot in Ptr toa Node) loop exitif (cur = NIL) print(cur^.data) cur <- cur^.next endloopendprocedure //Print_List
4 17
list_head
42
4
cur
An Iterative Traversal Exampleprocedure Print_List(cur iot in Ptr toa Node) loop exitif (cur = NIL) print(cur^.data) cur <- cur^.next endloopendprocedure //Print_List
4 17
list_head
42
4 17
cur
An Iterative Traversal Exampleprocedure Print_List(cur iot in Ptr toa Node) loop exitif (cur = NIL) print(cur^.data) cur <- cur^.next endloopendprocedure //Print_List
4 17
list_head
42
4 17
cur
An Iterative Traversal Exampleprocedure Print_List(cur iot in Ptr toa Node) loop exitif (cur = NIL) print(cur^.data) cur <- cur^.next endloopendprocedure //Print_List
4 17
list_head
42
4 17
cur
An Iterative Traversal Exampleprocedure Print_List(cur iot in Ptr toa Node) loop exitif (cur = NIL) print(cur^.data) cur <- cur^.next endloopendprocedure //Print_List
4 17
list_head
42
4 17
cur
An Iterative Traversal Exampleprocedure Print_List(cur iot in Ptr toa Node) loop exitif (cur = NIL) print(cur^.data) cur <- cur^.next endloopendprocedure //Print_List
4 17
list_head
42
4 17 42
cur
An Iterative Traversal Exampleprocedure Print_List(cur iot in Ptr toa Node) loop exitif (cur = NIL) print(cur^.data) cur <- cur^.next endloopendprocedure //Print_List
4 17
list_head
42
4 17 42
cur
An Iterative Traversal Exampleprocedure Print_List(cur iot in Ptr toa Node) loop exitif (cur = NIL) print(cur^.data) cur <- cur^.next endloopendprocedure //Print_List
4 17
list_head
42
4 17 42
cur
An Iterative Traversal Exampleprocedure Print_List(cur iot in Ptr toa Node) loop exitif (cur = NIL) print(cur^.data) cur <- cur^.next endloopendprocedure //Print_List
4 17
list_head
42
4 17 42
cur
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 <> NIL) then Do_Something( cur^.data ) Traverse( cur^.next ) endifendprocedure // Traverse
A Recursive Traversal Exampleprocedure Print_List(cur iot in Ptr toa Node) if (cur <> NIL) then print(cur^.data) Print_List(cur^.next) endifendprocedure //Print_List
4 17
list_head
42
Called via Print_List(list_head)
cur
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.
An Example
procedure DestroyList (cur iot in Ptr toa Node) if (cur <> nil) then cur^.data <- -1 DestroyList(cur^.next) endifendprocedure // DestroyList
48 17 142head //-1 -1 -1
Another Example
procedure DestroyList (cur iot in Ptr toa Node) cur^.next <- NILendprocedure // DestroyList
48 17 142head //48
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.
Questions?
Traversing a Binary Search Tree (BST)Pre-Order
Outline of Pre-Order Traversal
• Three principle steps:– Do work (Current)– Traverse Left– Traverse Right
• Work can be anything• Separate work from traversal
•Traverse the tree “Pre-order”:–Visit the current and do work–Visit the tree’s left sub-tree–Visit right sub-tree
Pre-Order Traversal Procedureprocedure 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 <> NIL ) then Do_Whatever( cur^.data ) Pre_Order( cur^.left_child ) Pre_Order( cur^.right_child ) endif endprocedure // Pre_Order
Proc PreOrderPrint(pointer) pointer NOT NIL? print(data) PreOrderPrint(left child) PreOrderPrint(right child)
22
root
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PLR
Questions?
Traversing a Binary Search Tree (BST)Post-Order
Outline of Post-Order Traversal
• Three principle steps:– Traverse Left– Traverse Right– Do work (Current)
• Work can be anything• Separate work from traversal
•Traverse the tree “Post order”:–Visit the tree’s left sub-tree–Visit right sub-tree–Visit the current and do work
Post-Order Traversal Procedureprocedure 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 <> NIL ) then Post_Order( cur^.left_child ) Post_Order( cur^.right_child ) Do_Whatever( cur^.data ) endif endprocedure // Post_Order
Proc PostOrderPrint(pointer) pointer NOT NIL? PostOrderPrint(left child) PostOrderPrint(right child) print(data)
22
root
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363 14
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LRP
Questions?
