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## PowerPoint Slideshow about ' Data Structures' - sanjiv

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### Sorting

### Linked Structures

### Dynamic Structures

Basic Concepts

Whether using arrays or lists…there has to be some way to:

Search the data structure

Sort the data structure

To do this, we use keys…

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Keys

- Should be:
- unique,
- short,
- easy to understand,
- easily recognizable, and
- have some inherent value**
- Some debate about this feature

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Keys cont.

- Can be used in searching for a record and for sorting records within a data structure such as arrays or files.
- Ex. There “should be” only one of each:
- Social Security Number
- Driver’s License Number
- Birth Certificate ID

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Sorting

- Where the elements are placed in some particular order.
- Sort order can be in by
- Ascending or (A-Z; Smallest to Largest)
- Descending (Z-A; Largest to Smallest)

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Types of Sorts

- Bubble (slow!)
- Selection Sort or Exchange Sort
- Insertion Sort
- Merge Sort
- Quick Sort
- Radix Sort
- Shell Sort (variation of Insertion Sort)

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Two Simple Concepts

- Compare
- A comparison is made between two pieces of data upon which a decision is made to move the data or not.

- Exchange
- An exchange is each time a piece of data is switched with another piece of data.
- The Swap

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The Swap Routine

Exchanges Two Values

Private Sub Swap(Array(J), Array(J+1))

If Array(J) > Array(J+1) Then

Temp = Array(J)

Array(J) = Array(J+1)

Array(J+1) = Temp

End If

End Sub

- Use the Call statement to access the Swap Routine.

Element

Index

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Bubble Sort

- One of the simplest to understand
- The Concept:
- Lower numbers “float” to the top of the array and larger numbers “sink” to the bottom of the array.

- The Concept:

1

2

3

4

5

6

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The Process

- Successively exchanges adjacent pairs of elements in a series of passes, repeating the process until the entire sequence is in sorted order.
- Each pass starts at one end of the array and works toward the other end, with each pair of elements that are out of order being exchanged.
- The entire sequence considers n-1 pieces of data
- With each succeeding pass one less piece of data than the previous pass needs to be considered.

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A Bubble Sort

- 390205205205205
- 205 390182182182
- 182 182390 45 45
- 45 4545390235
- 235235235235 390

Worst Case

Exchanges = 1/2 n(n - 1) = 1/2 5(5 - 1) = 10

Compares = 1/2 n(n - 1) = 1/2 5(5 - 1) = 10

Best Case Exchanges = 0 Compares = n - 1

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The Good and The Bad

- Advantage:
- If no exchanges are made during the first pass, the sequence is already in sorted order.

- Disadvantage:
- Is one of the slowest sorting algorithms and is probably only used because its logic is easily understood.

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Pseudocode Example

Get Array (or List) Input

For Index = 1 to ListLength

Input Value into Array(Index)

Next Index

‘Then Sort

For I = 1 to ListLength

For J = 1 to ListLength – 1

If Array(J)>Array(J+1) Then

Call Swap (Array(J), Array(J+1))

Next J

Next I

Call to Swap isnow insidethe IF statementrather than the IFStatement being inside the Swap Procedure

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The Revised Better Swap Routine

Exchange Two Values

Private Sub Swap(Array(J), Array(J+1))

Temp = Array(J)

Array(J) = Array(J+1)

Array(J+1) = Temp

End Sub

Use the Call statement to access the Swap Routine.

Swap

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The Selection Sort

- A rearrangement of data such that the data are in increasing (or decreasing) sequence.
The Algorithm for a Selection Sort

For Index 1 to ListLength-1 do

Find the position of the smallest element in list[1..ListLength].

If List(Index) is not the position of the smallest element then

Exchange the smallest element with the one at position List(Index)

Next Index

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The Process

- Selects the smallest (or largest) element from a sequence of elements.
- The values are moved into position by successively exchanging values in a list until the sequence is in sorted order.
- Only desirable property? Records of successively smaller keys are identified one by one, so that output of the sorted sequence can proceed virtually in parallel with the sort itself.

- The values are moved into position by successively exchanging values in a list until the sequence is in sorted order.

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The Selection Sort

- 39045 45 45 45 45
- 205 205182182 182182
- 182182205205205205
- 45 390 390 390235235
- 235 235 235 235390390

Exchanges = n - 1 = 5 - 1 = 4

Compares = 1/2 n(n - 1) = 1/2 5(5 - 1) = 10

Note: 390 is not included as it is the last item in list.

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The Good and The Bad

- Advantages
- Easiest to remember
- The only desirable property is that records of successively smaller keys are identified one by one, so that output (or processing) of the sorted sequence can proceed virtually in parallel with the sort itself.

- Disadvantages
- Still slow

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The Insertion Sort

- Inserts each element into a sequence of sorted elements so that the resulting sequence is still sorted.
- With arrays, a new array is used to insert the values from the old array
- On average, half of the array will have to be compared.