Miscellaneous BST Traversals
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
Proc MiscTraverse(pointer) pointer NOT NIL? print(data) MiscTraverse(right child) MiscTraverse(left child)
22
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P
LR
22
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9
Proc MiscTraverse(pointer) pointer NOT NIL? print(data) MiscTraverse(right child) MiscTraverse(left child)
P
LR
22
root
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363 14
447
94
971
9
P
Output: 22
Proc MiscTraverse(pointer) pointer NOT NIL? print(data) MiscTraverse(right child) MiscTraverse(left child)
P
LR
22
root
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363 14
447
94
971
9
P
Output: 22
Proc MiscTraverse(pointer) pointer NOT NIL? print(data) MiscTraverse(right child) MiscTraverse(left child)
P
LR R
22
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363 14
447
94
971
9
P
Output: 22 67
Proc MiscTraverse(pointer) pointer NOT NIL? print(data) MiscTraverse(right child) MiscTraverse(left child)
P
LR R
P
22
root
67
363 14
447
94
971
9
P
Output: 22 67
Proc MiscTraverse(pointer) pointer NOT NIL? print(data) MiscTraverse(right child) MiscTraverse(left child)
P
LR R
PR
22
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447
94
971
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P
Output: 22 67 94
Proc MiscTraverse(pointer) pointer NOT NIL? print(data) MiscTraverse(right child) MiscTraverse(left child)
P
LR R
PR
P
22
root
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363 14
447
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971
9
P
Output: 22 67 94
Proc MiscTraverse(pointer) pointer NOT NIL? print(data) MiscTraverse(right child) MiscTraverse(left child)
P
LR R
PR
PR
22
root
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363 14
447
94
971
9
P
Output: 22 67 94 97
Proc MiscTraverse(pointer) pointer NOT NIL? print(data) MiscTraverse(right child) MiscTraverse(left child)
P
LR R
PR
PR
P
22
root
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363 14
447
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9
P
Output: 22 67 94 97
Proc MiscTraverse(pointer) pointer NOT NIL? print(data) MiscTraverse(right child) MiscTraverse(left child)
P
LR R
PR
PR
PR
22
root
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363 14
447
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971
9
P
Output: 22 67 94 97 36 44 9 14 3 7 1
Proc MiscTraverse(pointer) pointer NOT NIL? print(data) MiscTraverse(right child) MiscTraverse(left child)
P
LR R
PR
PR
PRL
L
L
PR
PRL
L
L
PR
PRL
L
PR
PRL
L
PRL
Questions?
Depth First vs. Breadth First Traversal
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
Depth-First Traversal
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Breadth-First Traversal
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Depth First Traversal (In-Order)
Procedure DFT (current isoftype in Ptr toa Node)// In order DFT if(current <> NIL) then DFT(current^.left) print(current^.data) DFT(Current^.right) endifendprocedure
Depth First Traversal (Pre-Order)
Procedure DFT (current isoftype in Ptr toa Node)// In order DFT if(current <> NIL) then DFT(current^.left) print(current^.data) DFT(Current^.right) endifendprocedure
Move this hereto make aPreorder DFT
Depth First Traversal (Post-Order)
Procedure DFT (current isoftype in Ptr toa Node)// In order DFT if(current <> NIL) then DFT(current^.left) print(current^.data) DFT(Current^.right) endifendprocedure
Move this hereto make aPostorder DFT
Proc DFT(pointer) pointer NOT NIL? DFT(left child) print(data) DFT(right child)
22
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Breadth-First Traversal
Requires a queue to maintain which nodes to visit next.