- With lists, a new list is created from the values of the old list.

- With arrays, a new array is used to insert the values from the old array

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The Insertion Sort

Old List

- 390390 390390 45
- 205 205205 45 182
- 18218245182 205
- 4545182205235
- 235235235 235 390

New List

Exchanges = n - 1 = 5 - 1 = 4

Compares = 1/2 n(n - 1) = 1/2 5(5 - 1) = 10

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Sort Comparison Chart

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Sequential Search

- Searches the list for a specific item.
- If the list is not ordered, the entire list must be searched before a conclusion may be made that the item is not in the list.
- If the list is ordered, the list is searched only until a value is found that is larger than the search item.

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Searching an Unordered List

function ItemSearch (List : ListType;

Item : ComponentType ): Boolean;

var

Index : Integer;

begin

Index := 1;

List.Items[List.Length+1] := Item;

while List.Items[Index] <> Item do

Index := Index + 1;

ItemSearch := Index <> List.Length + 1

end;

Pascal Code

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Searching an Ordered List

function SeqSearch (List : ListType; Item : ComponentType ): Boolean;

var Index : Integer; Stop : Boolean;

begin

Index := 1; Stop := False; {Initialize}

List.Items[List.Length+1] := Item;

While Not Stop Do

{Item is not in List.Items[1]..List.Items[Index-1]}

If Item > List.Items[Index] then

Index := Index + 1

Else Stop := True; {Item is either found or not there}

SeqSearch := (Index <> List.Length + 1) and

(Item = List.Items[Index])

end;

Pascal Code

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The Binary Search

- Processes the list by dividing the list and then searching each half.
- List must be sorted.
- Much more efficient that a sequential search.
- In other words, a search of a 1000 element array (or list) would only take 10 compares opposed to 1000 using a sequential search

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The Concept

- Divides the List into 3 components
- List[1..Middle-1]
- List[Middle]
- List[Middle+1..Last]

[First] [Middle] [Last]

Item

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The Recursive Pseudocode Algorithm

Compute the subscript of the middle element.

If the Target is the middle value Then

Middle value is target location

Return with success

ElseIf the Target is less than the middle value Then

Search sublist with subscripts First..Middle-1

Else

Search sublist with subscripts Middle + 1..Last

End If

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Code Example

procedure BinarySearch (var List {Input} : IntArray;

Target {Input} : Integer; First, Last {Input} : Integer;

var Index {output} : Integer; var Found {output} : Boolean);

var Middle : Integer;

begin

Middle := (First + Last) div 2;

if First > Last then Found := False

else if Target = List[Middle] then

begin Found := True;

Index := Middle end

else if Target < List[Middle] then

BinarySearch (List, Target, First, Middle-1, Index, Found)

else

BinarySearch(List, Target, Middle+1, Last, Index, Found)

end;

Pascal’s Integer Division

Found the item

Pascal Code

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The Concept

- Each data structure element contains
- not only the element’s data value but
- also the addresses of one or more other data elements.

- Examples:
- Stacks
- Queues
- Trees “I Love Trees…”

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Linked List

- Probably the simplest linked structure
- Contain records

- Each element contains the address of the next list element.
- Are extremely flexible.
- They make it easy to add new information by creating a new node and inserting it between two existing nodes.
- It is also easy to delete a node.

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Node

Node nil

Linked List

Pointer

Pointer

Pointers- A data type whose values are the locations of values of other data types and are stored in memory.
- Considered a Referenced Variable
- A variable created and accessed not by a name but by a pointer variable -- a dynamic variable.

- Considered a Referenced Variable

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How the Link Works

- Linked Lists
- A connected group of dynamically allocated records.

- Nodes
- Records within a linked list.

key

data

Instance

of

Node

P

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Conceptual View of a Simple Linked List

As silly as it sounds…You always know where your head is…

head

current

nil

current

node

first

node

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Link Operations Inserting at the Head of a List Insertion at the End of a List

- List Head
- The first node in a list.

- Is more efficient and easier

- Less efficient because there is no specific pointer to the end of the list.
- The list must be followed from the head to the last list node and then perform the insertion.

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Link Operations cont. Traversing a List

- Deleting a Node
- Change the Link field of the node that points to its predecessor and point to the node’s successor.

- Processing the nodes in a list starting with the list head and ending with the last node following the trail of pointers.
- Head <> nil is typical for processing loops that process lists.

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Stacks

- Is a data structure in which only the top element can be accessed.
- Classic Example:
- Plates in a buffet line.
- Customer always takes the top-most plate.
- Plates are replaced from the top.

- Classic Example:
- LIFOLast-In First-Out Structure
- Last element stored is the first to be removed.

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pointer

head

pointer

4

3

2

1

4

3

2

1

a stack

a popped stack

head

pointer

5

4

3

2

1

a pushed stack

Push & Pop- Pushing Onto The Stack
- Placing a new top element on the stack.

- Popping The Stack
- Removing the top element of a stack.

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Queues

- A data structure in which elements are inserted at one end and removed from the other end.
- Classic Example:
- Customers in a Theater Ticket Line or a list of jobs waiting to be executed.

- Classic Example:
- FIFOFirst-In First-Out Structure
- First element stored is the first to be removed.
- Also Array Queues, Priority Queues, and Schedule Queues.

- First element stored is the first to be removed.

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pointer

tail

pointer

dog

head

pointer

head

pointer

an empty queue

after enqueuing an element

tail

pointer

cat

dog

tail

pointer

cat

head

pointer

head

pointer

after enqueuing

another element

after dequeuing an element

Enqueue & DequeueCOP1000

Trees

- Similar to a linked list, except that each element carries with it the addresses of 2 or more other elements, rather than just one.

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Binary Trees

- Similar to a linked list, except that each element carries with it the addresses of 2 or more other elements, rather than just one.

So…

Why is the treeupside down?

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Binary Trees

- Contains at most two subtrees(or two children).
- Each subtree is identified as being either the left subtree or the right subtree of its parent.
- It may be empty (a pointer with no successors).
- Each node in a binary tree can have 0, 1, or 2 successor nodes.

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Some Terms

- Root– a binary tree with at least one node at the top.
- Leaf Node – the nodes at the bottom of a binary tree node with zero successors.
- Left and Right subtrees– the two disjoint binary trees attached to the root of a binary tree.
- Disjoint subtrees– nodes cannot be on both a left and right subtree of the same node.

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More Terms

- Parent-child relationship – the relationship between a node and its successors.
- Parent– the predecessor of a node.
- Child – the successor of a node.
- Edge – line that connects two nodes
- Siblings – two children of the same parent node.

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More Terms

- Ancestors – all predecessors of a node, unless it is the root. The root has no ancestors.
- Descendants – all successors of a node.
- Balanced, Minimal Path – the difference between any two paths is at most 1.
- So…how are these terms used?

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A Balanced, Minimal Path,Inorder, Binary Search Tree

ancestors to

60 & 95

root

50

edge

descendant

to 50

descendant

to 50

left

child

right

child

25

75

ascendant

to 60 & 95

parent to

10 & 45

children

10

45

60

95

A subtree

to 50

with nodes

75, 60, 95

siblings (or twins) to 25

leaves (or terminal nodes)

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Remember the Binary Search?

- Now apply the concept to a Binary Tree…

[First] [Middle] [Last] Root M-1 M+1

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Binary Search Tree

- A tree structure that stores data in such a way that the data can be retrieved very efficiently.
- Each item in a binary search tree has a unique key.
- Is either empty or
- has the property that the item in its root has a larger key than each item in its left subtree and a smaller key than each item in its right subtree.
- Each subtree must be binary search trees.

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Traversal of Binary Search Trees

- A finite collection of objects means to process each object in the collection exactly once.
- Find the first node
- Find the next node.
- Determine when there are no more nodes.
- Process the current node.

It’s the order in which we place these statements that determines the type of Traversal.

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25

65

13

35

54

78

Notice the Order…This is calledInorder Traversal

There are two others

13 25 35 40 54 65 78

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2

6

1

3

5

7

Inorder Traversal- Each node is processed after all nodes in its left subtree but before any node in its right subtree. (left, center, right)
Private Sub InOrder (P : NodePointer)

If P <> nil Then

InOrder (P^.Left)

Process (P)

InOrder (P^.Right)

End If

End Sub

inorder

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2

5

3

4

6

7

Preorder Traversal- Each node is processed before any node in either of its subtrees.
(center, left, right)

Private Sub PreOrder (P : NodePointer)

If P <> nil Then

Process (P)

PreOrder (P^.Left)

PreOrder (P^.Right)

End If

End Sub

preorder

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3

6

1

2

4

5

Postorder Traversal- Each node is processed after all nodes in both of its subtrees. (left, right, center)
Private Sub PostOrder (P : NodePointer)

If P <> nil Then

PostOrder (P^.Left)

PostOrder (P^.Right)

Process (P)

End If

End Sub

postorder

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Examples

- Given a list
10, 15, 20, 25, 30, 35, 40

- Two trees can be created from this:
- A linear list
- A balanced, inorder, binary search tree(not necessarily minimal path)

- Why the difference?
- Depends on the instructions…

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Two Versions of the Trees

Linear List Balanced, Inorder, Binary Search Tree

10

25

15

20

35

15

25

Createdin orderfrom first tolast item inlist

10

20

30

40

30

35

Created from dividing list into twoand then dividing each side into twokeeping keys in order

40

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Other Types of Trees

- Patricia Trees – Example
- B-Trees – Example
- 23 Trees
- 234 Trees
- Tries
- Search T

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