Enqueue the root pointerLoop until no elements in the queue
Dequeue(current)Enqueue current’s left and right childrenDo work at current
Breadth-First Traversal
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
LB
Breadth-First Traversal
loop dequeue(temp, OK, Q) exitif(NOT OK) if(temp^.left <> NIL) then enqueue(temp^.left, Q)) endif if(temp^.right <> NIL) then enqueue(temp^.right, Q) endif print(temp^.data) // processing node endloopendprocedure
LB
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(data)endloop
22
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Q:
aNode:
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(data)endloop
22
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Q:22
aNode:
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(data)endloop
22
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Q:22
aNode:
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
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9
Q:
aNode:22
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
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9
Q:9 67
aNode:22
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
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363 14
447
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9
Q:9 67
aNode:22
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
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9
Q:9 67
aNode:22
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
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67
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447
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971
9
Q:67
aNode:9
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
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Q:67 3 14
aNode:9
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
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9
Q:67 3 14
aNode:9
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
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67
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447
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971
9
Q:67 3 14
aNode:9
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
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Q:3 14
aNode:67
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
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447
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9
Q:3 14 36 94
aNode:67
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
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9
Q:3 14 36 94
aNode:67
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
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447
94
971
9
Q:3 14 36 94
aNode:67
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
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363 14
447
94
971
9
Q:14 36 94
aNode:3
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
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363 14
447
94
971
9
Q:14 36 94 1 7
aNode:3
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
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363 14
447
94
971
9
Q:14 36 94 1 7
aNode:3
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:14 36 94 1 7
aNode:3
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
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363 14
447
94
971
9
Q:36 94 1 7
aNode:14
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
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363 14
447
94
971
9
Q:36 94 1 7
aNode:14
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
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363 14
447
94
971
9
Q:36 94 1 7
aNode:14
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
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363 14
447
94
971
9
Q:36 94 1 7
aNode:14
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:94 1 7
aNode:36
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
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363 14
447
94
971
9
Q:94 1 7 44
aNode:36
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:94 1 7 44
aNode:36
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:94 1 7 44
aNode:36
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:1 7 44
aNode:94
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:1 7 44 97
aNode:94
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:1 7 44 97
aNode:94
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:1 7 44 97
aNode:94
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:7 44 97
aNode:1
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:7 44 97
aNode:1
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:7 44 97
aNode:1
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:7 44 97
aNode:1
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:44 97
aNode:7
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:44 97
aNode:7
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:44 97
aNode:7
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:44 97
aNode:7
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:97
aNode:44
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:97
aNode:44
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:97
aNode:44
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:97
aNode:44
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:
aNode:97
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:
aNode:97
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
971
9
Q:
aNode:97
Enqueue(root)loop exitif Q empty dequeue(aNode) enqueue(children) print(aNode.data)endloop
22
root
67
363 14
447
94
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Q:
aNode:97
Queuestions?
Traversals of Arrays
The Scenario• We need to visit each element in the
array.• At each element, we do some work.• We’ll stop when we reach the end of
the array (when we reach MAX).
• Examples:– Printing all elements– Updating/changing the data of all
elements
Traversals
• A traversal visits every element in a collection.
• Two forms:– Recursive– Iterative (linear structures)
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 <= MAX) then Do_Something(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
Recursive Array Traversal Example
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call
if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 1
NumArray 12 43 11 9 98
1 2 3 4 5
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 1
12 43 11 9 98NumArray
1 2 3 4 5
12
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 1
12 43 11 9 98NumArray
1 2 3 4 5
12
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 1
12 43 11 9 98NumArray
1 2 3 4 5
12
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 2
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 1
12 43 11 9 98NumArray
1 2 3 4 5
12 43
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 2
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 1
12 43 11 9 98NumArray
1 2 3 4 5
12 43
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 2
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 1
12 43 11 9 98NumArray
1 2 3 4 5
12 43
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 2
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 3
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 1
12 43 11 9 98NumArray
1 2 3 4 5
12 43 11
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 2
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 3
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 1
12 43 11 9 98NumArray
1 2 3 4 5
12 43 11
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 2
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 3
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 1
12 43 11 9 98NumArray
1 2 3 4 5
12 43 11
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 2
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 3
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 4
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 1
12 43 11 9 98NumArray
1 2 3 4 5
12 43 11 9
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 2
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 3
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 4
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 1
12 43 11 9 98NumArray
1 2 3 4 5
12 43 11 9
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 2
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 3
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 4
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 1
12 43 11 9 98NumArray
1 2 3 4 5
12 43 11 9
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 2
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 3
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 4
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 5
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 1
12 43 11 9 98NumArray
1 2 3 4 5
12 43 11 9 98
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 2
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 3
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 4
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 5
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 1
12 43 11 9 98NumArray
1 2 3 4 5
12 43 11 9 98
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 2
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 3
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 4
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 5
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 1
12 43 11 9 98NumArray
1 2 3 4 5
12 43 11 9 98
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 2
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 3
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 4
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 5
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 6
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 1
12 43 11 9 98NumArray
1 2 3 4 5
12 43 11 9 98
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 2
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 3
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 4
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 5
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 1
12 43 11 9 98NumArray
1 2 3 4 5
12 43 11 9 98
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 2
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 3
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 4
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 1
12 43 11 9 98NumArray
1 2 3 4 5
12 43 11 9 98
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 2
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 3
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 1
12 43 11 9 98NumArray
1 2 3 4 5
12 43 11 9 98
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 2
procedure RecursiveTraverse (i iot in num, NumArray iot in NumArrayType)
// Pre: i = 1 on first call if (i <= MAX) then print(NumArray[i]) RecursiveTraverse (i+1, NumArray) endifendprocedure
i = 1
12 43 11 9 98NumArray
1 2 3 4 5
12 43 11 9 98
Iterative Traversal Templateprocedure 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 <- 1 loop exitif( i > MAX ) Do_Something( my_array[i] ) i <- i + 1 endloopendprocedure // Traverse
An Iterative Traversal ExampleTraverse the array and fill in every elementso that the ith element contains the value i.
1 2 3 4 5 6 7 8 9 101 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
data1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
data
i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
data
1i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
data
1i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
data
1i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 data
1i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 data
2i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 data
2i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 data
2i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 data
2i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 data
2i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 data
3i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 data
3i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 data
3i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 data
3i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 3 data
3i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 3 data
4i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 3 data
4i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 3 4 data
4i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 3 4 5 data
5i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 3 4 5 6 data
6i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 3 4 5 6 7 data
7i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 3 4 5 6 7 8 data
8i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 3 4 5 6 7 8 data
9i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 3 4 5 6 7 8 9 data
9i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 3 4 5 6 7 8 9 data
10i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 3 4 5 6 7 8 9 data
10i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 3 4 5 6 7 8 9 data
10i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 3 4 5 6 7 8 9 data
10i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 3 4 5 6 7 8 9 10data
10i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 3 4 5 6 7 8 9 10data
11i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 3 4 5 6 7 8 9 10data
11i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 3 4 5 6 7 8 9 10data
11i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 3 4 5 6 7 8 9 10data
11i
1 2 3 4 5 6 7 8 9 10
MAX is 10Vec_max definesa Array[1..MAX] of Numdata isoftype Vec_maxi isoftype Numi <- 1loop exitif(i > MAX) data[i] <- i i <- i + 1endloop
1 2 3 4 5 6 7 8 9 10data
11i
1 2 3 4 5 6 7 8 9 10
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.
A[1] = ‘A’A[2] = ‘Z’A[3] = ‘B’A[4] = ‘Y’A[5] = ‘C’A[6] = ‘X’
.
.
.A[23] = ‘L’A[24] = ‘O’A[25] = ‘M’A[26] = ‘N’
1 2 3 4 5. . .
23 24 25 26A 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.
A
A First TryALPHAS is 26A isoftype CharArrayTypei isoftype Numup isoftype chari <- 1up <- ‘A’loopexitif( i > ALPHAS )A[i] <- upup <- next_character(up)i <- i + 2
endloop
1 2 3 4 5. . .
23 24 25 26A 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.
A
ALPHAS is 26A isoftype Array_Typei isoftype Numup, down isoftype chari <- 1up <- ‘A’down <- ‘Z’loopexitif( i > ALPHAS )A[i] <- upA[i + 1] <- downup <- next_character(up)down <- previous_character(down)i <- i + 2
endloop
1 2 3 4 5. . .
23 24 25 26A 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.
A
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 it’s row position and column position
• (i.e. the element at position [i,j] should contain the value i*j)
1 2 3 4
123
i
j
1 2 3 42 4 6 83 6 9 12
MAX_COLS is 4Row_Type definesa Array[1..MAX_COLS] of Num
Sample isoftype Row_type
Sample:1 2 3 4
MAX_ROWS is 3MAX_COLS is 4Row_Type definesa Array[1..MAX_COLS] of NumMatrix definesa Array[1..MAX_ROWS] of Row_TypeData isoftype Matrix
Data:
1 2 3 4
1
2
3
MAX_ROWS is 3MAX_COLS is 4Row_Type definesa Array[1..MAX_COLS] of NumMatrix definesa Array[1..MAX_ROWS] of Row_TypeData isoftype MatrixRow, Col isoftype NumRow <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
MAX_ROWS is 3MAX_COLS is 4Row_Type definesa Array[1..MAX_COLS] of NumMatrix definesa Array[1..MAX_ROWS] of Row_TypeData isoftype MatrixRow, Col isoftype Num
Data:
Row: Col:
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
Data:
Row: 1Col:
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
Data:
Row: 1Col:
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
Data:
Row: 1Col:
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
Data:
Row: 1Col: 1
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
Data:
Row: 1Col: 1
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
Data:
Row: 1Col: 1
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
1
Data:
Row: 1Col: 1
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
1
Data:
Row: 1Col: 2
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
1
Data:
Row: 1Col: 2
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
1
Data:
Row: 1Col: 2
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
2 1
Data:
Row: 1Col: 2
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
2 1
Data:
Row: 1Col: 3
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
2 1
Data:
Row: 1Col: 3
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
2 1
Data:
Row: 1Col: 3
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
3 2 1
Data:
Row: 1Col: 3
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
3 2 1
Data:
Row: 1Col: 4
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
3 2 1
Data:
Row: 1Col: 4
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
3 2 1
Data:
Row: 1Col: 4
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
Data:
Row: 1Col: 4
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
Data:
Row: 1Col: 5
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
Data:
Row: 1Col: 5
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
Data:
Row: 1Col: 5
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
Data:
Row: 2Col: 5
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
Data:
Row: 2Col: 5
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
Data:
Row: 2Col: 5
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
Data:
Row: 2Col: 1
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
What would happenif we did this outside
first loop?
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
Data:
Row: 2Col: 1
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
Data:
Row: 2Col: 1
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
2
Data:
Row: 2Col: 1
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
4 2
Data:
Row: 2Col: 2
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
6 4 2
Data:
Row: 2Col: 3
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
8 6 4 2
Data:
Row: 2Col: 4
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
8 6 4 2
3
Data:
Row: 3Col: 1
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
8 6 4 2
6 3
Data:
Row: 3Col: 2
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
8 6 4 2
9 6 3
Data:
Row: 3Col: 3
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
8 6 4 2
12 9 6 3
Data:
Row: 3Col: 4
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
8 6 4 2
12 9 6 3
Data:
Row: 3Col: 5
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
8 6 4 2
12 9 6 3
Data:
Row: 3Col: 5
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
8 6 4 2
12 9 6 3
Data:
Row: 4Col: 5
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
8 6 4 2
12 9 6 3
Data:
Row: 4Col: 5
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
Row <- 1loop exitif(Row > MAX_ROWS) Col <- 1 loop exitif(Col > MAX_COLS) Data[Row][Col] <- Row * Col Col <- Col + 1 endloop Row <- Row + 1endloop
4 3 2 1
8 6 4 2
12 9 6 3
Data:
Row: 4Col: 5
MAX_ROWS is 3MAX_COLS is 4
1 2 3 4
1
2
3
and continue…
Summary
• Traversals involve visiting every element.– Recursion or iteration– Do some work at each element
• When doing iterative traversals on arrays– Use a loop for each dimension of the array– Embed loops– Typically perform work in innermost loop
Questions?
